From 828b652d3b9266e27ef7cf5a8a7fb82e3fd3133f Mon Sep 17 00:00:00 2001 From: Danila Fedorin Date: Sun, 18 Aug 2024 10:28:45 -1000 Subject: [PATCH] Rename 'a' to 'b' in fixedpoint algorithm proof Signed-off-by: Danila Fedorin --- Fixedpoint.agda | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/Fixedpoint.agda b/Fixedpoint.agda index 5e9f109..3f28c38 100644 --- a/Fixedpoint.agda +++ b/Fixedpoint.agda @@ -73,15 +73,15 @@ aᶠ≈faᶠ : aᶠ ≈ f aᶠ aᶠ≈faᶠ = proj₂ fix 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) (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 (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) - ... | yes _ = a₂≼a - ... | no _ = stepPreservesLess g' _ _ _ a a≈fa (≼-cong ≈-refl (≈-sym a≈fa) (Monotonicᶠ a₂≼a)) _ _ _ + ... | yes _ = a₂≼b + ... | 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≈fa = stepPreservesLess (suc h) 0 ⊥ᴬ ⊥ᴬ a a≈fa (⊥ᴬ≼ a) (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))