Documentation

Mathlib.Data.Sum.Lattice

Lexicographic sum of lattices #

This file proves that we can combine two lattices α and β into a lattice α ⊕ₗ β where everything in α is declared smaller than everything in β.

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instance Sum.Lex.instSemilatticeSup {α : Type u_1} {β : Type u_2} [SemilatticeSup α] [SemilatticeSup β] :
SemilatticeSup (Lex (α ⊕ β))
Equations
  • One or more equations did not get rendered due to their size.
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@[simp]
theorem Sum.Lex.inl_sup {α : Type u_1} {β : Type u_2} [SemilatticeSup α] [SemilatticeSup β] (a₁ a₂ : α) :
Sum.inlₗ (a₁ ⊔ a₂) = Sum.inlₗ a₁ ⊔ Sum.inlₗ a₂
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@[simp]
theorem Sum.Lex.inr_sup {α : Type u_1} {β : Type u_2} [SemilatticeSup α] [SemilatticeSup β] (b₁ b₂ : β) :
Sum.inrₗ (b₁ ⊔ b₂) = Sum.inrₗ b₁ ⊔ Sum.inrₗ b₂
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instance Sum.Lex.instSemilatticeInf {α : Type u_1} {β : Type u_2} [SemilatticeInf α] [SemilatticeInf β] :
SemilatticeInf (Lex (α ⊕ β))
Equations
  • One or more equations did not get rendered due to their size.
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@[simp]
theorem Sum.Lex.inl_inf {α : Type u_1} {β : Type u_2} [SemilatticeInf α] [SemilatticeInf β] (a₁ a₂ : α) :
Sum.inlₗ (a₁ ⊓ a₂) = Sum.inlₗ a₁ ⊓ Sum.inlₗ a₂
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@[simp]
theorem Sum.Lex.inr_inf {α : Type u_1} {β : Type u_2} [SemilatticeInf α] [SemilatticeInf β] (b₁ b₂ : β) :
Sum.inrₗ (b₁ ⊓ b₂) = Sum.inrₗ b₁ ⊓ Sum.inrₗ b₂
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instance Sum.Lex.instLattice {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] :
Lattice (Lex (α ⊕ β))
Equations
  • Sum.Lex.instLattice = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯
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def Sum.Lex.inlLatticeHom {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] :
LatticeHom α (Lex (α ⊕ β))

Sum.Lex.inlₗ as a lattice homomorphism.

Equations
  • Sum.Lex.inlLatticeHom = { toFun := Sum.inlₗ, map_sup' := ⋯, map_inf' := ⋯ }
Instances For
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    def Sum.Lex.inrLatticeHom {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] :
    LatticeHom β (Lex (α ⊕ β))

    Sum.Lex.inrₗ as a lattice homomorphism.

    Equations
    • Sum.Lex.inrLatticeHom = { toFun := Sum.inrₗ, map_sup' := ⋯, map_inf' := ⋯ }
    Instances For
      source
      instance Sum.Lex.instDistribLattice {α : Type u_1} {β : Type u_2} [DistribLattice α] [DistribLattice β] :
      DistribLattice (Lex (α ⊕ β))
      Equations