Introduction
We begin by defining the basic objects of study, including the algebraic definition of braids, the Lean theorem prover, and the subword reversing algorithm. At the end of this chapter, we give an overview of the entire project.
Mathematical Braids
Braids were given their first full definition and exposition as a mathematical object by Emil Artin in 1925 [2 ] . He gave both a topological and algebraic definition of these physical objects. A physical braid is a set of strings in three-dimensional space, affixed at top and bottom to the “floor” and “ceiling.” These strings can be twisted together or crossed, but must always flow downwards. No doubling-back is allowed. This is the sort of image to keep in mind:
Figure 6.
The topological definition lies outside the scope of this project. We work solely with Artin’s algebraic definition, occasionally using images of the corresponding physical objects to aid in exposition.
Definition 1.1 (Presented Group). Consider a set S S S R R R S S S F S F_S F S presented group ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩ F S F_S F S R R R R R R
Definition 1.2. Artin’s braid group on n n n
B n : = ⟨ σ 1 , … , σ n − 1 ∣ σ i σ j = σ j σ i for ∣ i − j ∣ ≥ 2 , σ i σ j σ i = σ j σ i σ j for ∣ i − j ∣ = 1 ⟩ B_n := \left\langle\sigma_1,\ldots,\sigma_{n-1} \mathrel{\left| \begin{aligned} \sigma_i\sigma_j &= \sigma_j\sigma_i && \text{for } |i-j| \ge2,\\ \sigma_i\sigma_j\sigma_i &= \sigma_j\sigma_i\sigma_j && \text{for } |i-j| = 1 \end{aligned}\right.}\right\rangle B n := ⟨ σ 1 , … , σ n − 1 σ i σ j σ i σ j σ i = σ j σ i = σ j σ i σ j for ∣ i − j ∣ ≥ 2 , for ∣ i − j ∣ = 1 ⟩ The infinite braid group B ∞ B_\infty B ∞ B ∞ : = ⋃ n ∈ N B n B_\infty:= \bigcup_{n\in\mathbb{N}} B_n B ∞ := ⋃ n ∈ N B n
B ∞ : = ⟨ σ 1 , σ 2 , σ 3 … σ i σ j = σ j σ i for ∣ i − j ∣ ≥ 2 , σ i σ j σ i = σ j σ i σ j for ∣ i − j ∣ = 1 ⟩ B_{\infty} := \left\langle\begin{array}{c|c} \sigma_1,\sigma_2,\sigma_3\ldots& \begin{aligned} \sigma_i\sigma_j&=\sigma_j\sigma_i && \text{for } |i-j|\ge2,\\ \sigma_i\sigma_j\sigma_i&=\sigma_j\sigma_i\sigma_j && \text{for } |i-j|=1 \end{aligned} \end{array} \right\rangle B ∞ := ⟨ σ 1 , σ 2 , σ 3 … σ i σ j σ i σ j σ i = σ j σ i = σ j σ i σ j for ∣ i − j ∣ ≥ 2 , for ∣ i − j ∣ = 1 ⟩ The generator σ i \sigma_i σ i i i i 1 ≤ i < n 1\le i<n 1 ≤ i < n
Figure 7 (PDF p. 4).
σ i − 1 \sigma_i^{-1} σ i − 1 i i i
Figure 8.
For example, the braid at the opening of this chapter is coded σ 1 − 1 σ 3 σ 2 − 1 σ 1 σ 2 − 1 σ 1 σ 3 σ 2 \sigma_1^{-1}\sigma_3\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_1\sigma_3\sigma_2 σ 1 − 1 σ 3 σ 2 − 1 σ 1 σ 2 − 1 σ 1 σ 3 σ 2
The physical representation shows how inverses untangle: here is σ i − 1 σ i \sigma_i^{-1}\sigma_i σ i − 1 σ i
Figure 9.
is equivalent to
Figure 10.
The defining relations of the braid group can be visualized as follows:
Crossings far apart can be slid up and down (if ∣ i − j ∣ ≥ 2 |i-j|\ge2 ∣ i − j ∣ ≥ 2 σ i σ j \sigma_i\sigma_j σ i σ j σ j σ i \sigma_j\sigma_i σ j σ i
Figure 11.
is equivalent to
Figure 12.
An adjacent strand can be pulled through a crossing (σ i σ i + 1 σ i \sigma_i\sigma_{i+1}\sigma_i σ i σ i + 1 σ i σ i + 1 σ i σ i + 1 \sigma_{i+1}\sigma_i\sigma_{i+1} σ i + 1 σ i σ i + 1
Figure 13.
is equivalent to
Figure 14.
Of course, to rigorously show that these relations are sufficient to capture equivalence in the physical objects, one must give a topological definition and the proof that the algebraic and topological definitions coincide. The interested reader can find a modern exposition in Patrick Dehornoy’s textbook “The Calculus of Braids”, chapters one and two [16 ] .
The Lean Theorem Prover
Lean is a functional programming language and interactive theorem prover based on dependent type theory. Leonardo de Moura launched Lean in 2013; the language is now on version 4 [8 ] . Mathematicians worldwide use Lean to formally verify proofs across mathematics, from undergraduate basics to the cutting edge of research. In Lean, one can implement algorithms and execute them using the #eval command along with stating and proving theorems about their properties, all in the same file. While algorithms of course must be computable in order to evaluate, Lean is not limited to computable functions or constructive proofs. One can choose to use Lean with constructive or classical logic. When proving correctness of an algorithm, for example, the algorithm must be computable but the proofs of correctness and termination need not be.
A large collaborative project to formalize mathematics in one central repository, named Mathlib, is ongoing, and currently counts 772 contributors [34 ] .1 2 [33 , 31 , 6 ] , including basic definitions in Lean3 4 [19 ] , compared to polynomial-time complexity for braid groups [13 ] . Thus, a connection to category theory does not simplify the project, and lies beyond the scope of this paper. Knots and a few invariants thereof were formalized in Isabelle/HOL by Prathamesh in 2015 [32 ] . However, the mathematics used to study braids (presented groups, localizations, orderings) differs from that used for knots (polynomial invariants). There are also a number of computational packages for working with braids, for example in MATLAB [35 ] , SAGE5 [29 ] . Lastly, two theorems about braid groups (unrelated to this project) recently appeared on Prove2Me,6 7
As to the formalization of a word problem decision procedure, existing similar work is limited. A recent paper on decidability in presented monoids does offer a (slow) Rocq algorithm (homog_dec) which can be used for any presented monoid whose relations preserve length [7 ] . This could be applied to determine equivalence in braid monoids but cannot be extended to braid groups.
Word Problem Algorithm
The word problem in a group is to determine whether two representatives represent the same equivalence class. In general, the word problem of a presented group is unsolvable [5 , 30 ] . For the braid group in particular, this problem is solvable in polynomial time. A large number of algorithms exist, in part prompted by the early-2000s hope that braid group problems could be used for post-quantum security protocols [27 , 10 ] . Early algorithms with exponential complexity included Artin combing [1 ] and Garside’s method of universal denominators [21 ] , although both methods have subsequently received updates which run in polynomial time [22 , 36 ] . A more modern solution by Birman, Ko, and Lee introduced an alternative presentation of the braid group, yielding a normal form with computational advantages [4 ] . Other methods such as Dynnikov coordinates [20 , 13 ] and the Lawrence-Krammer representation [26 , 3 ] give exceptionally simple algorithms which work quite efficiently, but the proofs of correctness involve advanced topology. Dehornoy’s most recent algorithm is handle reduction, which is experimentally quite fast (but there is no proof of polynomial-time complexity) [13 ] .
For this project, we have implemented Dehornoy’s earlier method: subword reversing (as introduced in a journal article [12 ] and elucidated in his textbook [16 ] ). While this may not be the fastest method, it runs in quadratic time, relies on a manageable amount of theory, and could be generalized to larger classes of groups [15 ] . Moreover, this solution builds infrastructure that could be reused in a future implementation of other algorithms, such as handle reduction.
The primary contribution lies in the treatment of this algorithm’s termination. Dehornoy wrote about this algorithm over a period of approximately twenty-five years. Most of Dehornoy’s work presents a compelling sketch of why subword reversing terminates via associating rectangular grid diagrams to individual instances of reversing, but does not give a proof for this method. Bridging this gap required the development of a new proof of termination, done in three stages. It does appear that in one of Dehornoy’s earliest papers he gives a different proof which may be fully rigorous as written. This proof is quite delicate (we will discuss this further in Chapter § 6 ). It seems that Dehornoy preferred the visual intuition and simplicity of the grid-based method, so we are pleased to have been able to fill in this method.
The heart of the algorithm is a reversing procedure on braid words. To underscore that we are working with words and not braid group elements, we use σ ˉ i \bar{\sigma}_i σ ˉ i σ i − 1 \sigma_i^{-1} σ i − 1 σ ˉ i σ j \bar{\sigma}_i\sigma_j σ ˉ i σ j
If ∣ i − j ∣ = 0 |i-j|=0 ∣ i − j ∣ = 0 σ ˉ i σ i \bar{\sigma}_i\sigma_i σ ˉ i σ i ε \varepsilon ε
If ∣ i − j ∣ > 1 |i-j|>1 ∣ i − j ∣ > 1 σ ˉ i σ j \bar{\sigma}_i\sigma_j σ ˉ i σ j σ j σ ˉ i \sigma_j\bar{\sigma}_i σ j σ ˉ i
If ∣ i − j ∣ = 1 |i-j|=1 ∣ i − j ∣ = 1 σ ˉ i σ j \bar{\sigma}_i\sigma_j σ ˉ i σ j σ j σ i σ ˉ j σ ˉ i \sigma_j\sigma_i\bar{\sigma}_j\bar{\sigma}_i σ j σ i σ ˉ j σ ˉ i
Note that these rewritings preserve equivalence under the braid relations. The proofs are straightforward algebraic manipulations; here is an instance of the third case, with numerical indices for readability:
σ ˉ 2 σ 3 ≡ σ ˉ 2 σ 3 σ 2 σ ˉ 2 (as ε ≡ σ 2 σ ˉ 2 ) ≡ σ ˉ 2 σ 3 σ 2 σ 3 σ ˉ 3 σ ˉ 2 (as ε ≡ σ 3 σ ˉ 3 ) ≡ σ ˉ 2 σ 2 σ ˉ 3 σ 2 σ 3 σ ˉ 2 (as σ 3 σ 2 σ 3 ≡ σ 2 σ 3 σ 2 ) ≡ σ 3 σ 2 σ ˉ 3 σ ˉ 2 (as σ ˉ 2 σ 2 ≡ ε ) \begin{aligned}
\bar{\sigma}_2\sigma_3 &\equiv\bar{\sigma}_2\sigma_3\sigma_2\bar{\sigma}_2 && \text{(as }\varepsilon\equiv\sigma_2\bar{\sigma}_2\text{)}\\
&\equiv\bar{\sigma}_2\sigma_3\sigma_2\sigma_3\bar{\sigma}_3\bar{\sigma}_2 && \text{(as }\varepsilon\equiv\sigma_3\bar{\sigma}_3\text{)}\\
&\equiv\bar{\sigma}_2\sigma_2\bar{\sigma}_3\sigma_2\sigma_3\bar{\sigma}_2 && \text{(as }\sigma_3\sigma_2\sigma_3\equiv\sigma_2\sigma_3\sigma_2\text{)}\\
&\equiv\sigma_3\sigma_2\bar{\sigma}_3\bar{\sigma}_2 && \text{(as }\bar{\sigma}_2\sigma_2\equiv\varepsilon\text{)}
\end{aligned} σ ˉ 2 σ 3 ≡ σ ˉ 2 σ 3 σ 2 σ ˉ 2 ≡ σ ˉ 2 σ 3 σ 2 σ 3 σ ˉ 3 σ ˉ 2 ≡ σ ˉ 2 σ 2 σ ˉ 3 σ 2 σ 3 σ ˉ 2 ≡ σ 3 σ 2 σ ˉ 3 σ ˉ 2 (as ε ≡ σ 2 σ ˉ 2 ) (as ε ≡ σ 3 σ ˉ 3 ) (as σ 3 σ 2 σ 3 ≡ σ 2 σ 3 σ 2 ) (as σ ˉ 2 σ 2 ≡ ε ) Now, let us show reversing in action. We begin with σ ‾ 1 σ 0 σ ‾ 2 σ 1 \overline{\sigma}_1\sigma_0\overline{\sigma}_2\sigma_1 σ 1 σ 0 σ 2 σ 1
σ ‾ 1 σ 0 σ ‾ 2 σ 1 σ 0 σ 1 σ ‾ 0 σ ‾ 1 σ ‾ 2 σ 1 σ 0 σ 1 σ ‾ 0 σ ‾ 1 σ 1 σ 2 σ ‾ 1 σ ‾ 2 σ 0 σ 1 σ ‾ 0 σ 2 σ 1 σ ‾ 2 σ 0 σ 1 σ 2 σ ‾ 0 σ ‾ 1 σ ‾ 2 \begin{aligned}
\overline{\sigma}_1\sigma_0\overline{\sigma}_2\sigma_1 \\
\sigma_0\sigma_1\overline{\sigma}_0\overline{\sigma}_1\overline{\sigma}_2\sigma_1 \\
\sigma_0\sigma_1\overline{\sigma}_0\overline{\sigma}_1\sigma_1\sigma_2\overline{\sigma}_1\overline{\sigma}_2 \\
\sigma_0\sigma_1\overline{\sigma}_0\sigma_2\sigma_1\overline{\sigma}_2 \\
\sigma_0\sigma_1\sigma_2\overline{\sigma}_0\overline{\sigma}_1\overline{\sigma}_2
\end{aligned} σ 1 σ 0 σ 2 σ 1 σ 0 σ 1 σ 0 σ 1 σ 2 σ 1 σ 0 σ 1 σ 0 σ 1 σ 1 σ 2 σ 1 σ 2 σ 0 σ 1 σ 0 σ 2 σ 1 σ 2 σ 0 σ 1 σ 2 σ 0 σ 1 σ 2 So, reversing σ ‾ 1 σ 0 σ ‾ 2 σ 1 \overline{\sigma}_1\sigma_0\overline{\sigma}_2\sigma_1 σ 1 σ 0 σ 2 σ 1 σ 0 σ 1 σ 2 σ ‾ 0 σ ‾ 1 σ ‾ 2 \sigma_0\sigma_1\sigma_2\overline{\sigma}_0\overline{\sigma}_1\overline{\sigma}_2 σ 0 σ 1 σ 2 σ 0 σ 1 σ 2
To determine if two braid group words are equivalent, we may simply check if a b − 1 ab^{-1} a b − 1
Outline
Throughout the text, clickable Lean icons in the righthand margin link to the relevant lines in the GitHub formalization. The repository, frozen at the time of this paper’s writing, can be found at https://github.com/hannahfechtner/braids_better/tree/098309403cda787eaf1fa954e07ff0ca3b2927ee . Information about the Lean and Mathlib versions needed to build the code are available in the README file.
Chapters 2–5 discuss formalization of known results, following Patrick Dehornoy’s textbook The Calculus of Braids [16 ] . Chapters 6–8 expand on Dehornoy’s work and contain new material necessary to implement his subword reversing algorithm in a theorem prover.
Chapter 2 develops the algebraic framework needed for the reversing algorithm, including the braid group, braid monoids, and Semi-Thue systems. Before we implement the algorithm itself, we spend the next five chapters preparing theorems which will be used to show its termination and correctness.
Chapters 3–5 prove the localization result that connects equivalence in the braid monoid with equivalence in the braid group, a connection that will be essential to the proof of correctness. Chapter 3 develops the necessary theory of Ore localization. Chapters 4 and 5 then show that the braid monoid satisfies the hypotheses required for this localization: Chapter 4 formalizes Dehornoy’s proof of the existence of common multiples, while Chapter 5 proves cancellativity. The proof of cancellativity introduces grid diagrams, which encode braid monoid equivalences and will play a central role throughout the remainder of the project.
Chapters 6 and 7 develop the structures and metrics needed to prove termination of the reversing algorithm. Chapter 6 extends the interpretation of grid diagrams by introducing partial grids and grid-style rewriting, and establishes the correspondence between these new structures and reversing rewrites. Chapter 7 defines metrics on these structures. The material in these two chapters will yield a well-founded measure that decreases as reversing proceeds, providing the basis for the termination proof. The constructions and results developed in these two chapters appear, to the best of my knowledge, to be novel.
Chapter 8 at last brings all these strands together in the implementation of the subword reversing algorithm, first for braid monoid words and then for braid group words. The results of the preceding chapters are combined to prove correctness and termination. The procedure is also extended to the finite braid group. Finally, Chapter 9 concludes with directions for future work.
Defining Braids
This chapter develops the algebraic structures that will be used throughout the remainder of the paper. We begin by defining Artin–Tits groups and specializing this construction to Artin’s infinite braid group. We then turn to two structures necessary for subword reversing. We begin with presented monoids, and we define the infinite braid monoid and establish its basic properties. We next discuss Semi-Thue systems, which we will use to model string rewriting. Together, these definitions and results provide the algebraic foundation for the subword reversing algorithm. Lastly, we return to all the structures discussed, this time in the finite setting. We introduce the finite braid groups and monoids and develop mappings between the infinite and finite braid group/monoids.
Artin-Tits Groups
Braid groups are a special case of Artin-Tits groups.
Definition 2.1. Given a semigroup G G G a , b ∈ G a,b \in G a , b ∈ G Artin-Tits relations are those of the form a b a … = b a b … aba \ldots= bab \ldots aba … = bab …
Definition 2.2. Artin-Tits groups are presented groups ⟨ S ∣ R ⟩ \langle S|R\rangle ⟨ S ∣ R ⟩ R R R m : ( S × S ) → N m : (S \times S) \to\mathbb{N} m : ( S × S ) → N
a b a ⋯ ⏟ length m ( a , b ) = b a b ⋯ ⏟ length m ( a , b ) \underbrace{aba\cdots}_{\text{length }m(a,b)} = \underbrace{bab\cdots}_{\text{length }m(a,b)} length m ( a , b ) aba ⋯ = length m ( a , b ) bab ⋯ for generators a , b ∈ S a,b \in S a , b ∈ S S S S a a a b b b m ( a , b ) : = 0 m(a,b) := 0 m ( a , b ) := 0 8
Artin-Tits groups are very similar to Coxeter groups, which have the additional property that for every generator a a a a 2 = 1 a^2 = 1 a 2 = 1 9
In Lean, following Jensen, Lee, and Álvarez, we give the set of generators as the type α \alpha α m m m ArtinTitsMatrix structure. The off_diagonal property ensures that different generators are distinct.
structure ArtinTitsMatrix (α : Type*) where
M : Matrix α α ℕ
isSymm : M.IsSymm := by decide
off_diagonal i i’ : i ≠ i’ → M i i’ ≠ 1 := by decideA simple recursive definition produces an alternating string of group elements.10
def Group.alternate [Monoid G] (a b : G) (n : ℕ) : G := match n with
| 0 => 1
| n’ + 1 => a * alternate b a n’We generate the Artin-Tits relation for any two generators with ArtinTits.relation; instead of giving an equality, we give a free group element equivalent to the identity:
( a b a ⋯ ⏟ length m ( a , b ) ) ⋅ ( b a b ⋯ ⏟ length m ( a , b ) ) − 1 \left( \underbrace{aba\cdots}_{\text{length }m(a,b)} \right) \cdot\left( \underbrace{bab\cdots}_{\text{length }m(a,b)} \right)^{-1} length m ( a , b ) aba ⋯ ⋅ length m ( a , b ) bab ⋯ − 1 The occurrences of FreeGroup.of in the below definition lift an element of a set to the corresponding free group element.
def ArtinTits.relation (M : ArtinTitsMatrix α) (a b : α) : FreeGroup (α) :=
Group.alternate (FreeGroup.of a) (FreeGroup.of b) (M a b) * (Group.alternate
(FreeGroup.of b) (FreeGroup.of a) (M a b))⁻¹We gather all the relations into a single set:
def ArtinTits.relation_set : Set (FreeGroup α) :=
Set.range (Function.uncurry (relation M))Finally, using the already-existing PresentedGroup structure from Mathlib, we define ArtinTitsGroup which will build the corresponding Artin-Tits group given a matrix m m m
def ArtinTitsGroup (M : ArtinTitsMatrix α) := PresentedGroup (relation_set M)At this point, we must decide whether or not to duplicate the PresentedGroup API, specializing it to Artin-Tits groups. If the construction were not canonical, we would duplicate the API so that their implementation would be opaque to users. Here, the definition as a presented group is canonical, and one can expect the user to know that an Artin-Tits group is a presented group. Often in this case no custom API is given, especially when the code is intended to go into Mathlib, for this leads to unnecessary extra code to maintain. There is a bit of leeway for non-Mathlib code, and also when the specialization leads to simpler hypotheses for certain theorems. The latter is true for the universal property for Artin-Tits groups. We recall the universal property for presented groups in general :
Theorem 2.3. (Universal Property for Presented Groups) Given a set of generators S S S R R R S ∗ S^* S ∗ ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩ G G G f : S → G f : S \to G f : S → G R R R g : ⟨ S ∣ R ⟩ → G g : \langle S \mid R\rangle\to G g : ⟨ S ∣ R ⟩ → G g ( ⟦ s ⟧ R ) = f ( s ) g(\llbracket s\rrbracket_R) = f(s) g ( [ [ s ] ] R ) = f ( s ) s ∈ S s \in S s ∈ S
Here, we decide to implement a custom universal property for the Artin-Tits groups, as we can simplify the hypothesis that the function to be lifted respects the relations.
Theorem 2.4. (Universal Property for Artin-Tits Groups) Given a set of generators S S S R R R m m m R R R ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩ G G G f : S → G f : S \to G f : S → G a , b ∈ S a,b \in S a , b ∈ S
f ( a ) f ( b ) f ( a ) ⋯ ⏟ length m ( a , b ) = f ( b ) f ( a ) f ( b ) ⋯ ⏟ length m ( a , b ) \underbrace{f(a)\,f(b)\,f(a)\cdots}_{\text{length }m(a,b)}
=
\underbrace{f(b)\,f(a)\,f(b)\cdots}_{\text{length }m(a,b)} length m ( a , b ) f ( a ) f ( b ) f ( a ) ⋯ = length m ( a , b ) f ( b ) f ( a ) f ( b ) ⋯ there exists a unique group homomorphism g : ⟨ S ∣ R ⟩ → G g : \langle S \mid R\rangle\to G g : ⟨ S ∣ R ⟩ → G g ( ⟦ s ⟧ R ) = f ( s ) g(\llbracket s\rrbracket_R) = f(s) g ( [ [ s ] ] R ) = f ( s ) s ∈ S s \in S s ∈ S
In Lean, we can phrase this property in terms of the matrix m m m Group.alternate, without needing to reference ArtinTits.relation and ArtinTits.relation_set; we simply require that the function f f f
def ArtinTits.IsLiftable (M : ArtinTitsMatrix α) [Group G] (f : α → G) : Prop :=
∀ i j, Group.alternate (f i) (f j) (M i j) = Group.alternate (f j) (f i) (M i j)Then we can prove that IsLiftable functions satisfy the liftability requirement for the presented group universal property. We do so by unpacking the definition of ArtinTits.relation_set and ArtinTits.relation; that way, they may remain opaque to the user. Then our custom universal property needs only an instance of IsLiftable:
def ArtinTits.toGroup {G : Type*} [Group G] {f : α → G} (M : ArtinTitsMatrix α)
(hf : IsLiftable M f) : ArtinTitsGroup M →* GBy contrast, we do not duplicate the rest of the PresentedGroup API (such as extensionality and induction principles) as doing so would not simplify theorems for the user.
The Braid Group
We now implement the specific case of the “infinite” braid group, hereafter simply referred to as the braid group. The set of generators is S = { σ i ∣ i ∈ N } S = \{\sigma_i \mid i \in\mathbb{N}\} S = { σ i ∣ i ∈ N } i i i index of the generator σ i \sigma_i σ i
Definition 2.5. The braid relation R B R_B R B i , j ∈ N i, j \in\mathbb{N} i , j ∈ N
If ∣ i − j ∣ = 1 \lvert i - j\rvert= 1 ∣ i − j ∣ = 1 R B ( σ i σ j σ i , σ j σ i σ j ) R_B(\sigma_i\sigma_j\sigma_i, \sigma_j\sigma_i\sigma_j) R B ( σ i σ j σ i , σ j σ i σ j )
If ∣ i − j ∣ ≥ 2 \lvert i - j\rvert\ge2 ∣ i − j ∣ ≥ 2 R B ( σ i σ j , σ j σ i ) R_B(\sigma_i\sigma_j, \sigma_j\sigma_i) R B ( σ i σ j , σ j σ i )
That is, if i = j i = j i = j ∣ i − j ∣ = 1 \lvert i - j\rvert= 1 ∣ i − j ∣ = 1
def BraidMatrixInf : ArtinTitsMatrix ℕ where
M := Matrix.of fun i j : ℕ ↦
if i = j then 0 else (if i.dist j = 1 then 3 else 2)
isSymm := by
grind [Matrix.IsSymm, Matrix.transpose, Matrix.of_apply,
EmbeddingLike.apply_eq_iff_eq, Nat.dist]
off_diagonal := by aesopdef BraidGroupInf := ArtinTitsGroup BraidMatrixInfIn our mathematical writeup, we have let σ i \sigma_i σ i S S S S S S R R R ⟦ σ i ⟧ R B \llbracket\sigma_i \rrbracket_{R_B} [ [ σ i ] ] R B (i : ℕ) as a generator. We write FreeGroup.of i for the free group element. Finally, (PresentedGroup.of i : BraidGroupInf) will give us the braid group element. We abbreviate (PresentedGroup.of i : BraidGroupInf) by σ \sigma σ
def σ (k : ℕ) : BraidGroupInf := PresentedGroup.of kWhile convenient in Lean, σ i \sigma_i σ i σ i \sigma_i σ i ( σ i ) (\sigma\ i) ( σ i ) ( σ i ) (\sigma_i) ( σ i )
For the braid group, we give a specialized API. Again, the construction of the braid group as a presented group is not meant to be hidden. The custom API will make the rest of the project move more smoothly — no need to constantly unfold the definition of BraidGroupInf. For example, for the universal property, we may omit mention of BraidMatrixInf and simplify the required hypothesis to
def BraidGroupInf.IsLiftable {G : Type*} [Group G] (f : N → G) : Prop :=
(∀ i j : N, i.dist j = 1 → f i * f j * f i = f j * f i * f j) ∧
(∀ i j : N, 2 ≤ i.dist j → f i * f j = f j * f i)Beyond specializations of the universal property, induction principles, et cetera, we moreover prove useful lemmas such as ⟦ σ i + 1 σ i σ i + 1 ⟧ = ⟦ σ i σ i + 1 σ i ⟧ \llbracket\sigma_{i+1}\sigma_i\sigma_{i+1}\rrbracket= \llbracket\sigma_i\sigma_{i+1}\sigma_i\rrbracket [ [ σ i + 1 σ i σ i + 1 ] ] = [ [ σ i σ i + 1 σ i ] ]
theorem BraidGroupInf.braid {i : N} :
σ i * σ (i+1) * σ i = σ (i+1) * σ i * σ (i+1)Such lemmas are mathematically immediate, but in Lean require careful traversal of the structures at play (quotient groups, congruences, setoids, etc.).
Presented Monoids
We next consider a structure akin to presented groups, but lacking inverses: presented monoids. These may be defined as a free monoid quotiented by the congruence generated by a given relation. That is the implementation chosen here, as we are able to build upon the Congruence structure which already is in Mathlib as well as the rich API surrounding quotients.
Definition 2.6 (Presented Monoid). We let ⟨ S ∣ R ⟩ + \langle S \mid R\rangle^{+} ⟨ S ∣ R ⟩ + S S S R R R S S S R R R R R R x ∈ S ∗ x \in S^{*} x ∈ S ∗ x x x ⟦ x ⟧ R + \llbracket x\rrbracket_{R}^{+} [ [ x ] ] R + R R R x , y ∈ S ∗ x,y \in S^{*} x , y ∈ S ∗ x ≡ R y x \mathrel{\equiv_{R}} y x ≡ R y R R R
In Lean, we first must give a formal definition of S S S FreeMonoid is the only type in Mathlib that has been given a Mul structure. So, we represent the S S S FreeMonoid instead of List.
Then, given a set of relations rels over a type α \alpha α PresentedMonoid constructs the monoid with generators x : α and relations rels as a quotient of a congruence relation over rels. Here conGen generates a congruence.
def PresentedMonoid (rels : FreeMonoid α → FreeMonoid α → Prop) := (conGen
rels).QuotientWe provide basic properties of the presented monoid, such as extensionality, the universal property, and lifts of a function f f f S ∗ S^{*} S ∗ ⟨ S ∣ R ⟩ + \langle S \mid R\rangle^{+} ⟨ S ∣ R ⟩ + f f f R R R
Presented monoids are also called abstract rewriting systems, string rewriting systems, and Thue systems. They model the idea of rewriting parts of a string according to a given symmetric relation. (Such derivations will be discussed more fully in the following section, so the following definition is not yet linked to Lean code).
Definition 2.7. [ R -derivation ] [R\text{-derivation}] [ R -derivation ] S S S R R R S ∗ S^{*} S ∗ R R R S S S R 0 , R 1 , … , R n R_0, R_1, \ldots, R_n R 0 , R 1 , … , R n k k k 0 ≤ k < n 0 \leq k < n 0 ≤ k < n ( x , y ) ∈ R (x,y) \in R ( x , y ) ∈ R R k = a x b R_k = axb R k = a x b R k + 1 = a y b R_{k+1} = ayb R k + 1 = a y b
Theorem 2.8. For any symmetric relation R R R R R R R 0 R_0 R 0 R n R_n R n [ [ R 0 ] ] R = [ [ R n ] ] R [\![R_0]\!]_R = [\![R_n]\!]_R [ [ R 0 ] ] R = [ [ R n ] ] R
A presented monoid is thus a symmetric rewriting system under another name. We give a robust API for a user looking to prove two words are equivalent in a presented monoid, with over a dozen lemmas. For example, we show:
Lemma 2.9. For x , y , z , w ∈ S ∗ x,y,z,w \in S^* x , y , z , w ∈ S ∗ ( x , y ) ∈ R (x,y) \in R ( x , y ) ∈ R z ≡ R + w z \equiv_R^+ w z ≡ R + w x z ≡ R + y w xz \equiv_R^+ yw x z ≡ R + y w
Beyond inspiration for future use cases, theorem 2.8 gives us an alternate induction principle for presented monoids:
Theorem 2.10. Consider a , b ∈ S ∗ a,b \in S^* a , b ∈ S ∗ a ≡ R + b a \equiv_R^+ b a ≡ R + b P P P S ∗ S^* S ∗ P ( a , b ) P(a,b) P ( a , b )
(reflexivity ) P ( a , a ) P(a,a) P ( a , a ) a ∈ S ∗ a \in S^* a ∈ S ∗
(one-step reduction ) If ( x , y ) ∈ R (x,y) \in R ( x , y ) ∈ R c , d ∈ S ∗ c,d \in S^* c , d ∈ S ∗ P ( c x d , c y d ) P(cxd,cyd) P ( c x d , cy d )
(symmetric one-step reduction ) If ( x , y ) ∈ R (x,y) \in R ( x , y ) ∈ R c , d ∈ S ∗ c,d \in S^* c , d ∈ S ∗ P ( c y d , c x d ) P(cyd,cxd) P ( cy d , c x d )
(transitivity ) For all a , b , c ∈ S ∗ a,b,c \in S^* a , b , c ∈ S ∗ P ( a , b ) P(a,b) P ( a , b ) P ( b , c ) P(b,c) P ( b , c ) P ( a , c ) P(a,c) P ( a , c )
Semi-Thue Systems
We return to the definition of R R R n n n n + 1 n+1 n + 1 A A A A A A → R \underset{R}{\to} R →
Definition 2.11. Consider a set A A A R R R A A A A A A c c c d d d R ( a , b ) R(a,b) R ( a , b ) c a d → R c b d cad \underset{R}{\to} cbd c a d R → c b d
Next, we take its reflexive, transitive closure.
Definition 2.12. Consider a set A A A R R R A A A → ∗ R \underset{R}{\xrightarrow{*}} R ∗
reflexivity : For all a ∈ A ∗ a \in A^* a ∈ A ∗ a → ∗ R a a \underset{R}{\xrightarrow{*}} a a R ∗ a
closure : For all a , b ∈ A ∗ a,b \in A^* a , b ∈ A ∗ a → R b a \underset{R}{\to} b a R → b a → ∗ R b a \underset{R}{\xrightarrow{*}} b a R ∗ b
transitivity : For all a , b , c ∈ A ∗ a,b,c \in A^* a , b , c ∈ A ∗ a → ∗ R b a \underset{R}{\xrightarrow{*}} b a R ∗ b b → ∗ R c b \underset{R}{\xrightarrow{*}} c b R ∗ c a → ∗ R c a \underset{R}{\xrightarrow{*}} c a R ∗ c
This relation is used in the traditional definition of a Semi-Thue system.
Definition 2.13. A Semi-Thue system consists of a tuple ( A , R ) (A,R) ( A , R ) A A A R R R A A A → ∗ R \underset{R}{\xrightarrow{*}} R ∗
In Lean, we call this SemiThue. For this project, it is convenient to provide a second definition of Semi-Thue systems. First, we define a new relation → R \underset{R}{\to} R →
Definition 2.14. Given a set A A A R R R A A A → R \xrightarrow{R} R
reflexivity : For all a ∈ A ∗ a \in A^* a ∈ A ∗ a → R a a \xrightarrow{R} a a R a
one-step transitivity : For all a , b 1 , x , b 2 , y ∈ A ∗ a,b_1,x,b_2,y \in A^* a , b 1 , x , b 2 , y ∈ A ∗ a → R b 1 x b 2 a \xrightarrow{R} b_1xb_2 a R b 1 x b 2 x R y xRy x R y a → R b 1 y b 2 a \xrightarrow{R} b_1yb_2 a R b 1 y b 2
Note that the second condition is equivalent to the following, which makes the pen-and-paper notation for recording derivations cleaner:
alternative one-step transitivity : For all a , b 1 , x , b 2 , y ∈ A ∗ a,b_1,x,b_2,y \in A^* a , b 1 , x , b 2 , y ∈ A ∗ a → R b 1 x b 2 a \xrightarrow{R} b_1xb_2 a R b 1 x b 2 b 1 x b 2 → R b 1 y b 2 b_1xb_2 \xrightarrow{R} b_1yb_2 b 1 x b 2 R b 1 y b 2 a → R b 1 y b 2 a \xrightarrow{R} b_1yb_2 a R b 1 y b 2
Whichever formulation of one-step transitivity we make, we note the initial word is always preserved throughout an → R \xrightarrow{R} R
Theorem 2.15. For all a , b ∈ A ∗ a,b \in A^* a , b ∈ A ∗ a → R b a \xrightarrow{R} b a R b a → R ∗ b a \xrightarrow{R^*} b a R ∗ b
Theorem 2.15 allows us to replace → R ∗ \xrightarrow{R^*} R ∗ → R \xrightarrow{R} R SemiThueDerivation in Lean.
The definition for both → R ∗ \xrightarrow{R^*} R ∗ → R \xrightarrow{R} R A A A List in Lean. The focus for Semi-Thue systems is on rewriting, which has a more computational flair. The API surrounding lists, with computationally-minded functions such as getLast, is far easier to use. The downside is that any work discussing both presented monoids and Semi-Thue systems requires repeated conversions between lists and free monoids. The occasional hassle of dealing with FreeMonoid.toList and List.toFreeMonoid is outweighed by the convenience of using List for Semi-Thue systems.
The “Braid” Monoid
With both presented monoids and Semi-Thue derivations in place, we can now specialize these general constructions to the braid generators and relations. At this point, we have referenced a proof that the braid group aligns with the topological definition of braids. Although it is tempting to think of a monoid in which the braid relations hold as a “braid monoid” (perhaps braids with only overcrossings), until we can connect such a monoid to the braid group mathematically, we have no geometric interpretation of the monoid. It is, for the moment, an arbitrarily defined mathematical structure with no connection to the braid group.
Definition 2.16. The “infinite” braid monoid B + B^+ B +
B + : = ⟨ σ 0 , σ 1 , σ 2 , … | σ i σ j = σ j σ i for ∣ i − j ∣ ≥ 2 , σ i σ j σ i = σ j σ i σ j for ∣ i − j ∣ = 1 ⟩ + B^+ := \left\langle\sigma_0,\sigma_1,\sigma_2,\ldots\ \middle| \ \begin{aligned} \sigma_i\sigma_j &= \sigma_j\sigma_i && \text{for } |i-j| \ge2, \\ \sigma_i\sigma_j\sigma_i &= \sigma_j\sigma_i\sigma_j && \text{for } |i-j| = 1 \end{aligned} \right\rangle^+ B + := ⟨ σ 0 , σ 1 , σ 2 , … σ i σ j σ i σ j σ i = σ j σ i = σ j σ i σ j for ∣ i − j ∣ ≥ 2 , for ∣ i − j ∣ = 1 ⟩ + Definition 2.17. The set of braid monoid words is
B W + : = ( ⋃ i ∈ N σ i ) ∗ BW^+ := \left(\bigcup_{i \in\mathbb{N}} \sigma_i\right)^* B W + := ( i ∈ N ⋃ σ i ) ∗ We denote the relation used in the definition of the braid monoid as R B + R_{B^+} R B +
The absolute-value condition in the relations leads to a doubling of cases in many proofs. Our Lean implementation of the braid monoids is direct; we do not generalize to some sort of “Artin-Tits monoid”, and thus there is no requirement for the relation to be symmetric. To simplify future proofs, we define a different relation for our formalization.
Definition 2.18. The simplified relation R B + s R^{s}_{B^{+}} R B + s
R B + s : = { σ i σ j = σ j σ i for j − i ≥ 2 , σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for j − i = 1 R^{s}_{B^{+}} := \begin{cases}
\sigma_i\sigma_j = \sigma_j\sigma_i & \text{for } j-i \ge2,\\
\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1} & \text{for } j-i = 1
\end{cases} R B + s := { σ i σ j = σ j σ i σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 for j − i ≥ 2 , for j − i = 1 Fact 2.19. ⟨ σ 0 , σ 1 , σ 2 , … ∣ R B + s ⟩ = B + \langle\sigma_0,\sigma_1,\sigma_2,\ldots\mid R^{s}_{B^{+}}\rangle= B^{+} ⟨ σ 0 , σ 1 , σ 2 , … ∣ R B + s ⟩ = B + R B n + s R^{s}_{B_n^{+}} R B n + s B n + B_n^{+} B n +
Proof. (Sketch) Presented monoids are generated by taking the congruence over the given relation (hence the symmetric closure), and R B + R_{B^{+}} R B + R B + s R^{s}_{B^{+}} R B + s
For our Lean definition of the relations for B + B^{+} B + j = i + 1 j=i+1 j = i + 1
inductive braid_monoid_rels_inf : FreeMonoid N → FreeMonoid N → Prop
| adjacent (i : N): braid_monoid_rels_inf (of i * of (i+1) * of i) (of (i+1) *
of i * of (i+1))
| separated (i j : N) (h : i + 2 ≤ j) : braid_monoid_rels_inf (of i * of j)
(of j * of i)This can be plugged in to our previously-defined PresentedMonoid definition, neatly giving us the braid monoid.
We next define a few properties of braid monoid elements.
Definition 2.20. The length of a braid monoid element b b b ∣ b ∣ |b| ∣ b ∣ ∣ [ a ] ∣ |\![a]\!| ∣ [ a ] ∣ a a a ε \varepsilon ε
Definition 2.21. The set of generators of a braid monoid element is given by the function gen \operatorname{gen} gen gen ( [ [ a ] ] ) \operatorname{gen}([\![a]\!]) gen ([ [ a ] ]) a a a
Definition 2.22. The index of a braid monoid word b b b b b b idx \operatorname{idx} idx idx ( b ) : = max { i ∣ σ i ∈ gen ( b ) } \operatorname{idx}(b) := \max\{i \mid\sigma_i \in\operatorname{gen}(b)\} idx ( b ) := max { i ∣ σ i ∈ gen ( b )}
Lemma 2.23. The reversing function on free monoid words, r r r a ≡ + b a \equiv^{+} b a ≡ + b r ( a ) ≡ + r ( b ) r(a) \equiv^{+} r(b) r ( a ) ≡ + r ( b )
We will thus need to define length, set of symbols in, and reversing for free monoid elements. Let us return for a moment to the choice made earlier about the type for S S S List is much richer than FreeMonoid. List already has functions length, toFinset, reverse, etc. Although it is simple to implement these for FreeMonoid, we must tread carefully. While free monoids are in fact defined as lists in Lean, they are meant to perform different roles. Free monoids generally bend more to mathematical theorems in abstract algebra, whereas lists are more used for computation. Even when the two share similarly-minded functionality, it is from a different viewpoint and with different implementation details. For example, if the idea is to apply a function to every element of our string and then combine them, we have FreeMonoid.lift but List.foldr. It is a balancing act to keep the two structures distinct, but also usable. Each has lemmas the other lacks.
We have decided to add definitions and lemmas to FreeMonoid if they will be able to be defined on the eventual presented monoid, the braid monoid. Such definitions seem reasonably “algebraic” since they can be defined for even more complex algebraic structures. Thus, we add in FreeMonoid.length, FreeMonoid.reverse, and FreeMonoid.symbols (the set of generators appearing in an element), because these play nicely with the braid monoid structure.
We take a different approach for List definitions that do not remain invariant in the braid structure, such as getLast (this returns the last element in a list). Equivalent braids need not have the same last generator — σ 1 σ 3 \sigma_1\sigma_3 σ 1 σ 3 σ 3 σ 1 \sigma_3\sigma_1 σ 3 σ 1 FreeMonoid.toList when needed. This is a bit cumbersome, but thankfully quite rare for this project.
We amass simple results of equivalence classes for words:
Lemma 2.24. If σ i ≡ + a \sigma_i \equiv_+ a σ i ≡ + a a = σ i a = \sigma_i a = σ i ⟦ σ i ⟧ = { σ i } \llbracket\sigma_i \rrbracket= \{\sigma_i\} [ [ σ i ] ] = { σ i }
Lemma 2.25. Let ( w , x ) ∈ R B + (w,x) \in R_{B+} ( w , x ) ∈ R B + w ≡ + a w \equiv_+ a w ≡ + a a = w a = w a = w a = x a = x a = x x x x ⟦ w ⟧ = { w , x } = ⟦ x ⟧ \llbracket w \rrbracket= \{w,x\} = \llbracket x \rrbracket [ [ w ] ] = { w , x } = [ [ x ] ]
Proof. For the case of the commutative relation, ( σ i σ j , σ j σ i ) ∈ R B + (\sigma_i\sigma_j,\sigma_j\sigma_i) \in R_{B+} ( σ i σ j , σ j σ i ) ∈ R B + ∣ i − j ∣ ≥ 2 \lvert i-j\rvert\ge2 ∣ i − j ∣ ≥ 2 ( σ i σ j σ i , σ j σ i σ j ) ∈ R B + (\sigma_i\sigma_j\sigma_i,\sigma_j\sigma_i\sigma_j) \in R_{B+} ( σ i σ j σ i , σ j σ i σ j ) ∈ R B + ∣ i − j ∣ = 1 \lvert i-j\rvert= 1 ∣ i − j ∣ = 1 a = σ j σ i σ i a = \sigma_j\sigma_i\sigma_i a = σ j σ i σ i w ≡ + a w \equiv_+ a w ≡ + a Theorem 2.10 .
For long elements of B W + BW^+ B W + rw, can exactly pattern-match the terms. Once we move to a specific presented monoid, we have concrete relations to work with. We can associate all parentheses to the left, and then use one of the following lemmas:
Lemma 2.26. For all x ∈ B W + x \in BW^+ x ∈ B W + ∣ i − j ∣ > 1 \lvert i-j\rvert> 1 ∣ i − j ∣ > 1
( x ⋅ ⟦ σ i ⟧ + ) ⋅ ⟦ σ j ⟧ + = ( x ⋅ ⟦ σ j ⟧ + ) ⋅ ⟦ σ i ⟧ + \left(x \cdot\llbracket\sigma_i \rrbracket^+\right) \cdot\llbracket\sigma_j \rrbracket^+ = \left(x \cdot\llbracket\sigma_j \rrbracket^+\right) \cdot\llbracket\sigma_i \rrbracket^+ ( x ⋅ [ [ σ i ] ] + ) ⋅ [ [ σ j ] ] + = ( x ⋅ [ [ σ j ] ] + ) ⋅ [ [ σ i ] ] + Lemma 2.27. For all x ∈ B W + x \in BW^+ x ∈ B W + ∣ i − j ∣ = 1 \lvert i-j\rvert= 1 ∣ i − j ∣ = 1
( ( x ⋅ ⟦ σ i ⟧ + ) ⋅ ⟦ σ j ⟧ + ) ⋅ ⟦ σ i ⟧ + = ( ( x ⋅ ⟦ σ j ⟧ + ) ⋅ ⟦ σ i ⟧ + ) ⋅ ⟦ σ j ⟧ + \left(\left(x \cdot\llbracket\sigma_i \rrbracket^+\right) \cdot\llbracket\sigma_j \rrbracket^+\right) \cdot\llbracket\sigma_i \rrbracket^+ = \left(\left(x \cdot\llbracket\sigma_j \rrbracket^+\right) \cdot\llbracket\sigma_i \rrbracket^+\right) \cdot\llbracket\sigma_j \rrbracket^+ ( ( x ⋅ [ [ σ i ] ] + ) ⋅ [ [ σ j ] ] + ) ⋅ [ [ σ i ] ] + = ( ( x ⋅ [ [ σ j ] ] + ) ⋅ [ [ σ i ] ] + ) ⋅ [ [ σ j ] ] + For example, say we are trying to transform the element ⟦ σ 2 σ 7 σ 3 σ 4 σ 5 σ 4 σ 0 σ 1 ⟧ \llbracket\sigma_2\sigma_7\sigma_3\sigma_4\sigma_5\sigma_4\sigma_0\sigma_1 \rrbracket [ [ σ 2 σ 7 σ 3 σ 4 σ 5 σ 4 σ 0 σ 1 ] ] ⟦ σ 2 σ 7 σ 5 σ 3 σ 4 σ 5 σ 0 σ 1 ⟧ \llbracket\sigma_2\sigma_7\sigma_5\sigma_3\sigma_4\sigma_5\sigma_0\sigma_1 \rrbracket [ [ σ 2 σ 7 σ 5 σ 3 σ 4 σ 5 σ 0 σ 1 ] ] ⟦ σ 2 ⟧ ⋅ ⟦ σ 7 ⟧ ⋅ ⟦ σ 3 ⟧ ⋅ ⟦ σ 4 ⟧ ⋅ ⟦ σ 5 ⟧ ⋅ ⟦ σ 4 ⟧ ⋅ ⟦ σ 0 ⟧ ⋅ ⟦ σ 1 ⟧ \llbracket\sigma_2 \rrbracket\cdot\llbracket\sigma_7 \rrbracket\cdot\llbracket\sigma_3 \rrbracket\cdot\llbracket\sigma_4 \rrbracket\cdot\llbracket\sigma_5 \rrbracket\cdot\llbracket\sigma_4 \rrbracket\cdot\llbracket\sigma_0 \rrbracket\cdot\llbracket\sigma_1 \rrbracket [ [ σ 2 ] ] ⋅ [ [ σ 7 ] ] ⋅ [ [ σ 3 ] ] ⋅ [ [ σ 4 ] ] ⋅ [ [ σ 5 ] ] ⋅ [ [ σ 4 ] ] ⋅ [ [ σ 0 ] ] ⋅ [ [ σ 1 ] ]
( ( ( ( ( ( ⟦ σ 2 ⟧ ⋅ ⟦ σ 7 ⟧ ) ⋅ ⟦ σ 3 ⟧ ) ⋅ ⟦ σ 4 ⟧ ) ⋅ ⟦ σ 5 ⟧ ) ⋅ ⟦ σ 4 ⟧ ) ⋅ ⟦ σ 0 ⟧ ) ⋅ ⟦ σ 1 ⟧ ((((((\llbracket\sigma_2 \rrbracket\cdot\llbracket\sigma_7 \rrbracket) \cdot\llbracket\sigma_3 \rrbracket) \cdot\llbracket\sigma_4 \rrbracket) \cdot\llbracket\sigma_5 \rrbracket) \cdot\llbracket\sigma_4 \rrbracket) \cdot\llbracket\sigma_0 \rrbracket) \cdot\llbracket\sigma_1 \rrbracket (((((( [ [ σ 2 ] ] ⋅ [ [ σ 7 ] ] ) ⋅ [ [ σ 3 ] ] ) ⋅ [ [ σ 4 ] ] ) ⋅ [ [ σ 5 ] ] ) ⋅ [ [ σ 4 ] ] ) ⋅ [ [ σ 0 ] ] ) ⋅ [ [ σ 1 ] ] Then, Lean’s rw tactic used with lemma 2.27 will assign ( ( ⟦ σ 2 ⟧ ⋅ ⟦ σ 7 ⟧ ) ⋅ ⟦ σ 3 ⟧ ) ((\llbracket\sigma_2 \rrbracket\cdot\llbracket\sigma_7 \rrbracket) \cdot\llbracket\sigma_3 \rrbracket) (( [ [ σ 2 ] ] ⋅ [ [ σ 7 ] ] ) ⋅ [ [ σ 3 ] ] ) x x x ( ( ( ( ⟦ σ 2 ⟧ ⋅ ⟦ σ 7 ⟧ ) ⋅ ⟦ σ 3 ⟧ ) ⋅ ⟦ σ 4 ⟧ ) ⋅ ⟦ σ 5 ⟧ ) ⋅ ⟦ σ 4 ⟧ ((((\llbracket\sigma_2 \rrbracket\cdot\llbracket\sigma_7 \rrbracket) \cdot\llbracket\sigma_3 \rrbracket) \cdot\llbracket\sigma_4 \rrbracket) \cdot\llbracket\sigma_5 \rrbracket) \cdot\llbracket\sigma_4 \rrbracket (((( [ [ σ 2 ] ] ⋅ [ [ σ 7 ] ] ) ⋅ [ [ σ 3 ] ] ) ⋅ [ [ σ 4 ] ] ) ⋅ [ [ σ 5 ] ] ) ⋅ [ [ σ 4 ] ] ( ( ( ( ⟦ σ 2 ⟧ ⋅ ⟦ σ 7 ⟧ ) ⋅ ⟦ σ 3 ⟧ ) ⋅ ⟦ σ 5 ⟧ ) ⋅ ⟦ σ 4 ⟧ ) ⋅ ⟦ σ 5 ⟧ ((((\llbracket\sigma_2 \rrbracket\cdot\llbracket\sigma_7 \rrbracket) \cdot\llbracket\sigma_3 \rrbracket) \cdot\llbracket\sigma_5 \rrbracket) \cdot\llbracket\sigma_4 \rrbracket) \cdot\llbracket\sigma_5 \rrbracket (((( [ [ σ 2 ] ] ⋅ [ [ σ 7 ] ] ) ⋅ [ [ σ 3 ] ] ) ⋅ [ [ σ 5 ] ] ) ⋅ [ [ σ 4 ] ] ) ⋅ [ [ σ 5 ] ]
( ( ( ( ( ( ⟦ σ 2 ⟧ ⋅ ⟦ σ 7 ⟧ ) ⋅ ⟦ σ 3 ⟧ ) ⋅ ⟦ σ 5 ⟧ ) ⋅ ⟦ σ 4 ⟧ ) ⋅ ⟦ σ 5 ⟧ ) ⋅ ⟦ σ 0 ⟧ ) ⋅ ⟦ σ 1 ⟧ ((((((\llbracket\sigma_2 \rrbracket\cdot\llbracket\sigma_7 \rrbracket) \cdot\llbracket\sigma_3 \rrbracket) \cdot\llbracket\sigma_5 \rrbracket) \cdot\llbracket\sigma_4 \rrbracket) \cdot\llbracket\sigma_5 \rrbracket) \cdot\llbracket\sigma_0 \rrbracket) \cdot\llbracket\sigma_1 \rrbracket (((((( [ [ σ 2 ] ] ⋅ [ [ σ 7 ] ] ) ⋅ [ [ σ 3 ] ] ) ⋅ [ [ σ 5 ] ] ) ⋅ [ [ σ 4 ] ] ) ⋅ [ [ σ 5 ] ] ) ⋅ [ [ σ 0 ] ] ) ⋅ [ [ σ 1 ] ] We then use rw with lemma 2.26, explicitly setting x x x ( ⟦ σ 2 ⟧ ⋅ ⟦ σ 7 ⟧ ) (\llbracket\sigma_2 \rrbracket\cdot\llbracket\sigma_7 \rrbracket) ( [ [ σ 2 ] ] ⋅ [ [ σ 7 ] ] )
( ( ( ( ( ( ⟦ σ 2 ⟧ ⋅ ⟦ σ 7 ⟧ ) ⋅ ⟦ σ 5 ⟧ ) ⋅ ⟦ σ 3 ⟧ ) ⋅ ⟦ σ 4 ⟧ ) ⋅ ⟦ σ 5 ⟧ ) ⋅ ⟦ σ 0 ⟧ ) ⋅ ⟦ σ 1 ⟧ ((((((\llbracket\sigma_2 \rrbracket\cdot\llbracket\sigma_7 \rrbracket) \cdot\llbracket\sigma_5 \rrbracket) \cdot\llbracket\sigma_3 \rrbracket) \cdot\llbracket\sigma_4 \rrbracket) \cdot\llbracket\sigma_5 \rrbracket) \cdot\llbracket\sigma_0 \rrbracket) \cdot\llbracket\sigma_1 \rrbracket (((((( [ [ σ 2 ] ] ⋅ [ [ σ 7 ] ] ) ⋅ [ [ σ 5 ] ] ) ⋅ [ [ σ 3 ] ] ) ⋅ [ [ σ 4 ] ] ) ⋅ [ [ σ 5 ] ] ) ⋅ [ [ σ 0 ] ] ) ⋅ [ [ σ 1 ] ] which can be simplified to ⟦ σ 2 σ 7 σ 5 σ 3 σ 4 σ 5 σ 0 σ 1 ⟧ \llbracket\sigma_2\sigma_7\sigma_5\sigma_3\sigma_4\sigma_5\sigma_0\sigma_1 \rrbracket [ [ σ 2 σ 7 σ 5 σ 3 σ 4 σ 5 σ 0 σ 1 ] ]
Finite Braid Groups and Finite Braid Monoids
Above, we have discussed only infinite braid groups and infinite braid monoids and the equivalences on them, ≡ \equiv ≡ ≡ + \equiv^{+} ≡ + n n n n − 1 n-1 n − 1 σ 0 , σ 1 , … σ n − 2 \sigma_0,\sigma_1,\ldots\sigma_{n-2} σ 0 , σ 1 , … σ n − 2
Definition 2.28. The finite braid group B n B_n B n n n n
B n : = ⟨ σ 0 , … , σ n − 2 | σ i σ j = σ j σ i for ∣ i − j ∣ ≥ 2 , σ i σ j σ i = σ j σ i σ j for ∣ i − j ∣ = 1 ⟩ B_n := \left\langle\sigma_0,\ldots,\sigma_{n-2}\ \middle|\ \begin{aligned} \sigma_i\sigma_j&=\sigma_j\sigma_i &&\text{for } |i-j|\ge2,\\ \sigma_i\sigma_j\sigma_i&=\sigma_j\sigma_i\sigma_j &&\text{for } |i-j|=1 \end{aligned}\right\rangle B n := ⟨ σ 0 , … , σ n − 2 σ i σ j σ i σ j σ i = σ j σ i = σ j σ i σ j for ∣ i − j ∣ ≥ 2 , for ∣ i − j ∣ = 1 ⟩ We denote the defining relation (on the righthand side) as R B n R_{B_n} R B n B n B_n B n ≡ n \equiv_n ≡ n
Note that B 0 B_0 B 0 B 1 B_1 B 1
Definition 2.29.
B n + : = ⟨ σ 0 , … , σ n − 2 | σ i σ j = σ j σ i for ∣ i − j ∣ ≥ 2 , σ i σ j σ i = σ j σ i σ j for ∣ i − j ∣ = 1 ⟩ + B_n^{+} := \left\langle\sigma_0,\ldots,\sigma_{n-2}\ \middle|\ \begin{aligned} \sigma_i\sigma_j&=\sigma_j\sigma_i &&\text{for } |i-j|\ge2,\\ \sigma_i\sigma_j\sigma_i&=\sigma_j\sigma_i\sigma_j &&\text{for } |i-j|=1 \end{aligned}\right\rangle^{+} B n + := ⟨ σ 0 , … , σ n − 2 σ i σ j σ i σ j σ i = σ j σ i = σ j σ i σ j for ∣ i − j ∣ ≥ 2 , for ∣ i − j ∣ = 1 ⟩ + Here, the relation is denoted R B n + R_{B_n^{+}} R B n + ≡ n + \equiv_n^{+} ≡ n +
Definition 2.30. We define the set of generators in B n B_n B n B n + B_n^{+} B n + S n = { σ i ∣ 0 ≤ i ≤ n − 2 } S_n=\{\sigma_i\mid0\le i\le n-2\} S n = { σ i ∣ 0 ≤ i ≤ n − 2 } B W n : = ( S n ∪ S n ‾ ) ∗ BW_n:=(S_n\cup\overline{S_n})^{*} B W n := ( S n ∪ S n ) ∗ B W n + : = S n ∗ BW_n^{+}:=S_n^{*} B W n + := S n ∗
Let us now move to the Lean implementation. For infinite braid groups and monoids, the generators are kept track of via their index, of type N \mathbb{N} N B n B_n B n B n + B_n^{+} B n + n − 1 n-1 n − 1 n n n k k k k < n − 1 k<n-1 k < n − 1
This is especially true as working with the finite braid groups/monoids is difficult in Lean. First, we are always juggling a hypothesis of the form k < n − 1 k<n-1 k < n − 1 i + 2 ≤ j i+2\le j i + 2 ≤ j
Theorem 2.31. For braid words a , b a,b a , b a ≡ n b a\equiv_n b a ≡ n b a ≡ m b a\equiv_m b a ≡ m b m > n m>n m > n a ≡ b a\equiv b a ≡ b a , b a,b a , b monoid words, and if a ≡ n + b a\equiv_n^{+}b a ≡ n + b a ≡ m + b a\equiv_m^{+}b a ≡ m + b m > n m>n m > n a ≡ + b a\equiv^{+}b a ≡ + b
There is a small hiccup here: when we say a ≡ n b a\equiv_n b a ≡ n b a , b ∈ B n a,b\in B_n a , b ∈ B n a ≡ k b a\equiv_k b a ≡ k b a , b ∈ B k a,b\in B_k a , b ∈ B k B W 0 ⊂ B W 1 ⊂ B W 2 … ⊂ B W BW_0\subset BW_1\subset BW_2\ldots\subset BW B W 0 ⊂ B W 1 ⊂ B W 2 … ⊂ B W N \mathbb{N} N
When can we transfer an equivalence from the infinite braid group or monoid to the corresponding finite structure? Let us begin with the monoid case.
Theorem 2.32. Consider a , b ∈ B W + a,b \in BW^{+} a , b ∈ B W + a ≡ + b a \equiv^{+} b a ≡ + b idx ( a ) = i a \operatorname{idx}(a)=i_a idx ( a ) = i a a a a i a i_a i a idx ( b ) = i b \operatorname{idx}(b)=i_b idx ( b ) = i b n ≥ max ( i a , i b ) n \ge\max(i_a,i_b) n ≥ max ( i a , i b ) a ≡ n + b a \equiv^{+}_{n} b a ≡ n + b
The proof proceeds by induction on the derivation a ≡ + b a \equiv^{+} b a ≡ + b
def FreeMonoid.mapNatToFin (n : ℕ) (a : FreeMonoid ℕ) (bound : ∀ x ∈ a, x < n) :
FreeMonoid (Fin n)The above theorems will eventually be used to apply our solver to the finite case and show that it is correct for finite braid group inputs. We leave the finite braid groups/monoids aside until the final chapter – the algorithm is built for the infinite braid group, and then proved to work for the finite group as well. We are not yet able to give a theorem analogous to Theorem 2.32 for the braid groups. In fact, as the very last step of our project, we will use the correctness specification of our solver to prove this.
In sum, we have defined the infinite and finite braid groups, as well as a number of structures needed to implement and reason about subword reversing (presented monoids and Semi-Thue systems).
Localization
We next elucidate the connection between the braid group and the “braid” monoid by constructing an embedding of B + B^{+} B + B B B
Ore Localization
Given a monoid M M M M 0 M_{0} M 0 M M M M 0 M_{0} M 0
For example, one may see that σ 1 σ 2 \sigma_{1}\sigma_{2} σ 1 σ 2 σ 2 σ 1 \sigma_{2}\sigma_{1} σ 2 σ 1 σ 1 σ 2 \sigma_{1}\sigma_{2} σ 1 σ 2
Figure 15 (PDF p. 17).
Instead, we use the Ore localization, which requires different monoid properties. We denote the (left)11 M M M M 0 M_{0} M 0 [ M 0 − 1 ] M [M_{0}^{-1}]M [ M 0 − 1 ] M
The following two conditions suffice for left Ore localization; we will refer to them as Ore conditions throughout this project:
Definition 3.1. We say that a monoid M M M M 0 M_0 M 0 M M M
Left Common Multiples: For any a ∈ M a \in M a ∈ M b ∈ M 0 b \in M_0 b ∈ M 0 c ∈ M 0 c \in M_0 c ∈ M 0 d ∈ M d \in M d ∈ M c a = d b ca=db c a = d b
Right Cancellativity: For a , b ∈ M a,b \in M a , b ∈ M c ∈ M 0 c \in M_0 c ∈ M 0 a c = b c ac=bc a c = b c a = b a=b a = b
The second condition can be slightly weakened.12 13
We begin with a monoid M M M M 0 ⊆ M M_0 \subseteq M M 0 ⊆ M ( x , y ) (x,y) ( x , y ) x ∈ M x \in M x ∈ M y ∈ M 0 y \in M_0 y ∈ M 0
Definition 3.2 (∼ \sim ∼ We say ( x 1 , y 1 ) ∼ ( x 2 , y 2 ) (x_1,y_1) \sim(x_2,y_2) ( x 1 , y 1 ) ∼ ( x 2 , y 2 ) a ∈ M a \in M a ∈ M b ∈ M 0 b \in M_0 b ∈ M 0 a x 1 = b x 2 ax_1=bx_2 a x 1 = b x 2 a y 1 = b y 2 ay_1=by_2 a y 1 = b y 2
Theorem 3.3. ∼ \sim ∼ ( x , y ) (x,y) ( x , y )
The proofs of the theorem follows directly from the definition of ∼ \sim ∼ ∼ \sim ∼ x y \frac{x}{y} y x ( x , y ) (x,y) ( x , y ) ∼ \sim ∼
Definition 3.4. Given a monoid M M M M 0 M_0 M 0 M M M [ M 0 − 1 ] M [M_0^{-1}]M [ M 0 − 1 ] M { x y ∣ x ∈ M , y ∈ M 0 } \left\{\frac{x}{y} \mid x \in M, y \in M_0\right\} { y x ∣ x ∈ M , y ∈ M 0 } 1 M 1 M 0 \frac{1_M}{1_{M_0}} 1 M 0 1 M 1 M 1_M 1 M 1 M 0 1_{M_0} 1 M 0 ∙ \bullet ∙ x y ∙ z w = a z b y \frac{x}{y} \bullet\frac{z}{w} = \frac{az}{by} y x ∙ w z = b y a z a ∈ M a \in M a ∈ M b ∈ M 0 b \in M_0 b ∈ M 0 a w = b x aw=bx a w = b x
The proof that the definition is independent of the choice of common multiples a a a b b b M M M [ M 0 − 1 ] M [M_0^{-1}]M [ M 0 − 1 ] M
Theorem 3.5. The function h h h M M M [ M 0 − 1 ] M [M_0^{-1}]M [ M 0 − 1 ] M h ( x ) : = x 1 h(x):=\frac{x}{1} h ( x ) := 1 x
Proof. We have h ( x 1 x 2 ) = x 1 x 2 1 h(x_1x_2)=\frac{x_1x_2}{1} h ( x 1 x 2 ) = 1 x 1 x 2 h ( x 1 ) ∙ h ( x 2 ) = x 1 1 ∙ x 2 1 h(x_1)\bullet h(x_2)=\frac{x_1}{1}\bullet\frac{x_2}{1} h ( x 1 ) ∙ h ( x 2 ) = 1 x 1 ∙ 1 x 2 x 1 x_1 x 1 1 1 1 1 ⋅ x 1 = x 1 ⋅ 1 1\cdot x_1=x_1\cdot1 1 ⋅ x 1 = x 1 ⋅ 1 a = x 1 a=x_1 a = x 1 b = 1 b=1 b = 1 x 1 1 ∙ x 2 1 = x 1 ⋅ x 2 1 \frac{x_1}{1}\bullet\frac{x_2}{1}=\frac{x_1\cdot x_2}{1} 1 x 1 ∙ 1 x 2 = 1 x 1 ⋅ x 2 h ( x 1 x 2 ) = h ( x 1 ) ∙ h ( x 2 ) h(x_1x_2)=h(x_1)\bullet h(x_2) h ( x 1 x 2 ) = h ( x 1 ) ∙ h ( x 2 )
We now add one more condition. Together with our original Ore conditions, these form the cancellative Ore conditions:
Left Cancellativity: For a , b ∈ M a,b \in M a , b ∈ M c ∈ M 0 c \in M_0 c ∈ M 0 c a = c b ca=cb c a = c b a = b a=b a = b
In the context of the cancellative Ore conditions, h h h M M M [ M 0 − 1 ] M [M_0^{-1}]M [ M 0 − 1 ] M
Theorem 3.6. Given a monoid M M M M 0 M_0 M 0 cancellative Ore conditions, the function h ( x ) : = x 1 h(x) := \frac{x}{1} h ( x ) := 1 x
Proof. Assume h ( x 1 ) = h ( x 2 ) h(x_1) = h(x_2) h ( x 1 ) = h ( x 2 ) x 1 1 = x 2 1 \frac{x_1}{1} = \frac{x_2}{1} 1 x 1 = 1 x 2 ( x 1 , 1 ) ∼ ( x 2 , 1 ) (x_1, 1) \sim(x_2, 1) ( x 1 , 1 ) ∼ ( x 2 , 1 ) a ∈ M a \in M a ∈ M b ∈ M 0 b \in M_0 b ∈ M 0 a x 1 = b x 2 ax_1 = bx_2 a x 1 = b x 2 a ⋅ 1 = b ⋅ 1 a \cdot1 = b \cdot1 a ⋅ 1 = b ⋅ 1 a = b a = b a = b b x 1 = b x 2 bx_1 = bx_2 b x 1 = b x 2 x 1 = x 2 x_1 = x_2 x 1 = x 2
We now lay the full cancellative Ore conditions aside for the moment. For this project, we first specialize to the case of localizing by the entire monoid, viewed as a submonoid of itself. Mathlib already contains the typeclass OreSet, which represents a pair of a monoid and a submonoid satisfying the (weakened) Ore conditions. Thus, we want to show that if a monoid has left common multiples and right cancellativity, then it paired with the full submonoid of itself is an instance of OreSet. To take in the hypotheses that the monoid M satisfies the left common multiple and right cancellativity criteria, we use mixins. These are typeclasses that extend a structure to add in an extra property. For example, we define
class IsCommonLeftMultipleMul (M : Type*) [Mul M] where
common_left_multiple : ∀ a b : M, ∃ c d : M, c * a = d * bThe above code is nonconstructive, as the existential quantifier merely asserts the existence of the common multiples. It does not provide them as data. This is in line with general Mathlib conventions. However, the existing Ore localization code is set up to be used constructively, because OreSet has fields for the common-multiple data. We take advantage of this design, and give both constructive and non-constructive versions for braids. The constructive version will be useful for a concurrent project, developing a widget to visualize braids in the Lean InfoView. The widget will use the data from constructive proofs to display common multiples, equivalences, canonical forms, etc. Here is the analogous constructive version of IsCommonLeftMultipleMul:
class HasCommonLeftMultipleData (M : Type*) [Mul M] where
cl1 : M → M → M
cl2 : M → M → M
cl_spec : ∀ a b : M, cl2 a b * a = cl1 a b * bWe continue our discussion in the non-data-carrying setting only. The nonconstructive version is used in the implementation of the braid word problem solver, for Ore localization is used only in the proof of correctness of the algorithm, not in its definition. As such, it need not be constructive. We thus use the following mixin hypotheses added to a monoid M M M
variable {M : Type*} [Monoid M] [IsRightCancelMul M] [IsCommonLeftMultipleMul M]With the above assumed, we are able to show that the full submonoid, ( ⊤ : Submonoid M ) (\top: \operatorname{Submonoid} M) ( ⊤ : Submonoid M )
noncomputable instance : OreLocalization.OreSet (⊤ : Submonoid M) where ...The proof uses Classical.choose to noncomputably provide the witnesses that the constructive OreSet requires, and Classical.choose_spec to assert that those witnesses have the desired property of being common multiples.
We now may define the Ore localization of M M M
abbrev "OreLocalizationSelf" => OreLocalization (⊤ : Submonoid M) MTheorem 3.7. For any monoid M M M [ M − 1 ] M [M^{-1}]M [ M − 1 ] M
Proof. We already have shown that [ M − 1 ] M [M^{-1}]M [ M − 1 ] M x y ∈ [ M − 1 ] M \frac{x}{y} \in[M^{-1}]M y x ∈ [ M − 1 ] M ( x y ) − 1 : = y x \left(\frac{x}{y}\right)^{-1} := \frac{y}{x} ( y x ) − 1 := x y y x \frac{y}{x} x y
Next, we consider presented monoids. We aim to plug in a presented monoid satisfying the Ore conditions to the existing Ore localization API. It is now clear why mixins are required. If we instead took in the hypotheses by defining an extension of the Monoid class satisfying the Ore hypotheses called OreMonoid, much like CancelMonoid, we would run into an issue. Both the PresentedMonoid and OreMonoid instances would carry with them a .toMonoid function. If we try to give Lean a type M with instances [PresentedMonoid M] [OreMonoid M], Lean can infer the Monoid instance on M in two different ways. This is referred to as a “bad diamond”.
Monoid
PresentedMonoid OreMonoid
M M M
Thus, we choose to use mixins to take in the hypotheses.
Theorem 3.8. Let M = ⟨ S ∣ R ⟩ + M = \langle S \mid R\rangle^{+} M = ⟨ S ∣ R ⟩ + [ M − 1 ] M [M^{-1}]M [ M − 1 ] M ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩
The proof will proceed by a careful inspection of the following universal properties of the Ore localization and of presented groups; we begin by defining those universal properties. For simplicity, we let PML ( S , R ) \operatorname{PML}(S,R) PML ( S , R ) ( ⟨ S ∣ R ⟩ + ) − 1 ⟨ S ∣ R ⟩ + (\langle S \mid R\rangle^{+})^{-1}\langle S \mid R\rangle^{+} (⟨ S ∣ R ⟩ + ) − 1 ⟨ S ∣ R ⟩ +
Definition 3.9 (Lift). Given a map f f f S S S M M M f f f S S S M M M f ‾ \overline{f} f f f f S S S f ‾ + \overline{f}^{+} f +
Theorem 3.10. [Universal property of the Ore localization of a presented monoid] Consider a presented monoid M = ⟨ S ∣ R ⟩ + M = \langle S \mid R\rangle^{+} M = ⟨ S ∣ R ⟩ + G G G f : S → G f : S \to G f : S → G R R R ( r 1 , r 2 ) ∈ R (r_{1},r_{2}) \in R ( r 1 , r 2 ) ∈ R f ‾ + ( r 1 ) = f ‾ + ( r 2 ) \overline{f}^{+}(r_{1}) = \overline{f}^{+}(r_{2}) f + ( r 1 ) = f + ( r 2 )
f ∗ : [ M − 1 ] M → G f^{*} : [M^{-1}]M \to G f ∗ : [ M − 1 ] M → G such that, for every s ∈ S s \in S s ∈ S
f ∗ ( [ s ] R + 1 ) = f ( s ) . f^{*}\left(\frac{[s]_{R}^{+}}{1}\right) = f(s). f ∗ ( 1 [ s ] R + ) = f ( s ) . Proof. This is merely a specialization of the universal property for Ore localizations, which is again a specialization of that for quotients.
Remark. Since the identity map on PML ( S , R ) \operatorname{PML}(S,R) PML ( S , R ) g : PML ( S , R ) → PML ( S , R ) g : \operatorname{PML}(S,R) \to\operatorname{PML}(S,R) g : PML ( S , R ) → PML ( S , R ) g ( [ s ] R + 1 ) = [ s ] R + 1 g\left(\frac{[s]_{R}^{+}}{1}\right) = \frac{[s]_{R}^{+}}{1} g ( 1 [ s ] R + ) = 1 [ s ] R + s ∈ S s \in S s ∈ S
Our next task will be to define two homomorphisms between the presented group ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩ ⟨ S ∣ R ⟩ + \langle S \mid R\rangle^{+} ⟨ S ∣ R ⟩ +
Definition 3.11. We define a homomorphism h 1 h_1 h 1 ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩ PML ( S , R ) \operatorname{PML}(S,R) PML ( S , R ) f ( s ) = ⟦ s ⟧ R + ⟨ S , R ⟩ + : S → PML ( S , R ) f(s)=\frac{\llbracket s\rrbracket_R^+}{\langle S,R\rangle^+}:S\to\operatorname{PML}(S,R) f ( s ) = ⟨ S , R ⟩ + [ [ s ] ] R + : S → PML ( S , R ) f f f h h h Theorem 3.5 (recall h ( s ) = s 1 h(s)=\frac{s}{1} h ( s ) = 1 s ⟦ ⋅ ⟧ R + \llbracket\cdot\rrbracket_R^+ [ [ ⋅ ] ] R + PresentedMonoid.of when applied to a single generator).
We define a homomorphism h 2 h_2 h 2 PML ( S , R ) \operatorname{PML}(S,R) PML ( S , R ) ⟨ S ∣ R ⟩ \langle S\mid R\rangle ⟨ S ∣ R ⟩ m ↦ ⟦ m ⟧ R m\mapsto\llbracket m\rrbracket_R m ↦ [ [ m ] ] R
Remark. Note that for any s ∈ S ∗ s\in S^* s ∈ S ∗ h 2 ( ⟦ s ⟧ R + 1 ) = ⟦ s ⟧ R h_2\left(\frac{\llbracket s\rrbracket_R^+}{1}\right)=\llbracket s\rrbracket_R h 2 ( 1 [ [ s ] ] R + ) = [ [ s ] ] R s s s s s s
Figure 1 (PDF p. 21).
Finally, we put the pieces together to complete the proof of Theorem 3.8 . Composing h 1 h_1 h 1 h 2 h_2 h 2 ⟨ S ∣ R ⟩ \langle S\mid R\rangle ⟨ S ∣ R ⟩ ⟨ S ∣ R ⟩ \langle S\mid R\rangle ⟨ S ∣ R ⟩ Theorem 2.3 ), that must be the identity! Similarly, composing h 2 h_2 h 2 h 1 h_1 h 1 Theorem 3.10 . Thus, h 1 h_1 h 1 h 2 h_2 h 2 PML ( S , R ) \operatorname{PML}(S,R) PML ( S , R ) ⟨ S ∣ R ⟩ \langle S\mid R\rangle ⟨ S ∣ R ⟩
As to the Lean implementation of the above proof, we must note that the API for presented monoids takes a relation as a function of two FreeMonoid arguments, while the API for presented groups requires the relations to be given as a set of FreeGroup elements. To translate from the former to the latter, we use the function
def free_group_set_of_function (rels : FreeMonoid α → FreeMonoid α → Prop) :
Set (FreeGroup α)We further provide a number of lemmas about the interaction of free_group_set_of_function with lifts and the relations. For example, consider a function f f f S S S G 1 G_1 G 1 f f f free_group_set_of_function rels equals one if and only if f + ( r 1 ) = f + ( r 2 ) f^+(r_1)=f^+(r_2) f + ( r 1 ) = f + ( r 2 ) ( r 1 , r 2 ) (r_1,r_2) ( r 1 , r 2 ) rels holds:
theorem free_group_set_of_function_lift_eq_one_iff {G₁ : Type} [Group G₁]
{rels : FreeMonoid α → FreeMonoid α → Prop} (f : α → G₁) :
(∀ r ∈ free_group_set_of_function rels, (FreeGroup.lift f) r = 1) ↔
(∀ r₁ r₂, rels r₁ r₂ → FreeMonoid.lift f r₁ = FreeMonoid.lift f r₂)Such lemmas help us prove the required hypotheses for the universal properties in the definition of h 1 h_1 h 1 h 2 h_2 h 2 h 1 h_1 h 1 h 2 h_2 h 2 PML ( S , R ) \operatorname{PML}(S,R) PML ( S , R ) ⟨ S ∣ R ⟩ \langle S\mid R\rangle ⟨ S ∣ R ⟩
Returning to the mathematical development, we aim to show that for a cancellative presented monoid, equivalence in the associated presented group of positive words implies equivalence in the presented monoid of those same words. For the rest of this section, we discuss only presented monoids which satisfy the cancellative Ore conditions. (We say that the monoid M M M M M M
Definition 3.12. We define the function ι : ⟨ S ∣ R ⟩ + → ⟨ S ∣ R ⟩ \iota: \langle S \mid R\rangle^{+} \to\langle S \mid R\rangle ι : ⟨ S ∣ R ⟩ + → ⟨ S ∣ R ⟩ ι : = h 2 ∘ h \iota:= h_{2} \circ h ι := h 2 ∘ h h h h Theorem 3.5 .
Theorem 3.13. For a monoid ⟨ S ∣ R ⟩ + \langle S \mid R\rangle^{+} ⟨ S ∣ R ⟩ + ι \iota ι ⟨ S ∣ R ⟩ + \langle S \mid R\rangle^{+} ⟨ S ∣ R ⟩ + ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩
Proof. By Theorem 3.6 , h h h h 2 h_{2} h 2 h 1 h_{1} h 1 ι \iota ι ⟨ S ∣ R ⟩ + \langle S \mid R\rangle^{+} ⟨ S ∣ R ⟩ + ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩
Finally, we may show that equivalence in the group implies equivalence in the monoid for positive words:
Theorem 3.14. Let ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩ a ≡ R b a \equiv_{R} b a ≡ R b positive words a , b ∈ S ∗ a,b \in S^{*} a , b ∈ S ∗ 14 a ≡ R + b a \equiv_{R}^{+} b a ≡ R + b
Proof. We know that ⟦ a ⟧ = ⟦ b ⟧ \llbracket a\rrbracket= \llbracket b\rrbracket [ [ a ] ] = [ [ b ] ] h 2 h_{2} h 2 ⟦ a ⟧ = h 2 ( ⟦ a ⟧ + 1 ) \llbracket a\rrbracket= h_{2}\left(\frac{\llbracket a\rrbracket^{+}}{1}\right) [ [ a ] ] = h 2 ( 1 [ [ a ] ] + ) ⟦ b ⟧ = h 2 ( ⟦ b ⟧ + 1 ) \llbracket b\rrbracket= h_{2}\left(\frac{\llbracket b\rrbracket^{+}}{1}\right) [ [ b ] ] = h 2 ( 1 [ [ b ] ] + ) ⟦ a ⟧ + 1 = ⟦ b ⟧ + 1 \frac{\llbracket a\rrbracket^{+}}{1} = \frac{\llbracket b\rrbracket^{+}}{1} 1 [ [ a ] ] + = 1 [ [ b ] ] + h h h h ( ⟦ a ⟧ + ) = h ( ⟦ b ⟧ + ) h(\llbracket a\rrbracket^{+}) = h(\llbracket b\rrbracket^{+}) h ( [ [ a ] ] + ) = h ( [ [ b ] ] + ) h h h ⟦ a ⟧ + = ⟦ b ⟧ + \llbracket a\rrbracket^{+} = \llbracket b\rrbracket^{+} [ [ a ] ] + = [ [ b ] ] + a ≡ R + b a \equiv_{R}^{+} b a ≡ R + b
The next two chapters will be dedicated to showing the braid monoid satisfies the cancellative Ore conditions. Once we have done so, we will be able to embed the braid monoid (no scare quotes now!) into the braid group. Then Theorem 3.14 can be used to state that if two positive braid words are equivalent, they are also equivalent in the braid monoid.
Theorem 3.15. For positive a , b ∈ B W a,b \in BW a , b ∈ B W a ≡ R B b a \equiv_{R_B} b a ≡ R B b a ≡ R B + + b a \equiv_{R_{B^{+}}}^{+} b a ≡ R B + + b
That is, there exists a proof of their equivalence in which undercrossings never appear.15
Definition 3.16. We implement the (infinite) braid monoid relation R M R_M R M
inductive braid_monoid_rels_inf_one_refl_symm : FreeMonoid ℕ → FreeMonoid ℕ → Prop
| adjacent (i j : ℕ) (h : i.dist j = 1) :
braid_monoid_rels_inf_one_refl_symm (of i * of j * of i) (of j * of i * of j)
| separated (i j : ℕ) (h : i.dist j ≥ 2) :
braid_monoid_rels_inf_one_refl_symm (of i * of j) (of j * of i)
| basic : braid_monoid_rels_inf_one_refl_symm 1 1Note that the basic case could be replaced by any instance of reflexivity.
Lemma 3.17. We have our desired condition that
free_group_set_of_function (braid_monoid_rels_inf_one_refl_symm) = braidRelationInf.The following (and final) theorem is not necessary for our project, but is a nice consequence of the above discussion:
Theorem 3.19. Let ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩ c c c ⟨ S ∣ R ⟩ \langle S \mid R\rangle ⟨ S ∣ R ⟩ a , b ∈ ⟨ S ∣ R ⟩ + a,b \in\langle S \mid R\rangle^{+} a , b ∈ ⟨ S ∣ R ⟩ + c = ( ι ( b ) ) − 1 ι ( a ) c=(\iota(b))^{-1}\iota(a) c = ( ι ( b ) ) − 1 ι ( a )
Proof. By surjectivity of h 2 h_2 h 2 x ∈ PML ( S , R ) x \in\operatorname{PML}(S,R) x ∈ PML ( S , R ) h 2 ( x ) = c h_2(x)=c h 2 ( x ) = c x = r s x=\frac{r}{s} x = s r r , s ∈ ⟨ S ∣ R ⟩ + r,s \in\langle S \mid R\rangle^{+} r , s ∈ ⟨ S ∣ R ⟩ + c = ( ι ( s ) ) − 1 ι ( r ) c=(\iota(s))^{-1}\iota(r) c = ( ι ( s ) ) − 1 ι ( r )
c = h 2 ( r s ) = h 2 ( 1 ⋅ r 1 ⋅ s ) = h 2 ( 1 s ⋅ r 1 ) letting a = 1 and b = 1 in definition 3.4 of ∙ = h 2 ( ( s 1 ) − 1 ⋅ r 1 ) definition of inverse in 3.7 = ( h 2 ( s 1 ) ) − 1 ⋅ h 2 ( r 1 ) h 2 is a homomorphism = ( h 2 ( h ( s ) ) ) − 1 ⋅ h 2 ( h ( r ) ) definition of h = ( ι ( s ) ) − 1 ι ( r ) definition of ι \begin{aligned}
c &= h_2\left(\frac{r}{s}\right) \\
&= h_2\left(\frac{1\cdot r}{1\cdot s}\right) \\
&= h_2\left(\frac{1}{s}\cdot\frac{r}{1}\right) && \text{letting }a=1\text{ and }b=1\text{ in definition }3.4\text{ of }\bullet\\
&= h_2\left(\left(\frac{s}{1}\right)^{-1}\cdot\frac{r}{1}\right) && \text{definition of inverse in }3.7 \\
&= \left(h_2\left(\frac{s}{1}\right)\right)^{-1}\cdot h_2\left(\frac{r}{1}\right) && h_2\text{ is a homomorphism} \\
&= \left(h_2(h(s))\right)^{-1}\cdot h_2(h(r)) && \text{definition of }h \\
&= (\iota(s))^{-1}\iota(r) && \text{definition of }\iota
\end{aligned} c = h 2 ( s r ) = h 2 ( 1 ⋅ s 1 ⋅ r ) = h 2 ( s 1 ⋅ 1 r ) = h 2 ( ( 1 s ) − 1 ⋅ 1 r ) = ( h 2 ( 1 s ) ) − 1 ⋅ h 2 ( 1 r ) = ( h 2 ( h ( s )) ) − 1 ⋅ h 2 ( h ( r )) = ( ι ( s ) ) − 1 ι ( r ) letting a = 1 and b = 1 in definition 3.4 of ∙ definition of inverse in 3.7 h 2 is a homomorphism definition of h definition of ι Thus, specialized to braids, we see that any braid is equivalent to one in positive-negative form (that is, all overcrossings appearing before any undercrossings).
Common Multiples
In this chapter, we show the first of the two required conditions of Ore localization: common multiples. The proof for the common-multiples property follows Dehornoy exactly.16
Definition 4.1. The σ i , j \sigma_{i,j} σ i , j i th i^{\text{th}} i th j j j i , j ∈ N i,j \in\mathbb{N} i , j ∈ N
σ i , j = { ε if i = j , σ i ⋅ σ i + 1 , j if i < j , σ i − 1 ⋅ σ i − 1 , j if i > j . \sigma_{i,j} =
\begin{cases}
\varepsilon& \text{if }i=j,\\
\sigma_i\cdot\sigma_{i+1,j} & \text{if }i<j,\\
\sigma_{i-1}\cdot\sigma_{i-1,j} & \text{if }i>j.
\end{cases} σ i , j = ⎩ ⎨ ⎧ ε σ i ⋅ σ i + 1 , j σ i − 1 ⋅ σ i − 1 , j if i = j , if i < j , if i > j . For example, σ 6 , 2 = σ 5 σ 4 σ 3 σ 2 \sigma_{6,2}=\sigma_5\sigma_4\sigma_3\sigma_2 σ 6 , 2 = σ 5 σ 4 σ 3 σ 2 17
Lemma 3.18. The presented monoids with generating relations R M R_M R M R B + S R^S_{B+} R B + S
Thus, we may return to Theorem 3.14 and safely apply it in this case.
Figure 16 (PDF p. 24).
Definition 4.2. The Δ n \Delta_n Δ n n + 1 n+1 n + 1
Δ 0 = ε \Delta_0=\varepsilon Δ 0 = ε Δ n + 1 = σ 0 , n + 1 Δ n \Delta_{n+1}=\sigma_{0,n+1}\Delta_n Δ n + 1 = σ 0 , n + 1 Δ n For example, Δ 4 = σ 0 σ 1 σ 2 σ 3 σ 0 σ 1 σ 2 σ 0 σ 1 σ 0 \Delta_4=\sigma_0\sigma_1\sigma_2\sigma_3\sigma_0\sigma_1\sigma_2\sigma_0\sigma_1\sigma_0 Δ 4 = σ 0 σ 1 σ 2 σ 3 σ 0 σ 1 σ 2 σ 0 σ 1 σ 0
Figure 17 (PDF p. 24).
We next note that any generator can be completed into a delta braid of sufficiently large index:
Theorem 4.3. For n , i ∈ N n,i\in\mathbb{N} n , i ∈ N 0 < i < n 0<i<n 0 < i < n B W + BW^+ B W + ∂ n ( i ) \partial_n(i) ∂ n ( i ) σ i ∂ n ( i ) ≡ + Δ n \sigma_i\partial_n(i)\equiv_+\Delta_n σ i ∂ n ( i ) ≡ + Δ n
Thus, for any k ∈ N k \in\mathbb{N} k ∈ N k > i k > i k > i k > j k > j k > j Δ k \Delta_k Δ k σ i \sigma_i σ i σ j \sigma_j σ j
Definition 4.4. The flip function f n f_n f n B W + BW^{+} B W + f n ( σ i ) = σ n − 1 − i f_n(\sigma_i)=\sigma_{n-1-i} f n ( σ i ) = σ n − 1 − i i < n i<n i < n f n f_n f n B W + BW^{+} B W + n n n f n f_n f n f n f_n f n
The flip function helps us move generators, and hence entire words in B W + BW^{+} B W +
Lemma 4.5. For any a ∈ B W + a \in BW^{+} a ∈ B W + n n n a Δ n ≡ + Δ n f n ( a ) a\Delta_n \equiv^{+} \Delta_n f_n(a) a Δ n ≡ + Δ n f n ( a )
We will use the flip function in the witness to the following theorem:
Theorem 4.6. Let a ∈ B W + a \in BW^{+} a ∈ B W + ℓ \ell ℓ a a a n n n p ℓ , n ( a ) p_{\ell,n}(a) p ℓ , n ( a ) a p ℓ , n ( a ) ≡ + ( Δ n ) ℓ a p_{\ell,n}(a) \equiv^{+} (\Delta_n)^\ell a p ℓ , n ( a ) ≡ + ( Δ n ) ℓ
Proof. (Sketch) Let a = σ a 1 σ a 2 … σ a ℓ − 1 σ a ℓ a=\sigma_{a_1}\sigma_{a_2}\ldots\sigma_{a_{\ell-1}}\sigma_{a_\ell} a = σ a 1 σ a 2 … σ a ℓ − 1 σ a ℓ
p ℓ , n ( a ) = p ℓ , n ( σ a 1 σ a 2 … σ a ℓ − 1 σ a ℓ ) = ∂ n ( a ℓ ) f n ( ∂ n ( a ℓ − 1 ) ) ∂ n ( a ℓ − 2 ) f n ( ∂ n ( a ℓ − 3 ) ) … p_{\ell,n}(a)=p_{\ell,n}(\sigma_{a_1}\sigma_{a_2}\ldots\sigma_{a_{\ell-1}}\sigma_{a_\ell})=\partial_n(a_\ell)f_n(\partial_n(a_{\ell-1}))\partial_n(a_{\ell-2})f_n(\partial_n(a_{\ell-3}))\ldots p ℓ , n ( a ) = p ℓ , n ( σ a 1 σ a 2 … σ a ℓ − 1 σ a ℓ ) = ∂ n ( a ℓ ) f n ( ∂ n ( a ℓ − 1 )) ∂ n ( a ℓ − 2 ) f n ( ∂ n ( a ℓ − 3 )) … 18
For example, when ℓ = 3 \ell=3 ℓ = 3
p 3 , n ( a ) = p 3 , n ( σ a 1 σ a 2 σ a 3 ) = ∂ n ( a 3 ) f n ( ∂ n ( a 2 ) ) ∂ n ( a 1 ) p_{3,n}(a)=p_{3,n}(\sigma_{a_1}\sigma_{a_2}\sigma_{a_3})=\partial_n(a_3)f_n(\partial_n(a_2))\partial_n(a_1) p 3 , n ( a ) = p 3 , n ( σ a 1 σ a 2 σ a 3 ) = ∂ n ( a 3 ) f n ( ∂ n ( a 2 )) ∂ n ( a 1 ) To show correctness of this particular witness, we calculate
a p 3 , n ( a ) = σ a 1 σ a 2 σ a 3 ∂ n ( a 3 ) f n ( ∂ n ( a 2 ) ) ∂ n ( a 1 ) ≡ + σ a 1 σ a 2 Δ n f n ( ∂ n ( a 2 ) ) ∂ n ( a 1 ) by Lemma 4.3 ≡ + σ a 1 σ a 2 ∂ n ( a 2 ) Δ n ∂ n ( a 1 ) by Lemma 4.5 ≡ + σ a 1 Δ n Δ n ∂ n ( a 1 ) by Lemma 4.3 = σ a 1 Δ n Δ n f n ( f n ( ∂ n ( a 1 ) ) ) f n is an involution ≡ + σ a 1 Δ i f n ( ∂ i ( a 1 ) ) Δ n by Lemma 4.5 ≡ + σ a 1 ∂ n ( a 1 ) Δ n Δ n by Lemma 4.5 ≡ + Δ n Δ n Δ n by Lemma 4.3 = ( Δ n ) 3 \begin{aligned}
a p_{3,n}(a) &= \sigma_{a_1}\sigma_{a_2}\sigma_{a_3}\partial_n(a_3)f_n(\partial_n(a_2))\partial_n(a_1) \\
&\equiv^{+} \sigma_{a_1}\sigma_{a_2}\Delta_n f_n(\partial_n(a_2))\partial_n(a_1) && \text{by Lemma 4.3} \\
&\equiv^{+} \sigma_{a_1}\sigma_{a_2}\partial_n(a_2)\Delta_n\partial_n(a_1) && \text{by Lemma 4.5} \\
&\equiv^{+} \sigma_{a_1}\Delta_n\Delta_n\partial_n(a_1) && \text{by Lemma 4.3} \\
&= \sigma_{a_1}\Delta_n\Delta_n f_n(f_n(\partial_n(a_1))) && f_n\text{ is an involution} \\
&\equiv^{+} \sigma_{a_1}\Delta_i f_n(\partial_i(a_1))\Delta_n && \text{by Lemma 4.5} \\
&\equiv^{+} \sigma_{a_1}\partial_n(a_1)\Delta_n\Delta_n && \text{by Lemma 4.5} \\
&\equiv^{+} \Delta_n\Delta_n\Delta_n && \text{by Lemma 4.3} \\
&= (\Delta_n)^3
\end{aligned} a p 3 , n ( a ) = σ a 1 σ a 2 σ a 3 ∂ n ( a 3 ) f n ( ∂ n ( a 2 )) ∂ n ( a 1 ) ≡ + σ a 1 σ a 2 Δ n f n ( ∂ n ( a 2 )) ∂ n ( a 1 ) ≡ + σ a 1 σ a 2 ∂ n ( a 2 ) Δ n ∂ n ( a 1 ) ≡ + σ a 1 Δ n Δ n ∂ n ( a 1 ) = σ a 1 Δ n Δ n f n ( f n ( ∂ n ( a 1 ))) ≡ + σ a 1 Δ i f n ( ∂ i ( a 1 )) Δ n ≡ + σ a 1 ∂ n ( a 1 ) Δ n Δ n ≡ + Δ n Δ n Δ n = ( Δ n ) 3 by Lemma 4.3 by Lemma 4.5 by Lemma 4.3 f n is an involution by Lemma 4.5 by Lemma 4.5 by Lemma 4.3 The full proof of correctness proceeds by induction on ℓ \ell ℓ
Theorem 4.7. (Right Common Multiples) For all a a a b b b B W + BW^{+} B W + c c c d d d B W + BW^{+} B W + a c ≡ + b d ac \equiv^{+} bd a c ≡ + b d
Proof. Set ℓ \ell ℓ a a a b b b n ∈ N n \in\mathbb{N} n ∈ N a a a b b b Theorem 4.6 we have
a p ℓ , n ( a ) ≡ + ( Δ n ) ℓ ≡ + b p ℓ , n ( b ) a p_{\ell,n}(a) \equiv^{+} (\Delta_n)^\ell\equiv^{+} b p_{\ell,n}(b) a p ℓ , n ( a ) ≡ + ( Δ n ) ℓ ≡ + b p ℓ , n ( b ) so p ℓ , n ( a ) p_{\ell,n}(a) p ℓ , n ( a ) p ℓ , n ( b ) p_{\ell,n}(b) p ℓ , n ( b )
While not needed for Ore localization, showing left common multiples is simple, thanks to the reversing function r r r Lemma 2.23 .
Theorem 4.8. (Left Common Multiples) For all a a a b b b B W + BW^{+} B W + c c c d d d B W + BW^{+} B W + c a ≡ + d b ca \mathrel{\equiv^{+}} db c a ≡ + d b
Proof. Apply Theorem 4.7 to r ( a ) r(a) r ( a ) r ( b ) r(b) r ( b ) c ′ c' c ′ d ′ d' d ′ r ( a ) ⋅ c ′ ≡ + r ( b ) ⋅ d ′ r(a)\cdot c' \mathrel{\equiv^{+}} r(b)\cdot d' r ( a ) ⋅ c ′ ≡ + r ( b ) ⋅ d ′ r ( c ′ ) r(c') r ( c ′ ) r ( d ′ ) r(d') r ( d ′ ) r ( c ′ ) ⋅ a ≡ + r ( d ′ ) ⋅ b r(c')\cdot a \mathrel{\equiv^{+}} r(d')\cdot b r ( c ′ ) ⋅ a ≡ + r ( d ′ ) ⋅ b Lemma 2.23 , it suffices to show that r ( r ( c ′ ) ⋅ a ) ≡ + r ( r ( d ′ ) ⋅ b ) r(r(c')\cdot a) \mathrel{\equiv^{+}} r(r(d')\cdot b) r ( r ( c ′ ) ⋅ a ) ≡ + r ( r ( d ′ ) ⋅ b ) r r r r ( a ) c ′ ≡ + r ( b ) d ′ r(a)c' \mathrel{\equiv^{+}} r(b)d' r ( a ) c ′ ≡ + r ( b ) d ′
We have thus shown the first of two Ore conditions for the braid monoid. Cancellativity is a bit trickier; it is proved using an alternative, diagrammatic representation of equivalence in the braid monoid, called a grid. This structure will be discussed in the following chapter.
Grids
We next prove the remaining condition for Ore localization – cancellativity of the braid monoid. To do so, following Dehornoy, we model braid monoid equivalence using rectangular grid diagrams19
Definition and Basic Properties
Here is an example of a grid, with source ( σ 2 σ 1 , σ 0 σ 1 ) (\sigma_2\sigma_1,\sigma_0\sigma_1) ( σ 2 σ 1 , σ 0 σ 1 ) ( σ 0 σ 1 σ 2 , σ 2 σ 1 σ 0 ) (\sigma_0\sigma_1\sigma_2,\sigma_2\sigma_1\sigma_0) ( σ 0 σ 1 σ 2 , σ 2 σ 1 σ 0 )
Figure 1 (PDF p. 26).
Intuitively, a grid is defined inductively as either a cell, or two grids abutted horizontally or vertically so that the relevant edges match. We give the data for a grid as a 4-tuple of braid monoid words, listing out the left, top, bottom, and right sides. We will later see that this uniquely determines the inner edges of the grid. We begin with the definition of cells:
Definition 5.1. The set of cells C C C ( B W + ) 4 (BW^{+})^{4} ( B W + ) 4
⟨ ε , ε , ε , ε ⟩ ∈ C \langle\varepsilon,\varepsilon,\varepsilon,\varepsilon\rangle\in C ⟨ ε , ε , ε , ε ⟩ ∈ C ε \varepsilon ε
Figure 2 (PDF p. 26).
⟨ ε , σ i , σ i , ε ⟩ ∈ C \langle\varepsilon,\sigma_i,\sigma_i,\varepsilon\rangle\in C ⟨ ε , σ i , σ i , ε ⟩ ∈ C i ∈ N i \in\mathbb{N} i ∈ N
ε → σ i ε ↓ ↓ ε → σ i ε \begin{array}{c}
\varepsilon\quad\xrightarrow{\ \sigma_i\ }\quad\varepsilon\\
\downarrow\hspace{5.1em}\downarrow\\
\varepsilon\quad\xrightarrow{\ \sigma_i\ }\quad\varepsilon
\end{array} ε σ i ε ↓ ↓ ε σ i ε ⟨ σ i , ε , ε , σ i ⟩ ∈ C \langle\sigma_i,\varepsilon,\varepsilon,\sigma_i\rangle\in C ⟨ σ i , ε , ε , σ i ⟩ ∈ C i ∈ N i \in\mathbb{N} i ∈ N
σ i → ε σ i ↓ ↓ σ i → ε σ i \begin{array}{c}
\sigma_i \quad\xrightarrow{\ \varepsilon\ }\quad\sigma_i\\
\downarrow\hspace{5.1em}\downarrow\\
\sigma_i \quad\xrightarrow{\ \varepsilon\ }\quad\sigma_i
\end{array} σ i ε σ i ↓ ↓ σ i ε σ i ⟨ σ i , σ i , ε , ε ⟩ ∈ C \langle\sigma_i,\sigma_i,\varepsilon,\varepsilon\rangle\in C ⟨ σ i , σ i , ε , ε ⟩ ∈ C i ∈ N i \in\mathbb{N} i ∈ N
σ i → σ i ε ↓ ↓ σ i → ε ε \begin{array}{c}
\sigma_i \quad\xrightarrow{\ \sigma_i\ }\quad\varepsilon\\
\downarrow\hspace{5.1em}\downarrow\\
\sigma_i \quad\xrightarrow{\ \varepsilon\ }\quad\varepsilon
\end{array} σ i σ i ε ↓ ↓ σ i ε ε For i , j ∈ N i,j \in\mathbb{N} i , j ∈ N ∣ i − j ∣ = 1 |i-j|=1 ∣ i − j ∣ = 1 ⟨ σ j , σ i , σ i σ j , σ j σ i ⟩ ∈ C \langle\sigma_j,\sigma_i,\sigma_i\sigma_j,\sigma_j\sigma_i\rangle\in C ⟨ σ j , σ i , σ i σ j , σ j σ i ⟩ ∈ C
σ j → σ i σ i ↓ σ j ↓ σ j σ j → σ i σ i σ j → σ j σ i \begin{array}{c}
\phantom{\sigma_j}\quad\xrightarrow{\ \sigma_i\ }\quad\phantom{\sigma_i}\\
\downarrow{\scriptstyle\sigma_j}\hspace{5.1em}\downarrow{\scriptstyle\sigma_j}\\
\phantom{\sigma_j}\quad\xrightarrow{\ \sigma_i\ }\quad\phantom{\sigma_i}\\[-0.2em]
\phantom{\sigma_j}\quad\xrightarrow{\ \sigma_j\ }\quad\phantom{\sigma_i}
\end{array} σ j σ i σ i ↓ σ j ↓ σ j σ j σ i σ i σ j σ j σ i For i , j ∈ N i,j \in\mathbb{N} i , j ∈ N ∣ i − j ∣ > 1 |i-j|>1 ∣ i − j ∣ > 1 ⟨ σ j , σ i , σ i , σ j ⟩ ∈ C \langle\sigma_j,\sigma_i,\sigma_i,\sigma_j\rangle\in C ⟨ σ j , σ i , σ i , σ j ⟩ ∈ C
σ j → σ i σ i ↓ σ i ↓ σ i σ j → σ i σ i \begin{array}{c}
\phantom{\sigma_j}\quad\xrightarrow{\ \sigma_i\ }\quad\phantom{\sigma_i}\\
\downarrow{\scriptstyle\sigma_i}\hspace{5.1em}\downarrow{\scriptstyle\sigma_i}\\
\phantom{\sigma_j}\quad\xrightarrow{\ \sigma_i\ }\quad\phantom{\sigma_i}
\end{array} σ j σ i σ i ↓ σ i ↓ σ i σ j σ i σ i Often we simplify our diagrams combining adjacent arrows and labels. This does not change the cell or structure; it is merely notation. Thus, for the second-to-last case of the above definition, we could draw:
σ j → σ i σ j σ i ↓ σ j ↓ σ j σ i σ i σ j → σ i σ j σ i \begin{array}{c}
\phantom{\sigma_j}\quad\xrightarrow{\ \sigma_i\ }\quad\phantom{\sigma_j\sigma_i}\\
\downarrow{\scriptstyle\sigma_j}\hspace{5.1em}\downarrow{\scriptstyle\sigma_j\sigma_i}\\
\phantom{\sigma_i\sigma_j}\quad\xrightarrow{\ \sigma_i\sigma_j\ }\quad\phantom{\sigma_i}
\end{array} σ j σ i σ j σ i ↓ σ j ↓ σ j σ i σ i σ j σ i σ j σ i Definition 5.2. The set of grids G G G
Horizontal abutting: If ⟨ a , b , c , d ⟩ , ⟨ d , e , f , g ⟩ ∈ G \langle a,b,c,d\rangle,\langle d,e,f,g\rangle\in G ⟨ a , b , c , d ⟩ , ⟨ d , e , f , g ⟩ ∈ G ⟨ a , b e , c f , g ⟩ ∈ G \langle a,be,cf,g\rangle\in G ⟨ a , b e , c f , g ⟩ ∈ G
Visually, if
b → b ↓ a ↓ d → c and e → e ↓ d ↓ g → f \begin{array}{c}
\phantom{b}\quad\xrightarrow{\ b\ }\quad\phantom{}\\
\downarrow{\scriptstyle a}\hspace{3em}\downarrow{\scriptstyle d}\\
\phantom{}\quad\xrightarrow{\ c\ }\quad\phantom{}
\end{array}
\qquad\text{and}\qquad
\begin{array}{c}
\phantom{e}\quad\xrightarrow{\ e\ }\quad\phantom{}\\
\downarrow{\scriptstyle d}\hspace{3em}\downarrow{\scriptstyle g}\\
\phantom{}\quad\xrightarrow{\ f\ }\quad\phantom{}
\end{array} b b ↓ a ↓ d c and e e ↓ d ↓ g f are grids, then so too is
b → b → e ↓ a ↓ d ↓ g → c → f \begin{array}{c}
\phantom{b}\quad\xrightarrow{\ b\ }\quad\phantom{}\quad\xrightarrow{\ e\ }\quad\phantom{}\\
\downarrow{\scriptstyle a}\hspace{3em}\downarrow{\scriptstyle d}\hspace{3em}\downarrow{\scriptstyle g}\\
\phantom{}\quad\xrightarrow{\ c\ }\quad\phantom{}\quad\xrightarrow{\ f\ }\quad\phantom{}
\end{array} b b e ↓ a ↓ d ↓ g c f Vertical abutting: If ⟨ a , b , c , d ⟩ , ⟨ e , c , f , g ⟩ ∈ G \langle a,b,c,d\rangle,\langle e,c,f,g\rangle\in G ⟨ a , b , c , d ⟩ , ⟨ e , c , f , g ⟩ ∈ G ⟨ a e , b , f , d g ⟩ ∈ G \langle ae,b,f,dg\rangle\in G ⟨ a e , b , f , d g ⟩ ∈ G
Visually, if
b → b ↓ a ↓ d → c and c → c ↓ e ↓ g → f \begin{array}{c}
\phantom{b}\quad\xrightarrow{\ b\ }\quad\phantom{}\\
\downarrow{\scriptstyle a}\hspace{3em}\downarrow{\scriptstyle d}\\
\phantom{}\quad\xrightarrow{\ c\ }\quad\phantom{}
\end{array}
\qquad\text{and}\qquad
\begin{array}{c}
\phantom{c}\quad\xrightarrow{\ c\ }\quad\phantom{}\\
\downarrow{\scriptstyle e}\hspace{3em}\downarrow{\scriptstyle g}\\
\phantom{}\quad\xrightarrow{\ f\ }\quad\phantom{}
\end{array} b b ↓ a ↓ d c and c c ↓ e ↓ g f are grids, then so too is
b → b ↓ a ↓ d → c ↓ e ↓ g → f \begin{array}{c}
\phantom{b}\quad\xrightarrow{\ b\ }\quad\phantom{}\\
\downarrow{\scriptstyle a}\hspace{3em}\downarrow{\scriptstyle d}\\
\phantom{}\quad\xrightarrow{\ c\ }\quad\phantom{}\\
\downarrow{\scriptstyle e}\hspace{3em}\downarrow{\scriptstyle g}\\
\phantom{}\quad\xrightarrow{\ f\ }\quad\phantom{}
\end{array} b b ↓ a ↓ d c ↓ e ↓ g f We may also draw a grid omitting its inner arrows, simply showing the four sides and the words labelling them. Later we will prove a uniqueness property for grids (Theorem 5.9 ), so this representation does not omit unrecoverable information. The last-drawn grid would thus be
⟶ b ↓ a e ↓ d g ⟶ f \begin{array}{c}
\underset{b}{\longrightarrow}\\[-2pt]
\downarrow^{ae}\qquad\qquad\downarrow^{dg}\\[-2pt]
\underset{f}{\longrightarrow}
\end{array} b ⟶ ↓ a e ↓ d g f ⟶ Definition 5.3. Given a grid ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G source (later also called spine) is ( a , b ) (a,b) ( a , b ) target is ( c , d ) (c,d) ( c , d )
For example, the above grid has source ( a e , b ) (ae,b) ( a e , b ) ( f , d g ) (f,dg) ( f , d g )
Theorem 5.4. If ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G a c ≡ + b d ac \equiv^{+} bd a c ≡ + b d
Proof. By structural induction on the grid.
Definition 5.5. Given a grid ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G length of its diagonal is the length of a c ac a c
Remark. The length of a grid’s diagonal is also equal to ∣ b d ∣ |bd| ∣ b d ∣ a c ≡ + b d ac \equiv^{+} bd a c ≡ + b d
We next amass simple properties of grids that will be helpful in proving more complex theorems. All proofs are via structural induction on the given grid.
Lemma 5.6. If ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G ⟨ b , a , d , c ⟩ ∈ G \langle b,a,d,c\rangle\in G ⟨ b , a , d , c ⟩ ∈ G
Visually, given
⟶ b ↓ a ↓ d ⟶ c \begin{array}{c}
\underset{b}{\longrightarrow}\\[-2pt]
\downarrow^{a}\qquad\qquad\downarrow^{d}\\[-2pt]
\underset{c}{\longrightarrow}
\end{array} b ⟶ ↓ a ↓ d c ⟶ we also have
⟶ a ↓ b ↓ c ⟶ d \begin{array}{c}
\underset{a}{\longrightarrow}\\[-2pt]
\downarrow^{b}\qquad\qquad\downarrow^{c}\\[-2pt]
\underset{d}{\longrightarrow}
\end{array} a ⟶ ↓ b ↓ c d ⟶ Lemma 5.7. (Splitting) Given ⟨ a 1 a 2 , b , c , d ⟩ ∈ G \langle a_1a_2,b,c,d\rangle\in G ⟨ a 1 a 2 , b , c , d ⟩ ∈ G ⟨ a 1 , b , m , d 1 ⟩ \langle a_1,b,m,d_1\rangle ⟨ a 1 , b , m , d 1 ⟩ ⟨ a 2 , m , c , d 2 ⟩ \langle a_2,m,c,d_2\rangle ⟨ a 2 , m , c , d 2 ⟩ d = d 1 d 2 d=d_1d_2 d = d 1 d 2
That is,
⟶ b ↓ a 1 ↓ a 2 ↓ d ⟶ c \begin{array}{c}
\underset{b}{\longrightarrow}\\[-2pt]
\downarrow^{a_1}\\[-2pt]
\downarrow^{a_2}\qquad\qquad\downarrow^{d}\\[-2pt]
\underset{c}{\longrightarrow}
\end{array} b ⟶ ↓ a 1 ↓ a 2 ↓ d c ⟶ can be split horizontally into two subgrids as so:
⟶ b ↓ a 1 ⟶ m ↓ d 1 ↓ a 2 ↓ d 2 ⟶ c \begin{array}{ccc}
&\underset{b}{\longrightarrow}&\\[-2pt]
\downarrow^{a_1}&\underset{m}{\longrightarrow}&\downarrow^{d_1}\\[-2pt]
\downarrow^{a_2}&&\downarrow^{d_2}\\[-2pt]
&\underset{c}{\longrightarrow}&
\end{array} ↓ a 1 ↓ a 2 b ⟶ m ⟶ c ⟶ ↓ d 1 ↓ d 2 Similarly, given ⟨ a , b 1 b 2 , c , d ⟩ ∈ G \langle a,b_1b_2,c,d\rangle\in G ⟨ a , b 1 b 2 , c , d ⟩ ∈ G ⟨ a , b 1 , c 1 , m ⟩ \langle a,b_1,c_1,m\rangle ⟨ a , b 1 , c 1 , m ⟩ ⟨ m , b 2 , c 2 , d ⟩ \langle m,b_2,c_2,d\rangle ⟨ m , b 2 , c 2 , d ⟩ c = c 1 c 2 c=c_1c_2 c = c 1 c 2
That is,
⟶ b 1 ⟶ b 2 ↓ a ↓ d ⟶ c \begin{array}{ccc}
\underset{b_1}{\longrightarrow}&\underset{b_2}{\longrightarrow}\\[-2pt]
\downarrow^{a}&&\downarrow^{d}\\[-2pt]
\underset{c}{\longrightarrow}
\end{array} b 1 ⟶ ↓ a c ⟶ b 2 ⟶ ↓ d can be split vertically into two subgrids so:
⟶ b 1 ⟶ b 2 ↓ a ↓ m ↓ d ⟶ c 1 ⟶ c 2 \begin{array}{ccc}
\underset{b_1}{\longrightarrow}&\underset{b_2}{\longrightarrow}\\[-2pt]
\downarrow^{a}&\downarrow^{m}&\downarrow^{d}\\[-2pt]
\underset{c_1}{\longrightarrow}&\underset{c_2}{\longrightarrow}
\end{array} b 1 ⟶ ↓ a c 1 ⟶ b 2 ⟶ ↓ m c 2 ⟶ ↓ d Note that Lemma 5.6 allows vertical splitting to be derived from horizontal splitting.
Lemma 5.8 (Determinative Spines). For braid monoid words a , b , c , d a,b,c,d a , b , c , d i , j i,j i , j
If ⟨ ε , b , c , d ⟩ ∈ G \langle\varepsilon,b,c,d\rangle\in G ⟨ ε , b , c , d ⟩ ∈ G c = b c=b c = b d = ε d=\varepsilon d = ε
If ⟨ a , ε , c , d ⟩ ∈ G \langle a,\varepsilon,c,d\rangle\in G ⟨ a , ε , c , d ⟩ ∈ G c = ε c=\varepsilon c = ε d = a d=a d = a
If ⟨ a , a , c , d ⟩ ∈ G \langle a,a,c,d\rangle\in G ⟨ a , a , c , d ⟩ ∈ G c = ε c=\varepsilon c = ε d = ε d=\varepsilon d = ε
If ⟨ σ i , σ j , c , d ⟩ ∈ G \langle\sigma_i,\sigma_j,c,d\rangle\in G ⟨ σ i , σ j , c , d ⟩ ∈ G ∣ i − j ∣ = 1 |i-j|=1 ∣ i − j ∣ = 1 c = σ j σ i c=\sigma_j\sigma_i c = σ j σ i d = σ i σ j d=\sigma_i\sigma_j d = σ i σ j
If ⟨ σ i , σ j , c , d ⟩ ∈ G \langle\sigma_i,\sigma_j,c,d\rangle\in G ⟨ σ i , σ j , c , d ⟩ ∈ G ∣ i − j ∣ > 1 |i-j|>1 ∣ i − j ∣ > 1 c = σ j c=\sigma_j c = σ j d = σ i d=\sigma_i d = σ i
Theorem 5.9. (Uniqueness of sinks) Any two grids with the same source must have the same sink.
Proof. (Sketch) Consider two grids g 1 = ⟨ a , b , c , d ⟩ g_1=\langle a,b,c,d\rangle g 1 = ⟨ a , b , c , d ⟩ g 2 = ⟨ a , b , c ′ , d ′ ⟩ g_2=\langle a,b,c',d'\rangle g 2 = ⟨ a , b , c ′ , d ′ ⟩ c = c ′ c=c' c = c ′ d = d ′ d=d' d = d ′ g 1 g_1 g 1 g 2 g_2 g 2
Stability
We define a property of grids called stability which will be essential to build grids from monoid equivalence, and thus to use grid properties to show cancellativity in the braid monoid.
Definition 5.10 (Stability). A grid ⟨ a , b , c , d ⟩ \langle a,b,c,d\rangle ⟨ a , b , c , d ⟩ stable if for all a ′ , b ′ ∈ B W + a',b' \in BW^{+} a ′ , b ′ ∈ B W + a ≡ + a ′ a \equiv^{+} a' a ≡ + a ′ b ≡ + b ′ b \equiv^{+} b' b ≡ + b ′ c ′ , d ′ ∈ B W + c',d' \in BW^{+} c ′ , d ′ ∈ B W + ⟨ a ′ , b ′ , c ′ , d ′ ⟩ \langle a',b',c',d'\rangle ⟨ a ′ , b ′ , c ′ , d ′ ⟩ c ≡ + c ′ c \equiv^{+} c' c ≡ + c ′ d ≡ + d ′ d \equiv^{+} d' d ≡ + d ′
The Lean proof that all grids are stable is done by a triple induction, first on the length of the grid’s diagonal. We then induct structurally on the equivalence between a a a a ′ a' a ′ b b b b ′ b' b ′ a ≡ a ′ a \equiv a' a ≡ a ′ b ≡ b ′ b \equiv b' b ≡ b ′
Lemma 5.11. Let ( w , x ) ∈ R B + (w,x) \in R_{B}^{+} ( w , x ) ∈ R B + k ∈ N k \in\mathbb{N} k ∈ N ( σ k , w ) (\sigma_k,w) ( σ k , w ) ( σ k , x ) (\sigma_k,x) ( σ k , x )
Proof. (sketch) We discuss the proof for ( σ k , w ) (\sigma_k,w) ( σ k , w ) ( σ k , x ) (\sigma_k,x) ( σ k , x ) g = ⟨ σ k , w , c , d ⟩ g = \langle\sigma_k,w,c,d\rangle g = ⟨ σ k , w , c , d ⟩ w w w k k k w w w w w w σ i σ j \sigma_i\sigma_j σ i σ j ∣ i − j ∣ ≥ 2 |i-j| \ge2 ∣ i − j ∣ ≥ 2 k k k ∣ i − k ∣ = 1 |i-k| = 1 ∣ i − k ∣ = 1 ∣ j − k ∣ = 1 |j-k| = 1 ∣ j − k ∣ = 1 i = 2 i = 2 i = 2 j = 4 j = 4 j = 4 k = 3 k = 3 k = 3 g ′ = ⟨ σ 3 , σ 2 σ 3 , σ 2 σ 3 σ 4 σ 3 σ 2 , σ 3 σ 4 σ 2 σ 3 ⟩ g' = \langle\sigma_3,\sigma_2\sigma_3,\sigma_2\sigma_3\sigma_4\sigma_3\sigma_2,\sigma_3\sigma_4\sigma_2\sigma_3\rangle g ′ = ⟨ σ 3 , σ 2 σ 3 , σ 2 σ 3 σ 4 σ 3 σ 2 , σ 3 σ 4 σ 2 σ 3 ⟩
Figure 18.
Thus, by uniqueness of sinks (Theorem 5.9 ) applied to g g g g ′ g' g ′ c = σ 2 σ 3 σ 4 σ 3 σ 2 c = \sigma_2\sigma_3\sigma_4\sigma_3\sigma_2 c = σ 2 σ 3 σ 4 σ 3 σ 2 d = σ 3 σ 4 σ 2 σ 3 d = \sigma_3\sigma_4\sigma_2\sigma_3 d = σ 3 σ 4 σ 2 σ 3 a ′ a' a ′ b ′ b' b ′ σ 3 ≡ + a ′ \sigma_3 \equiv^{+} a' σ 3 ≡ + a ′ σ 2 σ 4 ≡ + b ′ \sigma_2\sigma_4 \equiv^{+} b' σ 2 σ 4 ≡ + b ′ Lemma 2.24 , a ′ = σ 3 a' = \sigma_3 a ′ = σ 3 Lemma 2.25 , either b ′ = σ 2 σ 4 b' = \sigma_2\sigma_4 b ′ = σ 2 σ 4 b ′ = σ 4 σ 2 b' = \sigma_4\sigma_2 b ′ = σ 4 σ 2 c ′ = c c' = c c ′ = c d ′ = d d' = d d ′ = d ⟨ σ 3 , σ 2 σ 4 , c , d ⟩ \langle\sigma_3,\sigma_2\sigma_4,c,d\rangle ⟨ σ 3 , σ 2 σ 4 , c , d ⟩ c ′ = σ 4 σ 3 σ 2 σ 3 σ 4 c' = \sigma_4\sigma_3\sigma_2\sigma_3\sigma_4 c ′ = σ 4 σ 3 σ 2 σ 3 σ 4 d ′ = σ 3 σ 2 σ 4 σ 3 d' = \sigma_3\sigma_2\sigma_4\sigma_3 d ′ = σ 3 σ 2 σ 4 σ 3
::: A quick calculation verifies
c = σ 2 σ 3 σ 4 σ 3 σ 2 ≡ + σ 2 σ 4 σ 3 σ 4 σ 2 ≡ + σ 4 σ 2 σ 3 σ 4 σ 2 ≡ + σ 4 σ 2 σ 3 σ 2 σ 4 ≡ + σ 4 σ 3 σ 2 σ 3 σ 4 = c ′ \begin{aligned}
c &= \sigma_2\sigma_3\sigma_4\sigma_3\sigma_2\\
&\equiv^{+} \sigma_2\sigma_4\sigma_3\sigma_4\sigma_2\\
&\equiv^{+} \sigma_4\sigma_2\sigma_3\sigma_4\sigma_2\\
&\equiv^{+} \sigma_4\sigma_2\sigma_3\sigma_2\sigma_4\\
&\equiv^{+} \sigma_4\sigma_3\sigma_2\sigma_3\sigma_4\\
&=c'
\end{aligned} c = σ 2 σ 3 σ 4 σ 3 σ 2 ≡ + σ 2 σ 4 σ 3 σ 4 σ 2 ≡ + σ 4 σ 2 σ 3 σ 4 σ 2 ≡ + σ 4 σ 2 σ 3 σ 2 σ 4 ≡ + σ 4 σ 3 σ 2 σ 3 σ 4 = c ′ A similar calculation verifies d ≡ + d ′ d\equiv^{+}d' d ≡ + d ′
Theorem 5.12. All grids are stable.
For the proof of stability, we present one illustrative case, drawn from the inductive case of the first induction (on diagonal length). Our inductive hypothesis says that for some l l l l l l ⟨ a , b , c , d ⟩ \langle a,b,c,d\rangle ⟨ a , b , c , d ⟩ l l l
We select the case where a ≡ + a a\equiv^{+}a a ≡ + a a = a ′ a=a' a = a ′ b ≡ + b ′ b\equiv^{+}b' b ≡ + b ′ b = b 1 w b 2 b=b_1wb_2 b = b 1 w b 2 b ′ = b 1 x b 2 b'=b_1xb_2 b ′ = b 1 x b 2 ( w , x ) ∈ R B + (w,x)\in R_{B+} ( w , x ) ∈ R B +
where c = c 1 e c 2 c=c_1ec_2 c = c 1 e c 2 d 1 d_1 d 1 d 1 = σ k d 2 d_1=\sigma_kd_2 d 1 = σ k d 2
Note that f = f 1 f 2 f=f_1f_2 f = f 1 f 2 G 3 G_3 G 3 ∣ d 2 e ∣ |d_2e| ∣ d 2 e ∣ G 3 G_3 G 3
∣ d 2 e ∣ < ∣ σ k d 2 e ∣ ≤ ∣ b 1 σ k d 2 e c 2 ∣ = ∣ a c 1 e c 2 ∣ = ∣ a c ∣ = l |d_2e|<|\sigma_kd_2e|\leq|b_1\sigma_kd_2ec_2|=|ac_1ec_2|=|ac|=l ∣ d 2 e ∣ < ∣ σ k d 2 e ∣ ≤ ∣ b 1 σ k d 2 e c 2 ∣ = ∣ a c 1 e c 2 ∣ = ∣ a c ∣ = l G 3 G_3 G 3 G 4 G_4 G 4 20 Lemma 5.11 , we know that G 2 G_2 G 2
Unfortunately, lemmas showing preservation of stability under appending are as difficult to prove as Theorem 5.12 itself, as they require inductions on equivalences. Instead, we proceed manually, calculating the required grids and equivalences for this specific case.
First, by stability of G 2 G_2 G 2 G 2 ′ G_2' G 2 ′ ( σ k , x ) (\sigma_k,x) ( σ k , x ) g ′ , f 1 ′ g',f_1' g ′ , f 1 ′ g ≡ + g ′ g \equiv_+ g' g ≡ + g ′ f 1 ≡ + f 1 ′ f_1 \equiv_+ f_1' f 1 ≡ + f 1 ′ G 3 G_3 G 3 G 3 ′ G_3' G 3 ′ ( d 2 , g ′ ) (d_2,g') ( d 2 , g ′ ) ( e ′ , f 2 ′ ) (e',f_2') ( e ′ , f 2 ′ ) e ≡ + e ′ e \equiv_+ e' e ≡ + e ′ f 2 ≡ + f 2 ′ f_2 \equiv_+ f_2' f 2 ≡ + f 2 ′ G 4 G_4 G 4 f 1 f 2 ≡ + f 1 ′ f 2 ′ f_1f_2 \equiv_+ f_1'f_2' f 1 f 2 ≡ + f 1 ′ f 2 ′ G 4 G_4 G 4 G 4 ′ G_4' G 4 ′ ( f 1 ′ f 2 ′ , b 2 ) (f_1'f_2',b_2) ( f 1 ′ f 2 ′ , b 2 ) ( c 2 ′ , d ′ ) (c_2',d') ( c 2 ′ , d ′ ) c 2 ′ ≡ + c 2 c_2' \equiv_+ c_2 c 2 ′ ≡ + c 2 d ′ ≡ + d d' \equiv_+ d d ′ ≡ + d
→ b 1 → x → b 2 σ k ↓ G 2 ′ ↓ f 1 ′ G 1 ↓ G 4 ′ → c 1 → e ′ → c 2 ′ G 3 ′ ↓ d 2 \begin{array}{ccccc}
& \xrightarrow{\ b_1\ } & \xrightarrow{\ x\ } & \xrightarrow{\ b_2\ } & \\
\sigma_k & \downarrow& G_2' & \downarrow& f_1' \\
G_1 & & \downarrow& & G_4' \\
& \xrightarrow{\ c_1\ } & \xrightarrow{\ e'\ } & \xrightarrow{\ c_2'\ } & \\
& & G_3' & & \\
& & \downarrow& & \\
& & d_2 & & \\
\end{array} σ k G 1 b 1 ↓ c 1 x G 2 ′ ↓ e ′ G 3 ′ ↓ d 2 b 2 ↓ c 2 ′ f 1 ′ G 4 ′ We have just seen from the stability of G 4 G_4 G 4 d ≡ + d ′ d \equiv_+ d' d ≡ + d ′ c 1 ′ e ′ c 2 ′ ≡ + c c_1'e'c_2' \equiv_+ c c 1 ′ e ′ c 2 ′ ≡ + c
As we have seen in the above example, the proof can branch at the induction on the length and the inductions on the equivalences a ≡ + a ′ a \equiv_+ a' a ≡ + a ′ b ≡ + b ′ b \equiv_+ b' b ≡ + b ′
Theorem 5.13. There exists a grid with source ( a , b ) (a,b) ( a , b ) ( ε , ε ) (\varepsilon,\varepsilon) ( ε , ε ) a ≡ + b a \equiv_+ b a ≡ + b
Proof. The forward direction is immediate by Theorem 5.4 . For the backwards direction, assume that a ≡ + b a \equiv_+ b a ≡ + b a a a ⟨ a , a , ε , ε ⟩ \langle a,a,\varepsilon,\varepsilon\rangle ⟨ a , a , ε , ε ⟩ Theorem 5.12 , ⟨ a , a , ε , ε ⟩ \langle a,a,\varepsilon,\varepsilon\rangle ⟨ a , a , ε , ε ⟩ ⟨ a , b , c ′ , d ′ ⟩ \langle a,b,c',d'\rangle ⟨ a , b , c ′ , d ′ ⟩ c ′ ≡ + ε c' \equiv_+ \varepsilon c ′ ≡ + ε d ′ ≡ + ε d' \equiv_+ \varepsilon d ′ ≡ + ε c ′ = ε c'=\varepsilon c ′ = ε d ′ = ε d'=\varepsilon d ′ = ε ⟨ a , b , ε , ε ⟩ \langle a,b,\varepsilon,\varepsilon\rangle ⟨ a , b , ε , ε ⟩
We are now ready to prove cancellativity.
Cancellativity
Theorem 5.14. The braid monoid is left-cancellative; that is, if c a ≡ + c b ca \equiv_+ cb c a ≡ + c b a ≡ + b a \equiv_+ b a ≡ + b
Proof. Assume c a = c b ca=cb c a = c b Theorem 5.13 , there is a grid ⟨ c a , c b , ε , ε ⟩ \langle ca,cb,\varepsilon,\varepsilon\rangle ⟨ c a , c b , ε , ε ⟩
→ c → b c ↓ ε ↓ ε → ε → b a ↓ a ↓ ε → ε → ε \begin{array}{ccccc}
& \xrightarrow{\ c\ } & \xrightarrow{\ b\ } & & \\
c & \downarrow& \varepsilon& \downarrow& \varepsilon\\
& \xrightarrow{\ \varepsilon\ } & \xrightarrow{\ b\ } & & \\
a & \downarrow& a & \downarrow& \varepsilon\\
& \xrightarrow{\ \varepsilon\ } & \xrightarrow{\ \varepsilon\ } &
\end{array} c a c ↓ ε ↓ ε b ε b a ε ↓ ↓ ε ε The lower right subgrid gives us ⟨ a , b , ε , ε ⟩ ∈ G \langle a,b,\varepsilon,\varepsilon\rangle\in G ⟨ a , b , ε , ε ⟩ ∈ G Theorem 5.13 , we know a ≡ + b a \equiv^{+} b a ≡ + b
Theorem 5.15. The braid monoid is right-cancellative; that is, if a c ≡ + b c ac \equiv^{+} bc a c ≡ + b c a ≡ + b a \equiv^{+} b a ≡ + b Theorem 4.8 .
Thus, we have shown the necessary criteria to prove Theorem 3.15 , and may conclude that for positive a , b ∈ B W a,b \in BW a , b ∈ B W a ≡ R B b a \equiv_{R_B} b a ≡ R B b a ≡ R B + b a \equiv_{R_{B^{+}}} b a ≡ R B + b
Grid Existence
Beyond cancellativity, stability (via Theorem 5.13 ) is also used to show the existence of grids.
Theorem 5.16. For any a , b ∈ B W + a,b \in BW^{+} a , b ∈ B W + c , d ∈ B W + c,d \in BW^{+} c , d ∈ B W + ( a , b ) (a,b) ( a , b ) ( c , d ) (c,d) ( c , d )
Proof. Consider some a , b ∈ B W + a,b \in BW^{+} a , b ∈ B W + c 1 , d 1 c_{1},d_{1} c 1 , d 1 a c 1 ≡ + b d 1 ac_{1} \equiv^{+} bd_{1} a c 1 ≡ + b d 1 Theorem 4.7 . Then by Theorem 5.13 there is a grid ⟨ a c 1 , b d 1 , ε , ε ⟩ \langle ac_{1},bd_{1},\varepsilon,\varepsilon\rangle ⟨ a c 1 , b d 1 , ε , ε ⟩ c , d , e , f ∈ B W + c,d,e,f \in BW^{+} c , d , e , f ∈ B W +
b d 1 a ↓ ↓ c ⟶ f ⟶ c 1 ↓ ↓ ε ε \begin{array}{ccc}
& b & d_{1} \\
a & \downarrow& \downarrow\\
c & \longrightarrow& f \longrightarrow\\
c_{1} & \downarrow& \downarrow\\
& \varepsilon& \varepsilon
\end{array} a c c 1 b ↓ ⟶ ↓ ε d 1 ↓ f ⟶ ↓ ε The top-left grid gives us ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G
Grids and Rewriting
We have used grids fruitfully in the proof of the cancellativity in the braid monoid, as part of the localization procedure connecting the braid monoid to the braid group. This portion of the project is now complete.
We find continued use for grids in the next stage of the project, in which we discuss the reversing procedure, as informally discussed in the introduction. Recall the reversing relation:
Definition 6.1. The reversing relation R r e v R_{\mathrm{rev}} R rev
For all i ∈ N i \in\mathbb{N} i ∈ N ( σ i ‾ σ i , ε ) ∈ R r e v (\overline{\sigma_i}\sigma_i,\varepsilon) \in R_{\mathrm{rev}} ( σ i σ i , ε ) ∈ R rev
For all i , j ∈ N i,j \in\mathbb{N} i , j ∈ N ∣ i − j ∣ = 1 \lvert i-j\rvert=1 ∣ i − j ∣ = 1 ( σ i ‾ σ j , σ j σ i σ j ‾ σ i ‾ ) ∈ R r e v (\overline{\sigma_i}\sigma_j,\sigma_j\sigma_i\overline{\sigma_j}\overline{\sigma_i}) \in R_{\mathrm{rev}} ( σ i σ j , σ j σ i σ j σ i ) ∈ R rev
For all i , j ∈ N i,j \in\mathbb{N} i , j ∈ N ∣ i − j ∣ > 1 \lvert i-j\rvert>1 ∣ i − j ∣ > 1 ( σ i ‾ σ j , σ j σ i ‾ ) ∈ R r e v (\overline{\sigma_i}\sigma_j,\sigma_j\overline{\sigma_i}) \in R_{\mathrm{rev}} ( σ i σ j , σ j σ i ) ∈ R rev
Note that the reversing relation preserves equivalence under the braid relations – a quick induction shows the following:
Theorem 6.2. For any x , y ∈ B W x,y \in BW x , y ∈ B W x → ∗ R r e v y x \mathrel{\xrightarrow{*}_{R_{\mathrm{rev}}}} y x ∗ R rev y x ≡ y x \equiv y x ≡ y
We aim to connect reversing to grids in the following manner:
Theorem 6.3. There exists a grid ⟨ a , b , c , d ⟩ \langle a,b,c,d\rangle ⟨ a , b , c , d ⟩ a ‾ b → ∗ R r e v c d ‾ \overline{a}b \mathrel{\xrightarrow{*}_{R_{\mathrm{rev}}}} c\overline{d} a b ∗ R rev c d
We will thus be able to use properties of grids (such as existence and uniqueness) to prove properties of our reversing algorithm (such as, respectively, termination and correctness). In fact it turns out that the above theorem is not quite as precise as we will need for termination. We will need to define size metrics on Semi-Thue derivations and on grids to eventually provide a natural-number (and hence well-founded) measure that strictly decreases as we progress through our algorithm. We first present our proof of the theorem in its original form, and then augment it with the size metric in the following chapter. The astute reader will note that the linked code for this section occasionally contains material regarding size (in Lean, a suffix of the form .length for derivations and grids), and certain code structure (such as Type in place of Prop). This may be safely ignored for now, and will be discussed in detail in the following chapter.
Returning to Theorem 6.3 as we have presented it here, the forward direction is nearly immediate by induction on grids:
Theorem 6.4. If there exists a grid ⟨ a , b , c , d ⟩ \langle a,b,c,d\rangle ⟨ a , b , c , d ⟩ a b ‾ → R r e v ∗ c d ‾ \overline{ab} \xrightarrow[R_{\mathrm{rev}}]{*} \overline{cd} ab ∗ R rev c d
The reverse poses a much thornier problem. Patrick Dehornoy returned to the issue time and again. In one of his earliest papers, he took a different approach to showing termination for the subword reversing algorithm.
In his article “Deux propriétés des groupes de tresses ” from 1992 [12 ] , Dehornoy created an extended alphabet which grouped together letters from the original alphabet into single letters in the new alphabet. For example, he added to our S ∗ S^* S ∗ A = σ 1 σ 2 A=\sigma_1\sigma_2 A = σ 1 σ 2 B = σ 2 σ 1 B=\sigma_2\sigma_1 B = σ 2 σ 1 σ 1 σ 2 → R r e v ∗ σ 2 σ 1 σ 2 σ 1 = B A \sigma_1\sigma_2 \xrightarrow[R_{\mathrm{rev}}]{*} \sigma_2\sigma_1\sigma_2\sigma_1=BA σ 1 σ 2 ∗ R rev σ 2 σ 1 σ 2 σ 1 = B A
I have not attempted to implement this proof, and so I cannot speak to its rigor. It appears to be a solid, reliable proof. Nevertheless, there is still interest in the grid-based proof, due to its clarity of exposition and connection with other mathematical structures such as van Kampen diagrams. Clearly, Dehornoy preferred the grid representation, as nearly all of his following papers on the matter (approximately 10) discuss grids instead of the extended alphabet. He had a certain freedom in experimenting with different, intuitive options for grids, knowing that he had shown termination by a different method earlier.
In his 2019 textbook, he gives the following sketch [variable names changed to match ours]:
... when a b ‾ → R r e v ∗ c d ‾ \overline{ab} \xrightarrow[R_{\mathrm{rev}}]{*} \overline{cd} ab ∗ R rev c d a b ‾ \overline{ab} ab c d ‾ \overline{cd} c d
Let Γ 0 \Gamma_0 Γ 0 a a a b b b a b ‾ \overline{ab} ab Γ 0 \Gamma_0 Γ 0 Γ \Gamma Γ ( a , b ) (a,b) ( a , b ) ( c , d ) (c,d) ( c , d )
The sketch is compelling – one can easily envision the grid being built step-by-step. Earlier papers of Dehornoy’s which discuss reversing grids give additional ideas. In a 2011 paper [15 ] , he begins with the case of general van Kampen diagrams 21 :
Lemma 1.1 (folklore). If ( S , R ) (S,R) ( S , R ) w w w w ′ w' w ′ S ∗ S^* S ∗ R R R ( S , R ) (S,R) ( S , R ) ( w , w ′ ) (w,w') ( w , w ′ )
Proof (sketch). If ( w 0 , … , w p ) (w_0,\ldots,w_p) ( w 0 , … , w p ) R R R w w w w ′ w' w ′ w i w_i w i ( w , w ′ ) (w,w') ( w , w ′ ) K K K ( w , w ′ ) (w,w') ( w , w ′ ) R R R w w w w ′ w' w ′ K K K
Again, this feels intuitively justifiable, but does not suffice for a rigorous proof. In that same article (written before the textbook came out), he also gives a description of how one might extend this method to rectangular grids, using dotted lines to identify edges that would have touched in a regular van Kampen diagram (essentially, ε \varepsilon ε 22
Figure 4 (PDF p. 34). Reversing diagram associated with the last diagram in Figure 3 (PDF p. 50) ; the only difference is that we insist that all edges are horizontal oriented to the right or vertical oriented to bottom. Again we are stuck as there is no relation c … = d … c\ldots=d\ldots c … = d …
Figure 3 (PDF p. 50).
In 2006, Dehornoy returned to the idea of the extended alphabet, and showed grids where each cell edge corresponded to a letter of the extended alphabet [18 ] . The below figure is from page 10 of Dehornoy and Wiest’s “On Word Reversing in Braid Groups”:
Figure 5 (PDF p. 34). The reversing diagram (left) and the reversing grid (right) associated with the sequence σ 1 − 1 σ 2 σ 1 − 1 σ 3 ↷ σ 2 σ 1 σ 2 − 1 σ 1 − 1 σ 3 ↷ σ 2 σ 1 σ 2 − 1 σ 3 σ 1 − 1 ↷ σ 2 σ 1 σ 2 − 1 σ 3 σ 1 − 1 \sigma_1^{-1}\sigma_2\sigma_1^{-1}\sigma_3 \mathrel{\curvearrowright} \sigma_2\sigma_1\sigma_2^{-1}\sigma_1^{-1}\sigma_3 \mathrel{\curvearrowright} \sigma_2\sigma_1\sigma_2^{-1}\sigma_3\sigma_1^{-1} \mathrel{\curvearrowright} \sigma_2\sigma_1\sigma_2^{-1}\sigma_3\sigma_1^{-1} σ 1 − 1 σ 2 σ 1 − 1 σ 3 ↷ σ 2 σ 1 σ 2 − 1 σ 1 − 1 σ 3 ↷ σ 2 σ 1 σ 2 − 1 σ 3 σ 1 − 1 ↷ σ 2 σ 1 σ 2 − 1 σ 3 σ 1 − 1
This approach is tempting. However, it was not immediately applicable to our algorithm’s termination, as we are working over the original alphabet. The proof translating termination from one alphabet to the other is non-trivial, and a metric defined on the extended-alphabet grid would not increase with each one of our individual rewrites. In any case, Dehornoy’s sketch of extended-alphabet grids would have needed a more detailed treatment for formalization, and would have still involved the two largest parts of our argument in this chapter: translating one rewriting sequence for another and a definition of a grid-under-construction. In some sense, such extended-alphabet grids are less satisfying, as they hide many steps of the algorithm, all combined into a single cell.
Returning to our definition of grids, my early naive attempts based on keeping track of the cells filled in unfortunately fail to capture the complexity of the construction required. It certainly feels so simple, so intuitive, and just out of reach. Below, we give a novel (to the best of my knowledge) proof of this claim. The proof defines two intermediate structures: grid-style rewriting and partial grids. First, we convert the normal rewriting to grid-style rewriting. Then we show grid-style rewriting corresponds to a partial grid. Finally, we connect the idea of partial grids to regular grids.
Reading Grids Backwards
We begin by enlarging our definition of grids. At the moment, we have only discussed reading words off grids following the direction of the arrows. We can expand our interpretation of a grid structure by defining a procedure for reading against the direction of an arrow.
Instead of the labels on arrows representing elements of B W + BW^{+} B W + B W BW B W σ ˉ i \bar{\sigma}_i σ ˉ i σ i − 1 \sigma_i^{-1} σ i − 1 σ ˉ i σ i ≠ ε \bar{\sigma}_i\sigma_i \ne\varepsilon σ ˉ i σ i = ε σ ˉ i σ i ≡ ε \bar{\sigma}_i\sigma_i \equiv\varepsilon σ ˉ i σ i ≡ ε
We obtain such σ ˉ i \bar{\sigma}_i σ ˉ i σ ˉ i σ j \bar{\sigma}_i\sigma_j σ ˉ i σ j σ l σ ˉ k \sigma_l\bar{\sigma}_k σ l σ ˉ k
Figure 1 (PDF p. 35).
For a side labelled with a word of length > 1 >1 > 1
Figure 2 (PDF p. 35).
The upper bottom-left to top-right path would be read as σ ˉ k σ ˉ j σ ˉ i σ l \bar{\sigma}_k\bar{\sigma}_j\bar{\sigma}_i\sigma_l σ ˉ k σ ˉ j σ ˉ i σ l σ l σ ˉ k σ ˉ j σ ˉ i \sigma_l\bar{\sigma}_k\bar{\sigma}_j\bar{\sigma}_i σ l σ ˉ k σ ˉ j σ ˉ i w = σ a 0 σ a 1 … σ a n w=\sigma_{a_0}\sigma_{a_1}\ldots\sigma_{a_n} w = σ a 0 σ a 1 … σ a n w ˉ \bar{w} w ˉ σ ˉ a n … σ ˉ a 1 σ ˉ a 0 \bar{\sigma}_{a_n}\ldots\bar{\sigma}_{a_1}\bar{\sigma}_{a_0} σ ˉ a n … σ ˉ a 1 σ ˉ a 0
Notably, we have only given examples of reading a downwards-pointing arrow upwards. We could read left-to-right arrows backwards as well, using the same procedure. However, we will not find a use for this in the following algorithm.
In Lean, we implement such words as elements of List (Nat x Bool). For example, [(4, false), (3, true)] represents σ ˉ 4 σ 3 \bar{\sigma}_4\sigma_3 σ ˉ 4 σ 3 List over FreeMonoid – this is a nod to Theorem 6.3 , which discusses Semi-Thue rewriting, not presented monoids.
Empty Arrows
Recall that we omitted ε \varepsilon ε → σ 2 ↓ σ 0 ↓ σ 0 → σ 2 \begin{array}{ccc} & \xrightarrow{\sigma_2} & \\ \downarrow{\sigma_0} & & \downarrow{\sigma_0}\\ & \xrightarrow{\sigma_2} & \end{array} ↓ σ 0 σ 2 σ 2 ↓ σ 0 → σ 2 → ε ↓ σ 0 ↓ σ 0 ↓ σ 0 → σ 2 → ε \begin{array}{ccccc} & \xrightarrow{\sigma_2} & & \xrightarrow{\varepsilon} & \\ \downarrow{\sigma_0} & & \downarrow{\sigma_0} & & \downarrow{\sigma_0}\\ & \xrightarrow{\sigma_2} & & \xrightarrow{\varepsilon} & \end{array} ↓ σ 0 σ 2 σ 2 ↓ σ 0 ε ε ↓ σ 0
We add a new symbol 1 \mathbb{1} 1 1 \mathbb{1} 1 ε \varepsilon ε
Definition 6.5. The augmented braid generators are given by the set S 1 : = { 1 } ∪ ⋃ i ∈ N { σ i } S^{\mathbb{1}} := \{\mathbb{1}\} \cup\bigcup_{i\in\mathbb{N}}\{\sigma_i\} S 1 := { 1 } ∪ ⋃ i ∈ N { σ i }
The augmented braid alphabet is the set
A 1 : = S 1 ∪ S 1 ‾ = { 1 , 1 ‾ } ∪ ⋃ i ∈ N { σ i , σ i ‾ } \mathcal{A}^{\mathbb{1}} := S^{\mathbb{1}} \cup\overline{S^{\mathbb{1}}} = \{\mathbb{1},\overline{\mathbb{1}}\} \cup\bigcup_{i\in\mathbb{N}}\{\sigma_i,\overline{\sigma_i}\} A 1 := S 1 ∪ S 1 = { 1 , 1 } ∪ i ∈ N ⋃ { σ i , σ i } The set of augmented braid words B W 1 BW^{\mathbb{1}} B W 1
B W 1 : = ( A 1 ) ∗ = ( { 1 , 1 ‾ } ∪ ⋃ i ∈ N { σ i , σ i ‾ } ) ∗ BW^{\mathbb{1}} := \left(\mathcal{A}^{\mathbb{1}}\right)^* = \left(\{\mathbb{1},\overline{\mathbb{1}}\} \cup\bigcup_{i\in\mathbb{N}}\{\sigma_i,\overline{\sigma_i}\}\right)^* B W 1 := ( A 1 ) ∗ = ( { 1 , 1 } ∪ i ∈ N ⋃ { σ i , σ i } ) ∗ A subset of B W 1 BW^{\mathbb{1}} B W 1 positive augmented braid words B W 1 , + BW^{\mathbb{1},+} B W 1 , +
B W 1 , + : = ( S 1 ) ∗ BW^{\mathbb{1},+} := \left(S^{\mathbb{1}}\right)^* B W 1 , + := ( S 1 ) ∗ Remark. We occasionally find use for the terminology “negative augmented braid word”, which is an element of ( S 1 ‾ ) ∗ \left(\overline{S^{\mathbb{1}}}\right)^* ( S 1 ) ∗
In Lean, B W 1 BW^{\mathbb{1}} B W 1 List (Option N × Bool). The Bool keeps track of whether there is an overline on the generator; a value of none for Option N indicates the generator is a 1 \mathbb{1} 1 some n for Option N means the generator is σ n \sigma_n σ n
Definition 6.6. We define the following functions to map between B W BW B W B W 1 BW^{\mathbb{1}} B W 1
i 1 : B W → B W 1 i^{\mathbb{1}} : BW \to BW^{\mathbb{1}} i 1 : B W → B W 1
s 1 : B W 1 → B W s^{\mathbb{1}} : BW^{\mathbb{1}} \to BW s 1 : B W 1 → B W
f ( x ) = { σ i if x = σ i , ε if x = 1. f(x) =
\begin{cases}
\sigma_i & \text{if } x = \sigma_i,\\
\varepsilon& \text{if } x = \mathbb{1}.
\end{cases} f ( x ) = { σ i ε if x = σ i , if x = 1. Lemma 6.7. The function s 1 s^{\mathbb{1}} s 1 i 1 i^{\mathbb{1}} i 1
An Example
Recall the part of Theorem 6.3 to be proved: if a ‾ b ‾ → R r e v ∗ c ‾ d ‾ \overline{a}\overline{b} \xrightarrow{R_{\mathrm{rev}}^*} \overline{c}\overline{d} a b R rev ∗ c d ⟨ a , b , c , d ⟩ \langle a,b,c,d\rangle ⟨ a , b , c , d ⟩
To motivate the definitions and other decisions in the upcoming proof, we present an illustrative example. Let a = σ 2 σ 1 a = \sigma_2\sigma_1 a = σ 2 σ 1 b = σ 0 σ 1 b = \sigma_0\sigma_1 b = σ 0 σ 1 c = σ 0 σ 1 σ 2 c = \sigma_0\sigma_1\sigma_2 c = σ 0 σ 1 σ 2 d = σ 2 σ 1 σ 0 d = \sigma_2\sigma_1\sigma_0 d = σ 2 σ 1 σ 0 a ˉ → ∗ R r e v c ˉ d ˉ \bar{a} \xrightarrow{*}_{R_{\mathrm{rev}}} \bar{c}\bar{d} a ˉ ∗ R rev c ˉ d ˉ
σ 2 σ 1 ‾ σ 0 ‾ σ 0 σ 1 = σ 1 ‾ σ 2 σ 0 ‾ σ 1 ⟶ R r e v σ 1 ‾ σ 0 σ 2 ‾ σ 1 ⟶ R r e v σ 0 ‾ σ 1 σ 0 ‾ σ 1 ‾ σ 2 σ 1 ⟶ R r e v σ 0 ‾ σ 1 σ 0 ‾ σ 1 ‾ σ 2 σ 1 ‾ σ 2 ‾ ⟶ R r e v σ 0 ‾ σ 1 σ 0 ‾ σ 2 σ 1 ‾ σ 2 ‾ ⟶ R r e v σ 0 ‾ σ 1 σ 2 σ 0 ‾ σ 1 ‾ σ 2 ‾ = σ 0 σ 1 σ 2 σ 2 ‾ σ 1 ‾ σ 0 ‾ \overline{\sigma_2\sigma_1}\overline{\sigma_0}\sigma_0\sigma_1
= \overline{\sigma_1}\overline{\sigma_2\sigma_0}\sigma_1
\overset{R_{\mathrm{rev}}}{\longrightarrow} \overline{\sigma_1}\sigma_0\overline{\sigma_2}\sigma_1
\overset{R_{\mathrm{rev}}}{\longrightarrow} \overline{\sigma_0}\sigma_1\overline{\sigma_0}\overline{\sigma_1}\sigma_2\sigma_1
\overset{R_{\mathrm{rev}}}{\longrightarrow} \overline{\sigma_0}\sigma_1\overline{\sigma_0}\overline{\sigma_1}\sigma_2\overline{\sigma_1}\overline{\sigma_2}
\overset{R_{\mathrm{rev}}}{\longrightarrow} \overline{\sigma_0}\sigma_1\overline{\sigma_0}\sigma_2\overline{\sigma_1}\overline{\sigma_2}
\overset{R_{\mathrm{rev}}}{\longrightarrow} \overline{\sigma_0}\sigma_1\sigma_2\overline{\sigma_0}\overline{\sigma_1}\overline{\sigma_2}
= \sigma_0\sigma_1\sigma_2\overline{\sigma_2}\overline{\sigma_1}\overline{\sigma_0} σ 2 σ 1 σ 0 σ 0 σ 1 = σ 1 σ 2 σ 0 σ 1 ⟶ R rev σ 1 σ 0 σ 2 σ 1 ⟶ R rev σ 0 σ 1 σ 0 σ 1 σ 2 σ 1 ⟶ R rev σ 0 σ 1 σ 0 σ 1 σ 2 σ 1 σ 2 ⟶ R rev σ 0 σ 1 σ 0 σ 2 σ 1 σ 2 ⟶ R rev σ 0 σ 1 σ 2 σ 0 σ 1 σ 2 = σ 0 σ 1 σ 2 σ 2 σ 1 σ 0 We aim to construct a grid ⟨ σ 2 σ 1 , σ 0 σ 1 , σ 0 σ 1 σ 2 , σ 2 σ 1 σ 0 ⟩ \langle\sigma_2\sigma_1,\sigma_0\sigma_1,\sigma_0\sigma_1\sigma_2,\sigma_2\sigma_1\sigma_0\rangle ⟨ σ 2 σ 1 , σ 0 σ 1 , σ 0 σ 1 σ 2 , σ 2 σ 1 σ 0 ⟩
→ σ 0 σ 1 σ 2 σ 1 ↓ ↓ σ 2 σ 1 σ 0 → σ 0 σ 1 σ 2 \begin{CD}
@>{\sigma_0\sigma_1}>> \\
@V{\sigma_2\sigma_1}VV @VV{\sigma_2\sigma_1\sigma_0}V \\
@>{\sigma_0\sigma_1\sigma_2}>>
\end{CD} σ 2 σ 1 ↓ ⏐ σ 0 σ 1 σ 0 σ 1 σ 2 ↓ ⏐ σ 2 σ 1 σ 0 We begin by writing out the spine of the grid:
→ σ 0 → σ 1 σ 2 ↓ σ 1 ↓ \begin{CD}
@>{\sigma_0}>> @>{\sigma_1}>> \\
@V{\sigma_2}VV \\
@V{\sigma_1}VV
\end{CD} σ 2 ↓ ⏐ σ 0 σ 1 ↓ ⏐ σ 1 The first rewrite of σ 2 ‾ σ 0 ‾ \overline{\sigma_2}\overline{\sigma_0} σ 2 σ 0 σ 0 ‾ σ 2 ‾ \overline{\sigma_0}\overline{\sigma_2} σ 0 σ 2
→ σ 0 σ 2 ↓ ↓ σ 2 → σ 0 \begin{CD}
@>{\sigma_0}>> \\
@V{\sigma_2}VV @VV{\sigma_2}V \\
@>{\sigma_0}>>
\end{CD} σ 2 ↓ ⏐ σ 0 σ 0 ↓ ⏐ σ 2 We define the frontier of a partially-completed grid as its southeasternmost bottom-left-to-top-right path. The frontier of the above diagram is σ 1 ‾ σ 0 ‾ σ 2 σ 1 \overline{\sigma_1}\overline{\sigma_0}\sigma_2\sigma_1 σ 1 σ 0 σ 2 σ 1
This procedure continues easily for the next three steps of the derivation, lines 2 through 4, with corresponding cells fitting into the frontier of diagram neatly:
σ 0 σ 1 ⟶ ⟶ σ 2 ↓ σ 2 ↓ σ 0 ⟶ σ 1 ↓ σ 1 ↓ σ 0 ⟶ σ 1 ⟶ σ 0 σ 1 ⟶ ⟶ σ 2 ↓ σ 2 ↓ σ 2 ↓ σ 1 ↓ σ 0 ⟶ σ 1 ⟶ σ 2 ⟶ σ 1 ↓ σ 1 ↓ σ 0 ⟶ , and σ 0 σ 1 ⟶ ⟶ σ 2 ↓ σ 2 ↓ σ 2 ↓ σ 1 ↓ σ 0 ⟶ σ 1 ⟶ σ 2 ⟶ σ 1 ↓ σ 1 ↓ ε ↓ σ 0 ⟶ σ 1 ⟶ ε ⟶ \begin{array}{ccccc}
& \sigma_{0} & & \sigma_{1} & \\
& \longrightarrow& & \longrightarrow& \\
\sigma_{2}\downarrow& & \sigma_{2}\downarrow& & \\
& \sigma_{0}\longrightarrow& & \\
\sigma_{1}\downarrow& & \sigma_{1}\downarrow& & \\
& \sigma_{0}\longrightarrow& & \sigma_{1}\longrightarrow&
\end{array}
\qquad
\begin{array}{ccccc}
& \sigma_{0} & & \sigma_{1} & \\
& \longrightarrow& & \longrightarrow& \sigma_{2}\downarrow\\
\sigma_{2}\downarrow& & \sigma_{2}\downarrow& & \sigma_{1}\downarrow\\
& \sigma_{0}\longrightarrow& \sigma_{1}\longrightarrow& \sigma_{2}\longrightarrow& \\
\sigma_{1}\downarrow& & \sigma_{1}\downarrow& & \\
& & \sigma_{0}\longrightarrow& &
\end{array}
\qquad\text{, and}\qquad
\begin{array}{ccccc}
& \sigma_{0} & & \sigma_{1} & \\
& \longrightarrow& & \longrightarrow& \sigma_{2}\downarrow\\
\sigma_{2}\downarrow& & \sigma_{2}\downarrow& & \sigma_{1}\downarrow\\
& \sigma_{0}\longrightarrow& \sigma_{1}\longrightarrow& \sigma_{2}\longrightarrow& \\
\sigma_{1}\downarrow& & \sigma_{1}\downarrow& \varepsilon\downarrow& \\
& \sigma_{0}\longrightarrow& \sigma_{1}\longrightarrow& \varepsilon\longrightarrow&
\end{array} σ 2 ↓ σ 1 ↓ σ 0 ⟶ σ 0 ⟶ σ 0 ⟶ σ 2 ↓ σ 1 ↓ σ 1 ⟶ σ 1 ⟶ σ 2 ↓ σ 1 ↓ σ 0 ⟶ σ 0 ⟶ σ 2 ↓ σ 1 ⟶ σ 1 ↓ σ 0 ⟶ σ 1 ⟶ σ 2 ⟶ σ 2 ↓ σ 1 ↓ , and σ 2 ↓ σ 1 ↓ σ 0 ⟶ σ 0 ⟶ σ 0 ⟶ σ 2 ↓ σ 1 ⟶ σ 1 ↓ σ 1 ⟶ σ 1 ⟶ σ 2 ⟶ ε ↓ ε ⟶ σ 2 ↓ σ 1 ↓ We encounter an issue when moving to line 5. There, we replace σ ‾ 0 σ 2 \overline{\sigma}_{0}\sigma_{2} σ 0 σ 2 σ 2 σ ‾ 0 \sigma_{2}\overline{\sigma}_{0} σ 2 σ 0
σ 2 σ 0 ↓ ↓ σ 0 σ 2 ⟶ \begin{array}{ccc}
& \sigma_{2} & \\
\sigma_{0}\downarrow& & \downarrow\sigma_{0} \\
& \sigma_{2}\longrightarrow&
\end{array} σ 0 ↓ σ 2 σ 2 ⟶ ↓ σ 0 But this does not fit into the frontier! The frontier edges σ 0 \sigma_{0} σ 0
σ 2 σ 0 ↓ σ 2 ⟶ \begin{array}{ccc}
& \sigma_{2} & \\
\sigma_{0}\downarrow& & \\
& \sigma_{2}\longrightarrow&
\end{array} σ 0 ↓ σ 2 σ 2 ⟶ are separated by ε \varepsilon ε
To insert this cell, we must first add in the following two cells (which we will call trivial cells ), which do not correspond to a step of the derivation:
σ 0 σ 1 ⟶ ⟶ σ 2 ↓ σ 2 ↓ σ 2 ↓ σ 1 ↓ σ 0 ⟶ σ 1 ⟶ σ 2 ⟶ ε ↓ σ 1 ↓ ε ↓ σ 0 ⟶ σ 1 ⟶ ε ⟶ \begin{array}{ccccc}
& \sigma_{0} & & \sigma_{1} & \\
& \longrightarrow& & \longrightarrow& \sigma_{2}\downarrow\\
\sigma_{2}\downarrow& & \sigma_{2}\downarrow& & \sigma_{1}\downarrow\\
& \sigma_{0}\longrightarrow& \sigma_{1}\longrightarrow& \sigma_{2}\longrightarrow& \varepsilon\downarrow\\
\sigma_{1}\downarrow& & \varepsilon\downarrow& & \\
& \sigma_{0}\longrightarrow& \sigma_{1}\longrightarrow& \varepsilon\longrightarrow&
\end{array} σ 2 ↓ σ 1 ↓ σ 0 ⟶ σ 0 ⟶ σ 0 ⟶ σ 2 ↓ σ 1 ⟶ ε ↓ σ 1 ⟶ σ 1 ⟶ σ 2 ⟶ ε ⟶ σ 2 ↓ σ 1 ↓ ε ↓ Further, note that adding in trivial cells must be done recursively. To see why, consider the following snippet of a grid:
σ 1 ⟶ σ 2 ⟶ σ 1 ⟶ σ 1 ↓ ε ↓ ε ⟶ \begin{array}{cccc}
& \sigma_{1}\longrightarrow& \sigma_{2}\longrightarrow& \sigma_{1}\longrightarrow\\
\sigma_{1}\downarrow& & \varepsilon\downarrow& \\
& \varepsilon\longrightarrow& &
\end{array} σ 1 ↓ σ 1 ⟶ ε ⟶ σ 2 ⟶ ε ↓ σ 1 ⟶ When we add in the appropriate trivial cell
σ 2 ε ↓ ↓ ε σ 2 ⟶ \begin{array}{ccc}
& \sigma_{2} & \\
\varepsilon\downarrow& & \downarrow\varepsilon\\
& \sigma_{2}\longrightarrow&
\end{array} ε ↓ σ 2 σ 2 ⟶ ↓ ε we obtain the following grid snippet:
σ 1 ⟶ σ 2 ⟶ σ 1 ⟶ σ 1 ↓ ε ↓ ε ↓ ε ⟶ σ 2 ⟶ \begin{array}{cccc}
& \sigma_{1}\longrightarrow& \sigma_{2}\longrightarrow& \sigma_{1}\longrightarrow\\
\sigma_{1}\downarrow& & \varepsilon\downarrow& \varepsilon\downarrow\\
& \varepsilon\longrightarrow& \sigma_{2}\longrightarrow&
\end{array} σ 1 ↓ σ 1 ⟶ ε ⟶ σ 2 ⟶ ε ↓ σ 2 ⟶ σ 1 ⟶ ε ↓ This snippet now requires another trivial cell to be inserted.
Returning to our main example, we may now complete the grid following the rewrite in step (5), obtaining This has frontier σ 0 σ 1 1 σ 2 σ 0 1 ‾ σ 1 ‾ σ 2 ‾ \sigma_0\sigma_1 1\sigma_2\sigma_0\overline{1}\,\overline{\sigma_1}\,\overline{\sigma_2} σ 0 σ 1 1 σ 2 σ 0 1 σ 1 σ 2 s 1 ( σ 0 σ 1 1 σ 2 σ 0 1 ‾ σ 1 ‾ σ 2 ‾ ) = σ 0 σ 1 σ 2 σ 0 1 ‾ σ 1 ‾ σ 2 ‾ s^1(\sigma_0\sigma_1 1\sigma_2\sigma_0\overline{1}\,\overline{\sigma_1}\,\overline{\sigma_2})=\sigma_0\sigma_1\sigma_2\sigma_0\overline{1}\,\overline{\sigma_1}\,\overline{\sigma_2} s 1 ( σ 0 σ 1 1 σ 2 σ 0 1 σ 1 σ 2 ) = σ 0 σ 1 σ 2 σ 0 1 σ 1 σ 2
To give a rigorous proof, we must first give a definition of partially-constructed grids. Then, we must develop a procedure for inserting trivial cells. We could enact this procedure on the partial grids. It turns out to be simpler to translate the given derivation to one tracking ε \varepsilon ε
Grid-Style Rewriting
We define a relation R g r i d R_{\mathrm{grid}} R grid ε \varepsilon ε 1 1 1 R g r i d ( x , y ) R_{\mathrm{grid}}(x,y) R grid ( x , y ) x x x y y y
Definition 6.8. The relation R n o n t r i v R_{\mathrm{nontriv}} R nontriv R r e v R_{\mathrm{rev}} R rev R r e v R_{\mathrm{rev}} R rev σ i ‾ σ i \overline{\sigma_i}\sigma_i σ i σ i R n o n t r i v R_{\mathrm{nontriv}} R nontriv σ i ‾ σ i \overline{\sigma_i}\sigma_i σ i σ i 1 1 ‾ 1\overline{1} 1 1 R n o n t r i v R_{\mathrm{nontriv}} R nontriv
R n o n t r i v ( σ i ‾ σ i , 1 1 ‾ ) for all i ∈ N . R_{\mathrm{nontriv}}(\overline{\sigma_i}\sigma_i,1\overline{1})\quad\text{for all }i\in\mathbb{N}. R nontriv ( σ i σ i , 1 1 ) for all i ∈ N . This corresponds to the cell
σ i σ i ε ε \begin{array}{ccc}
& \sigma_i & \\
\sigma_i & \varepsilon& \varepsilon
\end{array} σ i σ i ε ε R n o n t r i v ( σ i ‾ σ j , σ j σ i σ j ‾ σ i ‾ ) when ∣ i − j ∣ = 1. R_{\mathrm{nontriv}}(\overline{\sigma_i}\sigma_j,\sigma_j\sigma_i\overline{\sigma_j}\,\overline{\sigma_i})\quad\text{when }|i-j|=1. R nontriv ( σ i σ j , σ j σ i σ j σ i ) when ∣ i − j ∣ = 1. This corresponds to the cell
σ j σ i σ j σ i \begin{array}{ccc}
& \sigma_j & \\
\sigma_i & \sigma_j & \sigma_i
\end{array} σ i σ j σ j σ i R n o n t r i v ( σ i ‾ σ j , σ j σ i ‾ ) when ∣ i − j ∣ ≥ 2. R_{\mathrm{nontriv}}(\overline{\sigma_i}\sigma_j,\sigma_j\overline{\sigma_i})\quad\text{when }|i-j|\ge2. R nontriv ( σ i σ j , σ j σ i ) when ∣ i − j ∣ ≥ 2. This corresponds to the cell
σ j σ i σ i \begin{array}{ccc}
& \sigma_j & \\
\sigma_i & \sigma_i
\end{array} σ i σ j σ i The relation R t r i v R_{\mathrm{triv}} R triv R r e v R_{\mathrm{rev}} R rev R t r i v R_{\mathrm{triv}} R triv
R t r i v ( 1 ‾ σ i , σ i 1 ‾ ) for all i ∈ N . R_{\mathrm{triv}}(\overline{1}\sigma_i,\sigma_i\overline{1})\quad\text{for all }i\in\mathbb{N}. R triv ( 1 σ i , σ i 1 ) for all i ∈ N . This corresponds to the cell
σ i ε ε σ i \begin{array}{ccc}
& \sigma_i & \\
\varepsilon& \varepsilon& \sigma_i
\end{array} ε σ i ε σ i R t r i v ( σ i ‾ 1 , 1 σ i ‾ ) R_{\mathrm{triv}}(\overline{\sigma_i}\,1,1\,\overline{\sigma_i}) R triv ( σ i 1 , 1 σ i ) i ∈ N i\in\mathbb{N} i ∈ N
σ i → ε σ i ↓ σ i ↓ σ i 1 → ε 1 \begin{array}{ccc}
\sigma_i & \xrightarrow{\varepsilon} & \sigma_i\\
\downarrow{\sigma_i} & & \downarrow{\sigma_i}\\
1 & \xrightarrow{\varepsilon} & 1
\end{array} σ i ↓ σ i 1 ε ε σ i ↓ σ i 1 R t r i v ( 1 ‾ 1 , 1 1 ‾ ) R_{\mathrm{triv}}(\overline{1}\,1,1\,\overline{1}) R triv ( 1 1 , 1 1 )
1 → ε 1 ↓ ε ↓ ε 1 → ε 1 \begin{array}{ccc}
1 & \xrightarrow{\varepsilon} & 1\\
\downarrow{\varepsilon} & & \downarrow{\varepsilon}\\
1 & \xrightarrow{\varepsilon} & 1
\end{array} 1 ↓ ε 1 ε ε 1 ↓ ε 1 Finally, R g r i d R_{\mathrm{grid}} R grid R t r i v R_{\mathrm{triv}} R triv R n o n t r i v R_{\mathrm{nontriv}} R nontriv
We next make precise the close relationship between R r e v R_{\mathrm{rev}} R rev R n o n t r i v R_{\mathrm{nontriv}} R nontriv
Theorem 6.9. If R r e v ( x , y ) R_{\mathrm{rev}}(x,y) R rev ( x , y ) y ′ ∈ B W 1 y'\in BW^1 y ′ ∈ B W 1 R n o n t r i v ( ι 1 ( x ) , y ′ ) R_{\mathrm{nontriv}}(\iota^1(x),y') R nontriv ( ι 1 ( x ) , y ′ ) y ′ y' y ′ s 1 ( y ′ ) = y s^1(y')=y s 1 ( y ′ ) = y
We next give terminology for augmented braid words which can or can not be reduced by the above relations.
Definition 6.10. An element b ∈ B W 1 b\in BW^1 b ∈ B W 1 b 1 x b 2 b_1xb_2 b 1 x b 2 ( x , y ) ∈ R t r i v (x,y)\in R_{\mathrm{triv}} ( x , y ) ∈ R triv
Otherwise, it is called trivially-irreducible . This is equivalent to saying b b b σ i ‾ 1 \overline{\sigma_i}1 σ i 1 1 σ i ‾ 1\overline{\sigma_i} 1 σ i 1 ‾ 1 \overline{1}1 1 1
We have analogous definitions for nontrivially-reducible and nontrivially-irreducible, and R g r i d R_{\mathrm{grid}} R grid R g r i d R_{\mathrm{grid}} R grid
Lemma 6.11. For any b ∈ B W b\in BW b ∈ B W ι 1 ( b ) \iota^1(b) ι 1 ( b )
We give a specific fact about irreducibility to be used in a later proof:
Lemma 6.12. If g g g s 1 ( g ) = c ‾ d ‾ s^1(g)=\overline{c}\overline{d} s 1 ( g ) = c d c , d ∈ B W + c,d\in BW^+ c , d ∈ B W + e , f ∈ B W 1 , + e,f\in BW^{1,+} e , f ∈ B W 1 , + g = e ‾ f ‾ g=\overline{e}\overline{f} g = e f
We name the important words of length two:
Definition 6.13. Given letters α , β ∈ S 1 \alpha,\beta\in S^1 α , β ∈ S 1 α ‾ β ‾ \overline{\alpha}\overline{\beta} α β open pair . A word in which at least one of α \alpha α β \beta β 1 1 1 trivial open pair . A word of the form σ i ‾ σ j ‾ \overline{\sigma_i}\overline{\sigma_j} σ i σ j sigma-sigma open pair . We use the adjective “open” to signify there is an opening for a grid cell in the corresponding diagram.
We have the analogous vocabulary for pairs of the form α ‾ β ‾ \overline{\alpha}\overline{\beta} α β
Building Grid-Style Rewriting
This section’s main result is the generation of an R g r i d R_{\mathrm{grid}} R grid R r e v R_{\mathrm{rev}} R rev
Theorem 6.14. Let a , b , c , d ∈ B W + a,b,c,d\in BW^+ a , b , c , d ∈ B W + a b ‾ ⟶ R r e v ∗ c d ‾ \overline{ab}\mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{rev}}}}\overline{cd} ab R rev ⟶ ∗ c d e , f e,f e , f ι 1 ( a ) ι 1 ( b ) ⟶ R g r i d ∗ e ‾ f ‾ \iota^1(a)\iota^1(b)\mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{grid}}}}\overline{e}\overline{f} ι 1 ( a ) ι 1 ( b ) R grid ⟶ ∗ e f 1 1 1 1 ‾ \overline{1} 1 e e e c c c s 1 ( e ) = c s^1(e)=c s 1 ( e ) = c s 1 ( f ) = d s^1(f)=d s 1 ( f ) = d
Before we begin the proof, let us return to our example from earlier in the chapter. Recall the derivation:
σ 2 ‾ σ 1 σ 0 ‾ σ 1 = σ 1 ‾ σ 2 ‾ σ 0 σ 1 \overline{\sigma_2}\sigma_1\overline{\sigma_0}\sigma_1=\overline{\sigma_1}\,\overline{\sigma_2}\sigma_0\sigma_1 σ 2 σ 1 σ 0 σ 1 = σ 1 σ 2 σ 0 σ 1 → R r e v σ 1 ‾ σ 0 σ 2 ‾ σ 1 (1) \to_{R_{\mathrm{rev}}}\overline{\sigma_1}\,\sigma_0\overline{\sigma_2}\sigma_1 \tag*{(1)} → R rev σ 1 σ 0 σ 2 σ 1 (1) → R r e v σ 0 σ 1 σ 0 ‾ σ 1 ‾ σ 2 ‾ σ 1 (2) \to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_0}\,\overline{\sigma_1}\,\overline{\sigma_2}\sigma_1 \tag*{(2)} → R rev σ 0 σ 1 σ 0 σ 1 σ 2 σ 1 (2) → R r e v σ 0 σ 1 σ 0 ‾ σ 1 ‾ σ 1 σ 2 σ 1 ‾ σ 2 ‾ (3) \to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_0}\,\overline{\sigma_1}\sigma_1\sigma_2\overline{\sigma_1}\,\overline{\sigma_2} \tag*{(3)} → R rev σ 0 σ 1 σ 0 σ 1 σ 1 σ 2 σ 1 σ 2 (3) → R r e v σ 0 σ 1 σ 0 ‾ σ 2 σ 1 ‾ σ 2 ‾ (4) \to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_0}\sigma_2\overline{\sigma_1}\,\overline{\sigma_2} \tag*{(4)} → R rev σ 0 σ 1 σ 0 σ 2 σ 1 σ 2 (4) → R r e v σ 0 σ 1 σ 2 σ 0 ‾ σ 1 ‾ σ 2 ‾ = σ 0 σ 1 σ 2 σ 2 ‾ σ 1 ‾ σ 0 ‾ (5) \begin{aligned}
\to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\sigma_2\overline{\sigma_0}\,\overline{\sigma_1}\,\overline{\sigma_2} \\
&=\sigma_0\sigma_1\sigma_2\overline{\sigma_2}\overline{\sigma_1}\overline{\sigma_0} \tag*{(5)}
\end{aligned} → R rev σ 0 σ 1 σ 2 σ 0 σ 1 σ 2 = σ 0 σ 1 σ 2 σ 2 σ 1 σ 0 (5) The following derivation will suffice for Theorem 6.14 , with e = σ 0 σ 1 1 ‾ σ 2 e=\sigma_0\sigma_1\overline{1}\sigma_2 e = σ 0 σ 1 1 σ 2 f = σ 2 σ 1 1 ‾ σ 0 f=\sigma_2\sigma_1\overline{1}\sigma_0 f = σ 2 σ 1 1 σ 0
σ 2 ‾ σ 1 σ 0 ‾ σ 1 = σ 1 ‾ σ 2 ‾ σ 0 σ 1 \overline{\sigma_2}\sigma_1\overline{\sigma_0}\sigma_1=\overline{\sigma_1}\,\overline{\sigma_2}\sigma_0\sigma_1 σ 2 σ 1 σ 0 σ 1 = σ 1 σ 2 σ 0 σ 1 → R r e v σ 1 ‾ σ 0 σ 2 ‾ σ 1 (6) \to_{R_{\mathrm{rev}}}\overline{\sigma_1}\,\sigma_0\overline{\sigma_2}\sigma_1 \tag*{(6)} → R rev σ 1 σ 0 σ 2 σ 1 (6) → R r e v σ 0 σ 1 σ 0 ‾ σ 1 ‾ σ 2 ‾ σ 1 (7) \to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_0}\,\overline{\sigma_1}\,\overline{\sigma_2}\sigma_1 \tag*{(7)} → R rev σ 0 σ 1 σ 0 σ 1 σ 2 σ 1 (7) → R r e v σ 0 σ 1 σ 0 ‾ σ 1 ‾ σ 1 σ 2 σ 1 ‾ σ 2 ‾ (8) \to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_0}\,\overline{\sigma_1}\sigma_1\sigma_2\overline{\sigma_1}\,\overline{\sigma_2} \tag*{(8)} → R rev σ 0 σ 1 σ 0 σ 1 σ 1 σ 2 σ 1 σ 2 (8) → R r e v σ 0 σ 1 σ 0 ‾ 1 ‾ 1 ‾ σ 2 σ 1 ‾ σ 2 ‾ (9) \to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_0}\,\overline{1}\overline{1}\sigma_2\overline{\sigma_1}\,\overline{\sigma_2} \tag*{(9)} → R rev σ 0 σ 1 σ 0 1 1 σ 2 σ 1 σ 2 (9) → R r e v σ 0 σ 1 σ 0 ‾ 1 ‾ σ 0 ‾ σ 2 ‾ σ 1 ‾ σ 2 ‾ (10) \to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_0}\,\overline{1}\overline{\sigma_0}\,\overline{\sigma_2}\overline{\sigma_1}\,\overline{\sigma_2} \tag*{(10)} → R rev σ 0 σ 1 σ 0 1 σ 0 σ 2 σ 1 σ 2 (10) → R r e v σ 0 σ 1 σ 0 ‾ σ 0 ‾ σ 2 ‾ 1 ‾ σ 1 ‾ σ 2 ‾ (11) \to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_0}\,\overline{\sigma_0}\overline{\sigma_2}\,\overline{1}\,\overline{\sigma_1}\,\overline{\sigma_2} \tag*{(11)} → R rev σ 0 σ 1 σ 0 σ 0 σ 2 1 σ 1 σ 2 (11) → R r e v σ 0 σ 1 σ 2 ‾ σ 0 ‾ 1 ‾ σ 1 ‾ σ 2 ‾ = σ 0 σ 1 1 ‾ σ 2 σ 2 ‾ σ 1 1 ‾ σ 0 (12) \begin{aligned}
\to_{R_{\mathrm{rev}}}\sigma_0\sigma_1\overline{\sigma_2}\overline{\sigma_0}\,\overline{1}\,\overline{\sigma_1}\,\overline{\sigma_2} \\
&=\sigma_0\sigma_1\overline{1}\sigma_2\overline{\sigma_2}\sigma_1\overline{1}\sigma_0 \tag*{(12)}
\end{aligned} → R rev σ 0 σ 1 σ 2 σ 0 1 σ 1 σ 2 = σ 0 σ 1 1 σ 2 σ 2 σ 1 1 σ 0 (12) Here, the work lies in moving the 1s in lines (4) and (4.1). Then, we are relieved to see that in line (4.2), we are poised to make the rewrite in line (5).
Moving on from this example, we begin by defining a general procedure for performing the trivial reductions (i.e. moving the 1 and 1 ‾ \overline{1} 1 23
We first define a helper function concatenate_reduction, which appends a single letter of our alphabet A 1 \mathcal{A}^{1} A 1 b ∈ B W 1 b\in BW^{1} b ∈ B W 1 b b b
Definition 6.15. Given a single letter α ∈ A 1 \alpha\in\mathcal{A}^{1} α ∈ A 1 b ∈ B W 1 b\in BW^{1} b ∈ B W 1 concatenate_reduction a b as
σ i < σ j \sigma_i<\sigma_j σ i < σ j i < j i<j i < j
σ i ‾ < σ j ‾ \overline{\sigma_i}<\overline{\sigma_j} σ i < σ j i < j i<j i < j
σ i < σ j ‾ \sigma_i<\overline{\sigma_j} σ i < σ j i , j i,j i , j
1 ‾ < 1 \overline{1}<1 1 < 1
σ i < 1 \sigma_i<1 σ i < 1 i i i
1 ‾ < σ i ‾ \overline{1}<\overline{\sigma_i} 1 < σ i i i i
We extend this to the associated shortlex ordering over strings. Although this method was not ultimately used in the project, it did lead to the implementation of the shortlex order, now in Mathlib.
if α \alpha α 1 ‾ \overline{1} 1 b = σ i b 1 b=\sigma_i b_1 b = σ i b 1 σ i ⋅ \sigma_i \cdot σ i ⋅ concatenate_reduction a b1. (This case rewrites 1 ‾ σ i \overline{1}\sigma_i 1 σ i σ i 1 ‾ \sigma_i\overline{1} σ i 1 α \alpha α
if α \alpha α S 1 ‾ \overline{S^1} S 1 b = 1 ‾ b 1 b=\overline{1}b_1 b = 1 b 1 1 ‾ ⋅ \overline{1}\cdot 1 ⋅ concatenate_reduction a b1. (In this case, α = σ i ‾ \alpha=\overline{\sigma_i} α = σ i α = 1 ‾ \alpha=\overline{1} α = 1 σ i ‾ 1 ‾ \overline{\sigma_i}\overline{1} σ i 1 1 ‾ σ i ‾ \overline{1}\overline{\sigma_i} 1 σ i 1 ‾ 1 ‾ \overline{1}\overline{1} 1 1 1 ‾ 1 ‾ \overline{1}\overline{1} 1 1 α \alpha α
otherwise, return α b \alpha b α b
Lemma 6.16. If b b b concatenate_reduction a b. The proof proceeds by induction on b b b α \alpha α b b b
Theorem 6.17. α b ⟶ R t r i v ∗ \alpha b \mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{triv}}}} α b R triv ⟶ ∗ concatenate_reduction a b. The proof proceeds in the same style as that of the above theorem.
We are now prepared to define the algorithm move_ones which trivially-reduces a given augmented braid word.
Definition 6.18. Given b ∈ B W 1 b\in BW^1 b ∈ B W 1 move_ones b as follows:
If b b b move_ones b returns the empty word ε \varepsilon ε
If b b b α b 1 \alpha b_1 α b 1 α ∈ A 1 \alpha\in\mathcal{A}^1 α ∈ A 1 b 1 ∈ B W 1 b_1\in BW^1 b 1 ∈ B W 1 c c c move_ones b1. Then we return concatenate_reduction a c.
In Lean, this reads as
def move_ones (L : List (Option N × Bool)) :=
match L with
| [] => []
| head :: tail => concatenate_reduction head (move_ones tail)Theorem 6.19. b ⟶ R t r i v ∗ b\mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{triv}}}} b R triv ⟶ ∗ move_ones b. The proof proceeds by induction, using Lemma 6.17.
Notably, as R t r i v ⊆ R g r i d R_{\mathrm{triv}}\subseteq R_{\mathrm{grid}} R triv ⊆ R grid b ⟶ R g r i d ∗ b\mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{grid}}}} b R grid ⟶ ∗ move_ones b
Theorem 6.20. move_ones does not affect the relative positions of any σ i \sigma_i σ i s 1 ( move_ones b ) = s 1 ( b ) s^1(\text{move\_ones }b)=s^1(b) s 1 ( move_ones b ) = s 1 ( b )
Theorem 6.21. move_ones produces a trivially-irreducible word; for any b ∈ B W 1 b\in BW^1 b ∈ B W 1 move_ones b is trivially-irreducible.
Let us take stock of where we are in terms of our example. The move_ones algorithm effects the work done in steps (4) − (4.2). We next noted that we were poised for the next step from the R r e v R_{\mathrm{rev}} R rev
We begin by defining a property pairAppears.
Definition 6.22. An element b ∈ B W 1 b\in BW^1 b ∈ B W 1 pairAppears property if for every substring σ i ‾ σ j ‾ \overline{\sigma_i}\overline{\sigma_j} σ i σ j s 1 ( b ) s^1(b) s 1 ( b ) σ i ‾ σ j ‾ \overline{\sigma_i}\overline{\sigma_j} σ i σ j b b b
This is not quite enough to define the desired property. For example, the string σ 1 ‾ 1 ‾ σ 2 ‾ 1 ‾ σ 1 ‾ σ 2 ‾ \overline{\sigma_1}\overline{1}\overline{\sigma_2}\overline{1}\overline{\sigma_1}\overline{\sigma_2} σ 1 1 σ 2 1 σ 1 σ 2 pairAppears, but the first occurrence of σ 1 ‾ σ 2 ‾ \overline{\sigma_1}\overline{\sigma_2} σ 1 σ 2 1 ‾ \overline{1} 1 pairAppears as follows:
Definition 6.23. An element b ∈ B W 1 b \in BW^1 b ∈ B W 1 b b b
Showing that an augmented braid word is trivially-irreducible is an easy way to show that it satisfies pairsTogether, thanks to the following theorem:
Theorem 6.24. Any trivially-irreducible b ∈ B W 1 b \in BW^1 b ∈ B W 1
Proof. (Sketch) We show that any irreducible b b b b b b
Note that pairsTogether is strictly weaker than trivial irreducibility. For example, σ 2 σ ˉ 1 \sigma_2\bar{\sigma}_1 σ 2 σ ˉ 1
We give two specific lemmas about open-pair infixes of trivially-irreducible strings, which will be used for the proof of Theorem 6.14 :
Lemma 6.25. Let w ∈ B W 1 w \in BW^1 w ∈ B W 1 s 1 ( w ) s^1(w) s 1 ( w ) σ ˉ i σ j \bar{\sigma}_i\sigma_j σ ˉ i σ j s 1 ( w ) = c σ ˉ i σ j d s^1(w)=c\bar{\sigma}_i\sigma_jd s 1 ( w ) = c σ ˉ i σ j d c , d ∈ B W 1 c,d \in BW^1 c , d ∈ B W 1 w 1 , w 2 ∈ B W 1 w_1,w_2 \in BW^1 w 1 , w 2 ∈ B W 1 w = w 1 σ ˉ i σ j w 2 w=w_1\bar{\sigma}_i\sigma_jw_2 w = w 1 σ ˉ i σ j w 2 c = s 1 ( w 1 ) c=s^1(w_1) c = s 1 ( w 1 ) d = s 1 ( w 2 ) d=s^1(w_2) d = s 1 ( w 2 )
Proof. (sketch) The proof proceeds by stacking induction on w w w w w w w w w w 1 , w 2 , w m ∈ B W 1 w_1,w_2,w_m \in BW^1 w 1 , w 2 , w m ∈ B W 1 w = w 1 w m w 2 w=w_1w_mw_2 w = w 1 w m w 2 s 1 ( w m ) = σ ˉ i σ j s^1(w_m)=\bar{\sigma}_i\sigma_j s 1 ( w m ) = σ ˉ i σ j
Lemma 6.26. Given trivially-irreducible a , b ∈ B W 1 a,b \in BW^1 a , b ∈ B W 1 c , d ∈ B W c,d \in BW c , d ∈ B W s 1 ( a ) σ ˉ i σ j s 1 ( b ) = c σ ˉ i σ j d s^1(a)\bar{\sigma}_i\sigma_js^1(b)=c\bar{\sigma}_i\sigma_jd s 1 ( a ) σ ˉ i σ j s 1 ( b ) = c σ ˉ i σ j d c c c s 1 ( a ) s^1(a) s 1 ( a )
Table 1.
Case 2: We have b 1 , b 2 ∈ B W 1 b_1,b_2 \in BW^1 b 1 , b 2 ∈ B W 1 b = b 1 σ ˉ i σ j b 2 b=b_1\bar{\sigma}_i\sigma_jb_2 b = b 1 σ ˉ i σ j b 2 c = s 1 ( a ) σ ˉ i σ j s 1 ( b 1 ) c=s^1(a)\bar{\sigma}_i\sigma_js^1(b_1) c = s 1 ( a ) σ ˉ i σ j s 1 ( b 1 ) d = s 1 ( b 2 ) d=s^1(b_2) d = s 1 ( b 2 )
Table 2.
Case 3: We have a 1 , a 2 ∈ B W 1 a_1,a_2 \in BW^1 a 1 , a 2 ∈ B W 1 a = a 1 σ ˉ i σ j a 2 a=a_1\bar{\sigma}_i\sigma_ja_2 a = a 1 σ ˉ i σ j a 2 c = s 1 ( a 1 ) c=s^1(a_1) c = s 1 ( a 1 ) d = s 1 ( a 2 ) σ ˉ i σ j s 1 ( b ) d=s^1(a_2)\bar{\sigma}_i\sigma_js^1(b) d = s 1 ( a 2 ) σ ˉ i σ j s 1 ( b )
Table 3.
Proof. (sketch) Because σ ˉ i ≠ σ j \bar{\sigma}_i \ne\sigma_j σ ˉ i = σ j c c c s 1 ( a ) ± 1 s^1(a) \pm1 s 1 ( a ) ± 1 Lemma 6.25 does the majority of the work.
With the necessary machinery set up, we prove a generalization of Theorem 6.14 .
Theorem 6.27. Given a , b ∈ B W a,b \in BW a , b ∈ B W a → ∗ R r e v b a \xrightarrow{*}_{R_{\mathrm{rev}}} b a ∗ R rev b g ∈ B W 1 g \in BW^{1} g ∈ B W 1 i 1 ( a ) → ∗ R r e v g i^{1}(a) \xrightarrow{*}_{R_{\mathrm{rev}}} g i 1 ( a ) ∗ R rev g s 1 ( g ) = b s^{1}(g)=b s 1 ( g ) = b
Proof. Assume a → ∗ R r e v b a \xrightarrow{*}_{R_{\mathrm{rev}}} b a ∗ R rev b a → R r e v b a \xrightarrow{}_{R_{\mathrm{rev}}} b a R rev b
Reflexivity: In this case, a = b a=b a = b g = i 1 ( b ) g=i^{1}(b) g = i 1 ( b ) i 1 ( a ) → ∗ R g r i d i 1 ( b ) i^{1}(a) \xrightarrow{*}_{R_{\mathrm{grid}}} i^{1}(b) i 1 ( a ) ∗ R grid i 1 ( b ) i 1 ( b ) i^{1}(b) i 1 ( b ) s 1 ( i 1 ( b ) ) = b s^{1}(i^{1}(b))=b s 1 ( i 1 ( b )) = b
Transitive one-step reduction: In this case, we have c , d , x , y ∈ B W c,d,x,y \in BW c , d , x , y ∈ B W a → R r e v c x d a \xrightarrow{}_{R_{\mathrm{rev}}} cxd a R rev c x d R r e v ( x , y ) R_{\mathrm{rev}}(x,y) R rev ( x , y ) b = c y d b=cyd b = cy d e ∈ B W 1 e \in BW^{1} e ∈ B W 1 i 1 ( a ) → R g r i d e i^{1}(a) \xrightarrow{}_{R_{\mathrm{grid}}} e i 1 ( a ) R grid e s 1 ( e ) = c x d s^{1}(e)=cxd s 1 ( e ) = c x d
As R r e v ( x , y ) R_{\mathrm{rev}}(x,y) R rev ( x , y ) x = σ ‾ i σ ‾ j x=\overline{\sigma}_{i}\overline{\sigma}_{j} x = σ i σ j i i i j j j y ′ ∈ B W 1 y' \in BW^{1} y ′ ∈ B W 1 R g r i d ( x , y ′ ) R_{\mathrm{grid}}(x,y') R grid ( x , y ′ ) Theorem 6.9 .
Thus, s 1 ( e ) = c σ ‾ i σ ‾ j d s^{1}(e)=c\overline{\sigma}_{i}\overline{\sigma}_{j}d s 1 ( e ) = c σ i σ j d e e e Theorem 6.24 . Hence σ ‾ i σ ‾ j \overline{\sigma}_{i}\overline{\sigma}_{j} σ i σ j e e e e = w σ ‾ i σ ‾ j t e=w\overline{\sigma}_{i}\overline{\sigma}_{j}t e = w σ i σ j t s 1 ( w σ ‾ i σ ‾ j t ) = c σ ‾ i σ ‾ j d s^{1}(w\overline{\sigma}_{i}\overline{\sigma}_{j}t)=c\overline{\sigma}_{i}\overline{\sigma}_{j}d s 1 ( w σ i σ j t ) = c σ i σ j d s 1 ( w ) σ ‾ i σ ‾ j s 1 ( t ) = c σ ‾ i σ ‾ j d s^{1}(w)\overline{\sigma}_{i}\overline{\sigma}_{j}s^{1}(t)=c\overline{\sigma}_{i}\overline{\sigma}_{j}d s 1 ( w ) σ i σ j s 1 ( t ) = c σ i σ j d w w w t t t e e e Lemma 6.26 we have the following three cases:
s 1 ( w ) = c s^{1}(w)=c s 1 ( w ) = c s 1 ( t ) = d s^{1}(t)=d s 1 ( t ) = d g = move_ones ( w + + y ′ + + t ) g=\texttt{move\_ones}(w\mathbin{++}y'\mathbin{++}t) g = move_ones ( w + + y ′ + + t ) i 1 ( a ) → R g r i d w x t i^{1}(a)\xrightarrow{}_{R_{\mathrm{grid}}}wxt i 1 ( a ) R grid w x t R g r i d ( x , y ′ ) R_{\mathrm{grid}}(x,y') R grid ( x , y ′ ) i 1 ( a ) → R g r i d w y ′ t i^{1}(a)\xrightarrow{}_{R_{\mathrm{grid}}}wy't i 1 ( a ) R grid w y ′ t i 1 ( a ) → ∗ R g r i d w y ′ t i^{1}(a)\xrightarrow{*}_{R_{\mathrm{grid}}}wy't i 1 ( a ) ∗ R grid w y ′ t Theorem 6.19 , w y ′ t → ∗ R g r i d move_ones ( w + + y ′ + + t ) wy't\xrightarrow{*}_{R_{\mathrm{grid}}}\texttt{move\_ones}(w\mathbin{++}y'\mathbin{++}t) w y ′ t ∗ R grid move_ones ( w + + y ′ + + t ) i 1 ( a ) → ∗ R g r i d move_ones ( w + + y ′ + + t ) i^{1}(a)\xrightarrow{*}_{R_{\mathrm{grid}}}\texttt{move\_ones}(w\mathbin{++}y'\mathbin{++}t) i 1 ( a ) ∗ R grid move_ones ( w + + y ′ + + t )
We thus have i 1 ( a ) → ∗ R g r i d g i^{1}(a)\xrightarrow{*}_{R_{\mathrm{grid}}}g i 1 ( a ) ∗ R grid g g = move_ones ( w + + y ′ + + t ) g=\texttt{move\_ones}(w\mathbin{++}y'\mathbin{++}t) g = move_ones ( w + + y ′ + + t ) Theorem 6.21 . Lastly,
s 1 ( g ) = s 1 ( move_ones ( w + + y ′ + + t ) ) = s 1 ( w y ′ t ) (by Theorem 6.20) = s 1 ( w ) s 1 ( y ′ ) s 1 ( t ) = c ⋅ s 1 ( y ′ ) ⋅ d = c y d (by Theorem 6.9) = b \begin{aligned}
s^{1}(g) &= s^{1}(\texttt{move\_ones}(w\mathbin{++}y'\mathbin{++}t)) \\
&= s^{1}(wy't) && \text{(by Theorem 6.20)} \\
&= s^{1}(w)s^{1}(y')s^{1}(t) \\
&= c\cdot s^{1}(y')\cdot d \\
&= cyd && \text{(by Theorem 6.9)} \\
&= b
\end{aligned} s 1 ( g ) = s 1 ( move_ones ( w + + y ′ + + t )) = s 1 ( w y ′ t ) = s 1 ( w ) s 1 ( y ′ ) s 1 ( t ) = c ⋅ s 1 ( y ′ ) ⋅ d = cy d = b (by Theorem 6.20) (by Theorem 6.9) as desired
In this case, t = t 1 σ ‾ i σ ‾ j t 2 t=t_{1}\overline{\sigma}_{i}\overline{\sigma}_{j}t_{2} t = t 1 σ i σ j t 2 c = s 1 ( w ) σ ‾ i σ ‾ j s 1 c=s^{1}(w)\overline{\sigma}_{i}\overline{\sigma}_{j}s^{1} c = s 1 ( w ) σ i σ j s 1 d = s 1 ( t 2 ) d=s^{1}(t_{2}) d = s 1 ( t 2 ) move_ones ( w + + x + + t 1 + + y ′ + + t 2 ) \texttt{move\_ones}(w\mathbin{++}x\mathbin{++}t_{1}\mathbin{++}y'\mathbin{++}t_{2}) move_ones ( w + + x + + t 1 + + y ′ + + t 2 )
Finally, we give the proof of Theorem 6.14 . Recall we must show that if a ˉ b ˉ → R r e v ∗ c ˉ d ˉ \bar{a}\bar{b} \xrightarrow{R_{\mathrm{rev}}^{*}} \bar{c}\bar{d} a ˉ b ˉ R rev ∗ c ˉ d ˉ e , f ∈ B W 1 , + e,f \in BW^{1,+} e , f ∈ B W 1 , + i 1 ( a ˉ ) i 1 ( b ) → R g r i d ∗ e ˉ f ˉ i^{1}(\bar{a})i^{1}(b) \xrightarrow{R_{\mathrm{grid}}^{*}} \bar{e}\bar{f} i 1 ( a ˉ ) i 1 ( b ) R grid ∗ e ˉ f ˉ s 1 ( e ) = c s^{1}(e)=c s 1 ( e ) = c s 1 ( f ) = d s^{1}(f)=d s 1 ( f ) = d
Proof. Let a , b , c , d ∈ B W + a,b,c,d \in BW^{+} a , b , c , d ∈ B W + a ˉ b ˉ → R r e v ∗ c ˉ d ˉ \bar{a}\bar{b} \xrightarrow{R_{\mathrm{rev}}^{*}} \bar{c}\bar{d} a ˉ b ˉ R rev ∗ c ˉ d ˉ Theorem 6.27 , there exists some trivially-irreducible g ∈ B W 1 g \in BW^{1} g ∈ B W 1 i 1 ( a ˉ b ˉ ) → R g r i d ∗ g i^{1}(\bar{a}\bar{b}) \xrightarrow{R_{\mathrm{grid}}^{*}} g i 1 ( a ˉ b ˉ ) R grid ∗ g s 1 ( g ) = c ˉ d ˉ s^{1}(g)=\bar{c}\bar{d} s 1 ( g ) = c ˉ d ˉ e , f ∈ B W 1 e,f \in BW^{1} e , f ∈ B W 1 g = e ˉ f ˉ g=\bar{e}\bar{f} g = e ˉ f ˉ i 1 ( a ˉ ) i 1 ( b ) → R g r i d ∗ e ˉ f ˉ i^{1}(\bar{a})i^{1}(b) \xrightarrow{R_{\mathrm{grid}}^{*}} \bar{e}\bar{f} i 1 ( a ˉ ) i 1 ( b ) R grid ∗ e ˉ f ˉ
Moreover, note that s 1 ( g ) = s 1 ( e ˉ f ˉ ) = s 1 ( e ) s 1 ( f ˉ ) s^{1}(g)=s^{1}(\bar{e}\bar{f})=s^{1}(e)s^{1}(\bar{f}) s 1 ( g ) = s 1 ( e ˉ f ˉ ) = s 1 ( e ) s 1 ( f ˉ ) s 1 ( e ) s^{1}(e) s 1 ( e ) s 1 ( f ˉ ) s^{1}(\bar{f}) s 1 ( f ˉ ) s 1 ( e ) s 1 ( f ˉ ) = c ˉ d ˉ s^{1}(e)s^{1}(\bar{f})=\bar{c}\bar{d} s 1 ( e ) s 1 ( f ˉ ) = c ˉ d ˉ s 1 ( e ) = c s^{1}(e)=c s 1 ( e ) = c s 1 ( f ˉ ) = d s^{1}(\bar{f})=d s 1 ( f ˉ ) = d
We now know that we may produce an R g r i d R_{\mathrm{grid}} R grid R r e v R_{\mathrm{rev}} R rev R g r i d R_{\mathrm{grid}} R grid ⟨ a , b , c , d ⟩ \langle a,b,c,d\rangle ⟨ a , b , c , d ⟩
Partial Grids
A partial grid generalizes the notion of a grid to include “unfinished” grids, which are missing cells in the bottom/right area. Let’s use the following as a motivating example:
Figure 19.
Partial grids will be defined as an inductive subset of ( B W 1 ) 5 (BW^{1})^{5} ( B W 1 ) 5 ⟨ σ ˉ 1 σ ˉ 1 , σ 2 σ 2 , σ 2 σ 1 , σ ˉ 2 σ ˉ 1 σ ˉ 1 , 1 , 1 σ ˉ 1 , σ ˉ 2 σ ˉ 1 ⟩ \langle\bar{\sigma}_{1}\bar{\sigma}_{1},\sigma_{2}\sigma_{2},\sigma_{2}\sigma_{1},\bar{\sigma}_{2}\bar{\sigma}_{1}\bar{\sigma}_{1},1,1\bar{\sigma}_{1},\bar{\sigma}_{2}\bar{\sigma}_{1}\rangle ⟨ σ ˉ 1 σ ˉ 1 , σ 2 σ 2 , σ 2 σ 1 , σ ˉ 2 σ ˉ 1 σ ˉ 1 , 1 , 1 σ ˉ 1 , σ ˉ 2 σ ˉ 1 ⟩
Figure 20.
In general, we read a partial grid as ⟨ a ˉ , b , c , m , d ˉ ⟩ \langle\bar{a}, b, c, m, \bar{d}\rangle ⟨ a ˉ , b , c , m , d ˉ ⟩
We draw a partial grid ⟨ a ˉ , b , c , m , d ˉ ⟩ \langle\bar{a}, b, c, m, \bar{d}\rangle ⟨ a ˉ , b , c , m , d ˉ ⟩
a ˉ → b ↓ d ˉ a ˉ → c ∼ m \begin{array}{c}
\bar{a}\ \ \xrightarrow{\ b\ }\ \ \downarrow\bar{d}\\[-2pt]
\phantom{\bar{a}}\xrightarrow{\ c\ }\ \sim m
\end{array} a ˉ b ↓ d ˉ a ˉ c ∼ m . We will soon show that both a a a d d d a ˉ \bar{a} a ˉ d ˉ \bar{d} d ˉ
Often, we speak of the (total) frontier of a partial grid; this is the concatenation of the bottom frontier, middle frontier, and right frontier. In our example, it would be σ 1 σ 2 ⋅ σ ˉ 2 σ ˉ 1 σ ˉ 1 σ ˉ 1 σ 1 σ 2 σ ˉ 1 ⋅ σ ˉ 2 σ ˉ 1 \sigma_1\sigma_2\cdot\bar{\sigma}_2\bar{\sigma}_1\bar{\sigma}_1\bar{\sigma}_1\sigma_1\sigma_2\bar{\sigma}_1\cdot\bar{\sigma}_2\bar{\sigma}_1 σ 1 σ 2 ⋅ σ ˉ 2 σ ˉ 1 σ ˉ 1 σ ˉ 1 σ 1 σ 2 σ ˉ 1 ⋅ σ ˉ 2 σ ˉ 1
We now give a formal, inductive definition of the set of partial grids G P ⊂ ( B W 1 ) 5 G_P\subset(BW^1)^5 G P ⊂ ( B W 1 ) 5
Definition 6.28. The set of partial grids G P G_P G P ( B W 1 ) 5 (BW^1)^5 ( B W 1 ) 5
(a cell) : For any a , b , c , d ∈ B W 1 , + a,b,c,d\in BW^{1,+} a , b , c , d ∈ B W 1 , + ( a ˉ b , c ˉ d ˉ ) ∈ R g r i d (\bar{a}b,\bar{c}\bar{d})\in R_{\mathrm{grid}} ( a ˉ b , c ˉ d ˉ ) ∈ R grid ⟨ a ˉ , b , c , ε , d ˉ ⟩ ∈ G P \langle\bar{a},b,c,\varepsilon,\bar{d}\rangle\in G_P ⟨ a ˉ , b , c , ε , d ˉ ⟩ ∈ G P
Visually,
a ˉ → b ↓ d ˉ a ˉ → c \begin{array}{c}
\bar{a}\ \ \xrightarrow{\ b\ }\ \ \downarrow\bar{d}\\[-2pt]
\phantom{\bar{a}}\xrightarrow{\ c\ }
\end{array} a ˉ b ↓ d ˉ a ˉ c (a spine) : For any a , b ∈ B W 1 , + ∖ { ε } a,b\in BW^{1,+}\setminus\{\varepsilon\} a , b ∈ B W 1 , + ∖ { ε } ⟨ a ˉ , b , ε , a ˉ b , ε ⟩ ∈ G P \langle\bar{a},b,\varepsilon,\bar{a}b,\varepsilon\rangle\in G_P ⟨ a ˉ , b , ε , a ˉ b , ε ⟩ ∈ G P
Visually,
a ˉ → b \begin{array}{c}
\bar{a}\ \ \xrightarrow{\ b\ }
\end{array} a ˉ b for non-empty a a a
a ˉ 1 → b 1 ↓ d ˉ 1 d ˉ 1 → b 2 ↓ d ˉ 2 a ˉ 1 → b 1 ⋅ b 2 ↓ d ˉ 2 a ˉ 1 → c 1 d ˉ 1 → c 2 ∼ m 2 a ˉ 1 → c 1 ⋅ c 2 ∼ m 2 \begin{array}{ccc}
\bar{a}_1\ \xrightarrow{\ b_1\ }\ \downarrow\bar{d}_1
&
\bar{d}_1\ \xrightarrow{\ b_2\ }\ \downarrow\bar{d}_2
&
\bar{a}_1\ \xrightarrow{\ b_1\cdot b_2\ }\ \downarrow\bar{d}_2\\
\phantom{\bar{a}_1}\xrightarrow{\ c_1\ }\ &
\phantom{\bar{d}_1}\xrightarrow{\ c_2\ }\ \sim m_2
&
\phantom{\bar{a}_1}\xrightarrow{\ c_1\cdot c_2\ }\ \sim m_2
\end{array} a ˉ 1 b 1 ↓ d ˉ 1 a ˉ 1 c 1 d ˉ 1 b 2 ↓ d ˉ 2 d ˉ 1 c 2 ∼ m 2 a ˉ 1 b 1 ⋅ b 2 ↓ d ˉ 2 a ˉ 1 c 1 ⋅ c 2 ∼ m 2 Given ⟨ a ˉ 1 , b 1 , c 1 , ε , d ˉ 1 ⟩ ∈ G P \langle\bar{a}_1,b_1,c_1,\varepsilon,\bar{d}_1\rangle\in G_P ⟨ a ˉ 1 , b 1 , c 1 , ε , d ˉ 1 ⟩ ∈ G P ⟨ d ˉ 1 , b 2 , c 2 , m 2 , d ˉ 2 ⟩ ∈ G P \langle\bar{d}_1,b_2,c_2,m_2,\bar{d}_2\rangle\in G_P ⟨ d ˉ 1 , b 2 , c 2 , m 2 , d ˉ 2 ⟩ ∈ G P ⟨ a ˉ 1 , b 1 ⋅ b 2 , c 1 ⋅ c 2 , m 2 , d ˉ 2 ⟩ ∈ G P \langle\bar{a}_1,b_1\cdot b_2,c_1\cdot c_2,m_2,\bar{d}_2\rangle\in G_P ⟨ a ˉ 1 , b 1 ⋅ b 2 , c 1 ⋅ c 2 , m 2 , d ˉ 2 ⟩ ∈ G P
a ˉ 1 → b 1 ↓ d ˉ 1 d ˉ 1 → b 2 ↓ d ˉ 2 a ˉ 1 → b 1 ⋅ b 2 ↓ d ˉ 2 a ˉ 1 → c 1 ∼ m 1 d ˉ 1 → c 2 ∼ m 2 a ˉ 1 → c 1 ∼ m 1 ⋅ c 2 ⋅ m 2 \begin{array}{ccc}
\bar{a}_1\ \xrightarrow{\ b_1\ }\ \downarrow\bar{d}_1
&
\bar{d}_1\ \xrightarrow{\ b_2\ }\ \downarrow\bar{d}_2
&
\bar{a}_1\ \xrightarrow{\ b_1\cdot b_2\ }\ \downarrow\bar{d}_2\\
\phantom{\bar{a}_1}\xrightarrow{\ c_1\ }\ \sim m_1
&
\phantom{\bar{d}_1}\xrightarrow{\ c_2\ }\ \sim m_2
&
\phantom{\bar{a}_1}\xrightarrow{\ c_1\ }\ \sim m_1\cdot c_2\cdot m_2
\end{array} a ˉ 1 b 1 ↓ d ˉ 1 a ˉ 1 c 1 ∼ m 1 d ˉ 1 b 2 ↓ d ˉ 2 d ˉ 1 c 2 ∼ m 2 a ˉ 1 b 1 ⋅ b 2 ↓ d ˉ 2 a ˉ 1 c 1 ∼ m 1 ⋅ c 2 ⋅ m 2 Given ⟨ a 1 , b 1 , c 1 , m 1 , d 1 ⟩ ∈ G P \langle a_1,b_1,c_1,m_1,d_1\rangle\in G_P ⟨ a 1 , b 1 , c 1 , m 1 , d 1 ⟩ ∈ G P ⟨ d 1 , b 2 , c 2 , m 2 , d 2 ⟩ ∈ G P \langle d_1,b_2,c_2,m_2,d_2\rangle\in G_P ⟨ d 1 , b 2 , c 2 , m 2 , d 2 ⟩ ∈ G P ⟨ a 1 , b 1 ⋅ b 2 , c 1 , m 1 ⋅ c 2 ⋅ m 2 , d 2 ⟩ ∈ G P \langle a_1,b_1\cdot b_2,c_1,m_1\cdot c_2\cdot m_2,d_2\rangle\in G_P ⟨ a 1 , b 1 ⋅ b 2 , c 1 , m 1 ⋅ c 2 ⋅ m 2 , d 2 ⟩ ∈ G P
a 1 ‾ ↓ d 1 ‾ ↓ ⟶ b 1 a 1 ‾ c 1 ⟶ a 2 ‾ ↓ d 2 ‾ ↓ ⟶ c 2 ↗ m 2 becomes ⟶ b 1 ↓ d 2 ⋅ d 1 ‾ ⟶ c 2 ↗ m 2 \begin{array}{c}
\overline{a_1}\quad\downarrow\qquad\overline{d_1}\quad\downarrow\\
\longrightarrow b_1 \\
\overline{\phantom{a_1}}\quad c_1\quad\longrightarrow\\
\overline{a_2}\quad\downarrow\qquad\overline{d_2}\quad\downarrow\\
\longrightarrow c_2\qquad\nearrow m_2
\end{array}
\qquad\text{becomes}\qquad
\begin{array}{c}
\longrightarrow b_1\\
\downarrow\qquad\overline{d_2\cdot d_1}\\
\longrightarrow c_2\qquad\nearrow m_2
\end{array} a 1 ↓ d 1 ↓ ⟶ b 1 a 1 c 1 ⟶ a 2 ↓ d 2 ↓ ⟶ c 2 ↗ m 2 becomes ⟶ b 1 ↓ d 2 ⋅ d 1 ⟶ c 2 ↗ m 2 Given ⟨ a 1 , b 1 , c 1 , ε , d 1 ⟩ ∈ G P \langle a_1,b_1,c_1,\varepsilon,d_1\rangle\in G_P ⟨ a 1 , b 1 , c 1 , ε , d 1 ⟩ ∈ G P ⟨ a 2 , c 1 , c 2 , m 2 , d 2 ⟩ ∈ G P \langle a_2,c_1,c_2,m_2,d_2\rangle\in G_P ⟨ a 2 , c 1 , c 2 , m 2 , d 2 ⟩ ∈ G P ⟨ a 2 ⋅ a 1 , b 1 , c 2 , m 2 , d 2 ⋅ d 1 ⟩ ∈ G P \langle a_2\cdot a_1,b_1,c_2,m_2,d_2\cdot d_1\rangle\in G_P ⟨ a 2 ⋅ a 1 , b 1 , c 2 , m 2 , d 2 ⋅ d 1 ⟩ ∈ G P a 2 ⋅ a 1 a_2\cdot a_1 a 2 ⋅ a 1 d 2 ⋅ d 1 d_2\cdot d_1 d 2 ⋅ d 1 against the direction of the arrow on the vertical side of the partial grid, so the order is flipped.
(m 1 m_1 m 1
a 1 ‾ ↓ d 1 ‾ ↓ ⟶ b 1 ↓ c 1 ⟶ m 1 a 2 ‾ ↓ d 2 ‾ ↓ ⟶ c 2 ↗ m 2 becomes ⟶ b 1 ↓ d 1 ‾ ↓ m 2 ⋅ d 2 ⋅ m 1 \begin{array}{c}
\overline{a_1}\quad\downarrow\qquad\overline{d_1}\quad\downarrow\\
\longrightarrow b_1\\
\phantom{\downarrow}\quad c_1\quad\longrightarrow\qquad m_1\\
\overline{a_2}\quad\downarrow\qquad\overline{d_2}\quad\downarrow\\
\longrightarrow c_2\qquad\nearrow m_2
\end{array}
\qquad\text{becomes}\qquad
\begin{array}{c}
\longrightarrow b_1\\
\downarrow\qquad\overline{d_1}\\
\phantom{\downarrow}\qquad m_2\cdot d_2\cdot m_1
\end{array} a 1 ↓ d 1 ↓ ⟶ b 1 ↓ c 1 ⟶ m 1 a 2 ↓ d 2 ↓ ⟶ c 2 ↗ m 2 becomes ⟶ b 1 ↓ d 1 ↓ m 2 ⋅ d 2 ⋅ m 1 Given ⟨ a 1 , b 1 , c 1 , m 1 , d 1 ⟩ ∈ G P \langle a_1,b_1,c_1,m_1,d_1\rangle\in G_P ⟨ a 1 , b 1 , c 1 , m 1 , d 1 ⟩ ∈ G P ⟨ a 2 , c 1 , c 2 , m 2 , d 2 ⟩ ∈ G P \langle a_2,c_1,c_2,m_2,d_2\rangle\in G_P ⟨ a 2 , c 1 , c 2 , m 2 , d 2 ⟩ ∈ G P ⟨ a 2 ⋅ a 1 , b 1 , c 2 , m 2 ⋅ d 2 ⋅ m 1 , d 1 ⟩ ∈ G P \langle a_2\cdot a_1,b_1,c_2,m_2\cdot d_2\cdot m_1,d_1\rangle\in G_P ⟨ a 2 ⋅ a 1 , b 1 , c 2 , m 2 ⋅ d 2 ⋅ m 1 , d 1 ⟩ ∈ G P
Definition 6.29. The total frontier (often referred to as merely “the frontier”) of a partial grid ⟨ a , b , c , m , d ⟩ ∈ G P \langle a,b,c,m,d\rangle\in G_P ⟨ a , b , c , m , d ⟩ ∈ G P c ⋅ m ⋅ d c\cdot m\cdot d c ⋅ m ⋅ d
We continue with three short lemmas about partial grids, all shown by structural induction. While simple in theory, the number of cases grows rapidly.
Lemma 6.30. Consider some partial grid ⟨ a , b , c , m , d ⟩ ∈ G P \langle a,b,c,m,d\rangle\in G_P ⟨ a , b , c , m , d ⟩ ∈ G P a a a d d d b b b c c c
Lemma 6.31. (Extending a partial grid’s spine) Consider a partial grid ⟨ a , b , c , m , d ⟩ \langle a,b,c,m,d\rangle ⟨ a , b , c , m , d ⟩ negative augmented grid word a ′ a' a ′ ⟨ a ′ ⋅ a , b , ε , a ′ ⋅ c ⋅ m , d ⟩ \langle a'\cdot a,b,\varepsilon,a'\cdot c\cdot m,d\rangle ⟨ a ′ ⋅ a , b , ε , a ′ ⋅ c ⋅ m , d ⟩
For any non-empty positive augmented grid word b ′ b' b ′ ⟨ a , b ⋅ b ′ , c , m ⋅ d ⋅ b ′ , ε ⟩ \langle a,b\cdot b',c,m\cdot d\cdot b',\varepsilon\rangle ⟨ a , b ⋅ b ′ , c , m ⋅ d ⋅ b ′ , ε ⟩
Figure 21 (PDF p. 48).
Figure 22 (PDF p. 48).
Lemma 6.32. The middle frontier of any partial grid is either empty or of the form α ˉ m ′ β \bar{\alpha}m'\beta α ˉ m ′ β α , β ∈ S 1 \alpha,\beta\in S^1 α , β ∈ S 1 m ′ ∈ B W 1 m' \in BW^1 m ′ ∈ B W 1 B W 1 BW^1 B W 1
The following two theorems are not used in the proof of Theorem 6.3 , but are basic properties of partial grids, so we present them now.
First, we can reflect a partial grid along its top-left to bottom-right diagonal.
Theorem 6.33. (Reflecting) Given a partial grid ⟨ a , b , c , m , d ⟩ \langle a,b,c,m,d\rangle ⟨ a , b , c , m , d ⟩ ⟨ b ˉ , a ˉ , d ˉ , m ˉ , c ˉ ⟩ \langle\bar{b},\bar{a},\bar{d},\bar{m},\bar{c}\rangle ⟨ b ˉ , a ˉ , d ˉ , m ˉ , c ˉ ⟩
Figure 23 (PDF p. 48).
This theorem shortens proofs significantly; for example, the second half of Lemma 6.31 becomes an immediate consequence of the first half. We also use this mirroring trick in the following theorem about splitting partial grids:
Theorem 6.34. (Vertical Splitting) Given p = ⟨ a , b 1 b 2 , c , m , d ⟩ ∈ G P p=\langle a,b_1b_2,c,m,d\rangle\in G_P p = ⟨ a , b 1 b 2 , c , m , d ⟩ ∈ G P b 1 , b 2 b_1,b_2 b 1 , b 2
p p p p 1 = ⟨ a , b 1 , c 1 , m 1 , d 1 ⟩ ∈ G P p_1=\langle a,b_1,c_1,m_1,d_1\rangle\in G_P p 1 = ⟨ a , b 1 , c 1 , m 1 , d 1 ⟩ ∈ G P p 2 = ⟨ d 1 , b 2 , c 2 , m 2 , d ⟩ ∈ G P p_2=\langle d_1,b_2,c_2,m_2,d\rangle\in G_P p 2 = ⟨ d 1 , b 2 , c 2 , m 2 , d ⟩ ∈ G P c ⋅ m = c 1 ⋅ m 1 ⋅ c 2 ⋅ m 2 c\cdot m=c_1\cdot m_1\cdot c_2\cdot m_2 c ⋅ m = c 1 ⋅ m 1 ⋅ c 2 ⋅ m 2
Figure 24 (PDF p. 48).
(note that c c c c 1 c_1 c 1 m 1 m_1 m 1 c = c 1 c 2 c = c_1c_2 c = c 1 c 2 d d d
p p p p 1 = ⟨ a , b 1 , c , m 1 , ε ⟩ ∈ G P p_1 = \langle a,b_1,c,m_1,\varepsilon\rangle\in G_P p 1 = ⟨ a , b 1 , c , m 1 , ε ⟩ ∈ G P b 2 b_2 b 2 m = m 1 ⋅ b 2 m = m_1 \cdot b_2 m = m 1 ⋅ b 2 d = ε d = \varepsilon d = ε m ⋅ d = m 1 ⋅ b 2 m \cdot d = m_1 \cdot b_2 m ⋅ d = m 1 ⋅ b 2
The proof is by induction on p p p
(Horizontal Splitting) We have the analogous property for a partial grid p ′ = ⟨ a 1 a 2 , b , c , m , d ⟩ p' = \langle a_1a_2,b,c,m,d\rangle p ′ = ⟨ a 1 a 2 , b , c , m , d ⟩ a 1 , a 2 a_1,a_2 a 1 , a 2 p ′ p' p ′
Building the Partial Grid
We next build a partial grid from an R g r i d R_{\mathrm{grid}} R grid
Theorem 6.35. For nonempty a , b ∈ B W 1 − , + a,b \in BW_1^{-,+} a , b ∈ B W 1 − , + a b → R g r i d ∗ c ab \xrightarrow[R_{\mathrm{grid}}]{*} c ab ∗ R grid c ( a , b ) (a,b) ( a , b )
We begin by defining a function called add_cell which adds a single cell to a partial grid.
Definition 6.36. add_cell is a function taking in some ( x , y ) ∈ R g r i d (x,y) \in R_{\mathrm{grid}} ( x , y ) ∈ R grid P 1 = ⟨ a , b , c , m 1 ⋅ x ⋅ m 2 , d ⟩ P_1 = \langle a,b,c,m_1 \cdot x \cdot m_2,d\rangle P 1 = ⟨ a , b , c , m 1 ⋅ x ⋅ m 2 , d ⟩ P 2 = ⟨ a , b , c ′ , m ′ , d ′ ⟩ P_2 = \langle a,b,c',m',d'\rangle P 2 = ⟨ a , b , c ′ , m ′ , d ′ ⟩ c ′ ⋅ m ′ ⋅ d ′ = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c' \cdot m' \cdot d' = c \cdot m_1 \cdot y \cdot m_2 \cdot d c ′ ⋅ m ′ ⋅ d ′ = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c c c c ′ c' c ′ d d d d ′ d' d ′ c ′ ⋅ m ′ ⋅ d ′ = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c' \cdot m' \cdot d' = c \cdot m_1 \cdot y \cdot m_2 \cdot d c ′ ⋅ m ′ ⋅ d ′ = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d
We define add_cell by recursion on P 1 P_1 P 1
a cell: This case would lead to a contradiction. Partial grids from the cell case have an empty middle frontier. As ( x , y ) ∈ R g r i d (x,y) \in R_{\mathrm{grid}} ( x , y ) ∈ R grid x x x P 1 P_1 P 1
a spine: Here c = d = ε c = d = \varepsilon c = d = ε m 1 ⋅ x ⋅ m 2 = a b m_1 \cdot x \cdot m_2 = ab m 1 ⋅ x ⋅ m 2 = ab ( x , y ) ∈ R g r i d (x,y) \in R_{\mathrm{grid}} ( x , y ) ∈ R grid α , β , γ , δ ∈ S 1 \alpha,\beta,\gamma,\delta\in S^1 α , β , γ , δ ∈ S 1 x = α β x = \alpha\beta x = α β y = γ δ y = \gamma\delta y = γ δ m 1 ⋅ α β ⋅ m 2 = a b m_1 \cdot\alpha\beta\cdot m_2 = ab m 1 ⋅ α β ⋅ m 2 = ab a a a b b b a = m 1 α a = m_1\alpha a = m 1 α b = β m 2 b = \beta m_2 b = β m 2 ( α β , γ δ ) ∈ R g r i d (\alpha\beta,\gamma\delta) \in R_{\mathrm{grid}} ( α β , γ δ ) ∈ R grid P 3 = ⟨ α , β , γ , ε , δ ⟩ ∈ G P P_3 = \langle\alpha,\beta,\gamma,\varepsilon,\delta\rangle\in G_P P 3 = ⟨ α , β , γ , ε , δ ⟩ ∈ G P m 1 m_1 m 1 m 2 m_2 m 2 P 3 P_3 P 3 P 4 = ⟨ m 1 α , β , ε , m 1 γ , δ ⟩ P_4 = \langle m_1\alpha,\beta,\varepsilon,m_1\gamma,\delta\rangle P 4 = ⟨ m 1 α , β , ε , m 1 γ , δ ⟩ P 4 P_4 P 4 P 5 = ⟨ m 1 α , β m 2 , ε , m 1 γ δ m 2 , ε ⟩ P_5 = \langle m_1\alpha,\beta m_2,\varepsilon,m_1\gamma\delta m_2,\varepsilon\rangle P 5 = ⟨ m 1 α , β m 2 , ε , m 1 γ δ m 2 , ε ⟩ ⟨ a , b , c , m 1 ⋅ y ⋅ m 2 , d ⟩ \langle a,b,c,m_1 \cdot y \cdot m_2,d\rangle ⟨ a , b , c , m 1 ⋅ y ⋅ m 2 , d ⟩ c = d = ε c = d = \varepsilon c = d = ε
In this case, we have b = b 1 b 2 b = b_1b_2 b = b 1 b 2 P 2 = ⟨ a , b 1 , c , n 1 , d 1 ⟩ P_2 = \langle a, b_1, c, n_1, d_1\rangle P 2 = ⟨ a , b 1 , c , n 1 , d 1 ⟩ P 3 = ⟨ d 1 , b 2 , c 2 , n 2 , d ⟩ P_3 = \langle d_1, b_2, c_2, n_2, d\rangle P 3 = ⟨ d 1 , b 2 , c 2 , n 2 , d ⟩ n 1 ⋅ c 2 ⋅ n 2 = m 1 ⋅ x ⋅ m 2 n_1 \cdot c_2 \cdot n_2 = m_1 \cdot x \cdot m_2 n 1 ⋅ c 2 ⋅ n 2 = m 1 ⋅ x ⋅ m 2 x x x n 1 n_1 n 1 n 2 n_2 n 2 n 1 = n 3 x n 4 n_1 = n_3xn_4 n 1 = n 3 x n 4 m 1 = n 3 m_1 = n_3 m 1 = n 3 m 2 = n 4 ⋅ c 2 ⋅ n 2 m_2 = n_4 \cdot c_2 \cdot n_2 m 2 = n 4 ⋅ c 2 ⋅ n 2
Figure 1 (PDF p. 50).
Here, recursively call add_cell on P 2 P_2 P 2 P 4 = ⟨ a , b 1 , c ′ , n ′ , d ′ ⟩ P_4 = \langle a, b_1, c', n', d'\rangle P 4 = ⟨ a , b 1 , c ′ , n ′ , d ′ ⟩ c ′ ⋅ n ′ ⋅ d 1 ′ = c ⋅ n 3 ⋅ y ⋅ n 4 ⋅ d 1 c' \cdot n' \cdot d'_1 = c \cdot n_3 \cdot y \cdot n_4 \cdot d_1 c ′ ⋅ n ′ ⋅ d 1 ′ = c ⋅ n 3 ⋅ y ⋅ n 4 ⋅ d 1 c c c c ′ c' c ′ d 1 d_1 d 1 d 1 ′ d'_1 d 1 ′ d 1 d_1 d 1 d 1 ′ d'_1 d 1 ′ d 2 d_2 d 2 d 1 ′ = d 1 d 2 d'_1 = d_1d_2 d 1 ′ = d 1 d 2 d 2 d_2 d 2 n ′ n' n ′
Figure 2 (PDF p. 50).
We extend the left side of P 3 P_3 P 3 P 5 = ⟨ d 1 ′ , b 2 , ε , d 2 c 2 n 2 , d ⟩ P_5 = \langle d'_1, b_2, \varepsilon, d_2c_2n_2, d\rangle P 5 = ⟨ d 1 ′ , b 2 , ε , d 2 c 2 n 2 , d ⟩
Next append P 4 P_4 P 4 P 5 P_5 P 5 P 6 = ⟨ a , b 1 b 2 , c ′ , n ′ d 2 c 2 n 2 , d ⟩ P_6 = \langle a, b_1b_2, c', n'd_2c_2n_2, d\rangle P 6 = ⟨ a , b 1 b 2 , c ′ , n ′ d 2 c 2 n 2 , d ⟩
Figure 25 (PDF p. 51).
To show that the first specification holds, we must show c ′ ⋅ n ′ ⋅ d 2 ⋅ c 2 ⋅ n 2 ⋅ d = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c' \cdot n' \cdot d_2 \cdot c_2 \cdot n_2 \cdot d = c \cdot m_1 \cdot y \cdot m_2 \cdot d c ′ ⋅ n ′ ⋅ d 2 ⋅ c 2 ⋅ n 2 ⋅ d = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c ′ ⋅ n ′ ⋅ d 1 ′ = c ⋅ n 3 ⋅ y ⋅ n 4 ⋅ d c' \cdot n' \cdot d'_1 = c \cdot n_3 \cdot y \cdot n_4 \cdot d c ′ ⋅ n ′ ⋅ d 1 ′ = c ⋅ n 3 ⋅ y ⋅ n 4 ⋅ d d 1 ′ = d 2 ⋅ d 1 d'_1 = d_2 \cdot d_1 d 1 ′ = d 2 ⋅ d 1 c ′ ⋅ n ′ ⋅ d 2 = c ⋅ n 3 ⋅ y ⋅ n 4 c' \cdot n' \cdot d_2 = c \cdot n_3 \cdot y \cdot n_4 c ′ ⋅ n ′ ⋅ d 2 = c ⋅ n 3 ⋅ y ⋅ n 4 c 2 ⋅ n 2 ⋅ d c_2 \cdot n_2 \cdot d c 2 ⋅ n 2 ⋅ d c ′ ⋅ n ′ ⋅ d 2 ⋅ c 2 ⋅ n 2 ⋅ d = c ⋅ n 3 ⋅ y ⋅ n 4 ⋅ c 2 ⋅ n 2 ⋅ d c' \cdot n' \cdot d_2 \cdot c_2 \cdot n_2 \cdot d = c \cdot n_3 \cdot y \cdot n_4 \cdot c_2 \cdot n_2 \cdot d c ′ ⋅ n ′ ⋅ d 2 ⋅ c 2 ⋅ n 2 ⋅ d = c ⋅ n 3 ⋅ y ⋅ n 4 ⋅ c 2 ⋅ n 2 ⋅ d m 1 = n 3 m_1 = n_3 m 1 = n 3 m 2 = n 4 ⋅ c 2 ⋅ n 2 m_2 = n_4 \cdot c_2 \cdot n_2 m 2 = n 4 ⋅ c 2 ⋅ n 2 c ′ ⋅ n ′ ⋅ d 2 ⋅ c 2 ⋅ n 2 ⋅ d = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c' \cdot n' \cdot d_2 \cdot c_2 \cdot n_2 \cdot d = c \cdot m_1 \cdot y \cdot m_2 \cdot d c ′ ⋅ n ′ ⋅ d 2 ⋅ c 2 ⋅ n 2 ⋅ d = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c c c c ′ c' c ′ d d d d d d
We are now prepared to prove Theorem 6.35 .
Proof. Consider nonempty, positive a , b ∈ B W 1 a,b \in BW^1 a , b ∈ B W 1 a b ‾ ⟶ R g r i d ∗ c \overline{ab} \mathrel{\mathop{\longrightarrow}\limits_{R_{\mathrm{grid}}}^{*}} c ab R grid ⟶ ∗ c a b ‾ ⟶ R g r i d c \overline{ab} \mathrel{\mathop{\longrightarrow}\limits_{R_{\mathrm{grid}}}} c ab R grid ⟶ c
reflexivity: In this case, we have c = a b ‾ c = \overline{ab} c = ab a a a b b b ⟨ a ‾ , b , ε , a b ‾ , ε ⟩ \langle\overline{a}, b, \varepsilon, \overline{ab}, \varepsilon\rangle ⟨ a , b , ε , ab , ε ⟩
transitive one-step reduction: We assume that a b ‾ ⟶ R r e v d x e ‾ \overline{ab} \mathrel{\mathop{\longrightarrow}\limits_{R_{\mathrm{rev}}}} \overline{dxe} ab R rev ⟶ d x e R r e v ( x , y ) R_{\mathrm{rev}}(x,y) R rev ( x , y ) c = d y e ‾ c = \overline{dye} c = d y e P 1 : = ⟨ a ‾ , b , c 1 , m 1 , d ‾ 1 ⟩ P_1 := \langle\overline{a}, b, c_1, m_1, \overline{d}_1 \rangle P 1 := ⟨ a , b , c 1 , m 1 , d 1 ⟩ c 1 m 1 d ‾ 1 = d x e ‾ c_1m_1\overline{d}_1 = \overline{dxe} c 1 m 1 d 1 = d x e x x x m 1 m_1 m 1 x x x c 1 c_1 c 1 d ‾ 1 \overline{d}_1 d 1 m 2 , m 3 ∈ B W 1 m_2,m_3 \in BW^1 m 2 , m 3 ∈ B W 1 m 1 = m 2 x m 3 m_1 = m_2xm_3 m 1 = m 2 x m 3 d ‾ = c 1 m 2 \overline{d} = c_1m_2 d = c 1 m 2 e = m 3 d ‾ 1 e = m_3\overline{d}_1 e = m 3 d 1 ( x , y ) (x,y) ( x , y ) P 1 P_1 P 1 P 2 : = ⟨ a ‾ , b , c 1 , m 2 y m 3 , d ‾ 1 ⟩ P_2 := \langle\overline{a}, b, c_1, m_2ym_3, \overline{d}_1 \rangle P 2 := ⟨ a , b , c 1 , m 2 y m 3 , d 1 ⟩ P 2 P_2 P 2 c 1 m 2 y m 3 d ‾ 1 = d y e ‾ = c c_1m_2ym_3\overline{d}_1 = \overline{dye} = c c 1 m 2 y m 3 d 1 = d y e = c
Empty Middle Frontiers
Finally, we connect partial grids to grids by proving that a partial grid with empty middle frontier is in fact a grid.
Theorem 6.37. Given ⟨ a ‾ , b , c , ε , d ‾ ⟩ ∈ G P \langle\overline{a}, b, c, \varepsilon, \overline{d} \rangle\in G_P ⟨ a , b , c , ε , d ⟩ ∈ G P ⟨ s 1 ( a ) , s 1 ( b ) , s 1 ( c ) , s 1 ( d ‾ ) ⟩ ∈ G \langle s^1(a),s^1(b),s^1(c),s^1(\overline{d}) \rangle\in G ⟨ s 1 ( a ) , s 1 ( b ) , s 1 ( c ) , s 1 ( d )⟩ ∈ G
The proof is by induction on the given partial grid.
We next note that any partial grid whose total frontier is in positive-negative form has an empty middle frontier:
Lemma 6.38. If a partial grid has total frontier c ‾ d ‾ \overline{c}\overline{d} c d c , d ∈ B W 1 , + c,d \in BW^{1,+} c , d ∈ B W 1 , + c c c ε \varepsilon ε d ‾ \overline{d} d
Conclusion
We are now prepared to give a proof of the remaining half of Theorem 6.3 , restated as follows:
Theorem 6.39. For any a , b , c , d ∈ B W + a,b,c,d \in BW^{+} a , b , c , d ∈ B W + a b ‾ → R r e v ∗ c d ‾ \overline{ab} \xrightarrow{R_{\mathrm{rev}}}^{*} \overline{cd} ab R rev ∗ c d ⟨ a , b , c , d ⟩ \langle a,b,c,d\rangle ⟨ a , b , c , d ⟩
Proof. Consider a , b , c , d ∈ B W + a,b,c,d \in BW^{+} a , b , c , d ∈ B W + a b ‾ → R r e v ∗ c d ‾ \overline{ab} \xrightarrow{R_{\mathrm{rev}}}^{*} \overline{cd} ab R rev ∗ c d Theorem 6.27 , we have e , f ∈ B W 1 , + e,f \in BW^{1,+} e , f ∈ B W 1 , + i 1 ( a ‾ ) i 1 ( b ) → R g r i d ∗ e f ‾ i^{1}(\overline{a})i^{1}(b) \xrightarrow{R_{\mathrm{grid}}}^{*} \overline{ef} i 1 ( a ) i 1 ( b ) R grid ∗ e f s 1 ( e ) = c s^{1}(e)=c s 1 ( e ) = c s 1 ( f ) = d s^{1}(f)=d s 1 ( f ) = d Theorem 6.35 , we have a partial grid P 1 P_{1} P 1 ( i 1 ( a ‾ ) , i 1 ( b ) ) (i^{1}(\overline{a}),i^{1}(b)) ( i 1 ( a ) , i 1 ( b )) e f ‾ \overline{ef} e f P 1 = ⟨ i 1 ( a ‾ ) , i 1 ( b ) , e , ε , f ⟩ P_{1}=\langle i^{1}(\overline{a}),i^{1}(b),e,\varepsilon,f\rangle P 1 = ⟨ i 1 ( a ) , i 1 ( b ) , e , ε , f ⟩ ⟨ s 1 ( i 1 ( a ‾ ) ) , s 1 ( i 1 ( b ) ) , s 1 ( e ) , s 1 ( f ) ⟩ = ⟨ a , b , c , d ⟩ \left\langle s^{1}(i^{1}(\overline{a})),s^{1}(i^{1}(b)),s^{1}(e),s^{1}(f)\right\rangle=\langle a,b,c,d\rangle ⟨ s 1 ( i 1 ( a )) , s 1 ( i 1 ( b )) , s 1 ( e ) , s 1 ( f ) ⟩ = ⟨ a , b , c , d ⟩
Theorem 6.3 can be used to give new, visual proofs of previous results. For example, note that properties of cancellativity and common multiples (the Ore conditions) are used for existence and uniqueness of grids, and hence for the proof of Theorem 6.3 . Thus, grids essentially capture Ore localization for a small subset of braid words (for those pairs of braid words which are the source and target of some grid). We thus have a slick proof of the following:
Theorem 6.40. For any a , b , c , d ∈ B W + a,b,c,d \in BW^{+} a , b , c , d ∈ B W + a b ‾ → R r e v ∗ c d ‾ \overline{ab} \xrightarrow{R_{\mathrm{rev}}}^{*} \overline{cd} ab R rev ∗ c d a c ≡ + b d ac \equiv^{+} bd a c ≡ + b d
Proof. Assume a b ‾ → R r e v ∗ c d ‾ \overline{ab} \xrightarrow{R_{\mathrm{rev}}}^{*} \overline{cd} ab R rev ∗ c d Theorem 6.3 , we have ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G a c ≡ + b d ac \equiv^{+} bd a c ≡ + b d
Alternatively, we could have noted that since a b ‾ → R r e v ∗ c d ‾ \overline{ab} \xrightarrow{R_{\mathrm{rev}}}^{*} \overline{cd} ab R rev ∗ c d a b ‾ ≡ c d ‾ \overline{ab} \equiv\overline{cd} ab ≡ c d a − 1 b ≡ c d − 1 a^{-1}b \equiv cd^{-1} a − 1 b ≡ c d − 1 a c ≡ b d ac \equiv bd a c ≡ b d a c ac a c b d bd b d Theorem 3.15 ), we know a c ≡ + b d ac \equiv^{+} bd a c ≡ + b d
Metrics
This chapter introduces metrics on grids, partial grids, and Semi-Thue derivations. In the Lean formalization, doing so requires us to distinguish between two ways in which an inductively defined mathematical object may be given: as a proposition asserting that the object exists, or as data recording how the object was constructed. A proposition in Lean has type Prop, while mathematical data lives in Type. To disambiguate the two, we begin by giving simple examples of Prop and Type.
Suppose that we have a predicate p : N → Prop p : \mathbb{N} \to\operatorname{Prop} p : N → Prop n n n p ( n ) p(n) p ( n ) ∃ n : N , p n \exists n : \mathbb{N}, p\ n ∃ n : N , p n Prop. A proof of this proposition establishes that some natural number satisfying p p p Classical.choose to give it a name, and Classical.choose_spec to assert that p p p Classical.choose establishes that a suitable n n n Classical.choose is marked noncomputable.
The corresponding Type-valued concept is the subtype { n : N / / p n } \{n : \mathbb{N} \mathbin{//} p\ n\} { n : N // p n } x : { n : N / / p n } x : \{n : \mathbb{N} \mathbin{//} p\ n\} x : { n : N // p n } x.val) with which we may compute, together with a proof that p p p x.val) holds (called x.prop). For example, a computable function can return x.val + 2. We can prove specifications about said computable function using x.prop. This subtype can also be written as the Sigma type Σ ( n : N ) , p ( n ) \Sigma(n : \mathbb{N}), p(n) Σ ( n : N ) , p ( n ) Type. Often in this formalization, we ignore the style guidelines to use subtypes, and rather favor sigma types. In the Lean infoview, a Sigma type (unlike a subtype) will show the type of n n n
Now let us discuss the case of inductive definitions. If an inductive definition takes values in Prop, Lean remembers only that the corresponding proposition has been proved; the particular constructor tree used to prove it is not computational data. If instead the inductive definition takes values in Type, its constructors form data. Accordingly, we will pass from several of our earlier Prop-valued relations to closely related Type-valued versions whenever their construction sequence is needed to define a metric.
Why not simply make all our structures Type-valued from the beginning? Working with unneeded data is unwieldy. Thus, the Mathlib library is designed to be Prop-valued, so we cannot benefit from its extensive codebase when working with data (for example, we had to implement our own small data-carrying library of facts about prefixes and suffixes of lists). Similarly, many automated tactics, such as aesop and grind do not work on Type-valued objects. Thus, we work in Prop as much as possible, and only pass to Type when the data is absolutely necessary.
Grids
Currently, grids are Prop-valued in Lean. Saying that ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G G G G ⟨ a , b , c , d ⟩ \langle a,b,c,d\rangle ⟨ a , b , c , d ⟩
b 1 b 2 a 1 → x d 1 a 2 → x d 2 c 1 c 2 and b 1 b 2 a 1 a 2 ↓ y ↓ d 1 d 2 c 1 c 2 are both b 1 b 2 a 1 a 2 ↓ d 1 d 2 c 1 c 2 \begin{array}{c}
\begin{array}{ccc}
& b_{1}b_{2} & \\
a_{1} & \xrightarrow{\phantom{x}} & d_{1}\\
a_{2} & \xrightarrow{\phantom{x}} & d_{2}\\
& c_{1}c_{2} &
\end{array}
\qquad\text{and}\qquad
\begin{array}{ccccc}
& b_{1} & & b_{2} & \\
a_{1}a_{2} & \downarrow& y & \downarrow& d_{1}d_{2}\\
& c_{1} & & c_{2} &
\end{array}
\qquad\text{are both}\qquad
\begin{array}{ccc}
& b_{1}b_{2} & \\
a_{1}a_{2} & \downarrow& d_{1}d_{2}\\
& c_{1}c_{2} &
\end{array}
\end{array} a 1 a 2 b 1 b 2 x x c 1 c 2 d 1 d 2 and a 1 a 2 b 1 ↓ c 1 y b 2 ↓ c 2 d 1 d 2 are both a 1 a 2 b 1 b 2 ↓ c 1 c 2 d 1 d 2 although they are constructed differently. In order to give a size metric on grids, we will need to track the data of how a given grid was constructed. It is entirely possible to give a definition of size which gives differing values for different construction sequences leading to the same frame. Our metric will provably give the same length to any pair of grids with matching sources and targets, but that is not the case for every metric. (In fact, I am not able to prove such a specification for the eventual size metric I will give on partial grids.)
We thus enhance our definition of grids as follows:
Definition 7.1. We define the set of grids with constructions G D G^{D} G D 24
The set of constructions K K K
trivial cell ∈ K \in K ∈ K
nontrivial cell ∈ K \in K ∈ K
horizontal ( g 1 , g 2 ) ∈ K (g_{1},g_{2}) \in K ( g 1 , g 2 ) ∈ K g 1 , g 2 ∈ G D g_{1},g_{2} \in G^{D} g 1 , g 2 ∈ G D g 1 g_{1} g 1 g 2 g_{2} g 2 g 1 g_{1} g 1 g 2 g_{2} g 2
vertical ( g 1 , g 2 ) ∈ K (g_{1},g_{2}) \in K ( g 1 , g 2 ) ∈ K g 1 , g 2 ∈ G D g_{1},g_{2} \in G^{D} g 1 , g 2 ∈ G D g 1 g_{1} g 1 g 2 g_{2} g 2 g 1 g_{1} g 1 g 2 g_{2} g 2
The set of grids with constructions G D G^{D} G D
The trivial cells:
⟨ ε , ε , ε , ε , trivial cell ⟩ ∈ G D \langle\varepsilon,\varepsilon,\varepsilon,\varepsilon,\text{trivial cell}\rangle\in G^{D} ⟨ ε , ε , ε , ε , trivial cell ⟩ ∈ G D
The trivial cells25
⟨ ε , ε , ε , ε , trivial cell ⟩ ∈ G D \langle\varepsilon,\varepsilon,\varepsilon,\varepsilon,\text{trivial cell}\rangle\in G^D ⟨ ε , ε , ε , ε , trivial cell ⟩ ∈ G D
For any i ∈ N i \in\mathbb{N} i ∈ N ⟨ σ i , ε , ε , σ i , trivial cell ⟩ ∈ G D \langle\sigma_i,\varepsilon,\varepsilon,\sigma_i,\text{trivial cell}\rangle\in G^D ⟨ σ i , ε , ε , σ i , trivial cell ⟩ ∈ G D
For any i ∈ N i \in\mathbb{N} i ∈ N ⟨ ε , σ i , σ i , ε , trivial cell ⟩ ∈ G D \langle\varepsilon,\sigma_i,\sigma_i,\varepsilon,\text{trivial cell}\rangle\in G^D ⟨ ε , σ i , σ i , ε , trivial cell ⟩ ∈ G D
The nontrivial cells:
For any i ∈ N i \in\mathbb{N} i ∈ N ⟨ σ i , σ i , ε , ε , nontrivial cell ⟩ ∈ G D \langle\sigma_i,\sigma_i,\varepsilon,\varepsilon,\text{nontrivial cell}\rangle\in G^D ⟨ σ i , σ i , ε , ε , nontrivial cell ⟩ ∈ G D
For i , j ∈ N i,j \in\mathbb{N} i , j ∈ N ∣ i − j ∣ = 1 \lvert i-j\rvert=1 ∣ i − j ∣ = 1 ⟨ σ j , σ i , σ i σ j , σ j σ i , nontrivial cell ⟩ ∈ G D \langle\sigma_j,\sigma_i,\sigma_i\sigma_j,\sigma_j\sigma_i,\text{nontrivial cell}\rangle\in G^D ⟨ σ j , σ i , σ i σ j , σ j σ i , nontrivial cell ⟩ ∈ G D
For i , j ∈ N i,j \in\mathbb{N} i , j ∈ N ∣ i − j ∣ > 1 \lvert i-j\rvert>1 ∣ i − j ∣ > 1 ⟨ σ j , σ i , σ i , σ j , nontrivial cell ⟩ ∈ G D \langle\sigma_j,\sigma_i,\sigma_i,\sigma_j,\text{nontrivial cell}\rangle\in G^D ⟨ σ j , σ i , σ i , σ j , nontrivial cell ⟩ ∈ G D
Horizontal abutting: If g 1 = ⟨ a , b , c , d , κ 1 ⟩ g_1=\langle a,b,c,d,\kappa_1\rangle g 1 = ⟨ a , b , c , d , κ 1 ⟩ g 2 = ⟨ d , e , f , g , κ 2 ⟩ g_2=\langle d,e,f,g,\kappa_2\rangle g 2 = ⟨ d , e , f , g , κ 2 ⟩ G D G^D G D ⟨ a , b e , c f , g , horizontal ( g 1 , g 2 ) ⟩ ∈ G D \langle a,be,cf,g,\operatorname{horizontal}(g_1,g_2)\rangle\in G^D ⟨ a , b e , c f , g , horizontal ( g 1 , g 2 )⟩ ∈ G D
Visually,
b ⟶ e ⟶ a ↓ d ↓ g c ⟶ f ⟶ \begin{array}{ccccc}
& b & \longrightarrow& e & \longrightarrow\\
a & \downarrow& d & \downarrow& g \\
& c & \longrightarrow& f & \longrightarrow
\end{array} a b ↓ c ⟶ d ⟶ e ↓ f ⟶ g ⟶ Vertical abutting: If g 1 = ⟨ a , b , c , d , κ 1 ⟩ g_1=\langle a,b,c,d,\kappa_1\rangle g 1 = ⟨ a , b , c , d , κ 1 ⟩ ⟨ e , c , f , g , κ 2 ⟩ \langle e,c,f,g,\kappa_2\rangle ⟨ e , c , f , g , κ 2 ⟩ G D G^D G D ⟨ a e , b , f , d g , vertical ( g 1 , g 2 ) ⟩ ∈ G D \langle ae,b,f,dg,\operatorname{vertical}(g_1,g_2)\rangle\in G^D ⟨ a e , b , f , d g , vertical ( g 1 , g 2 )⟩ ∈ G D
Visually,
b ⟶ a ↓ d ↓ c ⟶ e ↓ g ↓ f ⟶ \begin{array}{ccc}
& b & \longrightarrow\\
a & \downarrow& d \downarrow\\
& c & \longrightarrow\\
e & \downarrow& g \downarrow\\
& f & \longrightarrow
\end{array} a e b ↓ c ↓ f ⟶ d ↓ ⟶ g ↓ ⟶ In Lean, this is handled entirely for us by switching out Prop (which asserts the proposition that we have some grid) to Type, which preserves the data of the grid’s construction.
inductive GridData : FreeMonoid ℕ → FreeMonoid ℕ → FreeMonoid ℕ → FreeMonoid ℕ
→ TypeIt is immediate to delete the construction from a grid-with-construction and get a grid.
Theorem 7.2. If ⟨ a , b , c , d , κ ⟩ ∈ G D \langle a,b,c,d,\kappa\rangle\in G^D ⟨ a , b , c , d , κ ⟩ ∈ G D ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G
For any i ∈ N i \in\mathbb{N} i ∈ N ⟨ σ i , ε , ε , σ i , trivial cell ⟩ ∈ G D \langle\sigma_i,\varepsilon,\varepsilon,\sigma_i,\text{trivial cell}\rangle\in G^D ⟨ σ i , ε , ε , σ i , trivial cell ⟩ ∈ G D
For any i ∈ N i \in\mathbb{N} i ∈ N ⟨ ε , σ i , σ i , ε , trivial cell ⟩ ∈ G D \langle\varepsilon,\sigma_i,\sigma_i,\varepsilon,\text{trivial cell}\rangle\in G^D ⟨ ε , σ i , σ i , ε , trivial cell ⟩ ∈ G D
The nontrivial cells:
For any i ∈ N i \in\mathbb{N} i ∈ N ⟨ σ i , σ i , ε , ε , nontrivial cell ⟩ ∈ G D \langle\sigma_i,\sigma_i,\varepsilon,\varepsilon,\text{nontrivial cell}\rangle\in G^D ⟨ σ i , σ i , ε , ε , nontrivial cell ⟩ ∈ G D
For i , j ∈ N i,j \in\mathbb{N} i , j ∈ N ∣ i − j ∣ = 1 \lvert i-j\rvert=1 ∣ i − j ∣ = 1 ⟨ σ j , σ i , σ i σ j , σ j σ i , nontrivial cell ⟩ ∈ G D \langle\sigma_j,\sigma_i,\sigma_i\sigma_j,\sigma_j\sigma_i,\text{nontrivial cell}\rangle\in G^D ⟨ σ j , σ i , σ i σ j , σ j σ i , nontrivial cell ⟩ ∈ G D
For i , j ∈ N i,j \in\mathbb{N} i , j ∈ N ∣ i − j ∣ > 1 \lvert i-j\rvert>1 ∣ i − j ∣ > 1 ⟨ σ j , σ i , σ i , σ j , nontrivial cell ⟩ ∈ G D \langle\sigma_j,\sigma_i,\sigma_i,\sigma_j,\text{nontrivial cell}\rangle\in G^D ⟨ σ j , σ i , σ i , σ j , nontrivial cell ⟩ ∈ G D
Horizontal abutting: Given a , b , c , d , e , f , g ∈ B W 1 , + a,b,c,d,e,f,g \in BW_{1,+} a , b , c , d , e , f , g ∈ B W 1 , + κ 1 , κ 2 ∈ K \kappa_1,\kappa_2 \in K κ 1 , κ 2 ∈ K g 1 = ⟨ a , b , c , d , κ 1 ⟩ g_1=\langle a,b,c,d,\kappa_1\rangle g 1 = ⟨ a , b , c , d , κ 1 ⟩ g 2 = ⟨ d , e , f , g , κ 2 ⟩ g_2=\langle d,e,f,g,\kappa_2\rangle g 2 = ⟨ d , e , f , g , κ 2 ⟩ G D G^D G D ⟨ a , b e , c f , g , horizontal ( g 1 , g 2 ) ⟩ ∈ G D \langle a,be,cf,g,\operatorname{horizontal}(g_1,g_2)\rangle\in G^D ⟨ a , b e , c f , g , horizontal ( g 1 , g 2 )⟩ ∈ G D
Vertical abutting: Given a , b , c , d , e , f , g ∈ B W 1 , + a,b,c,d,e,f,g \in BW_{1,+} a , b , c , d , e , f , g ∈ B W 1 , + κ 1 , κ 2 ∈ K \kappa_1,\kappa_2 \in K κ 1 , κ 2 ∈ K g 1 = ⟨ a , b , c , d , κ 1 ⟩ g_1=\langle a,b,c,d,\kappa_1\rangle g 1 = ⟨ a , b , c , d , κ 1 ⟩ ⟨ e , c , f , g , κ 2 ⟩ \langle e,c,f,g,\kappa_2\rangle ⟨ e , c , f , g , κ 2 ⟩ G D G^D G D ⟨ a e , b , f , d g , vertical ( g 1 , g 2 ) ⟩ ∈ G D \langle ae,b,f,dg,\operatorname{vertical}(g_1,g_2)\rangle\in G^D ⟨ a e , b , f , d g , vertical ( g 1 , g 2 )⟩ ∈ G D
def GridData.to_grid (h : GridData a b c d) : grid a b c dCasting grids-with-constructions down to regular grids helps us convert many theorems (here, we show Theorem 5.4 and the Remark below Definition 5.5 ) as follows:
theorem braid_eq (h : GridData a b c d) : BraidMonoidInf.mk (a * c) =
BraidMonoidInf.mk (b * d) := Grid.braid_eq_of_grid (to_grid h)theorem diag_length_eq (h : GridData a b c d) : a.length + c.length = b.length +
d.length := Grid.diag_length_eq (to_grid h)Conversely, given a grid, we know that there must be at least one construction sequence giving that frame:
Theorem 7.3. If ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G κ \kappa κ ⟨ a , b , c , d , κ ⟩ ∈ G D \langle a,b,c,d,\kappa\rangle\in G^{D} ⟨ a , b , c , d , κ ⟩ ∈ G D
Note that the existential quantifier on κ \kappa κ Nonempty has the same meaning:
def GridData.of_grid (h : grid a b c d) : Nonempty (GridData a b c d)Thus, any theorems about grids can be translated to grids-with-constructions.
We began with Prop-valued grids because of the Mathlib support, and only move to the Type-valued grids (grids-with-constructions) when necessary. That occurs when we define a metric on grids-with-constructions. The metric is intended to represent the number of nontrivial cells occurring in the grid.
Definition 7.4. Define a metric on grids-with-constructions ℓ G : G D → N \ell_{G} : G^{D} \to\mathbb{N} ℓ G : G D → N
ℓ G ( ⟨ a , b , c , d , trivial cell ⟩ ) = 0 \ell_{G}(\langle a,b,c,d,\text{trivial cell}\rangle)=0 ℓ G (⟨ a , b , c , d , trivial cell ⟩) = 0
ℓ G ( ⟨ a , b , c , d , nontrivial cell ⟩ ) = 1 \ell_{G}(\langle a,b,c,d,\text{nontrivial cell}\rangle)=1 ℓ G (⟨ a , b , c , d , nontrivial cell ⟩) = 1
ℓ G ( ⟨ a , b , c , d , horizontal ( g 1 , g 2 ) ⟩ ) = ℓ G ( g 1 ) + ℓ G ( g 2 ) \ell_{G}(\langle a,b,c,d,\text{horizontal}(g_{1},g_{2})\rangle)=\ell_{G}(g_{1})+\ell_{G}(g_{2}) ℓ G (⟨ a , b , c , d , horizontal ( g 1 , g 2 )⟩) = ℓ G ( g 1 ) + ℓ G ( g 2 )
ℓ G ( ⟨ a , b , c , d , vertical ( g 1 , g 2 ) ⟩ ) = ℓ G ( g 1 ) + ℓ G ( g 2 ) \ell_{G}(\langle a,b,c,d,\text{vertical}(g_{1},g_{2})\rangle)=\ell_{G}(g_{1})+\ell_{G}(g_{2}) ℓ G (⟨ a , b , c , d , vertical ( g 1 , g 2 )⟩) = ℓ G ( g 1 ) + ℓ G ( g 2 )
We expand a number of theorems to discuss the grid size. For example, when splitting a grid, we show that the sum of the sizes of two new, smaller grids equals the size of the original grid. We run through the full proof of splittability given in Lemma 5.7 again, slightly augmented with facts about size, to obtain the following lemma:
Lemma 7.5. (Update to Lemma 5.7 ) Given G 0 = ⟨ a 1 a 2 , b , c , d , κ 0 ⟩ ∈ G G_{0}=\langle a_{1}a_{2},b,c,d,\kappa_{0}\rangle\in G G 0 = ⟨ a 1 a 2 , b , c , d , κ 0 ⟩ ∈ G G 1 = ⟨ a 1 , b , m , d 1 , κ 1 ⟩ G_{1}=\langle a_{1},b,m,d_{1},\kappa_{1}\rangle G 1 = ⟨ a 1 , b , m , d 1 , κ 1 ⟩ G 2 = ⟨ a 2 , m , c , d 2 , κ 2 ⟩ G_{2}=\langle a_{2},m,c,d_{2},\kappa_{2}\rangle G 2 = ⟨ a 2 , m , c , d 2 , κ 2 ⟩ d = d 1 d 2 d=d_{1}d_{2} d = d 1 d 2 ℓ G ( G 0 ) = ℓ G ( G 1 ) + ℓ G ( G 2 ) \ell_{G}(G_{0})=\ell_{G}(G_{1})+\ell_{G}(G_{2}) ℓ G ( G 0 ) = ℓ G ( G 1 ) + ℓ G ( G 2 )
Similarly, given G 3 = ⟨ a , b 1 b 2 , c , d , κ 3 ⟩ ∈ G G_{3}=\langle a,b_{1}b_{2},c,d,\kappa_{3}\rangle\in G G 3 = ⟨ a , b 1 b 2 , c , d , κ 3 ⟩ ∈ G G 4 = ⟨ a , b 1 , c 1 , m , κ 4 ⟩ G_{4}=\langle a,b_{1},c_{1},m,\kappa_{4}\rangle G 4 = ⟨ a , b 1 , c 1 , m , κ 4 ⟩ ⟨ m , b 2 , c 2 , d , κ 5 ⟩ \langle m,b_{2},c_{2},d,\kappa_{5}\rangle ⟨ m , b 2 , c 2 , d , κ 5 ⟩ c = c 1 c 2 c=c_{1}c_{2} c = c 1 c 2 ℓ G ( G 3 ) = ℓ G ( G 4 ) + ℓ G ( G 5 ) \ell_{G}(G_{3})=\ell_{G}(G_{4})+\ell_{G}(G_{5}) ℓ G ( G 3 ) = ℓ G ( G 4 ) + ℓ G ( G 5 )
Our main theorem is that the spine of a grid suffices to determine its length. We begin with a lemma discussing certain small cases:
Lemma 7.6. [Determinative Spines – Length] For a , b , c , d ∈ B W + a,b,c,d \in BW^{+} a , b , c , d ∈ B W + i , j ∈ N i,j \in\mathbb{N} i , j ∈ N
If ⟨ ε , b , c , d ⟩ ∈ G \langle\varepsilon,b,c,d\rangle\in G ⟨ ε , b , c , d ⟩ ∈ G ℓ G ( ⟨ ε , b , c , d ⟩ ) = 0 \ell_{G}(\langle\varepsilon,b,c,d\rangle)=0 ℓ G (⟨ ε , b , c , d ⟩) = 0
If ⟨ a , ε , c , d ⟩ ∈ G \langle a,\varepsilon,c,d\rangle\in G ⟨ a , ε , c , d ⟩ ∈ G ℓ G ( ⟨ a , ε , c , d ⟩ ) = 0 \ell_{G}(\langle a,\varepsilon,c,d\rangle)=0 ℓ G (⟨ a , ε , c , d ⟩) = 0
If ⟨ a , a , c , d ⟩ ∈ G \langle a,a,c,d\rangle\in G ⟨ a , a , c , d ⟩ ∈ G ℓ G ( ⟨ a , a , c , d ⟩ ) = ∣ a ∣ \ell_{G}(\langle a,a,c,d\rangle)=|a| ℓ G (⟨ a , a , c , d ⟩) = ∣ a ∣ ∣ a ∣ |a| ∣ a ∣ a a a
:::`
If ⟨ σ i , σ j , c , d ⟩ ∈ G \langle\sigma_i,\sigma_j,c,d\rangle\in G ⟨ σ i , σ j , c , d ⟩ ∈ G ∣ i − j ∣ = 1 |i-j|=1 ∣ i − j ∣ = 1 ℓ G ( ⟨ σ i , σ j , c , d ⟩ ) = 1 \ell_G(\langle\sigma_i,\sigma_j,c,d\rangle)=1 ℓ G (⟨ σ i , σ j , c , d ⟩) = 1
If ⟨ σ i , σ j , c , d ⟩ ∈ G \langle\sigma_i,\sigma_j,c,d\rangle\in G ⟨ σ i , σ j , c , d ⟩ ∈ G ∣ i − j ∣ > 1 |i-j|>1 ∣ i − j ∣ > 1 ℓ G ( ⟨ σ i , σ j , c , d ⟩ ) = 1 \ell_G(\langle\sigma_i,\sigma_j,c,d\rangle)=1 ℓ G (⟨ σ i , σ j , c , d ⟩) = 1
Now, thanks to uniqueness of grid spines (Theorem 5.9 ), we may show the full theorem.
Theorem 7.7. Given g 1 = ⟨ a , b , c , d , κ 1 ⟩ ∈ G D g_1=\langle a,b,c,d,\kappa_1\rangle\in G^D g 1 = ⟨ a , b , c , d , κ 1 ⟩ ∈ G D g 2 = ⟨ a , b , e , f , κ 2 ⟩ ∈ G D g_2=\langle a,b,e,f,\kappa_2\rangle\in G^D g 2 = ⟨ a , b , e , f , κ 2 ⟩ ∈ G D ℓ G ( g 1 ) = ℓ G ( g 2 ) \ell_G(g_1)=\ell_G(g_2) ℓ G ( g 1 ) = ℓ G ( g 2 )
Proof. (Sketch) We proceed by induction on g 1 g_1 g 1 g 1 g_1 g 1 g 2 g_2 g 2 g 1 g_1 g 1
Assume that g 1 = ⟨ a , b , c , d , horizontal ( g 3 , g 4 ) ⟩ g_1=\langle a,b,c,d,\operatorname{horizontal}(g_3,g_4)\rangle g 1 = ⟨ a , b , c , d , horizontal ( g 3 , g 4 )⟩ b = b 1 b 2 b=b_1b_2 b = b 1 b 2 c = c 1 c 2 c=c_1c_2 c = c 1 c 2 g 3 = ⟨ a , b 1 , c 1 , m , κ 1 ⟩ g_3=\langle a,b_1,c_1,m,\kappa_1\rangle g 3 = ⟨ a , b 1 , c 1 , m , κ 1 ⟩ g 4 = ⟨ m , b 2 , c 2 , d , κ 2 ⟩ g_4=\langle m,b_2,c_2,d,\kappa_2\rangle g 4 = ⟨ m , b 2 , c 2 , d , κ 2 ⟩ ℓ G ( g 1 ) = ℓ G ( g 3 ) + ℓ G ( g 4 ) \ell_G(g_1)=\ell_G(g_3)+\ell_G(g_4) ℓ G ( g 1 ) = ℓ G ( g 3 ) + ℓ G ( g 4 ) g 2 g_2 g 2 g 5 = ⟨ a , b 1 , c 5 , d 5 , κ 1 ⟩ g_5=\langle a,b_1,c_5,d_5,\kappa_1\rangle g 5 = ⟨ a , b 1 , c 5 , d 5 , κ 1 ⟩ g 6 = ⟨ d 5 , b 2 , c 6 , f , κ 2 ⟩ g_6=\langle d_5,b_2,c_6,f,\kappa_2\rangle g 6 = ⟨ d 5 , b 2 , c 6 , f , κ 2 ⟩ ℓ G ( g 2 ) = ℓ G ( g 5 ) + ℓ G ( g 6 ) \ell_G(g_2)=\ell_G(g_5)+\ell_G(g_6) ℓ G ( g 2 ) = ℓ G ( g 5 ) + ℓ G ( g 6 ) Theorem 5.9 combined with Theorem 7.2 ) applied to g 3 g_3 g 3 g 5 g_5 g 5 c 5 = c 1 c_5=c_1 c 5 = c 1 d 5 = m d_5=m d 5 = m ℓ G ( g 3 ) = ℓ G ( g 5 ) \ell_G(g_3)=\ell_G(g_5) ℓ G ( g 3 ) = ℓ G ( g 5 ) d 5 = m d_5=m d 5 = m g 6 = ⟨ m , b 2 , c 6 , f , κ 2 ⟩ g_6=\langle m,b_2,c_6,f,\kappa_2\rangle g 6 = ⟨ m , b 2 , c 6 , f , κ 2 ⟩ Theorem 5.9 to g 4 g_4 g 4 g 6 g_6 g 6 c 6 = c 2 c_6=c_2 c 6 = c 2 ℓ G ( g 4 ) = ℓ G ( g 6 ) \ell_G(g_4)=\ell_G(g_6) ℓ G ( g 4 ) = ℓ G ( g 6 )
We use the fact that the spine of a grid suffices to determine its size to inspire the following metric ℓ B \ell_B ℓ B
Definition 7.8. Given a , b ∈ B W + a,b\in BW^+ a , b ∈ B W + Theorem 5.16 and Theorem 7.3 that there exist c , d ∈ B W + c,d\in BW^+ c , d ∈ B W + κ \kappa κ g = ⟨ a , b , c , d , κ ⟩ ∈ G D g=\langle a,b,c,d,\kappa\rangle\in G^D g = ⟨ a , b , c , d , κ ⟩ ∈ G D
ℓ B ( a , b ) : = ℓ G ( g ) . \ell_B(a,b):=\ell_G(g). ℓ B ( a , b ) := ℓ G ( g ) . This is well-defined independently of the choice of c c c d d d κ \kappa κ Theorem 7.7 .
In Lean, we take a slightly different approach to well-definedness. We have noncomputable GridData.existence (via Prop-valued grid existence and Theorem 7.3 ). We then use the axiom of choice to select such a grid to serve as g g g ab_len to be the length of this selected grid. We do not use Theorem 7.7 and do not need to prove that our metric is independent of c c c d d d κ \kappa κ ℓ B \ell_B ℓ B c , d , κ c,d,\kappa c , d , κ ℓ B \ell_B ℓ B
Partial Grids
We similarly update the definition of partial grids to track the data of their construction. In fact, throughout our Lean formalization we have used Type-valued partial grids. Unlike grids, where there were significant theorems not mentioning size (such as stability, Theorem 5.12 ), almost every theorem relating to partial grids discusses size. So we had little use for Prop-valued partial grids, and hence did not implement them. Therefore, the reader should view the following definition of G P G_P G P G P D G_P^D G P D
Definition 7.9. (Update to Definition 6.28) We define the set of partial grids G P G_P G P 26
a trivial cell For any a , b , c , d ∈ B W 1 , + a,b,c,d \in BW^{1,+} a , b , c , d ∈ B W 1 , + ( a b ‾ , c d ‾ ) ∈ R t r i v i a l (\overline{ab},\overline{cd}) \in R_{\mathrm{trivial}} ( ab , c d ) ∈ R trivial ⟨ a ‾ , b , c , ε , d ‾ , t r i v i a l c e l l ⟩ ∈ G P \langle\overline{a},b,c,\varepsilon,\overline{d},\mathrm{trivial\ cell}\rangle\in G_P ⟨ a , b , c , ε , d , trivial cell ⟩ ∈ G P
a nontrivial cell For any a , b , c , d ∈ B W 1 , + a,b,c,d \in BW^{1,+} a , b , c , d ∈ B W 1 , + ( a b ‾ , c d ‾ ) ∈ R n o n t r i v i a l (\overline{ab},\overline{cd}) \in R_{\mathrm{nontrivial}} ( ab , c d ) ∈ R nontrivial ⟨ a ‾ , b , c , ε , d ‾ , n o n t r i v i a l c e l l ⟩ ∈ G P \langle\overline{a},b,c,\varepsilon,\overline{d},\mathrm{nontrivial\ cell}\rangle\in G_P ⟨ a , b , c , ε , d , nontrivial cell ⟩ ∈ G P
a spine For any a , b ∈ { B W 1 , + ∖ { ε } } a,b \in\{BW^{1,+}\setminus\{\varepsilon\}\} a , b ∈ { B W 1 , + ∖ { ε }} ⟨ a ‾ , b , ε , a b ‾ , ε , s p i n e ⟩ ∈ G P \langle\overline{a},b,\varepsilon,\overline{ab},\varepsilon,\mathrm{spine}\rangle\in G_P ⟨ a , b , ε , ab , ε , spine ⟩ ∈ G P
horizontal concatenation to a partial grid with empty middle frontier :
Given partial grids g 1 = ⟨ a 1 , b 1 , c 1 , ε , d 1 , κ 1 ⟩ ∈ G P g_1=\langle a_1,b_1,c_1,\varepsilon,d_1,\kappa_1\rangle\in G_P g 1 = ⟨ a 1 , b 1 , c 1 , ε , d 1 , κ 1 ⟩ ∈ G P g 2 = ⟨ d 1 , b 2 , c 2 , m 2 , d 2 , κ 2 ⟩ ∈ G P g_2=\langle d_1,b_2,c_2,m_2,d_2,\kappa_2\rangle\in G_P g 2 = ⟨ d 1 , b 2 , c 2 , m 2 , d 2 , κ 2 ⟩ ∈ G P ⟨ a 1 , b 1 ⋅ b 2 , c 1 ⋅ c 2 , m 2 , d 2 , h o r i z o n t a l ε ( g 1 , g 2 ) ⟩ ∈ G P \langle a_1,b_1\cdot b_2,c_1\cdot c_2,m_2,d_2,\mathrm{horizontal}_{\varepsilon}(g_1,g_2)\rangle\in G_P ⟨ a 1 , b 1 ⋅ b 2 , c 1 ⋅ c 2 , m 2 , d 2 , horizontal ε ( g 1 , g 2 )⟩ ∈ G P
horizontal concatenation to a partial grid with non-empty middle frontier (m 1 m_1 m 1
Given g 1 = ⟨ a 1 , b 1 , c 1 , m 1 , d 1 , κ 1 ⟩ ∈ G P g_1=\langle a_1,b_1,c_1,m_1,d_1,\kappa_1\rangle\in G_P g 1 = ⟨ a 1 , b 1 , c 1 , m 1 , d 1 , κ 1 ⟩ ∈ G P g 2 = ⟨ d 1 , b 2 , c 2 , m 2 , d 2 , κ 2 ⟩ ∈ G P g_2=\langle d_1,b_2,c_2,m_2,d_2,\kappa_2\rangle\in G_P g 2 = ⟨ d 1 , b 2 , c 2 , m 2 , d 2 , κ 2 ⟩ ∈ G P ⟨ a 1 , b 1 ⋅ b 2 , c 1 , m 1 ⋅ c 2 ⋅ m 2 , d 2 , h o r i z o n t a l ( g 1 , g 2 ) ⟩ ∈ G P \langle a_1,b_1\cdot b_2,c_1,m_1\cdot c_2\cdot m_2,d_2,\mathrm{horizontal}(g_1,g_2)\rangle\in G_P ⟨ a 1 , b 1 ⋅ b 2 , c 1 , m 1 ⋅ c 2 ⋅ m 2 , d 2 , horizontal ( g 1 , g 2 )⟩ ∈ G P
vertical concatenation to a partial grid with empty middle frontier :
Given g 1 = ⟨ a 1 , b 1 , c 1 , ε , d 1 , κ 1 ⟩ ∈ G P g_1=\langle a_1,b_1,c_1,\varepsilon,d_1,\kappa_1\rangle\in G_P g 1 = ⟨ a 1 , b 1 , c 1 , ε , d 1 , κ 1 ⟩ ∈ G P ⟨ a 2 , c 1 , c 2 , m 2 , d 2 , κ 2 ⟩ ∈ G P \langle a_2,c_1,c_2,m_2,d_2,\kappa_2\rangle\in G_P ⟨ a 2 , c 1 , c 2 , m 2 , d 2 , κ 2 ⟩ ∈ G P ⟨ a 2 ⋅ a 1 , b 1 , c 2 , m 2 , d 2 ⋅ d 1 , v e r t i c a l ε ( g 1 , g 2 ) ⟩ ∈ G P \langle a_2\cdot a_1,b_1,c_2,m_2,d_2\cdot d_1,\mathrm{vertical}_{\varepsilon}(g_1,g_2)\rangle\in G_P ⟨ a 2 ⋅ a 1 , b 1 , c 2 , m 2 , d 2 ⋅ d 1 , vertical ε ( g 1 , g 2 )⟩ ∈ G P
vertical concatenation to a partial grid with non-empty middle frontier
Given ⟨ a 1 , b 1 , c 1 , m 1 , d 1 , κ 1 ⟩ ∈ G P \langle a_1,b_1,c_1,m_1,d_1,\kappa_1\rangle\in G_P ⟨ a 1 , b 1 , c 1 , m 1 , d 1 , κ 1 ⟩ ∈ G P ⟨ a 2 , c 1 , c 2 , m 2 , d 2 , κ 2 ⟩ ∈ G P \langle a_2,c_1,c_2,m_2,d_2,\kappa_2\rangle\in G_P ⟨ a 2 , c 1 , c 2 , m 2 , d 2 , κ 2 ⟩ ∈ G P ⟨ a 2 ⋅ a 1 , b 1 , c 2 , m 2 ⋅ d 2 ⋅ m 1 , d 1 , v e r t i c a l ( g 1 , g 2 ) ⟩ ∈ G P \langle a_2\cdot a_1,b_1,c_2,m_2\cdot d_2\cdot m_1,d_1,\mathrm{vertical}(g_1,g_2)\rangle\in G_P ⟨ a 2 ⋅ a 1 , b 1 , c 2 , m 2 ⋅ d 2 ⋅ m 1 , d 1 , vertical ( g 1 , g 2 )⟩ ∈ G P
Definition 7.10. Define a metric on partial grids ℓ P \ell_P ℓ P
ℓ P ( ⟨ a , b , c , m , d , t r i v i a l c e l l ⟩ ) = 0 \ell_P(\langle a,b,c,m,d,\mathrm{trivial\ cell}\rangle)=0 ℓ P (⟨ a , b , c , m , d , trivial cell ⟩) = 0
ℓ P ( ⟨ a , b , c , m , d , n o n t r i v i a l c e l l ⟩ ) = 1 \ell_P(\langle a,b,c,m,d,\mathrm{nontrivial\ cell}\rangle)=1 ℓ P (⟨ a , b , c , m , d , nontrivial cell ⟩) = 1
ℓ P ( ⟨ a , b , c , m , d , s p i n e ⟩ ) = 0 \ell_P(\langle a,b,c,m,d,\mathrm{spine}\rangle)=0 ℓ P (⟨ a , b , c , m , d , spine ⟩) = 0
ℓ P ( ⟨ a , b , c , m , d , h o r i z o n t a l ε ( p 1 , p 2 ) ⟩ ) = ℓ P ( p 1 ) + ℓ P ( p 2 ) \ell_P(\langle a,b,c,m,d,\mathrm{horizontal}_{\varepsilon}(p_1,p_2)\rangle)=\ell_P(p_1)+\ell_P(p_2) ℓ P (⟨ a , b , c , m , d , horizontal ε ( p 1 , p 2 )⟩) = ℓ P ( p 1 ) + ℓ P ( p 2 )
ℓ P ( ⟨ a , b , c , m , d , h o r i z o n t a l ( p 1 , p 2 ) ⟩ ) = ℓ P ( p 1 ) + ℓ P ( p 2 ) \ell_P(\langle a,b,c,m,d,\mathrm{horizontal}(p_1,p_2)\rangle)=\ell_P(p_1)+\ell_P(p_2) ℓ P (⟨ a , b , c , m , d , horizontal ( p 1 , p 2 )⟩) = ℓ P ( p 1 ) + ℓ P ( p 2 )
ℓ P ( ⟨ a , b , c , m , d , v e r t i c a l ε ( p 1 , p 2 ) ⟩ ) = ℓ P ( p 1 ) + ℓ P ( p 2 ) \ell_P(\langle a,b,c,m,d,\mathrm{vertical}_{\varepsilon}(p_1,p_2)\rangle)=\ell_P(p_1)+\ell_P(p_2) ℓ P (⟨ a , b , c , m , d , vertical ε ( p 1 , p 2 )⟩) = ℓ P ( p 1 ) + ℓ P ( p 2 )
ℓ P ( ⟨ a , b , c , m , d , v e r t i c a l ( p 1 , p 2 ) ⟩ ) = ℓ P ( p 1 ) + ℓ P ( p 2 ) \ell_P(\langle a,b,c,m,d,\mathrm{vertical}(p_1,p_2)\rangle)=\ell_P(p_1)+\ell_P(p_2) ℓ P (⟨ a , b , c , m , d , vertical ( p 1 , p 2 )⟩) = ℓ P ( p 1 ) + ℓ P ( p 2 )
It is possible to extend a partial grid’s spine without affecting its size.
Lemma 7.11. (Update to Theorem Theorem 6.31) Consider a partial grid p = ⟨ a , b , c , m , d , κ ⟩ p=\langle a,b,c,m,d,\kappa\rangle p = ⟨ a , b , c , m , d , κ ⟩ negative augmented grid word a ′ a' a ′ p 1 = ⟨ a ′ ⋅ a , b , ε , a ′ ⋅ c ⋅ m , d , κ 1 ⟩ p_1=\langle a'\cdot a,b,\varepsilon,a'\cdot c\cdot m,d,\kappa_1\rangle p 1 = ⟨ a ′ ⋅ a , b , ε , a ′ ⋅ c ⋅ m , d , κ 1 ⟩ ℓ P ( p ) = ℓ P ( p 1 ) \ell_P(p)=\ell_P(p_1) ℓ P ( p ) = ℓ P ( p 1 ) positive augmented grid word b ′ b' b ′ p 2 = ⟨ a , b ⋅ b ′ , c , m ⋅ d ⋅ b ′ , ε , κ 2 ⟩ p_2=\langle a,b\cdot b',c,m\cdot d\cdot b',\varepsilon,\kappa_2\rangle p 2 = ⟨ a , b ⋅ b ′ , c , m ⋅ d ⋅ b ′ , ε , κ 2 ⟩ ℓ P ( p ) = ℓ P ( p 2 ) \ell_P(p)=\ell_P(p_2) ℓ P ( p ) = ℓ P ( p 2 )
Splittability also behaves as we might expect in regard to the size of the resulting partial grids.
Theorem 7.12. (Update to Theorem Theorem 6.34) Given p = ⟨ a , b 1 b 2 , c , m , d ⟩ ∈ G P p=\langle a,b_1b_2,c,m,d\rangle\in G_P p = ⟨ a , b 1 b 2 , c , m , d ⟩ ∈ G P b 1 , b 2 b_1,b_2 b 1 , b 2
p p p p 1 = ⟨ a , b 1 , c 1 , m 1 , d 1 ⟩ ∈ G P p_1=\langle a,b_1,c_1,m_1,d_1\rangle\in G_P p 1 = ⟨ a , b 1 , c 1 , m 1 , d 1 ⟩ ∈ G P p 2 = ⟨ d 1 , b 2 , c 2 , m 2 , d ⟩ ∈ G P p_2=\langle d_1,b_2,c_2,m_2,d\rangle\in G_P p 2 = ⟨ d 1 , b 2 , c 2 , m 2 , d ⟩ ∈ G P c ⋅ m = c 1 ⋅ m 1 ⋅ c 2 ⋅ m 2 c\cdot m=c_1\cdot m_1\cdot c_2\cdot m_2 c ⋅ m = c 1 ⋅ m 1 ⋅ c 2 ⋅ m 2 ℓ P ( p ) = ℓ P ( p 1 ) + ℓ P ( p 2 ) \ell_P(p)=\ell_P(p_1)+\ell_P(p_2) ℓ P ( p ) = ℓ P ( p 1 ) + ℓ P ( p 2 )
p p p p 1 = ⟨ a , b 1 , c , m 1 , ε ⟩ ∈ G P p_1 = \langle a,b_1,c,m_1,\varepsilon\rangle\in G_P p 1 = ⟨ a , b 1 , c , m 1 , ε ⟩ ∈ G P b 2 b_2 b 2 m = m 1 ⋅ b 2 m = m_1 \cdot b_2 m = m 1 ⋅ b 2 ℓ P ( p ) = ℓ P ( p 1 ) \ell_P(p) = \ell_P(p_1) ℓ P ( p ) = ℓ P ( p 1 )
We update our add_cell function to add in a fact about the size of the resulting partial grid:
Definition 7.13. add_cell is a function taking in some ( x , y ) ∈ R g r i d (x,y) \in R_{\mathrm{grid}} ( x , y ) ∈ R grid p 1 = ⟨ a , b , c , m 1 ⋅ x ⋅ m 2 , d , κ 1 ⟩ p_1 = \langle a,b,c,m_1 \cdot x \cdot m_2,d,\kappa_1\rangle p 1 = ⟨ a , b , c , m 1 ⋅ x ⋅ m 2 , d , κ 1 ⟩ p 2 = ⟨ a , b , c ′ , m ′ , d ′ , κ 2 ⟩ p_2 = \langle a,b,c',m',d',\kappa_2\rangle p 2 = ⟨ a , b , c ′ , m ′ , d ′ , κ 2 ⟩ c ′ ⋅ m ′ ⋅ d ′ = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c' \cdot m' \cdot d' = c \cdot m_1 \cdot y \cdot m_2 \cdot d c ′ ⋅ m ′ ⋅ d ′ = c ⋅ m 1 ⋅ y ⋅ m 2 ⋅ d c c c c ′ c' c ′ d d d d ′ d' d ′ ( x , y ) ∈ R t r i v (x,y) \in R_{\mathrm{triv}} ( x , y ) ∈ R triv ℓ P ( p 2 ) = ℓ P ( p 1 ) \ell_P(p_2) = \ell_P(p_1) ℓ P ( p 2 ) = ℓ P ( p 1 ) ( x , y ) ∈ R n o n t r i v (x,y) \in R_{\mathrm{nontriv}} ( x , y ) ∈ R nontriv ℓ P ( p 2 ) = ℓ P ( p 1 ) + 1 \ell_P(p_2) = \ell_P(p_1) + 1 ℓ P ( p 2 ) = ℓ P ( p 1 ) + 1
Note that we could separate out the final property about length, and prove it separately as a theorem about the output of add_cell (in contrast to, say, the prefix and suffix properties, which are required for the definition of add_cell). However, like the definition of add_cell, such a proof would have 17 cases27
Our main goal will be to show that given a grid and a partial grid sharing essentially the same spine (aside from any empty arrows), the size of the partial grid is at most that of the grid. Our proof requires us to strengthen the inductive statement with an auxiliary fact about prefixes and suffixes of the grid’s target (part (1) below).
Theorem 7.14. Let g = ⟨ a , b , c , d ⟩ ∈ G g = \langle a,b,c,d\rangle\in G g = ⟨ a , b , c , d ⟩ ∈ G p = ⟨ a 0 , b 0 , c 0 , m 0 , d 0 ⟩ ∈ G P p = \langle a_0,b_0,c_0,m_0,d_0\rangle\in G_P p = ⟨ a 0 , b 0 , c 0 , m 0 , d 0 ⟩ ∈ G P s 1 , + : B W 1 , + → B W + s^{1,+}: BW^{1,+} \to BW^+ s 1 , + : B W 1 , + → B W + 1 1 1 a = s 1 , + ( a 0 ) a = s^{1,+}(a_0) a = s 1 , + ( a 0 ) b = s 1 , + ( b 0 ) b = s^{1,+}(b_0) b = s 1 , + ( b 0 )
s 1 , + ( c 0 ) s^{1,+}(c_0) s 1 , + ( c 0 ) c c c s 1 , + ( d 0 ) s^{1,+}(d_0) s 1 , + ( d 0 ) d d d
ℓ P ( p ) ≤ ℓ G ( g ) \ell_P(p) \le\ell_G(g) ℓ P ( p ) ≤ ℓ G ( g )
Here is the image to keep in mind:
Figure 1 (PDF p. 58).
We note that we could prove (1) independently, and then use (1) to later prove (2). (We cannot proceed conversely; (2) cannot be proven without (1).) Due to the large number of cases in the proof, and the work to organize the hypotheses for each case, it is preferable to combine (1) and (2) as we have done, proving them simultaneously.
Proof. (Sketch) We proceed by induction on g g g g = ⟨ a , b , c , d , horizontal ( g 1 , g 2 ) ⟩ g = \langle a,b,c,d,\operatorname{horizontal}(g_1,g_2)\rangle g = ⟨ a , b , c , d , horizontal ( g 1 , g 2 )⟩ g 1 = ⟨ a , b 1 , c 1 , d 1 , κ 1 ⟩ g_1 = \langle a,b_1,c_1,d_1,\kappa_1\rangle g 1 = ⟨ a , b 1 , c 1 , d 1 , κ 1 ⟩ g 2 = ⟨ d 1 , b 2 , c 2 , d , κ 2 ⟩ g_2 = \langle d_1,b_2,c_2,d,\kappa_2\rangle g 2 = ⟨ d 1 , b 2 , c 2 , d , κ 2 ⟩ b = b 1 b 2 b = b_1b_2 b = b 1 b 2 c = c 1 c 2 c = c_1c_2 c = c 1 c 2 ℓ G ( g ) = ℓ G ( g 1 ) + ℓ G ( g 2 ) \ell_G(g) = \ell_G(g_1) + \ell_G(g_2) ℓ G ( g ) = ℓ G ( g 1 ) + ℓ G ( g 2 ) s 1 , + ( b 0 ) = b = b 1 b 2 s^{1,+}(b_0) = b = b_1b_2 s 1 , + ( b 0 ) = b = b 1 b 2 b 3 , b 4 ∈ B W 1 , + b_3,b_4 \in BW^{1,+} b 3 , b 4 ∈ B W 1 , + b 0 = b 3 b 4 b_0 = b_3b_4 b 0 = b 3 b 4 b 1 = s 1 , + ( b 3 ) b_1 = s^{1,+}(b_3) b 1 = s 1 , + ( b 3 ) b 2 = s 1 , + ( b 4 ) b_2 = s^{1,+}(b_4) b 2 = s 1 , + ( b 4 ) p = ⟨ a 0 , b 3 b 4 , c 0 , m 0 , d 0 ⟩ p = \langle a_0,b_3b_4,c_0,m_0,d_0\rangle p = ⟨ a 0 , b 3 b 4 , c 0 , m 0 , d 0 ⟩ p p p p 1 = ⟨ a 0 , b 3 , c 3 , m 3 , d 3 ⟩ p_1 = \langle a_0,b_3,c_3,m_3,d_3\rangle p 1 = ⟨ a 0 , b 3 , c 3 , m 3 , d 3 ⟩ p 2 = ⟨ d 3 , b 4 , c 4 , m 4 , d 0 ⟩ p_2 = \langle d_3,b_4,c_4,m_4,d_0\rangle p 2 = ⟨ d 3 , b 4 , c 4 , m 4 , d 0 ⟩ c 3 m 3 c 4 m 4 = c 0 m 0 c_3m_3c_4m_4 = c_0m_0 c 3 m 3 c 4 m 4 = c 0 m 0 ℓ P ( p ) = ℓ P ( p 1 ) + ℓ P ( p 2 ) \ell_P(p) = \ell_P(p_1) + \ell_P(p_2) ℓ P ( p ) = ℓ P ( p 1 ) + ℓ P ( p 2 ) m 3 m_3 m 3 c 0 = c 3 c_0 = c_3 c 0 = c 3 m 0 = m 3 c 4 m 4 m_0 = m_3c_4m_4 m 0 = m 3 c 4 m 4 g 1 g_1 g 1 p 1 p_1 p 1 s 1 , + ( c 3 ) s^{1,+}(c_3) s 1 , + ( c 3 ) c 1 c_1 c 1 s 1 , + ( c 0 ) s^{1,+}(c_0) s 1 , + ( c 0 ) c = c 1 c 2 c = c_1c_2 c = c 1 c 2 s 1 , + ( d 3 ) s^{1,+}(d_3) s 1 , + ( d 3 ) d 1 d_1 d 1 ℓ P ( p 1 ) ≤ ℓ G ( g 1 ) \ell_P(p_1) \le\ell_G(g_1) ℓ P ( p 1 ) ≤ ℓ G ( g 1 ) d 5 d_5 d 5 d 1 = s 1 , + ( d 5 ⋅ d 3 ) d_1 = s^{1,+}(d_5 \cdot d_3) d 1 = s 1 , + ( d 5 ⋅ d 3 ) p 2 p_2 p 2 p 3 = ⟨ d 5 ⋅ d 3 , b 4 , ε , d 5 ⋅ c 4 ⋅ m 4 , d 0 ⟩ p_3 = \langle d_5 \cdot d_3, b_4, \varepsilon, d_5 \cdot c_4 \cdot m_4, d_0\rangle p 3 = ⟨ d 5 ⋅ d 3 , b 4 , ε , d 5 ⋅ c 4 ⋅ m 4 , d 0 ⟩ ℓ P ( p 3 ) = ℓ P ( p 2 ) \ell_P(p_3) = \ell_P(p_2) ℓ P ( p 3 ) = ℓ P ( p 2 ) d 1 = s 1 , + ( d 5 ⋅ d 3 ) d_1 = s^{1,+}(d_5 \cdot d_3) d 1 = s 1 , + ( d 5 ⋅ d 3 ) g 2 g_2 g 2 p 3 p_3 p 3 s 1 , + ( d 0 ) s^{1,+}(d_0) s 1 , + ( d 0 ) d d d ℓ P ( p 3 ) ≤ ℓ G ( g 2 ) \ell_P(p_3) \le\ell_G(g_2) ℓ P ( p 3 ) ≤ ℓ G ( g 2 ) ℓ P ( p ) ≤ ℓ G ( g ) \ell_P(p) \le\ell_G(g) ℓ P ( p ) ≤ ℓ G ( g )
Semi-Thue Derivations
We return to our two definitions of Semi-Thue systems, defining data-carrying versions so that we may define a metric on them. In Lean, this once again corresponds to switching the output type from Prop to Type.
Definition 7.15. Consider a set A A A R R R A A A → ∗ R \xrightarrow{*}_{R} ∗ R
reflexivity : For all a ∈ A ∗ a \in A^* a ∈ A ∗ ⟨ a → ∗ R a , r e f l ⟩ \langle a \xrightarrow{*}_{R} a,\mathrm{refl}\rangle ⟨ a ∗ R a , refl ⟩
closure : For all a , b ∈ A ∗ a,b \in A^* a , b ∈ A ∗ a → R b a \xrightarrow{R} b a R b ⟨ a → ∗ R b , c l o s u r e ( a , b ) ⟩ \langle a \xrightarrow{*}_{R} b,\mathrm{closure}(a,b)\rangle ⟨ a ∗ R b , closure ( a , b )⟩
transitivity : For all a , b , c ∈ A ∗ a,b,c \in A^* a , b , c ∈ A ∗ s 0 = ⟨ a → ∗ R b , κ 0 ⟩ s_0 = \langle a \xrightarrow{*}_{R} b,\kappa_0\rangle s 0 = ⟨ a ∗ R b , κ 0 ⟩ s 1 = ⟨ b → ∗ R c , κ 1 ⟩ s_1 = \langle b \xrightarrow{*}_{R} c,\kappa_1\rangle s 1 = ⟨ b ∗ R c , κ 1 ⟩ ⟨ a → ∗ R c , t r a n s ( s 0 , s 1 ) ⟩ \langle a \xrightarrow{*}_{R} c,\mathrm{trans}(s_0,s_1)\rangle ⟨ a ∗ R c , trans ( s 0 , s 1 )⟩
Definition 7.16. We define a metric on Semi-Thue systems ℓ S R \ell_{SR} ℓ S R R R R ℓ R : R → N \ell_R : R \to\mathbb{N} ℓ R : R → N A A A R R R ℓ R \ell_R ℓ R
ℓ S ( ⟨ _ , r e f l ⟩ ) = 0 \ell_S(\langle\_,\mathrm{refl}\rangle) = 0 ℓ S (⟨ _ , refl ⟩) = 0
ℓ S R ( ⟨ _ , c l o s u r e ( a , b ) ⟩ ) = ℓ R ( a , b ) \ell_{SR}(\langle\_,\mathrm{closure}(a,b)\rangle) = \ell_R(a,b) ℓ S R (⟨ _ , closure ( a , b )⟩) = ℓ R ( a , b )
ℓ S R ( ⟨ _ , t r a n s ( s 0 , s 1 ) ⟩ ) = ℓ S R ( s 0 ) + ℓ S R ( s 1 ) \ell_{SR}(\langle\_,\mathrm{trans}(s_0,s_1)\rangle) = \ell_{SR}(s_0) + \ell_{SR}(s_1) ℓ S R (⟨ _ , trans ( s 0 , s 1 )⟩) = ℓ S R ( s 0 ) + ℓ S R ( s 1 )
Definition 7.17. Given a set A A A R R R A A A → R \xrightarrow{}_{R} R
reflexivity : For all a ∈ A ∗ a \in A^* a ∈ A ∗ ⟨ a → R a , r e f l ⟩ \langle a \xrightarrow{}_{R} a,\mathrm{refl}\rangle ⟨ a R a , refl ⟩
one-step transitivity : For all a , b 1 , x , b 2 , y ∈ A ∗ a,b_1,x,b_2,y \in A^* a , b 1 , x , b 2 , y ∈ A ∗ d 0 = ⟨ a → R b 1 x b 2 , κ 0 ⟩ d_0 = \langle a \xrightarrow{}_{R} b_1xb_2,\kappa_0\rangle d 0 = ⟨ a R b 1 x b 2 , κ 0 ⟩ x R y xRy x R y ⟨ a → R b 1 y b 2 , s t e p t r a n s i t i v i t y ( d 0 , x , y ) ⟩ \langle a \xrightarrow{}_{R} b_1yb_2,\mathrm{step\ transitivity}(d_0,x,y)\rangle ⟨ a R b 1 y b 2 , step transitivity ( d 0 , x , y )⟩
Definition 7.18. We define a metric on derivation-style Semi-Thue systems ℓ D R \ell_{DR} ℓ D R R R R ℓ R \ell_R ℓ R
ℓ D R ( ⟨ _ , r e f l ⟩ ) = 0 \ell_{DR}(\langle\_,\mathrm{refl}\rangle) = 0 ℓ D R (⟨ _ , refl ⟩) = 0
ℓ D R ( ⟨ _ , s t e p t r a n s i t i v i t y ( d 0 , x , y ) ⟩ ) = ℓ D R ( d 0 ) + ℓ R ( x , y ) \ell_{DR}(\langle\_,\mathrm{step\ transitivity}(d_0,x,y)\rangle) = \ell_{DR}(d_0) + \ell_R(x,y) ℓ D R (⟨ _ , step transitivity ( d 0 , x , y )⟩) = ℓ D R ( d 0 ) + ℓ R ( x , y )
Theorem 7.19. Given s 0 = ⟨ a → R ∗ b , κ 0 ⟩ s_0 = \langle a \mathop{\to}\limits_{R}^{*} b, \kappa_0\rangle s 0 = ⟨ a R → ∗ b , κ 0 ⟩ d 0 = ⟨ a → R b , κ 1 ⟩ d_0 = \langle a \mathop{\to}\limits_{R} b, \kappa_1\rangle d 0 = ⟨ a R → b , κ 1 ⟩ ℓ S R ( s 0 ) = ℓ D R ( d 0 ) \ell_{SR}(s_0) = \ell_{DR}(d_0) ℓ S R ( s 0 ) = ℓ D R ( d 0 )
With the basics set up, we now move on to discuss our specific relations.
Definition 7.20. We define ℓ R r e v \ell_{R_{\mathrm{rev}}} ℓ R rev ( x , y ) ∈ R r e v (x,y) \in R_{\mathrm{rev}} ( x , y ) ∈ R rev ℓ R r e v ( x , y ) = 1 \ell_{R_{\mathrm{rev}}}(x,y) = 1 ℓ R rev ( x , y ) = 1
ℓ R g r i d \ell_{R_{\mathrm{grid}}} ℓ R grid
For all ( x , y ) ∈ R t r i v (x,y) \in R_{\mathrm{triv}} ( x , y ) ∈ R triv ℓ R g r i d ( x , y ) = 0 \ell_{R_{\mathrm{grid}}}(x,y) = 0 ℓ R grid ( x , y ) = 0
For all ( x , y ) ∈ R n o n t r i v (x,y) \in R_{\mathrm{nontriv}} ( x , y ) ∈ R nontriv ℓ R g r i d ( x , y ) = 1 \ell_{R_{\mathrm{grid}}}(x,y) = 1 ℓ R grid ( x , y ) = 1
Next, we update our results from the previous chapter with facts about our metrics. The proofs proceed in an entirely analogous manner. Here are the main conclusions (of course, each of these requires a number of lemmas to be updated as well; we omit those statements and proofs):
Theorem 7.21. (Update to the backwards direction of Theorem 6.3) Given a partial grid p = ⟨ a , b , c , m , d , κ ⟩ p = \langle a,b,c,m,d,\kappa\rangle p = ⟨ a , b , c , m , d , κ ⟩ s = ⟨ a b ‾ → R r e v ∗ c m d ‾ , κ ′ ⟩ s = \langle\overline{ab} \mathop{\to}\limits_{R_{\mathrm{rev}}}^{*} \overline{cmd},\kappa'\rangle s = ⟨ ab R rev → ∗ c m d , κ ′ ⟩ ℓ P ( p ) = ℓ S R r e v ( s ) \ell_P(p) = \ell_{SR_{\mathrm{rev}}}(s) ℓ P ( p ) = ℓ S R rev ( s )
Theorem 7.22. (Update to Theorem 6.14) Let a , b , c , d ∈ B W 1 , + a,b,c,d \in BW^{1,+} a , b , c , d ∈ B W 1 , + s 0 = ⟨ a b ‾ → R r e v ∗ c d ‾ , κ 0 ⟩ s_0 = \langle\overline{ab} \mathop{\to}\limits_{R_{\mathrm{rev}}}^{*} \overline{cd},\kappa_0\rangle s 0 = ⟨ ab R rev → ∗ c d , κ 0 ⟩ e , f ∈ B W 1 , + e,f \in BW^{1,+} e , f ∈ B W 1 , + s 1 = ⟨ i 1 ( a b ‾ ) → R g r i d ∗ e f ‾ , κ 1 ⟩ s_1 = \langle i^1(\overline{ab}) \mathop{\to}\limits_{R_{\mathrm{grid}}}^{*} \overline{ef},\kappa_1\rangle s 1 = ⟨ i 1 ( ab ) R grid → ∗ e f , κ 1 ⟩ s 1 ( e ) = c s^1(e)=c s 1 ( e ) = c s 1 ( f ) = d s^1(f)=d s 1 ( f ) = d ℓ S R r e v ( s 0 ) = ℓ S R g r i d ( d 1 ) \ell_{SR_{\mathrm{rev}}}(s_0) = \ell_{SR_{\mathrm{grid}}}(d_1) ℓ S R rev ( s 0 ) = ℓ S R grid ( d 1 )
Theorem 7.23. (Update to Theorem 6.35) For nonempty a , b ∈ B W 1 , + a,b \in BW^{1,+} a , b ∈ B W 1 , + s 0 = ⟨ a b ‾ → R g r i d ∗ c , κ 0 ⟩ s_0 = \langle\overline{ab} \mathop{\to}\limits_{R_{\mathrm{grid}}}^{*} c,\kappa_0\rangle s 0 = ⟨ ab R grid → ∗ c , κ 0 ⟩ p p p ( a , b ) (a,b) ( a , b ) c c c ℓ S R r e v ( s 0 ) = ℓ P ( p ) \ell_{SR_{\mathrm{rev}}}(s_0) = \ell_P(p) ℓ S R rev ( s 0 ) = ℓ P ( p )
Alternative Choices
We have made a number of seemingly arbitrary decisions, mainly in the precise definition of partial grids, as well as that of their size. Here we present their shortfalls, and discuss alternative strategies with their own benefits and shortfalls. As the definitions and theorems here do not form part of the final project, the Lean formalizations linked may be a bit rougher, and we take a few more liberties in our presentation, deviating from the exact Lean formalization for the sake of exposition.
Let us begin with the shortfalls of our current definitions. There is one tantalizing theorem we have not been able to prove using structural induction. It would be nice to know that the frame of a partial grid determines its length:
Conjecture 7.24. Given partial grids p 1 = ⟨ a , b , c , m , d , κ 1 ⟩ p_1 = \langle a,b,c,m,d,\kappa_1\rangle p 1 = ⟨ a , b , c , m , d , κ 1 ⟩ p 2 = ⟨ a , b , c , m , d , κ 2 ⟩ p_2 = \langle a,b,c,m,d,\kappa_2\rangle p 2 = ⟨ a , b , c , m , d , κ 2 ⟩ ℓ P ( p 1 ) = ℓ P ( p 2 ) \ell_P(p_1) = \ell_P(p_2) ℓ P ( p 1 ) = ℓ P ( p 2 )
The obvious strategy of inducting on one of either p 1 p_1 p 1 p 2 p_2 p 2 p 1 p_1 p 1
Figure 26. p 1 p_1 p 1 p 3 p_3 p 3 p 4 p_4 p 4 a a a b b b c c c d d d m m m b 3 b_3 b 3 b 4 b_4 b 4 c 4 c_4 c 4 d 3 d_3 d 3 m 3 m_3 m 3 m 4 m_4 m 4
That is, we know there are nonempty b 3 , b 4 b_3,b_4 b 3 , b 4 b = b 3 b 4 b=b_3b_4 b = b 3 b 4 m 3 m_3 m 3 c 4 , m 4 c_4,m_4 c 4 , m 4 m = m 3 c 4 m 4 m=m_3c_4m_4 m = m 3 c 4 m 4 p 3 = ⟨ a , b 3 , c , m 3 , d 3 , κ 3 ⟩ p_3=\langle a,b_3,c,m_3,d_3,\kappa_3\rangle p 3 = ⟨ a , b 3 , c , m 3 , d 3 , κ 3 ⟩ p 4 = ⟨ d 3 , b 4 , c 4 , m 4 , d , κ 4 ⟩ p_4=\langle d_3,b_4,c_4,m_4,d,\kappa_4\rangle p 4 = ⟨ d 3 , b 4 , c 4 , m 4 , d , κ 4 ⟩ p 1 = ⟨ a , b 3 ⋅ b 4 , c , m 3 ⋅ c 4 ⋅ m 4 , d , horizontal ( p 3 , p 4 ) ⟩ p_1=\langle a,b_3\cdot b_4,c,m_3\cdot c_4\cdot m_4,d,\operatorname{horizontal}(p_3,p_4)\rangle p 1 = ⟨ a , b 3 ⋅ b 4 , c , m 3 ⋅ c 4 ⋅ m 4 , d , horizontal ( p 3 , p 4 )⟩ p 2 p_2 p 2 Theorem 7.12 , in hopes of applying our inductive hypothesis for p 3 p_3 p 3 p 4 p_4 p 4 p 2 p_2 p 2 p 5 , p 6 p_5,p_6 p 5 , p 6
Figure 1 (PDF p. 61).
Here, we have partial grids p 5 = ⟨ a , b 3 , c 5 , m 5 , d 5 ⟩ p_5=\langle a,b_3,c_5,m_5,d_5\rangle p 5 = ⟨ a , b 3 , c 5 , m 5 , d 5 ⟩ p 6 = ⟨ d 5 , b 4 , c 6 , m 6 , d ⟩ p_6=\langle d_5,b_4,c_6,m_6,d\rangle p 6 = ⟨ d 5 , b 4 , c 6 , m 6 , d ⟩ c ⋅ m = c 5 ⋅ m 5 ⋅ c 6 ⋅ m 6 c\cdot m=c_5\cdot m_5\cdot c_6\cdot m_6 c ⋅ m = c 5 ⋅ m 5 ⋅ c 6 ⋅ m 6 m 5 m_5 m 5 c = c 5 c=c_5 c = c 5 m = m 5 c 6 m 6 m=m_5c_6m_6 m = m 5 c 6 m 6 p 5 p_5 p 5 p 3 p_3 p 3 m 3 m_3 m 3 m 5 m_5 m 5 d 3 d_3 d 3 d 5 d_5 d 5 m 3 ⋅ c 4 ⋅ m 4 = m 5 ⋅ c 6 ⋅ m 6 m_3\cdot c_4\cdot m_4=m_5\cdot c_6\cdot m_6 m 3 ⋅ c 4 ⋅ m 4 = m 5 ⋅ c 6 ⋅ m 6 m 3 m_3 m 3 m 5 m_5 m 5 m 3 m_3 m 3 m 5 m_5 m 5 28 d 3 d_3 d 3 d 5 d_5 d 5
The situation is even worse for p 4 p_4 p 4 p 6 p_6 p 6 d 5 d_5 d 5 d 3 d_3 d 3 p 6 p_6 p 6 d 3 d_3 d 3
If d 5 d_5 d 5 d 3 d_3 d 3 p 4 p_4 p 4 p 4 p_4 p 4 p 6 p_6 p 6
Thus, Conjecture 7.24 remains inconclusive. It is possible to complete the project without Conjecture 7.24, but it is certainly displeasing. Intuitively, partial grids with the same frame contain the same (multi)set of cells, and should have the same size. We wonder if perhaps it is our definition of a partial grid, with its complicated division of the frontier into bottom, middle, and right segments, that is to blame. Here is an alternative, simpler definition of a partial grid:
Definition 7.25. The set of frontier-style partial grids G F G_F G F ( B W 1 ) 3 (BW^1)^3 ( B W 1 ) 3
a spine: For any positive, nonempty a , b ∈ B W 1 a,b\in BW^1 a , b ∈ B W 1 ⟨ a ˉ , b , a ˉ b , spine ⟩ ∈ G F \langle\bar{a},b,\bar{a}b,\operatorname{spine}\rangle\in G_F ⟨ a ˉ , b , a ˉ b , spine ⟩ ∈ G F
cell addition: Given ( x , y ) ∈ R g r i d (x,y) \in R_{\mathrm{grid}} ( x , y ) ∈ R grid f 1 = ⟨ a , b , c 1 x c 2 , κ 1 ⟩ ∈ G F f_1 = \langle a,b,c_1xc_2,\kappa_1\rangle\in G_F f 1 = ⟨ a , b , c 1 x c 2 , κ 1 ⟩ ∈ G F ⟨ a , b , c 1 y c 2 , cell addition ( x , y , f 1 ) ⟩ ∈ G F \langle a,b,c_1yc_2,\operatorname{cell\ addition}(x,y,f_1)\rangle\in G_F ⟨ a , b , c 1 y c 2 , cell addition ( x , y , f 1 )⟩ ∈ G F
Definition 7.26. We define the length metric ℓ G F : G F → N \ell_{G_F}:G_F \to\mathbb{N} ℓ G F : G F → N
a spine: ℓ G F ( ⟨ _ , _ , _ , spine ⟩ ) = 0 \ell_{G_F}(\langle\_,\_,\_,\operatorname{spine}\rangle)=0 ℓ G F (⟨ _ , _ , _ , spine ⟩) = 0
cell addition: ℓ G F ( ⟨ _ , _ , _ , cell addition ( x , y , f 1 ) ⟩ ) = ℓ G F ( f 0 ) + ℓ R g r i d ( x , y ) \ell_{G_F}\left(\langle\_,\_,\_,\operatorname{cell\ addition}(x,y,f_1)\rangle\right)=\ell_{G_F}(f_0)+\ell_{R_{\mathrm{grid}}}(x,y) ℓ G F ( ⟨ _ , _ , _ , cell addition ( x , y , f 1 )⟩ ) = ℓ G F ( f 0 ) + ℓ R grid ( x , y )
Alas, we again encounter issues trying to prove Conjecture 7.24, even with G F G_F G F G P G_P G P
Fallacy 7.27. Given ( x , y ) ∈ R g r i d (x,y) \in R_{\mathrm{grid}} ( x , y ) ∈ R grid f 1 = ⟨ a , b , c 1 y c 2 , κ 1 ⟩ f_1=\langle a,b,c_1yc_2,\kappa_1\rangle f 1 = ⟨ a , b , c 1 y c 2 , κ 1 ⟩ f 0 = ⟨ a , b , c 1 x c 2 , κ 0 ⟩ f_0=\langle a,b,c_1xc_2,\kappa_0\rangle f 0 = ⟨ a , b , c 1 x c 2 , κ 0 ⟩ ℓ F ( f 1 ) = ℓ F ( f 0 ) + ℓ R g r i d ( x , y ) \ell_F(f_1)=\ell_F(f_0)+\ell_{R_{\mathrm{grid}}}(x,y) ℓ F ( f 1 ) = ℓ F ( f 0 ) + ℓ R grid ( x , y )
In fact, Fallacy 7.27 does not even hold for the base cases. Consider the very simple frontier-style partial grid f 1 = ⟨ 1 ‾ , 1 , 11 ‾ , cell addition ( 11 ‾ , 11 ‾ , ⟨ 1 , 1 , 11 ‾ , spine ⟩ ) ⟩ f_1=\langle\overline{1},1,\overline{11},\operatorname{cell\ addition}(\overline{11},\overline{11},\langle1,1,\overline{11},\operatorname{spine}\rangle)\rangle f 1 = ⟨ 1 , 1 , 11 , cell addition ( 11 , 11 , ⟨ 1 , 1 , 11 , spine ⟩)⟩
We know that, say, ( σ 3 σ 3 ‾ , 11 ‾ ) ∈ R g r i d (\overline{\sigma_3\sigma_3},\overline{11})\in R_{\mathrm{grid}} ( σ 3 σ 3 , 11 ) ∈ R grid f 0 = ⟨ 1 ‾ , 1 , σ 3 σ 3 ‾ , κ 0 ⟩ f_0=\langle\overline{1},1,\overline{\sigma_3\sigma_3},\kappa_0\rangle f 0 = ⟨ 1 , 1 , σ 3 σ 3 , κ 0 ⟩
Fact 7.28. Given ( x , y ) ∈ R g r i d (x,y)\in R_{\mathrm{grid}} ( x , y ) ∈ R grid f 1 = ⟨ a , b , c 1 y c 2 , κ 1 ⟩ f_1=\langle a,b,c_1yc_2,\kappa_1\rangle f 1 = ⟨ a , b , c 1 y c 2 , κ 1 ⟩ f 0 = ⟨ a , b , c 1 z c 2 , κ 0 ⟩ f_0=\langle a,b,c_1zc_2,\kappa_0\rangle f 0 = ⟨ a , b , c 1 z c 2 , κ 0 ⟩ z ∈ B W 1 z\in BW^1 z ∈ B W 1 ( z , y ) ∈ R g r i d (z,y)\in R_{\mathrm{grid}} ( z , y ) ∈ R grid ℓ F ( f 1 ) = ℓ F ( f 0 ) + ℓ R g r i d ( z , y ) \ell_F(f_1)=\ell_F(f_0)+\ell_{R_{\mathrm{grid}}}(z,y) ℓ F ( f 1 ) = ℓ F ( f 0 ) + ℓ R grid ( z , y )
Due to the lack of a unique destructor, we are unable to prove that the frame of a frontier-style partial grid suffices to determine its size.
Conjecture 7.29. Given frontier-style partial grids f 1 = ⟨ a , b , c , κ 1 ⟩ f_1 = \langle a, b, c, \kappa_1 \rangle f 1 = ⟨ a , b , c , κ 1 ⟩ f 2 = ⟨ a , b , c , κ 2 ⟩ f_2 = \langle a, b, c, \kappa_2 \rangle f 2 = ⟨ a , b , c , κ 2 ⟩ ℓ F ( f 1 ) = ℓ F ( f 2 ) \ell_F(f_1) = \ell_F(f_2) ℓ F ( f 1 ) = ℓ F ( f 2 )
Again, it is instructive to see where the straightforward proof attempt fails. Say we proceed by induction on f 1 f_1 f 1 c = c 1 ⋅ 11 ‾ ⋅ c 2 c = c_1 \cdot\overline{1 1} \cdot c_2 c = c 1 ⋅ 11 ⋅ c 2 f 0 = ⟨ a , b , c 1 ⋅ σ 3 σ 3 ‾ ⋅ c 2 ⟩ f_0 = \langle a, b, c_1 \cdot\overline{\sigma_3 \sigma_3} \cdot c_2 \rangle f 0 = ⟨ a , b , c 1 ⋅ σ 3 σ 3 ⋅ c 2 ⟩ f 1 = ⟨ a , b , c 1 ⋅ 11 ‾ ⋅ c 2 , cell appending ( σ 3 σ 3 ‾ , 11 ‾ , f 0 ) ⟩ f_1 = \langle a, b, c_1 \cdot\overline{1 1} \cdot c_2, \operatorname{cell\ appending}(\overline{\sigma_3 \sigma_3}, \overline{1 1}, f_0) \rangle f 1 = ⟨ a , b , c 1 ⋅ 11 ⋅ c 2 , cell appending ( σ 3 σ 3 , 11 , f 0 )⟩ f 2 f_2 f 2 f 3 = ⟨ a , b , c 1 ⋅ σ 42 σ 42 ‾ ⋅ c 2 , κ 3 ⟩ f_3 = \langle a, b, c_1 \cdot\overline{\sigma_{42}\sigma_{42}} \cdot c_2, \kappa_3 \rangle f 3 = ⟨ a , b , c 1 ⋅ σ 42 σ 42 ⋅ c 2 , κ 3 ⟩ f 2 = ⟨ a , b , c 1 ⋅ 11 ‾ , cell appending ( σ 42 σ 42 ‾ , 11 ‾ , f 3 ) ⟩ f_2 = \langle a, b, c_1 \cdot\overline{1 1}, \operatorname{cell\ appending}(\overline{\sigma_{42}\sigma_{42}}, \overline{1 1}, f_3) \rangle f 2 = ⟨ a , b , c 1 ⋅ 11 , cell appending ( σ 42 σ 42 , 11 , f 3 )⟩
So, frontier-style partial grids suffer the same shortcoming as partial grids — we have not been able to prove that their frame determines their size. Do they improve on our original definition of partial grids in any substantial way? In some ways, yes. For example, the add_cell function would be nearly trivial to implement for frontier-style partial grids, as compared to the seventeen involved cases for our original definition. In other ways, certainly not. I could not find a proof using frontier-style partial grids for Theorem 6.37 (if a partial grid’s frontier is in negative-positive form, then it is in fact a grid). Thus, we would need to keep both structures and transfer theorems between them. The work of setting up a (size-preserving) bijection between G P G_P G P G F G_F G F add_cell directly on G P G_P G P
The enterprising reader is most welcome to formulate his own definition for partial grids. Here are three theorems such a definition must satisfy:
Theorem 6.37: if a partial grid’s frontier is in negative-positive form, then it is in fact a grid.
Definition 7.13 : there exists a function with the same specification as add_cell.
Theorem 7.23 : if a b ‾ \overline{ab} ab c c c ( a , b ) (a,b) ( a , b ) c c c
The ideal definition would also satisfy Conjecture 7.24; that is, the frame of the partial grid determines its size.
This is not the only unwieldy issue with the partial grids we have defined. We do not have a rigorous definition of a sub-partial-grid, which could provide a method of strong induction on partial grids. Surprisingly, there is not a single proof in this project requiring strong induction on partial grids. Nevertheless, one might find it easier to use than structural induction in some cases, or perhaps give a new theorem, or a proof of a conjecture, requiring strong induction.
Here are a few simple ideas for inducting on a partial grid ⟨ a , b , c , m , d ⟩ \langle a,b,c,m,d\rangle ⟨ a , b , c , m , d ⟩
σ i → σ j σ i σ i ↓ ↓ σ j σ j → σ i σ j \begin{CD}
\sigma_i @>{\sigma_j}>> \sigma_i \\
@V{\sigma_i}VV @VV{\sigma_j}V \\
\sigma_j @>{\sigma_i}>> \sigma_j
\end{CD} σ i σ i ↓ ⏐ σ j σ j σ i σ i ↓ ⏐ σ j σ j this metric often increases when we split a partial grid. The issue persists even if we ignored the empty cells by using ∣ s 1 ( a ) ∣ + ∣ s 1 ( b ) ∣ |s^1(a)| + |s^1(b)| ∣ s 1 ( a ) ∣ + ∣ s 1 ( b ) ∣
We might try to induct on the diagonal length of the partial grid, ∣ a ∣ + ∣ c ∣ |a| + |c| ∣ a ∣ + ∣ c ∣ ∣ b ∣ + ∣ d ∣ |b| + |d| ∣ b ∣ + ∣ d ∣
We might try to induct on the size of the partial grid using ℓ P \ell_P ℓ P
The last option could be amended to be workable: we could change our metric to ℓ P ′ \ell'_P ℓ P ′
Definition 7.30. Define an alternative metric on partial grids ℓ P ′ \ell'_P ℓ P ′ ℓ P \ell_P ℓ P
ℓ P ( ⟨ a , b , c , m , d , trivial cell ⟩ ) = 1 \ell_P(\langle a,b,c,m,d,\text{trivial cell}\rangle)=1 ℓ P (⟨ a , b , c , m , d , trivial cell ⟩) = 1
This would almost work for strong induction, except trimming off tails (such as arises in the second case of splittability) does not decrease ℓ P ′ \ell'_P ℓ P ′ ⟨ ℓ P ′ , spine length ⟩ \langle\ell'_P,\text{spine length}\rangle ⟨ ℓ P ′ , spine length ⟩ ℓ P ′ \ell'_P ℓ P ′ ℓ S r e v \ell_{S_{\mathrm{rev}}} ℓ S rev
So, even the simplest methods for defining strong induction are reasonably complex (in order to derive strong induction from the just-mentioned lexicographic order, one would have to consider a grid g g g g 1 g_1 g 1 g 1 g_1 g 1 ℓ P ′ \ell'_P ℓ P ′
In sum, we have defined metrics on our main structures (Semi-Thue derivations, grids, and partial grids). We have been able to show a number of properties of each metric, properties which will let us prove termination of the subword reversing algorithm in the following chapter.
The Reversing Procedure
We are now prepared to implement Dehornoy’s subword reversing algorithm, which determines whether a braid group element is equivalent to the identity element. (Of course, to check equivalence between elements a a a b b b a − 1 b a^{-1}b a − 1 b B W BW B W R r e v R_{\mathrm{rev}} R rev
Monoid Solver
We will first implement a solver for the braid monoid word problem. Much of the machinery will be expanded and re-used for the braid group word problem solver. We begin by defining simple reversing, in which we iteratively apply reversing rewrites to a ‾ b \overline{a}b a b a , b ∈ B W + a,b\in BW^+ a , b ∈ B W + + + +
To aid in exposition, we give terminology to the first element of any R r e v R_{\mathrm{rev}} R rev
Definition 8.1. For any i , j ∈ N i,j\in\mathbb{N} i , j ∈ N σ i ‾ σ j \overline{\sigma_i}\sigma_j σ i σ j open pair .
As a first attempt, we might define said iterative rewriting rev 1 + : B W → B W \operatorname{rev}_{1}^{+}: BW \to BW rev 1 + : B W → B W c 1 , c 2 ∈ B W c_1,c_2 \in BW c 1 , c 2 ∈ B W i , j ∈ N i,j \in\mathbb{N} i , j ∈ N c = c 1 ⋅ σ ˉ i σ j ⋅ c 2 c=c_1\cdot\bar{\sigma}_i\sigma_j\cdot c_2 c = c 1 ⋅ σ ˉ i σ j ⋅ c 2 c 1 c_1 c 1
rev 1 + ( c ) = rev 1 + ( c 1 ⋅ σ ˉ i σ j ⋅ c 2 ) : = { rev 1 + ( c 1 ⋅ c 2 ) i = j , rev 1 + ( c 1 ⋅ σ j σ ˉ i ⋅ c 2 ) ∣ i − j ∣ ≥ 2 , rev 1 + ( c 1 ⋅ σ j σ i σ ˉ j σ ˉ i ⋅ c 2 ) ∣ i − j ∣ = 1. \operatorname{rev}_{1}^{+}(c)=\operatorname{rev}_{1}^{+}(c_1\cdot\bar{\sigma}_i\sigma_j\cdot c_2):=
\begin{cases}
\operatorname{rev}_{1}^{+}(c_1\cdot c_2) & i=j,\\
\operatorname{rev}_{1}^{+}(c_1\cdot\sigma_j\bar{\sigma}_i\cdot c_2) & |i-j|\ge2,\\
\operatorname{rev}_{1}^{+}(c_1\cdot\sigma_j\sigma_i\bar{\sigma}_j\bar{\sigma}_i\cdot c_2) & |i-j|=1.
\end{cases} rev 1 + ( c ) = rev 1 + ( c 1 ⋅ σ ˉ i σ j ⋅ c 2 ) := ⎩ ⎨ ⎧ rev 1 + ( c 1 ⋅ c 2 ) rev 1 + ( c 1 ⋅ σ j σ ˉ i ⋅ c 2 ) rev 1 + ( c 1 ⋅ σ j σ i σ ˉ j σ ˉ i ⋅ c 2 ) i = j , ∣ i − j ∣ ≥ 2 , ∣ i − j ∣ = 1. Otherwise,
rev 1 + ( c ) = c . \operatorname{rev}_{1}^{+}(c)=c. rev 1 + ( c ) = c . The immediate problem is to show well-definedness of this function – that is, that the recursive calls in rev 1 + \operatorname{rev}_{1}^{+} rev 1 + 29 a ˉ b ˉ \bar{a}\bar{b} a ˉ b ˉ a ˉ b ˉ \bar{a}\bar{b} a ˉ b ˉ R r e v R_{\mathrm{rev}} R rev rev 1 + \operatorname{rev}_{1}^{+} rev 1 +
a ˉ b ˉ → R r e v rev 1 + ( a ˉ b ˉ ) . \bar{a}\bar{b}\mathrel{\xrightarrow{R_{\mathrm{rev}}}}\operatorname{rev}_{1}^{+}(\bar{a}\bar{b}). a ˉ b ˉ R rev rev 1 + ( a ˉ b ˉ ) . That is immediately circular – how can we speak about the output of a function before we know it terminates? Instead, we build this fact into the definition of the function by restricting the domain and codomain of rev + \operatorname{rev}^{+} rev +
Definition 8.2. Define the sets T T T a , b ∈ B W + a,b\in BW^{+} a , b ∈ B W + 30
T a , b = { c ∈ B W ∣ i ˉ + ( a ) i ˉ + ( b ) → R r e v c } . T_{a,b}=\{c\in BW\mid\bar{i}^{+}(a)\bar{i}^{+}(b)\mathrel{\xrightarrow{R_{\mathrm{rev}}}}c\}. T a , b = { c ∈ B W ∣ i ˉ + ( a ) i ˉ + ( b ) R rev c } . Here we are briefly explicit about i + i^{+} i + → R r e v \mathrel{\xrightarrow{R_{\mathrm{rev}}}} R rev T a , b T_{a,b} T a , b ( x , y ) (x,y) ( x , y ) x ∈ B W x\in BW x ∈ B W y : = ⟨ i ˉ + ( a ) i ˉ + ( b ) → R r e v x , κ ⟩ y:=\langle\bar{i}^{+}(a)\bar{i}^{+}(b)\mathrel{\xrightarrow{R_{\mathrm{rev}}}}x,\kappa\rangle y := ⟨ i ˉ + ( a ) i ˉ + ( b ) R rev x , κ ⟩
Then we inductively define the family of functions
rev a , b + : T a , b → T a , b \operatorname{rev}_{a,b}^{+}:T_{a,b}\to T_{a,b} rev a , b + : T a , b → T a , b as follows; note that the only differences from the first definition are the name of the function and the domain and range (and hence the added derivation).
Definition 8.3. Given a , b ∈ B W + a,b\in BW^{+} a , b ∈ B W + rev a , b + : T a , b → T a , b \operatorname{rev}_{a,b}^{+}:T_{a,b}\to T_{a,b} rev a , b + : T a , b → T a , b
(Recursive case) If there exist x 1 , x 2 ∈ B W x_1,x_2\in BW x 1 , x 2 ∈ B W i , j ∈ N i,j\in\mathbb{N} i , j ∈ N x = x 1 ⋅ σ ˉ i σ j ⋅ x 2 x=x_1\cdot\bar{\sigma}_i\sigma_j\cdot x_2 x = x 1 ⋅ σ ˉ i σ j ⋅ x 2 x i x_i x i
rev a , b + ( x 1 ⋅ σ i ‾ σ j ⋅ x 2 , y ) = { rev a , b + ( x 1 ⋅ x 2 , ⟨ i + ( a ) ‾ i + ( b ) → R r e v x 1 ⋅ x 2 , step transitivity ( y , σ i ‾ σ j , ε ) ⟩ ) , i = j , rev a , b + ( x 1 ⋅ σ j σ i σ j ‾ σ i ‾ ⋅ x 2 , ⟨ i + ( a ) ‾ i + ( b ) → R r e v x 1 ⋅ σ j σ i σ j ‾ σ i ‾ ⋅ x 2 , step transitivity ( y , σ i ‾ σ j , σ j σ i σ j ‾ σ i ‾ ) ⟩ ) , ∣ i − j ∣ = 1 , rev a , b + ( x 1 ⋅ σ j σ i ‾ ⋅ x 2 , ⟨ i + ( a ) ‾ i + ( b ) → R r e v x 1 ⋅ σ j σ i ‾ ⋅ x 2 , step transitivity ( y , σ i ‾ σ j , σ j σ i ‾ ) ⟩ ) , ∣ i − j ∣ > 1. \operatorname{rev}^{+}_{a,b}(x_1\cdot\overline{\sigma_i}\sigma_j\cdot x_2,y)=\begin{cases}
\operatorname{rev}^{+}_{a,b}\left(x_1\cdot x_2,\left\langle\overline{i^{+}(a)}i^{+}(b)\mathrel{\to}_{R_{\mathrm{rev}}}x_1\cdot x_2,\operatorname{step\ transitivity}\left(y,\overline{\sigma_i}\sigma_j,\varepsilon\right)\right\rangle\right), & i=j,\\
\operatorname{rev}^{+}_{a,b}\left(x_1\cdot\sigma_j\sigma_i\overline{\sigma_j}\overline{\sigma_i}\cdot x_2,\left\langle\overline{i^{+}(a)}i^{+}(b)\mathrel{\to}_{R_{\mathrm{rev}}}x_1\cdot\sigma_j\sigma_i\overline{\sigma_j}\overline{\sigma_i}\cdot x_2,\operatorname{step\ transitivity}\left(y,\overline{\sigma_i}\sigma_j,\sigma_j\sigma_i\overline{\sigma_j}\overline{\sigma_i}\right)\right\rangle\right), & \lvert i-j\rvert=1,\\
\operatorname{rev}^{+}_{a,b}\left(x_1\cdot\sigma_j\overline{\sigma_i}\cdot x_2,\left\langle\overline{i^{+}(a)}i^{+}(b)\mathrel{\to}_{R_{\mathrm{rev}}}x_1\cdot\sigma_j\overline{\sigma_i}\cdot x_2,\operatorname{step\ transitivity}\left(y,\overline{\sigma_i}\sigma_j,\sigma_j\overline{\sigma_i}\right)\right\rangle\right), & \lvert i-j\rvert>1.
\end{cases} rev a , b + ( x 1 ⋅ σ i σ j ⋅ x 2 , y ) = ⎩ ⎨ ⎧ rev a , b + ( x 1 ⋅ x 2 , ⟨ i + ( a ) i + ( b ) → R rev x 1 ⋅ x 2 , step transitivity ( y , σ i σ j , ε ) ⟩ ) , rev a , b + ( x 1 ⋅ σ j σ i σ j σ i ⋅ x 2 , ⟨ i + ( a ) i + ( b ) → R rev x 1 ⋅ σ j σ i σ j σ i ⋅ x 2 , step transitivity ( y , σ i σ j , σ j σ i σ j σ i ) ⟩ ) , rev a , b + ( x 1 ⋅ σ j σ i ⋅ x 2 , ⟨ i + ( a ) i + ( b ) → R rev x 1 ⋅ σ j σ i ⋅ x 2 , step transitivity ( y , σ i σ j , σ j σ i ) ⟩ ) , i = j , ∣ i − j ∣ = 1 , ∣ i − j ∣ > 1. (Non-recursive case) Otherwise,
rev a , b + ( x , y ) = ( x , y ) . \operatorname{rev}^{+}_{a,b}(x,y)=(x,y). rev a , b + ( x , y ) = ( x , y ) . Verifying the domain and range for the different cases in the definition of rev a , b + \operatorname{rev}^{+}_{a,b} rev a , b + FindOpenPair, to find the first occurrence of a substring of the form σ i ‾ σ j \overline{\sigma_i}\sigma_j σ i σ j List.Infix infrastructure in Mathlib as it is non-data-carrying. We thus develop a small repository of data-carrying proofs about basic List properties.
Next, we must show termination of rev a , b + \operatorname{rev}^{+}_{a,b} rev a , b + T a , b T_{a,b} T a , b
Definition 8.4. We define the family of functions ℓ T a , b : T a , b → N \ell_{T_{a,b}}:T_{a,b}\to\mathbb{N} ℓ T a , b : T a , b → N ℓ T \ell_T ℓ T a a a b b b ( x , y ) ∈ T a , b (x,y)\in T_{a,b} ( x , y ) ∈ T a , b x ∈ B W x\in BW x ∈ B W y = x → R r e v i + ( a ) ‾ i + ( b ) y=x\mathrel{\to}_{R_{\mathrm{rev}}}\overline{i^{+}(a)}i^{+}(b) y = x → R rev i + ( a ) i + ( b )
ℓ T ( x , y ) : = ℓ B ( a , b ) − ℓ S ( y ) \ell_T(x,y):=\ell_B(a,b)-\ell_S(y) ℓ T ( x , y ) := ℓ B ( a , b ) − ℓ S ( y ) Theorem 8.5. For any ( x , y ) ∈ T a , b (x,y)\in T_{a,b} ( x , y ) ∈ T a , b ℓ S ( y ) ≤ ℓ B ( a , b ) \ell_S(y)\leq\ell_B(a,b) ℓ S ( y ) ≤ ℓ B ( a , b )
Proof. Consider first the case for nonempty a , b a,b a , b y = i + ( a ) ‾ i + ( b ) → R r e v x y=\overline{i^{+}(a)}i^{+}(b)\mathrel{\to}_{R_{\mathrm{rev}}}x y = i + ( a ) i + ( b ) → R rev x Theorem 6.35 we know there is a partial grid p = ⟨ a , b , c , m , d , κ ⟩ p=\langle a,b,c,m,d,\kappa\rangle p = ⟨ a , b , c , m , d , κ ⟩ x = s 1 ( c m d ) x=s^1(cmd) x = s 1 ( c m d ) ℓ R ( p ) = ℓ S ( y ) \ell_R(p)=\ell_S(y) ℓ R ( p ) = ℓ S ( y ) ℓ P ( p ) ≤ ℓ B ( a , b ) \ell_P(p)\leq\ell_B(a,b) ℓ P ( p ) ≤ ℓ B ( a , b ) ℓ B \ell_B ℓ B Theorem 7.14 . Hence, ℓ S ( y ) ≤ ℓ B ( a , b ) \ell_S(y)\leq\ell_B(a,b) ℓ S ( y ) ≤ ℓ B ( a , b )
When either of a a a b b b i + ( a ) ‾ i + ( b ) \overline{i^{+}(a)}i^{+}(b) i + ( a ) i + ( b ) ℓ S ( y ) = 0 \ell_S(y)=0 ℓ S ( y ) = 0
Theorem 8.6. The metric ℓ T \ell_T ℓ T rev a , b + \operatorname{rev}^{+}_{a,b} rev a , b +
Proof. By Theorem 8.5 , it suffices to show that ℓ S ( y ) \ell_S(y) ℓ S ( y ) ℓ S ( y ) \ell_S(y) ℓ S ( y ) ℓ R r e v \ell_{R_{\mathrm{rev}}} ℓ R rev ℓ S ( y ) \ell_S(y) ℓ S ( y ) ℓ B ( a , b ) − ℓ S ( y ) \ell_B(a,b)-\ell_S(y) ℓ B ( a , b ) − ℓ S ( y )
As ℓ T \ell_T ℓ T rev + \operatorname{rev}^{+} rev +
Definition 8.7. We define simple reversing of a monoid pair r + : B W + → B W + → B W r^{+}: BW^{+} \to BW^{+} \to BW r + : B W + → B W + → B W
r + ( a , b ) = rev a , b + ( a b ‾ , ⟨ a b ‾ → R r e v a b ‾ , r e f l ⟩ ) r^{+}(a,b)=\operatorname{rev}^{+}_{a,b}\left(\overline{ab},\left\langle\overline{ab}\mathrel{\to_{R_{\mathrm{rev}}}}\overline{ab},\mathrm{refl}\right\rangle\right) r + ( a , b ) = rev a , b + ( ab , ⟨ ab → R rev ab , refl ⟩ ) We define a monoid solver solve + : ( B W + ) 2 → B o o l \operatorname{solve}^{+}: (BW^{+})^{2} \to\mathrm{Bool} solve + : ( B W + ) 2 → Bool solve + ( a , b ) = t r u e \operatorname{solve}^{+}(a,b)=\mathrm{true} solve + ( a , b ) = true r + ( a , b ) = ε r^{+}(a,b)=\varepsilon r + ( a , b ) = ε r + r^{+} r + solve + \operatorname{solve}^{+} solve +
Lemma 8.8. a b ‾ ⟶ R r e v ∗ r + ( a , b ) \overline{ab}\mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{rev}}}}r^{+}(a,b) ab R rev ⟶ ∗ r + ( a , b )
We next show that the output of r + ( a , b ) r^{+}(a,b) r + ( a , b )
Lemma 8.9. For a , b ∈ B W + a,b\in BW^{+} a , b ∈ B W + r + ( a , b ) r^{+}(a,b) r + ( a , b ) σ i ‾ σ j \overline{\sigma_i}\sigma_j σ i σ j
Proof. We show that for any x ∈ B W x\in BW x ∈ B W y = ⟨ a b ‾ → R r e v x , κ ⟩ y=\langle\overline{ab}\to_{R_{\mathrm{rev}}}x,\kappa\rangle y = ⟨ ab → R rev x , κ ⟩ rev a , b + ( x , y ) \operatorname{rev}^{+}_{a,b}(x,y) rev a , b + ( x , y ) ( x , y ) ∈ T a , b (x,y)\in T_{a,b} ( x , y ) ∈ T a , b ℓ T a , b \ell_{T_{a,b}} ℓ T a , b rev a , b + \operatorname{rev}^{+}_{a,b} rev a , b + x = x 1 ⋅ σ i ‾ σ j ⋅ x 2 x=x_1\cdot\overline{\sigma_i}\sigma_j\cdot x_2 x = x 1 ⋅ σ i σ j ⋅ x 2 ∣ i − j ∣ > 1 |i-j|>1 ∣ i − j ∣ > 1 rev a , b + ( x , y ) = rev a , b + ( x 1 ⋅ σ j σ i ‾ ⋅ x 2 , ⟨ i + ( a ) i + ( b ) ‾ → R r e v x 1 ⋅ σ j σ i ‾ ⋅ x 2 , step trans ( y , σ i ‾ σ j , σ j σ i ‾ ) ⟩ ) \operatorname{rev}^{+}_{a,b}(x,y)=\operatorname{rev}^{+}_{a,b}\left(x_1\cdot\sigma_j\overline{\sigma_i}\cdot x_2,\left\langle\overline{i^{+}(a)i^{+}(b)}\to_{R_{\mathrm{rev}}}x_1\cdot\sigma_j\overline{\sigma_i}\cdot x_2,\operatorname{step\ trans}(y,\overline{\sigma_i}\sigma_j,\sigma_j\overline{\sigma_i})\right\rangle\right) rev a , b + ( x , y ) = rev a , b + ( x 1 ⋅ σ j σ i ⋅ x 2 , ⟨ i + ( a ) i + ( b ) → R rev x 1 ⋅ σ j σ i ⋅ x 2 , step trans ( y , σ i σ j , σ j σ i ) ⟩ )
Note that ℓ T a , b ( x , y ) = 1 + ℓ T a , b ( ( x 1 ⋅ σ j σ i ‾ ⋅ x 2 , ⟨ i + ( a ) i + ( b ) ‾ → R r e v x 1 ⋅ σ j σ i ‾ ⋅ x 2 , step trans ( y , σ i ‾ σ j , σ j σ i ‾ ) ⟩ ) ) \ell_{T_{a,b}}(x,y)=1+\ell_{T_{a,b}}\left(\left(x_1\cdot\sigma_j\overline{\sigma_i}\cdot x_2,\left\langle\overline{i^{+}(a)i^{+}(b)}\to_{R_{\mathrm{rev}}}x_1\cdot\sigma_j\overline{\sigma_i}\cdot x_2,\operatorname{step\ trans}(y,\overline{\sigma_i}\sigma_j,\sigma_j\overline{\sigma_i})\right\rangle\right)\right) ℓ T a , b ( x , y ) = 1 + ℓ T a , b ( ( x 1 ⋅ σ j σ i ⋅ x 2 , ⟨ i + ( a ) i + ( b ) → R rev x 1 ⋅ σ j σ i ⋅ x 2 , step trans ( y , σ i σ j , σ j σ i ) ⟩ ) ) rev a , b + ( x 1 ⋅ σ j σ i ‾ ⋅ x 2 , ⟨ i + ( a ) i + ( b ) ‾ → R r e v x 1 ⋅ σ j σ i ‾ ⋅ x 2 , step trans ( y , σ i ‾ σ j , σ j σ i ‾ ) ⟩ ) \operatorname{rev}^{+}_{a,b}\left(x_1\cdot\sigma_j\overline{\sigma_i}\cdot x_2,\left\langle\overline{i^{+}(a)i^{+}(b)}\to_{R_{\mathrm{rev}}}x_1\cdot\sigma_j\overline{\sigma_i}\cdot x_2,\operatorname{step\ trans}(y,\overline{\sigma_i}\sigma_j,\sigma_j\overline{\sigma_i})\right\rangle\right) rev a , b + ( x 1 ⋅ σ j σ i ⋅ x 2 , ⟨ i + ( a ) i + ( b ) → R rev x 1 ⋅ σ j σ i ⋅ x 2 , step trans ( y , σ i σ j , σ j σ i ) ⟩ ) rev a , b + ( x , y ) \operatorname{rev}^{+}_{a,b}(x,y) rev a , b + ( x , y ) r + ( a , b ) r^{+}(a,b) r + ( a , b )
Lemma 8.10. If a word w ∈ B W w\in BW w ∈ B W c , d ∈ B W + c,d\in BW^{+} c , d ∈ B W + w = c ‾ d w=\overline{c}d w = c d w w w
Finally, we put together the two previous lemmas to obtain our desired theorem:
Theorem 8.11. For any a , b ∈ B W + a,b\in BW^{+} a , b ∈ B W + c , d ∈ B W + c,d\in BW^{+} c , d ∈ B W + r + ( a , b ) = c ‾ d r^{+}(a,b)=\overline{c}d r + ( a , b ) = c d
We can now prove a correctness theorem for our monoid solver:
Theorem 8.12. solve + ( a , b ) = t r u e \operatorname{solve}^{+}(a,b)=\mathrm{true} solve + ( a , b ) = true a ≡ + b a\equiv^{+}b a ≡ + b
Proof. Forward direction: Assume solve + ( a , b ) = t r u e \operatorname{solve}^{+}(a,b)=\mathrm{true} solve + ( a , b ) = true r + ( a , b ) = ε r^{+}(a,b)=\varepsilon r + ( a , b ) = ε Lemma 8.8 , a b ‾ ⟶ R r e v ∗ ε \overline{ab}\mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{rev}}}}\varepsilon ab R rev ⟶ ∗ ε a b ‾ ≡ ε \overline{ab}\equiv\varepsilon ab ≡ ε a ≡ b a\equiv b a ≡ b a ≡ + b a\equiv^{+}b a ≡ + b
Backward direction: Assume a ≡ + b a\equiv^{+}b a ≡ + b Theorem 5.13 , there is a grid with source ( a , b ) (a,b) ( a , b ) ( ε , ε ) (\varepsilon,\varepsilon) ( ε , ε ) Theorem 8.11 we know that for some c , d ∈ B W + c,d\in BW^{+} c , d ∈ B W + r + ( a , b ) = c ‾ d r^{+}(a,b)=\overline{c}d r + ( a , b ) = c d Lemma 8.8 , a b ‾ ⟶ R r e v ∗ c ‾ d \overline{ab}\mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{rev}}}}\overline{c}d ab R rev ⟶ ∗ c d Theorem 6.3 , there is a grid with source ( a , b ) (a,b) ( a , b ) ( c , d ) (c,d) ( c , d ) Theorem 5.9 ), c = ε c=\varepsilon c = ε d = ε d=\varepsilon d = ε r + ( a , b ) = ε r^{+}(a,b)=\varepsilon r + ( a , b ) = ε solve + ( a , b ) = t r u e \operatorname{solve}^{+}(a,b)=\mathrm{true} solve + ( a , b ) = true
In fact, for the forward direction, there is a method which does not use Ore’s theorem, but rather works through grids. Since a b ‾ → R r e v ∗ ε ( = ε ε ) \overline{ab} \xrightarrow[\ R_{\mathrm{rev}}]{*} \varepsilon(=\varepsilon\varepsilon) ab ∗ R rev ε ( = εε ) Theorem 6.3 , there is a grid from ( a , b ) (a,b) ( a , b ) ( ε , ε ) (\varepsilon,\varepsilon) ( ε , ε ) Theorem 5.4 a ≡ + b a \equiv_{+} b a ≡ + b
We also have an alternative proof for the backward direction, which uses a sort of restricted confluence for Semi-Thue systems over R r e v R_{\mathrm{rev}} R rev 1 1 1 1 ‾ \overline{1} 1
Theorem 8.13. Let g = ⟨ a , b , c , d ⟩ ∈ G g=\langle a,b,c,d\rangle\in G g = ⟨ a , b , c , d ⟩ ∈ G p = ⟨ a 0 , b 0 , c 0 , m 0 , d 0 ⟩ ∈ G P p=\langle a_0,b_0,c_0,m_0,d_0\rangle\in G_P p = ⟨ a 0 , b 0 , c 0 , m 0 , d 0 ⟩ ∈ G P s 1 s^1 s 1 1 1 1 1 ‾ \overline{1} 1 a = s 1 , + ( a 0 ‾ ) a=s^{1,+}(\overline{a_0}) a = s 1 , + ( a 0 ) b = s 1 , + ( b 0 ) b=s^{1,+}(b_0) b = s 1 , + ( b 0 ) c 1 , d 1 ∈ B W + c_1,d_1\in BW^+ c 1 , d 1 ∈ B W + c = s 1 , + ( c 0 ) ⋅ c 1 c=s^{1,+}(c_0)\cdot c_1 c = s 1 , + ( c 0 ) ⋅ c 1 d = s 1 , + ( d 0 ‾ ) ⋅ d 1 d=s^{1,+}(\overline{d_0})\cdot d_1 d = s 1 , + ( d 0 ) ⋅ d 1 s 1 ( m 0 ) → R r e v ∗ c 1 ⋅ d 1 ‾ s^1(m_0)\xrightarrow[\ R_{\mathrm{rev}}]{*}c_1\cdot\overline{d_1} s 1 ( m 0 ) ∗ R rev c 1 ⋅ d 1
Here is the image to keep in mind – we claim, roughly, that the purple section rewrites under R r e v R_{\mathrm{rev}} R rev
Figure 1 (PDF p. 68).
The proof follows the strategy of Theorem 7.14 , inducting on the grid and proceeding by extensive casework in each inductive case, splitting the partial grid as appropriate, and using prefix-suffix fact (1) from Theorem 7.14 to help apply the inductive hypothesis.
From the above theorem, we can immediately deduce our second needed theorem, that the middle frontier of a partial grid reverses to the sink of the grid in which it is nested:
Theorem 8.14. Let g = ⟨ a , b , c , d ⟩ ∈ G g=\langle a,b,c,d\rangle\in G g = ⟨ a , b , c , d ⟩ ∈ G p = ⟨ a 0 , b 0 , c 0 , m 0 , d 0 ⟩ ∈ G P p=\langle a_0,b_0,c_0,m_0,d_0\rangle\in G_P p = ⟨ a 0 , b 0 , c 0 , m 0 , d 0 ⟩ ∈ G P a = s 1 , + ( a 0 ‾ ) a=s^{1,+}(\overline{a_0}) a = s 1 , + ( a 0 ) b = s 1 , + ( b 0 ) b=s^{1,+}(b_0) b = s 1 , + ( b 0 ) s 1 ( c 0 m 0 d 0 ) → R r e v ∗ c d ‾ s^1(c_0m_0d_0)\xrightarrow[\ R_{\mathrm{rev}}]{*}\overline{cd} s 1 ( c 0 m 0 d 0 ) ∗ R rev c d
Now we are prepared to state and prove restricted confluence.
Theorem 8.15. Let a , b ∈ B W + a,b\in BW^+ a , b ∈ B W + w 1 , w 2 ∈ B W w_1,w_2\in BW w 1 , w 2 ∈ B W a b ‾ → R r e v ∗ w 1 \overline{ab}\xrightarrow[\ R_{\mathrm{rev}}]{*}w_1 ab ∗ R rev w 1 a b ‾ → R r e v ∗ w 2 \overline{ab}\xrightarrow[\ R_{\mathrm{rev}}]{*}w_2 ab ∗ R rev w 2 w ∈ B W w\in BW w ∈ B W w 1 → R r e v ∗ w w_1\xrightarrow[\ R_{\mathrm{rev}}]{*}w w 1 ∗ R rev w w 2 → R r e v ∗ w w_2\xrightarrow[\ R_{\mathrm{rev}}]{*}w w 2 ∗ R rev w
Proof. First consider the case of nonempty a a a b b b c , d ∈ B W + c,d\in BW^+ c , d ∈ B W + ⟨ a , b , c , d ⟩ ∈ G \langle a,b,c,d\rangle\in G ⟨ a , b , c , d ⟩ ∈ G Theorem 5.16 ). Let w = c d ‾ w=\overline{cd} w = c d a b ‾ → R r e v ∗ w 1 \overline{ab}\xrightarrow[\ R_{\mathrm{rev}}]{*}w_1 ab ∗ R rev w 1 ( a , b ) (a,b) ( a , b ) t 1 t_1 t 1 s 1 = w 1 s^1=w_1 s 1 = w 1 Theorem 8.14 , s 1 ( t 1 ) → R r e v ∗ c d ‾ s^1(t_1)\xrightarrow[\ R_{\mathrm{rev}}]{*}\overline{cd} s 1 ( t 1 ) ∗ R rev c d w 1 → R r e v ∗ c d ‾ w_1\xrightarrow[\ R_{\mathrm{rev}}]{*}\overline{cd} w 1 ∗ R rev c d w 2 → R r e v ∗ c d ‾ w_2\xrightarrow[\ R_{\mathrm{rev}}]{*}\overline{cd} w 2 ∗ R rev c d
In the case where either a a a b b b a b ‾ \overline{ab} ab a b ‾ = w 1 \overline{ab}=w_1 ab = w 1 a b ‾ = w 2 \overline{ab}=w_2 ab = w 2 w 1 w_1 w 1 w w w
Now we can return to our second proof of the reverse direction of the monoid solver’s correctness from Theorem 8.12 . Assume that a ≡ + b a\equiv_{+}b a ≡ + b Theorem 5.13 , there is a grid with source ( a , b ) (a,b) ( a , b ) ( ε , ε ) (\varepsilon,\varepsilon) ( ε , ε ) Theorem 6.3 , a b ‾ → R r e v ∗ ε \overline{ab}\xrightarrow[\ R_{\mathrm{rev}}]{*}\varepsilon ab ∗ R rev ε a b ‾ → R r e v ∗ r + ( a , b ) \overline{ab}\xrightarrow[\ R_{\mathrm{rev}}]{*}r^+(a,b) ab ∗ R rev r + ( a , b ) Theorem 8.15 , there exists some c ∈ B W c\in BW c ∈ B W ε → R r e v ∗ c \varepsilon\xrightarrow[\ R_{\mathrm{rev}}]{*}c ε ∗ R rev c c = ε c=\varepsilon c = ε r + ( a , b ) → R r e v ∗ c r^+(a,b)\xrightarrow[\ R_{\mathrm{rev}}]{*}c r + ( a , b ) ∗ R rev c r + ( a , b ) r^+(a,b) r + ( a , b ) r + ( a , b ) = c = ε r^+(a,b)=c=\varepsilon r + ( a , b ) = c = ε
This second proof is not very different from the first, but substitutes one instance of grids for restricted confluence.
We thus have created both a solver for the braid monoid (solve+ ^{+} + + ^{+} +
Group Solver
We now work to implement a solver for the word problem in the braid group. This again largely relies on a method to iteratively apply reversing rewrites to a braid word, akin to rev+ ^{+} +
Definition 8.16. We define the set of braid words in positive-negative form as B W P N BW^{PN} B W P N w ∈ B W w \in BW w ∈ B W c ‾ d ‾ \overline{c}\overline{d} c d c , d ∈ B W + c,d \in BW^{+} c , d ∈ B W +
Definition 8.17. We define the reversing function rev : B W → B W P N \operatorname{rev}:BW \to BW^{PN} rev : B W → B W P N 31 w ∈ B W w \in BW w ∈ B W u 0 , v 0 ∈ B W + u_0,v_0 \in BW^{+} u 0 , v 0 ∈ B W + w ′ ∈ B W w' \in BW w ′ ∈ B W w = u 0 ‾ v 0 ‾ w ′ w=\overline{u_0}\overline{v_0}w' w = u 0 v 0 w ′ 32 w ′ w' w ′ w w w
Let c , d ∈ B W + c,d \in BW^{+} c , d ∈ B W + r + ( u 0 , v 0 ) = c ‾ d ‾ r^{+}(u_0,v_0)=\overline{c}\overline{d} r + ( u 0 , v 0 ) = c d rev ( w ′ ) \operatorname{rev}(w') rev ( w ′ ) e ‾ f ‾ \overline{e}\overline{f} e f rev ( w ) = c ‾ ⋅ r + ( d ‾ , e ‾ ) ⋅ f ‾ \operatorname{rev}(w)=\overline{c}\cdot r^{+}(\overline{d},\overline{e})\cdot\overline{f} rev ( w ) = c ⋅ r + ( d , e ) ⋅ f
We can show that rev preserves equivalence via induction and appeals to the behavior of r + r^{+} r +
Theorem 8.18. For any w ∈ B W w \in BW w ∈ B W w ⟶ R r e v ∗ rev ( w ) w \mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{rev}}}} \operatorname{rev}(w) w R rev ⟶ ∗ rev ( w )
For our Lean formalization, it is easier (but not strictly necessary) to bundle this fact into the definition of rev. Thus, we define the codomain of rev as follows:
structure ReverseResult (L : List (N × Bool)) where
result : List (N × Bool)
ordered : SignedList.PosNegData result
derivation : SemiThueData reversing L resultThat is, the codomain is the set of braid words (result) in positive-negative order (ordered) such that w ⟶ R r e v ∗ rev ( w ) w \mathrel{\mathop{\longrightarrow}\limits^{*}_{R_{\mathrm{rev}}}} \operatorname{rev}(w) w R rev ⟶ ∗ rev ( w )
With reversing for words in B W BW B W
Definition 8.19. We define a procedure solve : B W 2 → B o o l \operatorname{solve}:BW^{2} \to\mathrm{Bool} solve : B W 2 → Bool w 1 , w 2 ∈ B W w_1,w_2 \in BW w 1 , w 2 ∈ B W rev ( w 1 ‾ w 2 ) = c ‾ d ‾ \operatorname{rev}(\overline{w_1}w_2)=\overline{c}\overline{d} rev ( w 1 w 2 ) = c d solve ( w 1 , w 2 ) = t r u e \operatorname{solve}(w_1,w_2)=\mathrm{true} solve ( w 1 , w 2 ) = true solve + ( d , c ) = t r u e \operatorname{solve}^{+}(d,c)=\mathrm{true} solve + ( d , c ) = true r + ( d , c ) = ε r^{+}(d,c)=\varepsilon r + ( d , c ) = ε
Next we show correctness of this procedure.
Theorem 8.20. For any w 1 , w 2 ∈ B W w_1,w_2 \in BW w 1 , w 2 ∈ B W solve ( w 1 , w 2 ) = t r u e \operatorname{solve}(w_1,w_2)=\mathrm{true} solve ( w 1 , w 2 ) = true w 1 ≡ w 2 w_1 \equiv w_2 w 1 ≡ w 2
Proof. Forward direction : Assume solve ( w 1 , w 2 ) = t r u e \operatorname{solve}(w_1,w_2)=\mathrm{true} solve ( w 1 , w 2 ) = true rev ( w 1 ‾ w 2 ) = c ‾ d ‾ \operatorname{rev}(\overline{w_1}w_2)=\overline{c}\overline{d} rev ( w 1 w 2 ) = c d solve + ( d , c ) = t r u e \operatorname{solve}^{+}(d,c)=\mathrm{true} solve + ( d , c ) = true solve + \operatorname{solve}^{+} solve + c ≡ + d c \equiv^{+} d c ≡ + d c ≡ d c \equiv d c ≡ d c ‾ d ‾ ≡ ε \overline{c}\overline{d} \equiv\varepsilon c d ≡ ε rev ( w 1 w 2 ‾ ) = c d ‾ \operatorname{rev}(\overline{w_1w_2})=\overline{cd} rev ( w 1 w 2 ) = c d Theorems 8.18 and 6.2 that w 1 w 2 ‾ ≡ c d ‾ \overline{w_1w_2}\equiv\overline{cd} w 1 w 2 ≡ c d w 1 w 2 ‾ ≡ ε \overline{w_1w_2}\equiv\varepsilon w 1 w 2 ≡ ε
Reverse direction : Assume w 1 ≡ w 2 w_1\equiv w_2 w 1 ≡ w 2 w 1 w 2 ‾ ≡ 1 \overline{w_1w_2}\equiv1 w 1 w 2 ≡ 1 rev ( w 1 w 2 ‾ ) = c ‾ \operatorname{rev}(\overline{w_1w_2})=\overline{c} rev ( w 1 w 2 ) = c w 1 w 2 ‾ ≡ c ‾ \overline{w_1w_2}\equiv\overline{c} w 1 w 2 ≡ c Theorems 8.18 and 6.2 , and so c ‾ ≡ 1 \overline{c}\equiv1 c ≡ 1 c ≡ d c\equiv d c ≡ d c ≡ + d c\equiv^{+}d c ≡ + d solve + ( d , c ) = true \operatorname{solve}^{+}(d,c)=\text{true} solve + ( d , c ) = true solve ( w 1 , w 2 ) = solve + ( d , c ) \operatorname{solve}(w_1,w_2)=\operatorname{solve}^{+}(d,c) solve ( w 1 , w 2 ) = solve + ( d , c )
Et voila! Thus, we have a terminating and correct algorithm to solve the braid word problem! Given two words in the braid group, we can run braid_group_word_solver. For example,
#eval braid_word_solver [(4, false), (1, true), (2, true), (1, true)]
[(2, true), (1, true), (2, true), (4, false)]returns true, as the two words represent equal elements in the group.
Or given two elements in the group B B B Quotient.exists_rep to build braid_solver. For example, we can run
#eval braid_solver (σi 1 * σi 2 * σi 1) (σi 2 * σi 1 * σi 2 * (σi 3)⁻¹* (σi 1 * σi 3)⁻¹)This comes with the specification that
theorem braid_solver_correct {a b : BraidGroupInf} : braid_solver a b ↔ a = bResults have been instantaneous using #eval on small cases (approximately .33 seconds using the #time command). Comparing two elements of B 8 B_8 B 8 SignedList.PosNegData or SemiThueData are computed, and so slow down the calculation. The main contribution of this project is the mathematical theory, and so we have not spent significant time attempting to speed up code runtime (especially as it would come at the cost of much more work to set up the proofs).
Finite Braid Group Solver
We now return at last to the case of the finite braid groups. Note that we have been able to avoid the work of implementing Ore localization and grids for finite braid groups – we haven’t even shown simple monoid properties like common multiples nor cancellativity.
The solver is incredibly simple : we cast the finite braid group words to infinite braid group words, and run our braid group solver. In this section, unlike § 2.6 , we are careful to keep track of the following casts:
Definition 8.21. Let i n i_n i n B W n BW_n B W n B W BW B W s n s_n s n B W BW B W B W n BW_n B W n B W + BW^{+} B W + B W n + BW_n^{+} B W n + B W BW B W B W n BW_n B W n
Definition 8.22. We define the finite braid group word solver as solve n ( a , b ) : = solve ( i n ( a ) , i n ( b ) ) \operatorname{solve}_n(a,b):=\operatorname{solve}(i_n(a),i_n(b)) solve n ( a , b ) := solve ( i n ( a ) , i n ( b )) a , b ∈ B W n a,b\in BW_n a , b ∈ B W n
Termination is thus immediate. Correctness requires one main lemma about reversing derivations, shown by induction :
Lemma 8.23. Given c ∈ B W c \in BW c ∈ B W c c c n − 1 n-1 n − 1 c → R r e v ∗ d c \xrightarrow{R_{\mathrm{rev}}^{*}} d c R rev ∗ d d d d n − 1 n-1 n − 1 s n ( c ) ≡ n s n ( d ) s_n(c) \equiv_n s_n(d) s n ( c ) ≡ n s n ( d )
Theorem 8.24. Given a , b ∈ B W n a,b \in BW_n a , b ∈ B W n solve n ( a , b ) = t r u e \operatorname{solve}_n(a,b)=\mathrm{true} solve n ( a , b ) = true a ≡ n b a \equiv_n b a ≡ n b
Proof. Forward direction : Assume solve n ( a , b ) = t r u e \operatorname{solve}_n(a,b)=\mathrm{true} solve n ( a , b ) = true solve ( i n ( a ) , i n ( b ) ) = t r u e \operatorname{solve}(i_n(a),i_n(b))=\mathrm{true} solve ( i n ( a ) , i n ( b )) = true rev ( ( i n ( a ) , i n ( b ) ) ‾ ) = c d ‾ \operatorname{rev}(\overline{(i_n(a),i_n(b))})=\overline{cd} rev ( ( i n ( a ) , i n ( b )) ) = c d solve + ( c , d ) = t r u e \operatorname{solve}^{+}(c,d)=\mathrm{true} solve + ( c , d ) = true solve + \operatorname{solve}^{+} solve + c ≡ + d c \equiv^{+} d c ≡ + d rev ( ( i n ( a ) , i n ( b ) ) ‾ ) = c d ‾ \operatorname{rev}(\overline{(i_n(a),i_n(b))})=\overline{cd} rev ( ( i n ( a ) , i n ( b )) ) = c d ( i n ( a ) , i n ( b ) ) ‾ → R r e v ∗ c d ‾ \overline{(i_n(a),i_n(b))} \xrightarrow{R_{\mathrm{rev}}^{*}} \overline{cd} ( i n ( a ) , i n ( b )) R rev ∗ c d
We know that the index of the largest generator in ( i n ( a ) , i n ( b ) ) ‾ \overline{(i_n(a),i_n(b))} ( i n ( a ) , i n ( b )) n − 1 n-1 n − 1 c d ‾ \overline{cd} c d c c c d d d s n ( c ) ≡ n + s n ( d ) s_n(c) \equiv_n^{+} s_n(d) s n ( c ) ≡ n + s n ( d ) c ≡ d c \equiv d c ≡ d s n ( c d ‾ ) ≡ ε s_n(\overline{cd}) \equiv\varepsilon s n ( c d ) ≡ ε ( i n ( a ) , i n ( b ) ) ‾ → R r e v ∗ c d ‾ \overline{(i_n(a),i_n(b))} \xrightarrow{R_{\mathrm{rev}}^{*}} \overline{cd} ( i n ( a ) , i n ( b )) R rev ∗ c d s n ( ( i n ( a ) , i n ( b ) ) ‾ ) ≡ n s n ( c d ‾ ) s_n(\overline{(i_n(a),i_n(b))}) \equiv_n s_n(\overline{cd}) s n ( ( i n ( a ) , i n ( b )) ) ≡ n s n ( c d ) s n ( c d ‾ ) ≡ ε s_n(\overline{cd}) \equiv\varepsilon s n ( c d ) ≡ ε a b ‾ ≡ n ε \overline{ab} \equiv_n \varepsilon ab ≡ n ε a ≡ n b a \equiv_n b a ≡ n b
Reverse direction : Assume a ≡ n b a \equiv_n b a ≡ n b i n ( a ) ≡ i n ( b ) i_n(a) \equiv i_n(b) i n ( a ) ≡ i n ( b ) solve ( i n ( a ) , i n ( b ) ) = t r u e \operatorname{solve}(i_n(a),i_n(b))=\mathrm{true} solve ( i n ( a ) , i n ( b )) = true solve n ( a , b ) = t r u e \operatorname{solve}_n(a,b)=\mathrm{true} solve n ( a , b ) = true
As a consequence of the correctness of this algorithm, we may now prove one elusive fact from the end of § 2.6 :
Theorem 8.25. If a , b ∈ B W a,b \in BW a , b ∈ B W n − 1 n-1 n − 1 a ≡ b a \equiv b a ≡ b s n ( a ) ≡ n s n ( b ) s_n(a) \equiv_n s_n(b) s n ( a ) ≡ n s n ( b )
Proof. Assume a ≡ b a \equiv b a ≡ b solve ( a , b ) = t r u e \operatorname{solve}(a,b)=\mathrm{true} solve ( a , b ) = true a = i n ( s n ( a ) ) a=i_n(s_n(a)) a = i n ( s n ( a )) b b b solve ( i n ( s n ( a ) ) , i n ( s n ( b ) ) ) = t r u e \operatorname{solve}(i_n(s_n(a)),i_n(s_n(b)))=\mathrm{true} solve ( i n ( s n ( a )) , i n ( s n ( b ))) = true solve n \operatorname{solve}_n solve n s n ( a ) ≡ n s n ( b ) s_n(a) \equiv_n s_n(b) s n ( a ) ≡ n s n ( b )
Future Work
The most tempting work to be completed is a proof of Conjecture 7.24 (any two partial grids with the same frame have the same size). On the topic of grids, another improvement would be to generalize their definition to one parametrized by a set of relations. Further, we could classify which relations lead to grid existence and/or uniqueness, following Dehornoy’s many papers which provide criteria such as right complemented presentations, the cube criterion, homogeneity, etc [9 , 11 , 14 , 17 ] . Such work could eventually lead to a generalization of the word problem solvers presented here to Garside groups and others.
There are also other algorithms to solve the word problem which rely on the results developed in this project. For example, Garside’s method of universal denominators (as mentioned in the introduction) could now be implemented fairly easily. We have also completed initial work necessary for more complex, popular, and efficient-in-practice methods such as Dehornoy’s handle reduction. A formalization of this algorithm could also be useful to other projects, such as Hurtig’s generalization of braid group for machine knitting, which uses the normal form from handle reduction as part of his algorithm to determine equivalence in a certain braided monoidal category [23 ] . (There is a plan to complete a Lean formalization of the mathematics in this paper).33 34
A larger goal is to formalize topological definitions of the braid group (for example, one under homotopy equivalence and another under ambient isotopy), and prove that the algebraic definition matches the topological ones. (There is some emerging, unpublished activity from AI theorem provers on a related problem using the configuration-space definition of braid groups.)
Beyond formalization of proofs, I am also working on a Lean widget to visualize braids. Given a formalized algorithm, the widget can show a braid transforming as the algorithm is applied to it (such as building its fraction form). In the reverse order, moving from imagery to Lean code, the strings in the widget will be able to be clicked and dragged in order to generate proofs.
Acknowledgements
I would like to thank Jeremy Avigad for his thoughtful feedback and careful edits throughout the project. I am grateful to Patrick Massot for his advice on the formalization of Ore localization for presented monoids. I thank Jim McCann, Jenny Lin, and Nat Hurtig for their brainstorming contributions concerning the definitions of partial grids, and Bhavik Mehta for his enthusiastic, spirited discussions of alternative definitions of partial grids, which led to the inclusion of § 7.4 . I also thank the Mathlib community (especially Yaël Dillies, Johann Commelin, and numerous other maintainers) for their suggestions and assistance, particularly regarding the definitions of presented monoids and braid groups. Lastly, this project has been greatly bolstered by the congenial atmosphere of the Hoskinson Center for Formal Mathematics built by all the students and visitors who have gathered together there to talk Lean and support one another.
The first version of the codebase was completed without AI assistance. During refinements, Github’s Copilot’s line-completion tool was occasionally used (every output was immediately checked by hand). Claude Opus 5.5 was asked to search for duplicate code, summarize the contents of files, and suggest alternatives for theorem and function names.
The prose of the paper used no AI tools. ChatGPT 5 and 6 assisted in Latex formatting. Claude Opus 5.5 was sometimes used to search the codebase for formalized statements given a Latex statement (it provided the name of the theorem and the file in which it was found; the Github links were copied by me).
This work was supported in part by NSF Grant DMS-2434614 and DARPA ExpMath Grant HR0011262E028.
References
Maps
Here, we summarize the many maps used between braid structures in this project. First, we give a diagram of functions between the various infinite braid structures. In the paper, some are often left implicit, such as i + i^{+} i +
Figure 27.
Table 4.
Note: In Lean, the two inclusions are denoted using SignedList.is_true.
We next consider functions between the finite braid structures, as well as maps to and from infinite braid structures. We define B W b o u n d e d n BW_{\mathrm{bounded}_n} B W bounded n n − 1 n-1 n − 1
Figure 28.
Table 5.
Note: To implement the inclusion maps in Lean, we use SignedList.bounded to define B W b o u n d e d n BW_{\mathrm{bounded}_n} B W bounded n ∀ x ∈ w \forall x \in w ∀ x ∈ w x < n . p r e d x < n.\mathrm{pred} x < n . pred B W b o u n d e d n + BW^+_{\mathrm{bounded}_n} B W bounded n +
Dependency Graph
Here is the dependency graph for the project:
Figure 1 (PDF p. 78).
The repository currently contains about 15,000 lines of code.
