#3265 ⟨a, b | ababbbbabba=1⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 11
Infinite non-Abelian group
Inverses of generators:
- a^-1 = b⁴cb
- b^-1 = bd
- c^-1 = bab⁴
- d^-1 = b²

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

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

#	Rule	Proof
1.	db ⇒ bd	[18]
2.	b²d ⇒ 1	[21]
3.	cba ⇒ d²	[24]
4.	ab⁴c ⇒ bd	[19]
5.	cbc ⇒ da²	[5]
6.	cb³c ⇒ d²abab³	[32]
7.	cb⁴c ⇒ ab²abd	[20]
8.	cb⁵c ⇒ bd²ab⁴ab	[50]
9.	ab²a² ⇒ c	[2]
10.	a²ba ⇒ b²cbd²	[26]
11.	ab⁴ab²a ⇒ b³cb	[39]
12.	abab⁴a ⇒ b⁴cbd	[38]
13.	ab²ac ⇒ cb²a²	[4]
14.	a²bc ⇒ b²cbda²	[23]
15.	(ab²)²c ⇒ cabab³	[34]
16.	ab⁴ab²c ⇒ b³cb³a²	[41]
17.	a²b³c ⇒ b²cbd²abab³	[49]
18.	ab²ab³c ⇒ cb³(bab)²d	[29]
19.	a²b⁵c ⇒ b²cdab⁴ab	[51]
20.	ab⁴ab⁵c ⇒ b³cb⁴(bab)²d	[45]
21.	abab⁶c ⇒ b⁴cbdabab³	[53]
22.	ab⁴ab⁶c ⇒ b³cb²ab⁴ab	[46]
23.	abab⁷c ⇒ b⁴cb(b²a)²bd	[40]
24.	abab⁸c ⇒ b⁴cab⁴ab	[55]

# ab:ababbbbabba=1 reversed:db/ac abbaa=c,cbabb=d morph:5/0,5/0
db=bd
bbd=1
cba=dd
abbbbc=bd
cbc=daa
cbbbc=ddababbb
cbbbbc=abbabd
cbbbbbc=bddabbbbab
abbaa=c
aaba=bbcbdd
abbbbabba=bbbcb
ababbbba=bbbbcbd
abbac=cbbaa
aabc=bbcbdaa
abbabbc=cababbb
abbbbabbc=bbbcbbbaa
aabbbc=bbcbddababbb
abbabbbc=cbbbbabbabd
aabbbbbc=bbcdabbbbab
abbbbabbbbbc=bbbcbbbbbabbabd
ababbbbbbc=bbbbcbdababbb
abbbbabbbbbbc=bbbcbbabbbbab
ababbbbbbbc=bbbbcbbbabbabd
ababbbbbbbbc=bbbbcabbbbab