#2087 ⟨a, b | abbbaaab=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 | aaabbbab^-1a^-1b⟩

Embedding: φ(a) = b, φ(b) = a

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 = abb
2. d = cbaaa
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.	cd³b ⇒ d	[5]
2.	(cd²)²bc ⇒ db	[10]
3.	(cd²)³bd ⇒ d⁴b	[15]
4.	c²(d²c)²dcd²bc² ⇒ dcb	[20]
5.	db² ⇒ cd²bc	[9]
6.	dbd³b ⇒ (cd²)²bd	[12]
7.	dcbd³b ⇒ c²(d²c)²dcd²bcd	[22]
8.	cd²bcb ⇒ bc	[11]
9.	d³bcb ⇒ c(d²c)²dcd²bc²	[18]
10.	d⁴bd²b ⇒ cd(dcd²c)²d²bd	[17]
11.	(cd²)²bdb ⇒ dbd²cd²bc	[13]
12.	(cd²)²bdcb ⇒ dbc(d²c)²dcd²bc²	[21]
13.	(cd²)²bd⁴b ⇒ dbd²(cd²)²bd	[16]
14.	c²(d²c)²dcd²bcdb ⇒ dcbd²cd²bc	[23]
15.	c²(d²c)²dcd²bdcb ⇒ dcb(d²c)²dcd²bc²	[26]
16.	c²(d²c)²dcd²bcdcb ⇒ dcbc(d²c)²dcd²bc²	[27]
17.	c²(d²c)²dcd²bcd⁴b ⇒ dcbd²(cd²)²bd	[25]
18.	dbd²bcb ⇒ cd(dc)²d²bc²	[14]
19.	dcbd²bcb ⇒ c²(d²c)²dbc²	[24]
20.	d⁴bdbcb ⇒ (cd²)²(cdcd²)²bc²	[19]
21.	ab² ⇒ c	[2]
22.	ca ⇒ abdb	[6]
23.	da ⇒ cd⁴b	[8]
24.	ba ⇒ db	[4]

# ab:abbbaaab=ba cd/b/a abb=c,cbaaa=d morph:3/2,5/6
cdddb=d
cddcddbc=db
cddcddcddbd=ddddb
ccddcddcdcddbcc=dcb
dbb=cddbc
dbdddb=cddcddbd
dcbdddb=ccddcddcdcddbcd
cddbcb=bc
dddbcb=cddcddcdcddbcc
ddddbddb=cddcddcdcddcddbd
cddcddbdb=dbddcddbc
cddcddbdcb=dbcddcddcdcddbcc
cddcddbddddb=dbddcddcddbd
ccddcddcdcddbcdb=dcbddcddbc
ccddcddcdcddbdcb=dcbddcddcdcddbcc
ccddcddcdcddbcdcb=dcbcddcddcdcddbcc
ccddcddcdcddbcddddb=dcbddcddcddbd
dbddbcb=cddcdcddbcc
dcbddbcb=ccddcddcdbcc
ddddbdbcb=cddcddcdcddcdcddbcc
abb=c
ca=abdb
da=cddddb
ba=db