33 lines
1.1 KiB
Lean4
33 lines
1.1 KiB
Lean4
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/-
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Port of `Lattice/Unit.agda`.
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The lattice structure itself (`_⊔_`, `_⊓_`, all semilattice/lattice laws) is
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lifted into mathlib: `PUnit.instLinearOrder` provides `Lattice PUnit`.
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What remains is the fixed-height structure: the unit lattice has height 0.
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-/
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import Spa.Lattice
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namespace Spa
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/-- Chains in a subsingleton order are bounded by any `n` (Agda: the `bounded`
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field of `Lattice/Unit.agda`'s `fixedHeight`, generalized). -/
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theorem boundedChains_of_subsingleton (α : Type*) [Preorder α] [Subsingleton α]
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(n : ℕ) : BoundedChains α n := fun c => by
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by_contra hc
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push_neg at hc
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exact (c.step ⟨0, by omega⟩).ne (Subsingleton.elim _ _)
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/-- Agda: `Lattice/Unit.agda`'s `fixedHeight`/`isFiniteHeightLattice`. -/
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instance : FiniteHeightLattice PUnit where
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height := 0
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fixedHeight :=
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{ bot := PUnit.unit
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top := PUnit.unit
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longestChain := RelSeries.singleton _ PUnit.unit
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head_longestChain := rfl
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last_longestChain := rfl
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length_longestChain := rfl
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bounded := boundedChains_of_subsingleton PUnit 0 }
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end Spa
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