#3277 ⟨a, b | abbaabaaaab=1⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 11
Infinite non-Abelian group
Inverses of generators:
- a^-1 = da
- b^-1 = aca⁴
- c^-1 = a⁴ba
- d^-1 = a²

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

#	Rule	Proof
1.	ad ⇒ da	[13]
2.	da² ⇒ 1	[14]
3.	bac ⇒ d²	[16]
4.	ca⁴b ⇒ da	[8]
5.	cac ⇒ b²d	[17]
6.	ca³c ⇒ a³babd²	[26]
7.	ca⁴c ⇒ daba²b	[10]
8.	ca⁵c ⇒ aba⁴bd²a	[44]
9.	b²a²b ⇒ c	[2]
10.	bab² ⇒ d²aca²	[19]
11.	ba²ba⁴b ⇒ aca³	[33]
12.	ba⁴bab ⇒ daca⁴	[32]
13.	cba²b ⇒ b²a²c	[5]
14.	cab² ⇒ b²daca²	[20]
15.	c(a²b)² ⇒ a³babc	[28]
16.	ca²ba⁴b ⇒ b²a³ca³	[35]
17.	ca³b² ⇒ a³babd²aca²	[43]
18.	ca(a²b)² ⇒ daba²ba⁴c	[23]
19.	ca⁵b² ⇒ aba⁴bdca²	[45]
20.	ca(a⁴b)² ⇒ daba²ba⁵ca³	[37]
21.	ca⁶bab ⇒ a³babdaca⁴	[47]
22.	ca⁶ba⁴b ⇒ aba⁴ba²ca³	[40]
23.	ca⁷bab ⇒ daba²ba³ca⁴	[34]
24.	ca⁸bab ⇒ aba⁴bca⁴	[49]

# ab:abbaabaaaab=1 da/bc bbaab=c,bacaa=d morph:5/0,5/0
ad=da
daa=1
bac=dd
caaaab=da
cac=bbd
caaac=aaababdd
caaaac=dabaab
caaaaac=abaaaabdda
bbaab=c
babb=ddacaa
baabaaaab=acaaa
baaaabab=dacaaaa
cbaab=bbaac
cabb=bbdacaa
caabaab=aaababc
caabaaaab=bbaaacaaa
caaabb=aaababddacaa
caaabaab=dabaabaaaac
caaaaabb=abaaaabdcaa
caaaaabaaaab=dabaabaaaaacaaa
caaaaaabab=aaababdacaaaa
caaaaaabaaaab=abaaaabaacaaa
caaaaaaabab=dabaabaaacaaaa
caaaaaaaabab=abaaaabcaaaa