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Implement the fixed point algorithm

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2023-09-16 00:23:44 -07:00
parent 866bc9124a
commit c338fa3ee5
1 changed files with 37 additions and 6 deletions
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43
Lattice.agda
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@ -452,7 +452,9 @@ module FixedHeightLatticeIsBounded {a} {A : Set a} {h : ℕ}
open IsFiniteHeightLattice lA using (_≼_; _≺_; fixedHeight; ≈-equiv; ≈-refl; ≈-sym; ≈-trans; ≼-refl; ≼-cong; ≺-cong; ≈-⊔-cong; absorb-⊔-⊓; ⊔-comm; ⊓-comm)
open IsDecidable decA using () renaming (R-dec to ≈-dec)
open NatProps using (+-comm; m+1+n≰m)
module ChainA = Chain _≈_ ≈-equiv _≺_ ≺-cong
private
module ChainA = Chain _≈_ ≈-equiv _≺_ ≺-cong
A-BoundedBelow : Σ A (λ ⊥ᴬ → ∀ (a : A) → ⊥ᴬ ≼ a)
A-BoundedBelow = (⊥ᴬ , ⊥ᴬ≼)
@ -466,7 +468,7 @@ module FixedHeightLatticeIsBounded {a} {A : Set a} {h : ℕ}
... | yes a≈⊥ᴬ = ≼-cong a≈⊥ᴬ ≈-refl (≼-refl a)
... | no a̷≈⊥ᴬ with ≈-dec ⊥ᴬ (a ⊓ ⊥ᴬ)
... | yes ⊥ᴬ≈a⊓⊥ᴬ = (a , ≈-trans (⊔-comm ⊥ᴬ a) (≈-trans (≈-⊔-cong (≈-refl {a}) ⊥ᴬ≈a⊓⊥ᴬ) (absorb-⊔-⊓ a ⊥ᴬ)))
... | no ⊥ᴬ̷≈a⊓⊥ᴬ = absurd (m+1+n≰m h h+1≤h)
... | no ⊥ᴬ̷≈a⊓⊥ᴬ = absurd (ChainA.Bounded-suc-n (proj₂ fixedHeight) (ChainA.step x≺⊥ᴬ ≈-refl (proj₂ (proj₁ fixedHeight))))
where
⊥ᴬ⊓a̷≈⊥ᴬ : ¬ (⊥ᴬ ⊓ a) ≈ ⊥ᴬ
⊥ᴬ⊓a̷≈⊥ᴬ = λ ⊥ᴬ⊓a≈⊥ᴬ → ⊥ᴬ̷≈a⊓⊥ᴬ (≈-trans (≈-sym ⊥ᴬ⊓a≈⊥ᴬ) (⊓-comm _ _))
@ -474,14 +476,43 @@ module FixedHeightLatticeIsBounded {a} {A : Set a} {h : ℕ}
x≺⊥ᴬ : (⊥ᴬ ⊓ a) ≺ ⊥ᴬ
x≺⊥ᴬ = ((⊥ᴬ , ≈-trans (⊔-comm _ _) (≈-trans (≈-refl {⊥ᴬ ⊔ (⊥ᴬ ⊓ a)}) (absorb-⊔-⊓ ⊥ᴬ a))) , ⊥ᴬ⊓a̷≈⊥ᴬ)
h+1≤h : h + 1 ≤ h
h+1≤h rewrite (+-comm h 1) = proj₂ fixedHeight (ChainA.step x≺⊥ᴬ ≈-refl (proj₂ (proj₁ fixedHeight)))
module FixedPoint {a} {A : Set a}
(h : ℕ)
(_≈_ : A → A → Set a)
(_≈_ : A → A → Set a) (decA : IsDecidable _≈_)
(_⊔_ : A → A → A) (_⊓_ : A → A → A)
(isFiniteHeightLattice : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_)
(f : A → A) (Monotonicᶠ : Monotonic (IsFiniteHeightLattice._≼_ isFiniteHeightLattice)
(IsFiniteHeightLattice._≼_ isFiniteHeightLattice) f) where
open IsDecidable decA using () renaming (R-dec to ≈-dec)
open IsFiniteHeightLattice isFiniteHeightLattice
open FixedHeightLatticeIsBounded decA isFiniteHeightLattice
open NatProps using (+-suc; +-comm)
module ChainA = Chain _≈_ ≈-equiv _≺_ ≺-cong
private
⊥ᴬ : A
⊥ᴬ = proj₁ A-BoundedBelow
⊥ᴬ≼ : ∀ (a : A) → ⊥ᴬ ≼ a
⊥ᴬ≼ = proj₂ A-BoundedBelow
-- using 'g', for gas, here helps make sure the function terminates.
-- since A forms a fixed-height lattice, we must find a solution after
-- enough 'h' steps at most. Gas is set up such that as soon as it runs
-- out, we have exceeded h steps, which shouldn't be possible.
doStep : ∀ (g hᶜ : ℕ) (a₁ a₂ : A) (c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ ≡ suc h) (a₂≼fa₂ : a₂ ≼ f a₂) → Σ A (λ a → a ≈ f a)
doStep 0 hᶜ a₁ a₂ c g+hᶜ≡sh a₂≼fa₂ rewrite g+hᶜ≡sh = absurd (ChainA.Bounded-suc-n (proj₂ fixedHeight) c)
doStep (suc g') hᶜ a₁ a₂ c g+hᶜ≡sh a₂≼fa₂ rewrite sym (+-suc g' hᶜ)
with ≈-dec a₂ (f a₂)
... | yes a₂≈fa₂ = (a₂ , a₂≈fa₂)
... | no a₂̷≈fa₂ = doStep g' (suc hᶜ) a₁ (f a₂) c' g+hᶜ≡sh (Monotonicᶠ a₂≼fa₂)
where
a₂≺fa₂ : a₂ ≺ f a₂
a₂≺fa₂ = (a₂≼fa₂ , a₂̷≈fa₂)
c' : ChainA.Chain a₁ (f a₂) (suc hᶜ)
c' rewrite +-comm 1 hᶜ = ChainA.concat c (ChainA.step a₂≺fa₂ ≈-refl (ChainA.done (≈-refl {f a₂})))
fix : Σ A (λ a → a ≈ f a)
fix = doStep (suc h) 0 ⊥ᴬ ⊥ᴬ (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))