#2073 ⟨a, b | abbaaaab=ba⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 10
Infinite cancellative non-commutative monoid
Submonoid of enveloping group: ⟨a, b | aaaabbab^-1a^-1b⟩

Embedding: φ(a) = a^-1, φ(b) = abaaa

Rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Auxiliary generators:
1. c = baaaa
2. d = ab
Reduction order:
- Left-to-right recursive path with deg(c) = deg(d) = 0, c < d; deg(b) = 1; deg(a) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	dc²b ⇒ cd	[10]
2.	(dc)³bd ⇒ cb	[9]
3.	(dc)⁴b ⇒ c	[5]
4.	d²(cd)³(dc)²bd² ⇒ cdb	[19]
5.	d(cd)³(dc)³bc ⇒ c²(dc)³b	[16]
6.	cbc²b ⇒ (dc)³bcd	[14]
7.	cbc(dc)³b ⇒ (dc)³bc	[11]
8.	c²b² ⇒ d(cd)³(dc)²bd²	[15]
9.	cdbc²b ⇒ d²(cd)³(dc)²bdcd	[24]
10.	cdbc(dc)³b ⇒ d²(cd)³(dc)²bdc	[21]
11.	dcb² ⇒ bd	[7]
12.	c³bc(dc)²b ⇒ dc(d(cd)²)³cbc	[29]
13.	c²(dc)²bdb ⇒ (dc)³(d(dc)²)²bd²	[28]
14.	(dc)³bcb ⇒ cbc(dc)²bd	[13]
15.	(dc)³bcdb ⇒ cbd(cd)³(dc)²bd²	[20]
16.	(dc)³bc³b ⇒ cb(cd)³(dc)³bcd	[18]
17.	(dc)³bc²(dc)³b ⇒ cb(cd)³(dc)³bc	[17]
18.	d²(cd)³(dc)²bcdb ⇒ cdb(cd)³(dc)²bd²	[25]
19.	d(dc)³d²c(dcb)² ⇒ cdbc(dc)²bd	[23]
20.	d²(cd)³(dc)²bdcdb ⇒ cdbd(cd)³(dc)²bd²	[26]
21.	d²(cd)³(dc)²bdc³b ⇒ cdb(cd)³(dc)³bcd	[32]
22.	d²(cd)³(dc)²bdc²(dc)³b ⇒ cdb(cd)³(dc)³bc	[31]
23.	cbcb² ⇒ (dc)²bd²	[12]
24.	cbc(dc)²bdb ⇒ dc(dcd)²cbd²	[27]
25.	cdbcb² ⇒ d²(cd)³(dc)²(bd)²	[22]
26.	cdbc(dc)²bdb ⇒ d(dc)³d²cbd²	[33]
27.	c³bcdcbdb ⇒ (dc)³(d(dc)²)³bd²	[30]
28.	ab ⇒ d	[3]
29.	ca ⇒ dc²	[8]
30.	da ⇒ adcb	[6]
31.	ba ⇒ dcb	[4]

# ab:abbaaaab=ba cd/b/a baaaa=c,ab=d morph:5/1,2/4
dccb=cd
dcdcdcbd=cb
dcdcdcdcb=c
ddcdcdcddcdcbdd=cdb
dcdcdcddcdcdcbc=ccdcdcdcb
cbccb=dcdcdcbcd
cbcdcdcdcb=dcdcdcbc
ccbb=dcdcdcddcdcbdd
cdbccb=ddcdcdcddcdcbdcd
cdbcdcdcdcb=ddcdcdcddcdcbdc
dcbb=bd
cccbcdcdcb=dcdcdcddcdcddcdcdcbc
ccdcdcbdb=dcdcdcddcdcddcdcbdd
dcdcdcbcb=cbcdcdcbd
dcdcdcbcdb=cbdcdcdcddcdcbdd
dcdcdcbcccb=cbcdcdcddcdcdcbcd
dcdcdcbccdcdcdcb=cbcdcdcddcdcdcbc
ddcdcdcddcdcbcdb=cdbcdcdcddcdcbdd
ddcdcdcddcdcbdcb=cdbcdcdcbd
ddcdcdcddcdcbdcdb=cdbdcdcdcddcdcbdd
ddcdcdcddcdcbdcccb=cdbcdcdcddcdcdcbcd
ddcdcdcddcdcbdccdcdcdcb=cdbcdcdcddcdcdcbc
cbcbb=dcdcbdd
cbcdcdcbdb=dcdcddcdcbdd
cdbcbb=ddcdcdcddcdcbdbd
cdbcdcdcbdb=ddcdcdcddcbdd
cccbcdcbdb=dcdcdcddcdcddcdcddcdcbdd
ab=d
ca=dcc
da=adcb
ba=dcb