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>

lean/Spa/Isomorphism.lean

@@ -1,58 +1,27 @@
-/-
-Port of `Isomorphism.agda` (`TransportFiniteHeight`).
-
-With propositional equality this module shrinks dramatically: the Agda
-hypotheses `f-preserves-≈`, `g-preserves-≈` are free, and `f-⊔-distr` /
-`g-⊔-distr` (which in the setoid world encoded monotonicity of `f` and `g`
-w.r.t. the derived order) become plain `Monotone` hypotheses. The chain
-transport `portChain₁` / `portChain₂` is mathlib's `LTSeries.map`, using that
-a monotone injective map between partial orders is strictly monotone.
-
-Correspondence:
-  IsInverseˡ / IsInverseʳ     ↦ explicit inverse hypotheses `hfg` / `hgf`
-  f-Injective / g-Injective   ↦ local `Function.LeftInverse.injective`
-  portChain₁ / portChain₂     ↦ LTSeries.map
-  instance fixedHeight        ↦ Spa.FixedHeight.transport
-  isFiniteHeightLattice,
-  finiteHeightLattice         ↦ Spa.FiniteHeightLattice.transport
--/
 import Spa.Lattice
 
 namespace Spa
 
-namespace FixedHeight
-
-variable {α β : Type*} [PartialOrder α] [PartialOrder β] {h : ℕ}
-
-/-- Agda: `TransportFiniteHeight.fixedHeight`. Transport a `FixedHeight`
-structure along a monotone inverse pair `f : α → β`, `g : β → α`. -/
-def transport (fh : FixedHeight α h) (f : α → β) (g : β → α)
-    (hf : Monotone f) (hg : Monotone g)
-    (hgf : ∀ a, g (f a) = a) (hfg : ∀ b, f (g b) = b) :
-    FixedHeight β h where
-  bot := f fh.bot
-  top := f fh.top
-  longestChain :=
-    fh.longestChain.map f
-      (hf.strictMono_of_injective (Function.LeftInverse.injective hgf))
-  head_longestChain := by
-    rw [LTSeries.head_map, fh.head_longestChain]
-  last_longestChain := by
-    rw [LTSeries.last_map, fh.last_longestChain]
-  length_longestChain := fh.length_longestChain
-  bounded := fun c =>
-    fh.bounded
-      (c.map g (hg.strictMono_of_injective (Function.LeftInverse.injective hfg)))
-
-end FixedHeight
-
-/-- Agda: `TransportFiniteHeight.finiteHeightLattice`. -/
+/-- Agda: `TransportFiniteHeight.finiteHeightLattice`. Transport a
+`FiniteHeightLattice` structure along a monotone inverse pair `f : α → β`,
+`g : β → α`. -/
 def FiniteHeightLattice.transport {α β : Type*} [Lattice α] [Lattice β]
-    (I : FiniteHeightLattice α) (f : α → β) (g : β → α)
+    [I : FiniteHeightLattice α] (f : α → β) (g : β → α)
     (hf : Monotone f) (hg : Monotone g)
     (hgf : ∀ a, g (f a) = a) (hfg : ∀ b, f (g b) = b) :
     FiniteHeightLattice β where
+  bot := f ⊥
+  top := f ⊤
   height := I.height
-  fixedHeight := I.fixedHeight.transport f g hf hg hgf hfg
+  longest_chain :=
+    { series :=
+        I.longest_chain.series.map f
+          (hf.strictMono_of_injective (Function.LeftInverse.injective hgf))
+      head_series := congrArg f I.longest_chain.head_series
+      last_series := congrArg f I.longest_chain.last_series
+      length_series := I.longest_chain.length_series }
+  chains_bounded := fun c =>
+    I.chains_bounded
+      (c.map g (hg.strictMono_of_injective (Function.LeftInverse.injective hfg)))
 
 end Spa