Danila Fedorin
2ee32580a2
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>
53 lines
1.8 KiB
Lean4
53 lines
1.8 KiB
Lean4
import Spa.Lattice
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namespace Spa.Fixedpoint
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open FiniteHeightLattice (height)
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variable {α : Type*} [Lattice α] [DecidableEq α] [FiniteHeightLattice α]
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def doStep (f : α → α) (hf : Monotone f) :
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∀ (g : ℕ) (c : LTSeries α), c.length + g = height (α := α) + 1 →
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c.last ≤ f c.last → {a : α // a = f a}
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| 0, c, hlen, _ =>
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absurd (FiniteHeightLattice.chains_bounded c) (by omega)
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| g + 1, c, hlen, hle =>
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if heq : c.last = f c.last then
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⟨c.last, heq⟩
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else
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doStep f hf g (c.snoc (f c.last) (lt_of_le_of_ne hle heq))
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(by simp [RelSeries.snoc]; omega)
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(by rw [RelSeries.last_snoc]; exact hf hle)
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def fix (f : α → α) (hf : Monotone f) : {a : α // a = f a} :=
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doStep f hf (height (α := α) + 1) (RelSeries.singleton _ ⊥)
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(by simp)
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(by simpa [RelSeries.last_singleton]
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using FiniteHeightLattice.bot_le α (f ⊥))
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def aFix (f : α → α) (hf : Monotone f) : α :=
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(fix f hf).1
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theorem aFix_eq (f : α → α) (hf : Monotone f) :
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aFix f hf = f (aFix f hf) :=
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(fix f hf).2
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theorem doStep_le (f : α → α) (hf : Monotone f)
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{b : α} (hb : b = f b) :
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∀ (g : ℕ) (c : LTSeries α) (hlen : c.length + g = height (α := α) + 1)
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(hle : c.last ≤ f c.last), c.last ≤ b →
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(doStep f hf g c hlen hle : α) ≤ b
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| 0, c, hlen, _ => fun _ => absurd (FiniteHeightLattice.chains_bounded c) (by omega)
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| g + 1, c, hlen, hle => fun hcb => by
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rw [doStep]
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split
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· exact hcb
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· exact doStep_le f hf hb g _ _ _
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(by rw [RelSeries.last_snoc]; exact le_of_le_of_eq (hf hcb) hb.symm)
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theorem aFix_le (f : α → α) (hf : Monotone f)
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{a : α} (ha : a = f a) : aFix f hf ≤ a :=
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doStep_le f hf ha _ _ _ _ (by simpa using FiniteHeightLattice.bot_le α a)
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end Spa.Fixedpoint
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