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Fix definition of 'less than' to not involve a third variable.

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin
2024-02-07 21:04:13 -08:00
parent 9646096c75
commit 512cd22be5
5 changed files with 38 additions and 30 deletions
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33
Lattice/AboveBelow.agda
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@@ -282,29 +282,26 @@ module Plain where
open IsLattice isLattice using (_≼_; _≺_)
⊥≺[x] : ∀ (x : A) → ⊥ ≺ [ x ]
⊥≺[x] x = (([ x ] , ≈-refl) , λ ())
⊥≺[x] x = (≈-refl , λ ())
x≺[y]⇒y≡⊥ : ∀ (x : AboveBelow) (y : A) → x ≺ [ y ] → x ≡ ⊥
x≺[y]⇒y≡⊥ x y ((d , x⊔d≈[y]) , x̷≈[y]) with d
... | ⊥ rewrite x⊔⊥≡x x with ≈-lift a≈y ← x⊔d≈[y] = ⊥-elim (x̷≈[y] (≈-lift a≈y))
... | ⊤ rewrite x⊔⊤≡⊤ x with () <- x⊔d≈[y]
... | [ a ] with x
... | ⊥ = refl
... | ⊤ with () <- x⊔d≈[y]
... | [ b ] with ≈₁-dec b a
... | yes _ with ≈-lift b≈y ← x⊔d≈[y] = ⊥-elim (x̷≈[y] (≈-lift b≈y))
... | no _ with () <- x⊔d≈[y]
x≺[y]⇒x≡⊥ : ∀ (x : AboveBelow) (y : A) → x ≺ [ y ] → x ≡ ⊥
x≺[y]⇒x≡⊥ x y ((x⊔[y]≈[y]) , x̷≈[y]) with x
... | ⊥ = refl
... | ⊤ with () ← x⊔[y]≈[y]
... | [ b ] with ≈₁-dec b y
... | yes b≈y = ⊥-elim (x̷≈[y] (≈-lift b≈y))
... | no _ with () ← x⊔[y]≈[y]
[x]≺⊤ : ∀ (x : A) → [ x ] ≺ ⊤
[x]≺⊤ x rewrite x⊔⊤≡⊤ [ x ] = ((⊤ , ≈-⊤-⊤) , λ ())
[x]≺⊤ x rewrite x⊔⊤≡⊤ [ x ] = (≈-⊤-⊤ , λ ())
[x]≺y⇒y≡⊤ : ∀ (x : A) (y : AboveBelow) → [ x ] ≺ y → y ≡ ⊤
[x]≺y⇒y≡⊤ x y ((d , [x]⊔d≈y) , [x]̷≈y) with d
... | ⊥ rewrite x⊔⊥≡x [ x ] with ≈-lift x≈a ← [x]⊔d≈y = ⊥-elim ([x]̷≈y (≈-lift x≈a))
... | ⊤ rewrite x⊔⊤≡⊤ [ x ] with ≈-⊤-⊤ ← [x]⊔d≈y = refl
[x]≺y⇒y≡⊤ x y ([x]⊔y≈y , [x]̷≈y) with y
... | ⊥ with () ← [x]⊔y≈y
... | ⊤ = refl
... | [ a ] with ≈₁-dec x a
... | yes _ with ≈-lift x≈a ← [x]⊔d≈y = ⊥-elim ([x]̷≈y (≈-lift x≈a))
... | no _ with ≈-⊤-⊤ ← [x]⊔d≈y = refl
... | yes x≈a = ⊥-elim ([x]̷≈y (≈-lift x≈a))
... | no _ with () ← [x]⊔y≈y
open Chain _≈_ ≈-equiv (IsLattice._≺_ isLattice) (IsLattice.≺-cong isLattice)
@@ -313,7 +310,7 @@ module Plain where
longestChain = step (⊥≺[x] x) ≈-refl (step ([x]≺⊤ x) ≈-⊤-⊤ (done ≈-⊤-⊤))
¬-Chain-⊤ : ∀ {ab : AboveBelow} {n : ℕ} → ¬ Chain ⊤ ab (suc n)
¬-Chain-⊤ (step ((d , ⊤⊔d≈x) , ⊤̷≈x) _ _) rewrite ⊤⊔x≡⊤ d = ⊥-elim (⊤̷≈x ⊤⊔d≈x)
¬-Chain-⊤ {x} (step (⊤⊔x≈x , ⊤̷≈x) _ _) rewrite ⊤⊔x≡⊤ x = ⊥-elim (⊤̷≈x ⊤⊔x≈x)
isLongest : ∀ {ab₁ ab₂ : AboveBelow} {n : ℕ} → Chain ab₁ ab₂ n → n ≤ 2
isLongest (done _) = z≤n