Make 'isLattice' for simple types be an instance
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -18,8 +18,9 @@ private
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≡-⊓-cong : ∀ {a₁ a₂ a₃ a₄} → a₁ ≡ a₂ → a₃ ≡ a₄ → (a₁ ⊓ a₃) ≡ (a₂ ⊓ a₄)
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≡-⊓-cong a₁≡a₂ a₃≡a₄ rewrite a₁≡a₂ rewrite a₃≡a₄ = refl
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isMaxSemilattice : IsSemilattice ℕ _≡_ _⊔_
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isMaxSemilattice = record
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instance
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isMaxSemilattice : IsSemilattice ℕ _≡_ _⊔_
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isMaxSemilattice = record
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{ ≈-equiv = record
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{ ≈-refl = refl
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; ≈-sym = sym
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@ -31,8 +32,8 @@ isMaxSemilattice = record
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; ⊔-idemp = ⊔-idem
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}
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isMinSemilattice : IsSemilattice ℕ _≡_ _⊓_
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isMinSemilattice = record
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isMinSemilattice : IsSemilattice ℕ _≡_ _⊓_
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isMinSemilattice = record
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{ ≈-equiv = record
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{ ≈-refl = refl
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; ≈-sym = sym
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@ -74,16 +75,17 @@ private
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helper : x ⊔ (x ⊓ y) ≤ x ⊔ x → x ⊔ x ≡ x → x ⊔ (x ⊓ y) ≤ x
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helper x⊔x⊓y≤x⊔x x⊔x≡x rewrite x⊔x≡x = x⊔x⊓y≤x⊔x
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isLattice : IsLattice ℕ _≡_ _⊔_ _⊓_
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isLattice = record
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instance
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isLattice : IsLattice ℕ _≡_ _⊔_ _⊓_
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isLattice = record
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{ joinSemilattice = isMaxSemilattice
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; meetSemilattice = isMinSemilattice
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; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
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; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
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}
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lattice : Lattice ℕ
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lattice = record
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lattice : Lattice ℕ
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lattice = record
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{ _≈_ = _≡_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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@ -45,8 +45,9 @@ tt ⊓ tt = tt
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⊔-idemp : (x : ⊤) → (x ⊔ x) ≈ x
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⊔-idemp tt = Eq.refl
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isJoinSemilattice : IsSemilattice ⊤ _≈_ _⊔_
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isJoinSemilattice = record
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instance
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isJoinSemilattice : IsSemilattice ⊤ _≈_ _⊔_
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isJoinSemilattice = record
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{ ≈-equiv = ≈-equiv
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; ≈-⊔-cong = ≈-⊔-cong
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; ⊔-assoc = ⊔-assoc
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@ -66,8 +67,9 @@ isJoinSemilattice = record
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⊓-idemp : (x : ⊤) → (x ⊓ x) ≈ x
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⊓-idemp tt = Eq.refl
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isMeetSemilattice : IsSemilattice ⊤ _≈_ _⊓_
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isMeetSemilattice = record
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instance
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isMeetSemilattice : IsSemilattice ⊤ _≈_ _⊓_
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isMeetSemilattice = record
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{ ≈-equiv = ≈-equiv
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; ≈-⊔-cong = ≈-⊓-cong
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; ⊔-assoc = ⊓-assoc
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@ -75,22 +77,17 @@ isMeetSemilattice = record
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; ⊔-idemp = ⊓-idemp
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}
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absorb-⊔-⊓ : (x y : ⊤) → (x ⊔ (x ⊓ y)) ≈ x
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absorb-⊔-⊓ tt tt = Eq.refl
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absorb-⊓-⊔ : (x y : ⊤) → (x ⊓ (x ⊔ y)) ≈ x
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absorb-⊓-⊔ tt tt = Eq.refl
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isLattice : IsLattice ⊤ _≈_ _⊔_ _⊓_
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isLattice = record
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instance
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isLattice : IsLattice ⊤ _≈_ _⊔_ _⊓_
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isLattice = record
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{ joinSemilattice = isJoinSemilattice
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; meetSemilattice = isMeetSemilattice
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; absorb-⊔-⊓ = absorb-⊔-⊓
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; absorb-⊓-⊔ = absorb-⊓-⊔
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; absorb-⊔-⊓ = λ { tt tt → Eq.refl }
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; absorb-⊓-⊔ = λ { tt tt → Eq.refl }
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}
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lattice : Lattice ⊤
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lattice = record
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lattice : Lattice ⊤
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lattice = record
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{ _≈_ = _≈_
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; _⊔_ = _⊔_
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; _⊓_ = _⊓_
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@ -107,22 +104,23 @@ private
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isLongest {tt} {tt} (step (tt⊔tt≈tt , tt̷≈tt) _ _) = ⊥-elim (tt̷≈tt refl)
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isLongest (done _) = z≤n
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fixedHeight : IsLattice.FixedHeight isLattice 0
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fixedHeight = record
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instance
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fixedHeight : IsLattice.FixedHeight isLattice 0
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fixedHeight = record
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{ ⊥ = tt
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; ⊤ = tt
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; longestChain = longestChain
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; bounded = isLongest
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}
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isFiniteHeightLattice : IsFiniteHeightLattice ⊤ 0 _≈_ _⊔_ _⊓_
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isFiniteHeightLattice = record
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isFiniteHeightLattice : IsFiniteHeightLattice ⊤ 0 _≈_ _⊔_ _⊓_
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isFiniteHeightLattice = record
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{ isLattice = isLattice
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; fixedHeight = fixedHeight
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}
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finiteHeightLattice : FiniteHeightLattice ⊤
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finiteHeightLattice = record
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finiteHeightLattice : FiniteHeightLattice ⊤
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finiteHeightLattice = record
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{ height = 0
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; _≈_ = _≈_
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; _⊔_ = _⊔_
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