#3182 ⟨a, b | abaaaabbaba=1⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 11
Infinite non-Abelian group
Inverses of generators:
- a^-1 = da²
- b^-1 = ca⁴
- c^-1 = a⁴b
- d^-1 = a³

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(a) = deg(d) = 0, a < d; deg(c) = deg(b) = 1, c < b
Auxiliary generators:
1. babaab=c
2. abc=d
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	ad ⇒ da	[22]
2.	da³ ⇒ 1	[23]
3.	bc ⇒ d²a²	[24]
4.	ca⁴b ⇒ 1	[32]
5.	b² ⇒ d²(a²c)²a³	[36]
6.	bab ⇒ c²a²	[21]
7.	ba²b ⇒ da²ca⁴c	[34]
8.	ba⁴b ⇒ daca³ca⁴	[47]
9.	ca⁴c² ⇒ abda	[41]
10.	c²a²c ⇒ bd	[26]
11.	ca²ca³c ⇒ a⁴bd²a²	[56]
12.	ca³ca⁴c ⇒ a²b	[39]
13.	ca⁴cb ⇒ abdaca²ca³	[46]
14.	ca⁴cab ⇒ aba²c²a²	[37]
15.	c²a²b ⇒ bdca²ca³	[38]
16.	ca⁴ca²b ⇒ abaca⁴c	[43]
17.	c²a³b ⇒ bac²a²	[27]
18.	ca²ca³b ⇒ a⁴bd²(a²c)²a³	[58]
19.	ca²ca⁵b ⇒ a⁴bda²ca⁴c	[54]
20.	ca³ca⁵b ⇒ a²ba⁴c²a²	[49]
21.	c²a⁶b ⇒ bda²ca³ca⁴	[51]
22.	ca³ca⁶b ⇒ a²ba³ca⁴c	[45]
23.	ca²ca⁷b ⇒ a⁴bdaca³ca⁴	[59]
24.	ca³ca⁸b ⇒ a²ba²ca³ca⁴	[50]

# ab:abaaaabbaba=1 reversed:ad/cb babaab=c,abc=d morph:6/0,3/0
ad=da
daaa=1
bc=ddaa
caaaab=1
bb=ddaacaacaaa
bab=ccaa
baab=daacaaaac
baaaab=dacaaacaaaa
caaaacc=abda
ccaac=bd
caacaaac=aaaabddaa
caaacaaaac=aab
caaaacb=abdacaacaaa
caaaacab=abaaccaa
ccaab=bdcaacaaa
caaaacaab=abacaaaac
ccaaab=baccaa
caacaaab=aaaabddaacaacaaa
caacaaaaab=aaaabdaacaaaac
caaacaaaaab=aabaaaaccaa
ccaaaaaab=bdaacaaacaaaa
caaacaaaaaab=aabaaacaaaac
caacaaaaaaab=aaaabdacaaacaaaa
caaacaaaaaaaab=aabaacaaacaaaa