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Prove that the 'join' transformation is monotonic

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin
2024-03-09 23:06:47 -08:00
parent 1b1b80465c
commit f84a1c923c
4 changed files with 78 additions and 15 deletions
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25
Lattice/FiniteMap.agda
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@@ -132,23 +132,24 @@ module WithKeys (ks : List A) where
; isLattice = isLattice
}
module _ {l} {L : Set l}
module GeneralizedUpdate
{l} {L : Set l}
{_≈ˡ_ : L → L → Set l} {_⊔ˡ_ : L → L → L} {_⊓ˡ_ : L → L → L}
(lL : IsLattice L _≈ˡ_ _⊔ˡ_ _⊓ˡ_) where
(lL : IsLattice L _≈ˡ_ _⊔ˡ_ _⊓ˡ_)
(f : L → FiniteMap) (f-Monotonic : Monotonic (IsLattice._≼_ lL) _≼_ f)
(g : A → L → B) (g-Monotonicʳ : ∀ k → Monotonic (IsLattice._≼_ lL) _≼₂_ (g k))
(ks : List A) where
open IsLattice lL using () renaming (_≼_ to _≼ˡ_)
module _ (f : L → FiniteMap) (f-Monotonic : Monotonic _≼ˡ_ _≼_ f)
(g : A → L → B) (g-Monotonicʳ : ∀ k → Monotonic _≼ˡ_ _≼₂_ (g k))
(ks : List A) where
updater : L → A → B
updater l k = g k l
updater : L → A → B
updater l k = g k l
f' : L → FiniteMap
f' l = (f l) updating ks via (updater l)
f' : L → FiniteMap
f' l = (f l) updating ks via (updater l)
f'-Monotonic : Monotonic _≼ˡ_ _≼_ f'
f'-Monotonic {l₁} {l₂} l₁≼l₂ = f'-Monotonicᵐ lL (proj₁ ∘ f) f-Monotonic g g-Monotonicʳ ks l₁≼l₂
f'-Monotonic : Monotonic _≼ˡ_ _≼_ f'
f'-Monotonic {l₁} {l₂} l₁≼l₂ = f'-Monotonicᵐ lL (proj₁ ∘ f) f-Monotonic g g-Monotonicʳ ks l₁≼l₂
all-equal-keys : ∀ (fm₁ fm₂ : FiniteMap) → (Map.keys (proj₁ fm₁) ≡ Map.keys (proj₁ fm₂))
all-equal-keys (fm₁ , km₁≡ks) (fm₂ , km₂≡ks) = trans km₁≡ks (sym km₂≡ks)