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Lax194892.ChildrenOfChildGraph

A for-transducer reads the children off a child configuration graph

concepts/Lax194892/ChildrenOfChildGraph.lean · lax-194892

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

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    In the paper

    Theorem

    There is a for-transducer which inputs the string representation of a child configuration graph and outputs the concatenation of the string representations of the corresponding child configurations (Claim D.2.7 of Transducers). A two-pebble transducer walks along the path of the graph, one pebble marking the current child and the other printing its representation, and pebble transducers are for-transducers (Theorem D.2.4); the book instead proceeds by induction on the width of the graph, as for snake graphs.

    Lean source view on GitHub

    1import Lax194892.ForTransducers
    2import Lax194892.ChildConfigurationGraphs
    3
    4/-!
    5---
    6title: A for-transducer reads the children off a child configuration graph
    7type: theorem
    8---
    9There is a for-transducer which inputs the string representation of a child
    10configuration graph and outputs the concatenation of the string representations
    11of the corresponding child configurations (Claim D.2.7 of *Transducers*). A
    12two-pebble transducer walks along the path of the graph, one pebble marking the
    13current child and the other printing its representation, and pebble transducers
    14are for-transducers (Theorem D.2.4); the book instead proceeds by induction on
    15the width of the graph, as for snake graphs.
    16
    17# Formalization notes
    18
    19The output is characterised by `CGOutIs`, which determines it; the
    20for-transducer is required to agree with it on every string that represents a
    21run of children. The input alphabet and the state space are assumed finite.
    22-/
    23
    24namespace Lax194892.ChildrenOfChildGraph
    25
    26open Lax194892.ChildConfigurationGraphs Lax194892.ForTransducers
    27
    28/-- A for-transducer maps the representation of a child configuration graph to the
    29concatenation of the representations of the children. -/
    30axiom exists_forTransducer_cgOut {A Q : Type} [Finite A] [Finite Q] (k : ℕ) :
    31 ∃ f : List (CGLetter A Q k) → List (ConfLetter A Q k),
    32 IsForTransducer f ∧ ∀ u v, CGOutIs u v → f u = v
    33
    34end Lax194892.ChildrenOfChildGraph
    35
    Show Proof

    Formalization notes

    The output is characterised by CGOutIsCGOutIs, which determines it; the for-transducer is required to agree with it on every string that represents a run of children. The input alphabet and the state space are assumed finite.

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