76 lines
3.0 KiB
Lean4
76 lines
3.0 KiB
Lean4
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/-
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Port of `Fixedpoint.agda`.
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Same gas-based algorithm: iterate `f` starting at the chain-bottom `⊥`; since
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the lattice has fixed height `h`, a fixed point must be reached within `h + 1`
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steps, or we would build a `<`-chain longer than the longest one. We
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deliberately do *not* use mathlib's `OrderHom.lfp` (different proof approach,
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and not computable).
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Correspondence:
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doStep ↦ Spa.Fixedpoint.doStep (the chain argument now carries
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`a₁ = ⊥` and its length in the
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`LTSeries` structure itself)
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fix ↦ Spa.Fixedpoint.fix
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aᶠ ↦ Spa.Fixedpoint.aFix
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aᶠ≈faᶠ ↦ Spa.Fixedpoint.aFix_eq
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stepPreservesLess ↦ Spa.Fixedpoint.doStep_le
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aᶠ≼ ↦ Spa.Fixedpoint.aFix_le
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-/
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import Spa.Lattice
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namespace Spa.Fixedpoint
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variable {α : Type*} [Lattice α] [DecidableEq α] {h : ℕ}
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/-- Agda: `doStep`. `g` is gas; the invariant `c.length + g = h + 1` guarantees
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that when gas runs out the chain contradicts boundedness. -/
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def doStep (fh : FixedHeight α h) (f : α → α) (hf : Monotone f) :
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∀ (g : ℕ) (c : LTSeries α), c.length + g = h + 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 (fh.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 fh 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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/-- Agda: `fix`. Start iterating from `⊥`. -/
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def fix (fh : FixedHeight α h) (f : α → α) (hf : Monotone f) : {a : α // a = f a} :=
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doStep fh f hf (h + 1) (RelSeries.singleton _ fh.bot)
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(by simp)
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(by simpa [RelSeries.last_singleton] using fh.bot_le (f fh.bot))
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/-- Agda: `aᶠ`. -/
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def aFix (fh : FixedHeight α h) (f : α → α) (hf : Monotone f) : α :=
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(fix fh f hf).1
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/-- Agda: `aᶠ≈faᶠ`. -/
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theorem aFix_eq (fh : FixedHeight α h) (f : α → α) (hf : Monotone f) :
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aFix fh f hf = f (aFix fh f hf) :=
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(fix fh f hf).2
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/-- Agda: `stepPreservesLess` — iteration stays below any fixed point. -/
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theorem doStep_le (fh : FixedHeight α h) (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 = h + 1)
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(hle : c.last ≤ f c.last), c.last ≤ b →
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(doStep fh f hf g c hlen hle : α) ≤ b
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| 0, c, hlen, _ => fun _ => absurd (fh.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 fh 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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/-- Agda: `aᶠ≼` — `aFix` is below every fixed point of `f`. -/
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theorem aFix_le (fh : FixedHeight α h) (f : α → α) (hf : Monotone f)
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{a : α} (ha : a = f a) : aFix fh f hf ≤ a :=
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doStep_le fh f hf ha _ _ _ _ (by simpa using fh.bot_le a)
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end Spa.Fixedpoint
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