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Rename 'a' to 'b' in fixedpoint algorithm proof

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin
2024-08-18 10:28:45 -10:00
parent 12971450e3
commit 828b652d3b
1 changed files with 5 additions and 5 deletions
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Fixedpoint.agda
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@@ -73,15 +73,15 @@ aᶠ≈faᶠ : aᶠ ≈ f aᶠ
aᶠ≈faᶠ = proj₂ fix aᶠ≈faᶠ = proj₂ fix
private private
stepPreservesLess : ∀ (g hᶜ : ℕ) (a₁ a₂ a : A) (a≈fa : a ≈ f a) (a₂≼a : a₂ ≼ a) stepPreservesLess : ∀ (g hᶜ : ℕ) (a₁ a₂ b : A) (b≈fb : b ≈ f b) (a₂≼a : a₂ ≼ b)
(c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ ≡ suc h) (c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ ≡ suc h)
(a₂≼fa₂ : a₂ ≼ f a₂) → (a₂≼fa₂ : a₂ ≼ f a₂) →
proj₁ (doStep g hᶜ a₁ a₂ c g+hᶜ≡h a₂≼fa₂) ≼ a proj₁ (doStep g hᶜ a₁ a₂ c g+hᶜ≡h a₂≼fa₂) ≼ b
stepPreservesLess 0 _ _ _ _ _ _ c g+hᶜ≡sh _ rewrite g+hᶜ≡sh = ⊥-elim (ChainA.Bounded-suc-n boundedᴬ c) stepPreservesLess 0 _ _ _ _ _ _ c g+hᶜ≡sh _ rewrite g+hᶜ≡sh = ⊥-elim (ChainA.Bounded-suc-n boundedᴬ c)
stepPreservesLess (suc g') hᶜ a₁ a₂ a a≈fa a₂≼a c g+hᶜ≡sh a₂≼fa₂ rewrite sym (+-suc g' hᶜ) stepPreservesLess (suc g') hᶜ a₁ a₂ b b≈fb a₂≼b c g+hᶜ≡sh a₂≼fa₂ rewrite sym (+-suc g' hᶜ)
with ≈-dec a₂ (f a₂) with ≈-dec a₂ (f a₂)
... | yes _ = a₂≼a ... | yes _ = a₂≼b
... | no _ = stepPreservesLess g' _ _ _ a a≈fa (≼-cong ≈-refl (≈-sym a≈fa) (Monotonicᶠ a₂≼a)) _ _ _ ... | no _ = stepPreservesLess g' _ _ _ b b≈fb (≼-cong ≈-refl (≈-sym b≈fb) (Monotonicᶠ a₂≼b)) _ _ _
aᶠ≼ : ∀ (a : A) → a ≈ f a → aᶠ ≼ a aᶠ≼ : ∀ (a : A) → a ≈ f a → aᶠ ≼ a
aᶠ≼ a a≈fa = stepPreservesLess (suc h) 0 ⊥ᴬ ⊥ᴬ a a≈fa (⊥ᴬ≼ a) (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ)) aᶠ≼ a a≈fa = stepPreservesLess (suc h) 0 ⊥ᴬ ⊥ᴬ a a≈fa (⊥ᴬ≼ a) (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))