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agda-spa/lean/Spa/Lattice/Unit.lean

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Lean migration: Phase 3 (Unit, Prod, AboveBelow lattices) - Spa.Lattice.Unit: PUnit fixed height 0 (lattice lifted from mathlib) - Spa.Lattice.Prod: chain unzip + FixedHeight (h1+h2) on products (componentwise lattice lifted from mathlib's Prod.instLattice) - Spa.Lattice.AboveBelow: flat lattice via Lattice.mk' (mirrors the Agda semilattices+absorption construction), boundedness via rank into Nat, Plain x ↦ plainFixedHeight x, height 2 Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
2026-06-09 18:48:02 -07:00
import Spa.Lattice
Write more documentation Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2026-06-25 19:36:14 -05:00
/-!
# Unit Lattice
This file provides a proof that in addition to being a lattice,
`PUnit` is a `Spa.FiniteHeightLattice`. This is fairly trivial result,
but the unit is used as a placeholder in various contexts (e.g.,
as a base case for the iterated product `Spa/Lattice/IterProd.lean`). -/
Lean migration: Phase 3 (Unit, Prod, AboveBelow lattices) - Spa.Lattice.Unit: PUnit fixed height 0 (lattice lifted from mathlib) - Spa.Lattice.Prod: chain unzip + FixedHeight (h1+h2) on products (componentwise lattice lifted from mathlib's Prod.instLattice) - Spa.Lattice.AboveBelow: flat lattice via Lattice.mk' (mirrors the Agda semilattices+absorption construction), boundedness via rank into Nat, Plain x ↦ plainFixedHeight x, height 2 Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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namespace Spa
Write more documentation Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2026-06-25 19:36:14 -05:00
/-- Since a singleton type's preorder has no nonempty `<` chains,
they are vacuously bounded by any minimum height. -/
Adopt lemma as the default keyword Convert every theorem to lemma (mathlib's default) except the headline results a reader of each module seeks out: analyze_correct (Forward/Sign/Constant), aFix_eq/aFix_le (Fixedpoint), trace (Language), and Stmt.cfg_sufficient (Language/Properties). lemma and theorem are interchangeable keywords, so no references change. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
2026-06-25 13:59:08 -05:00
lemma boundedChains_of_subsingleton (α : Type*) [Preorder α] [Subsingleton α]
Lean migration: Phase 3 (Unit, Prod, AboveBelow lattices) - Spa.Lattice.Unit: PUnit fixed height 0 (lattice lifted from mathlib) - Spa.Lattice.Prod: chain unzip + FixedHeight (h1+h2) on products (componentwise lattice lifted from mathlib's Prod.instLattice) - Spa.Lattice.AboveBelow: flat lattice via Lattice.mk' (mirrors the Agda semilattices+absorption construction), boundedness via rank into Nat, Plain x ↦ plainFixedHeight x, height 2 Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
2026-06-09 18:48:02 -07:00
(n : ℕ) : BoundedChains α n := fun c => by
by_contra hc
push_neg at hc
exact (c.step ⟨0, by omega⟩).ne (Subsingleton.elim _ _)
Lean migration cleanup: collapse FixedHeight struct into FiniteHeightLattice typeclass The fable-based migration left a two-layer design (a standalone `FixedHeight α h` struct, height carried as a type index, plus a `FiniteHeightLattice` wrapper). This collapses it to the single `FiniteHeightLattice` typeclass (height as a plain field, `⊥`/`⊤` via `extends Bot`/`Top`), and fixes the fallout so the whole project builds again (`lake build` green). - Lattice: repair `FixedHeight.bot_le` (compute the `▸` motive via a forward `rw`, drop the leftover `fh.length_longestChain`) and the `bot_le` alias. - Isomorphism: transport rewritten directly onto `FiniteHeightLattice`, taking the source as an instance argument. - Lattice/Prod, AboveBelow: `FixedHeight`-producing def + wrapper instance collapsed into one `FiniteHeightLattice` instance. `head`/`last` proofs use term-mode `congrArg` to bridge the `Bot`/`Top` defeq through the under-construction instance projection (where `rw`+`rfl` cannot). - Lattice/IterProd: `fixedHeight` recursion now yields a `FiniteHeightLattice` (no height index, so the `.cast (by ring)` reassociations vanish); `bot_fixedHeight` reprojected onto the def's own `.bot`. - Lattice/FiniteMap: `fixedHeight`/`bot_contains_bots` go through transport with the IterProd instance resolved by typeclass search; `punitFixedHeight` replaced by the `PUnit` instance. - Analysis/Forward/Lattices: `botV` uses `⊥` instead of the deleted `FiniteHeightLattice.bot` accessor. - Analysis/Sign: `num` case used unimported `ring`; the goal is a pure ℕ→ℤ cast identity, closed with `norm_cast`. Also fixes the missing `show` in `AboveBelow.monotone₂_of_strict` that left un-beta-reduced redexes. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
2026-06-22 18:33:48 -05:00
instance : FiniteHeightLattice PUnit where
Make FiniteHeightLattice extend Lattice and derive Top/Bot Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
2026-06-25 18:42:28 -05:00
toLattice := inferInstance
longestChain := RelSeries.singleton _ PUnit.unit
Lean migration cleanup: collapse FixedHeight struct into FiniteHeightLattice typeclass The fable-based migration left a two-layer design (a standalone `FixedHeight α h` struct, height carried as a type index, plus a `FiniteHeightLattice` wrapper). This collapses it to the single `FiniteHeightLattice` typeclass (height as a plain field, `⊥`/`⊤` via `extends Bot`/`Top`), and fixes the fallout so the whole project builds again (`lake build` green). - Lattice: repair `FixedHeight.bot_le` (compute the `▸` motive via a forward `rw`, drop the leftover `fh.length_longestChain`) and the `bot_le` alias. - Isomorphism: transport rewritten directly onto `FiniteHeightLattice`, taking the source as an instance argument. - Lattice/Prod, AboveBelow: `FixedHeight`-producing def + wrapper instance collapsed into one `FiniteHeightLattice` instance. `head`/`last` proofs use term-mode `congrArg` to bridge the `Bot`/`Top` defeq through the under-construction instance projection (where `rw`+`rfl` cannot). - Lattice/IterProd: `fixedHeight` recursion now yields a `FiniteHeightLattice` (no height index, so the `.cast (by ring)` reassociations vanish); `bot_fixedHeight` reprojected onto the def's own `.bot`. - Lattice/FiniteMap: `fixedHeight`/`bot_contains_bots` go through transport with the IterProd instance resolved by typeclass search; `punitFixedHeight` replaced by the `PUnit` instance. - Analysis/Forward/Lattices: `botV` uses `⊥` instead of the deleted `FiniteHeightLattice.bot` accessor. - Analysis/Sign: `num` case used unimported `ring`; the goal is a pure ℕ→ℤ cast identity, closed with `norm_cast`. Also fixes the missing `show` in `AboveBelow.monotone₂_of_strict` that left un-beta-reduced redexes. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
2026-06-22 18:33:48 -05:00
chains_bounded := boundedChains_of_subsingleton PUnit 0
Lean migration: Phase 3 (Unit, Prod, AboveBelow lattices) - Spa.Lattice.Unit: PUnit fixed height 0 (lattice lifted from mathlib) - Spa.Lattice.Prod: chain unzip + FixedHeight (h1+h2) on products (componentwise lattice lifted from mathlib's Prod.instLattice) - Spa.Lattice.AboveBelow: flat lattice via Lattice.mk' (mirrors the Agda semilattices+absorption construction), boundedness via rank into Nat, Plain x ↦ plainFixedHeight x, height 2 Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
2026-06-09 18:48:02 -07:00
end Spa
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