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feat(SymbolicDynamics): add monotonicity lemma for LanguageOn #39256

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8f2c7a8
Add monotonicity lemma for LanguageOn
Sfgangloff May 12, 2026
723fd79
fix: naming convention isMonotone_LanguageOn
Sfgangloff May 12, 2026
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10 changes: 9 additions & 1 deletion Mathlib/Dynamics/SymbolicDynamics/Basic.lean
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Expand Up @@ -579,7 +579,7 @@ end DefSubshiftByForbidden

section Language

variable {A : Type*} [Fintype A] [Inhabited A]
variable {A : Type*} [Inhabited A]
variable {G : Type*}

/-- Patterns with support exactly `U` form a finite set. -/
Expand Down Expand Up @@ -616,6 +616,14 @@ This is the set of all finite patterns obtained by restricting some configuratio
def LanguageOn (X : Set (G → A)) (U : Finset G) : Set (Pattern A G) :=
{ p | ∃ x ∈ X, Pattern.fromConfig x U = p }

/-- Considering two sets of configurations `X` and `Y` such that `X ⊆ Y`, the language
of `X` on (finite) shape `U` is included in the language of `Y` on shape `U`. -/
lemma isMonotone_LanguageOn {X Y : Set (G → A)} (h : X ⊆ Y) (U : Finset G) :
LanguageOn X U ⊆ LanguageOn Y U := by
intro p hp
rcases hp with ⟨x, hxX, rfl⟩
exact ⟨x, h hxX, rfl⟩

/-- The language of a subshift `Y` on a finite shape `U`. -/
def MulSubshift.languageOn {A G} [TopologicalSpace A] [Inhabited A] [Monoid G]
(Y : MulSubshift A G) (U : Finset G) : Set (Pattern A G) :=
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