:: LANG1 semantic presentation

definition

attr c₁ is strict;
struct DTConstrStr -> 1-sorted ;
aggr DTConstrStr(# carrier, Rules #) -> DTConstrStr ;
sel Rules c₁ -> Relation of the carrier of c₁,the carrier of c₁ * ;

end;

registration

cluster non empty strict DTConstrStr ;
existence

proof end;

end;

definition

attr c₁ is strict;
struct GrammarStr -> DTConstrStr ;
aggr GrammarStr(# carrier, InitialSym, Rules #) -> GrammarStr ;
sel InitialSym c₁ -> Element of the carrier of c₁;

end;

registration

cluster non empty GrammarStr ;
existence

proof end;

end;

definition

let G be DTConstrStr ;
mode Symbol of G is Element of G;
mode String of G is Element of the carrier of G * ;

end;

definition

let D be set ;
let p, q be Element of D * ;
:: original: ^
redefine func p ^ q -> Element of D * ;
coherence

proof end;

end;

registration

let D be set ;
cluster empty Element of D * ;
existence

proof end;

end;

definition

let D be set ;
:: original: <*>
redefine func <*> D -> empty Element of D * ;
coherence by FINSEQ_1:def 11;

end;

definition

let D be non empty set ;
let d be Element of D;
:: original: <*
redefine func <*d*> -> Element of D * ;
coherence

proof end;

let e be Element of D;
:: original: <*
redefine func <*d,e*> -> Element of D * ;
coherence

proof end;

end;

definition

let G be non empty DTConstrStr ;
let s be Symbol of G;
let n be FinSequence;
pred s ==> n means :Def1: :: LANG1:def 1
[s,n] in the Rules of G;

end;

:: deftheorem Def1 defines ==> LANG1:def 1 :

definition

let G be non empty DTConstrStr ;
func Terminals G -> set equals :: LANG1:def 2
{ s where s is Symbol of G : for n being FinSequence holds not s ==> n } ;
coherence ;
func NonTerminals G -> set equals :: LANG1:def 3
{ s where s is Symbol of G : ex n being FinSequence st s ==> n } ;
coherence ;

end;

:: deftheorem defines Terminals LANG1:def 2 :

:: deftheorem defines NonTerminals LANG1:def 3 :

theorem :: LANG1:1

for G being non empty DTConstrStr holds (Terminals G) \/ (NonTerminals G) = the carrier of G

proof end;

definition

let G be non empty DTConstrStr ;
let n, m be String of G;
pred n ==> m means :: LANG1:def 4
ex n1, n2, n3 being String of G ex s being Symbol of G st
( n = (n1 ^ <*s*>) ^ n2 & m = (n1 ^ n3) ^ n2 & s ==> n3 );

end;

:: deftheorem defines ==> LANG1:def 4 :

theorem :: LANG1:2

for G being non empty DTConstrStr
for s being Symbol of G
for n, n1, n2 being String of G st s ==> n holds
(n1 ^ <*s*>) ^ n2 ==> (n1 ^ n) ^ n2

proof end;

theorem Th3: :: LANG1:3

for G being non empty DTConstrStr
for s being Symbol of G
for n being String of G st s ==> n holds
<*s*> ==> n

proof end;

theorem Th4: :: LANG1:4

for G being non empty DTConstrStr
for s being Symbol of G
for n being String of G st <*s*> ==> n holds
s ==> n

proof end;

theorem Th5: :: LANG1:5

for G being non empty DTConstrStr
for n, n1, n2 being String of G st n1 ==> n2 holds
( n ^ n1 ==> n ^ n2 & n1 ^ n ==> n2 ^ n )

proof end;

definition

let G be non empty DTConstrStr ;
let n, m be String of G;
pred m is_derivable_from n means :Def5: :: LANG1:def 5
ex p being FinSequence st
( len p >= 1 & p . 1 = n & p . (len p) = m & ( for i being Nat st i >= 1 & i < len p holds
ex a, b being String of G st
( p . i = a & p . (i + 1) = b & a ==> b ) ) );

end;

:: deftheorem Def5 defines is_derivable_from LANG1:def 5 :

theorem Th6: :: LANG1:6

for G being non empty DTConstrStr
for n being String of G holds n is_derivable_from n

proof end;

theorem Th7: :: LANG1:7

for G being non empty DTConstrStr
for n, m being String of G st n ==> m holds
m is_derivable_from n

proof end;

theorem Th8: :: LANG1:8

for G being non empty DTConstrStr
for n1, n2, n3 being String of G st n1 is_derivable_from n2 & n2 is_derivable_from n3 holds
n1 is_derivable_from n3

proof end;

definition

let G be non empty GrammarStr ;
func Lang G -> set equals :: LANG1:def 6
{ a where a is Element of the carrier of G * : ( rng a c= Terminals G & a is_derivable_from <*the InitialSym of G*> ) } ;
coherence ;

end;

:: deftheorem defines Lang LANG1:def 6 :

theorem :: LANG1:9

for G being non empty GrammarStr
for n being String of G holds
( n in Lang G iff ( rng n c= Terminals G & n is_derivable_from <*the InitialSym of G*> ) )

proof end;

definition

let D, E be non empty set ;
let a be Element of [:D,E:];
:: original: {
redefine func {a} -> Relation of D,E;
coherence

proof end;

let b be Element of [:D,E:];
:: original: {
redefine func {a,b} -> Relation of D,E;
coherence

proof end;

end;

definition

let a be set ;
func EmptyGrammar a -> strict GrammarStr means :Def7: :: LANG1:def 7
( the carrier of it = {a} & the Rules of it = {[a,{} ]} );
existence

proof end;

uniqueness

proof end;

let b be set ;
func SingleGrammar a,b -> strict GrammarStr means :Def8: :: LANG1:def 8
( the carrier of it = {a,b} & the InitialSym of it = a & the Rules of it = {[a,<*b*>]} );
existence

proof end;

uniqueness ;
func IterGrammar a,b -> strict GrammarStr means :Def9: :: LANG1:def 9
( the carrier of it = {a,b} & the InitialSym of it = a & the Rules of it = {[a,<*b,a*>],[a,{} ]} );
existence

proof end;

uniqueness ;

end;

:: deftheorem Def7 defines EmptyGrammar LANG1:def 7 :

:: deftheorem Def8 defines SingleGrammar LANG1:def 8 :

:: deftheorem Def9 defines IterGrammar LANG1:def 9 :

registration

let a be set ;
cluster EmptyGrammar a -> non empty strict ;
coherence

proof end;

let b be set ;
cluster SingleGrammar a,b -> non empty strict ;
coherence

proof end;

cluster IterGrammar a,b -> non empty strict ;
coherence

proof end;

end;

definition

let D be non empty set ;
func TotalGrammar D -> strict GrammarStr means :Def10: :: LANG1:def 10
( the carrier of it = D \/ {D} & the InitialSym of it = D & the Rules of it = { [D,<*d,D*>] where d is Element of D : d = d } \/ {[D,{} ]} );
existence

proof end;

uniqueness ;

end;

:: deftheorem Def10 defines TotalGrammar LANG1:def 10 :

registration

let D be non empty set ;
cluster TotalGrammar D -> non empty strict ;
coherence

proof end;

end;

theorem Th10: :: LANG1:10

for a being set holds Terminals (EmptyGrammar a) = {}

proof end;

theorem Th11: :: LANG1:11

for a being set holds Lang (EmptyGrammar a) = {{} }

proof end;

theorem Th12: :: LANG1:12

for a, b being set st a <> b holds
Terminals (SingleGrammar a,b) = {b}

proof end;

theorem :: LANG1:13

for a, b being set st a <> b holds
Lang (SingleGrammar a,b) = {<*b*>}

proof end;

theorem Th14: :: LANG1:14

for a, b being set st a <> b holds
Terminals (IterGrammar a,b) = {b}

proof end;

theorem :: LANG1:15

for a, b being set st a <> b holds
Lang (IterGrammar a,b) = {b} *

proof end;

theorem Th16: :: LANG1:16

for D being non empty set holds Terminals (TotalGrammar D) = D

proof end;

theorem :: LANG1:17

for D being non empty set holds Lang (TotalGrammar D) = D *

proof end;

definition

let IT be non empty GrammarStr ;
attr IT is efective means :Def11: :: LANG1:def 11
( not Lang IT is empty & the InitialSym of IT in NonTerminals IT & ( for s being Symbol of IT st s in Terminals IT holds
ex p being String of IT st
( p in Lang IT & s in rng p ) ) );

end;

:: deftheorem Def11 defines efective LANG1:def 11 :

definition

let IT be GrammarStr ;
attr IT is finite means :: LANG1:def 12
the Rules of IT is finite;

end;

:: deftheorem defines finite LANG1:def 12 :

registration

cluster non empty efective finite GrammarStr ;
existence

proof end;

end;

definition

let G be non empty efective GrammarStr ;
:: original: NonTerminals
redefine func NonTerminals G -> non empty Subset of G;
coherence

proof end;

end;

definition

let X, Y be non empty set ;
let p be FinSequence of X;
let f be Function of X,Y;
:: original: *
redefine func f * p -> Element of Y * ;
coherence

proof end;

end;

definition

let X, Y be non empty set ;
let f be Function of X,Y;
canceled;
func f * -> Function of X * ,Y * means :: LANG1:def 14
for p being Element of X * holds it . p = f * p;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem LANG1:def 13 :

:: deftheorem defines * LANG1:def 14 :

theorem :: LANG1:18

for R being Relation holds R c= R [*]

proof end;

definition

let X be non empty set ;
let R be Relation of X;
:: original: [*]
redefine func R [*] -> Relation of X;
coherence

proof end;

end;

definition

let G be non empty GrammarStr ;
let X be non empty set ;
let f be Function of the carrier of G,X;
func G . f -> strict GrammarStr equals :: LANG1:def 15
GrammarStr(# X,(f . the InitialSym of G),(((f ~ ) * the Rules of G) * (f * )) #);
correctness ;

end;

:: deftheorem defines . LANG1:def 15 :