begin
theorem Th1:
theorem Th2:
theorem Th3:
theorem Th4:
begin
:: deftheorem Def1 defines XFinSequence-yielding REWRITE2:def 1 :
for f being Function holds
( f is XFinSequence-yielding iff for x being set st x in dom f holds
f . x is XFinSequence );
begin
:: deftheorem Def2 defines ^+ REWRITE2:def 2 :
for s being XFinSequence
for p, b3 being XFinSequence-yielding Function holds
( b3 = s ^+ p iff ( dom b3 = dom p & ( for x being set st x in dom p holds
b3 . x = s ^ (p . x) ) ) );
:: deftheorem Def3 defines +^ REWRITE2:def 3 :
for s being XFinSequence
for p, b3 being XFinSequence-yielding Function holds
( b3 = p +^ s iff ( dom b3 = dom p & ( for x being set st x in dom p holds
b3 . x = (p . x) ^ s ) ) );
theorem Th5:
theorem
theorem
theorem
theorem
begin
begin
:: deftheorem Def4 defines -->. REWRITE2:def 4 :
for E being set
for S being semi-Thue-system of E
for s, t being Element of E ^omega holds
( s -->. t,S iff [s,t] in S );
:: deftheorem Def5 defines ==>. REWRITE2:def 5 :
for E being set
for S being semi-Thue-system of E
for s, t being Element of E ^omega holds
( s ==>. t,S iff ex v, w, s1, t1 being Element of E ^omega st
( s = (v ^ s1) ^ w & t = (v ^ t1) ^ w & s1 -->. t1,S ) );
theorem Th10:
theorem
theorem Th12:
theorem Th13:
theorem Th14:
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
begin
definition
let E be
set ;
let S be
semi-Thue-system of
E;
func ==>.-relation S -> Relation of
(E ^omega ) means :
Def6:
for
s,
t being
Element of
E ^omega holds
(
[s,t] in it iff
s ==>. t,
S );
existence
ex b1 being Relation of (E ^omega ) st
for s, t being Element of E ^omega holds
( [s,t] in b1 iff s ==>. t,S )
uniqueness
for b1, b2 being Relation of (E ^omega ) st ( for s, t being Element of E ^omega holds
( [s,t] in b1 iff s ==>. t,S ) ) & ( for s, t being Element of E ^omega holds
( [s,t] in b2 iff s ==>. t,S ) ) holds
b1 = b2
end;
:: deftheorem Def6 defines ==>.-relation REWRITE2:def 6 :
for E being set
for S being semi-Thue-system of E
for b3 being Relation of (E ^omega ) holds
( b3 = ==>.-relation S iff for s, t being Element of E ^omega holds
( [s,t] in b3 iff s ==>. t,S ) );
theorem Th22:
theorem Th23:
theorem
theorem Th25:
theorem Th26:
theorem Th27:
theorem Th28:
theorem Th29:
theorem Th30:
theorem Th31:
begin
:: deftheorem Def7 defines ==>* REWRITE2:def 7 :
for E being set
for S being semi-Thue-system of E
for s, t being Element of E ^omega holds
( s ==>* t,S iff ==>.-relation S reduces s,t );
theorem Th32:
theorem Th33:
theorem
theorem Th35:
theorem Th36:
theorem Th37:
theorem
theorem
theorem Th40:
theorem Th41:
theorem Th42:
theorem Th43:
theorem Th44:
theorem Th45:
begin
:: deftheorem defines Lang REWRITE2:def 8 :
for E being set
for S being semi-Thue-system of E
for w being Element of E ^omega holds Lang w,S = { s where s is Element of E ^omega : w ==>* s,S } ;
theorem Th46:
theorem Th47:
theorem Th48:
theorem Th49:
theorem Th50:
theorem
begin
:: deftheorem Def9 defines are_equivalent_wrt REWRITE2:def 9 :
for E being set
for S, T being semi-Thue-system of E
for w being Element of E ^omega holds
( S,T are_equivalent_wrt w iff Lang w,S = Lang w,T );
theorem
theorem
theorem
theorem
theorem Th56:
theorem Th57:
theorem Th58:
theorem Th59:
theorem Th60:
theorem
theorem