Prove that a finite height lattice is bounded below

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2023-09-15 21:07:14 -07:00
parent 5cab39ca82
commit 266c3dd81e

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@ -442,3 +442,46 @@ module IsFiniteHeightLatticeInstances where
(proj₂ (IsFiniteHeightLattice.fixedHeight lB) b₁b₂))
)
}
module FixedHeightLatticeIsBounded {a} {A : Set a} {h : }
{_≈_ : A A Set a}
{_⊔_ : A A A} {_⊓_ : A A A}
(decA : IsDecidable _≈_)
(lA : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_) where
open IsFiniteHeightLattice lA using (_≼_; _≺_; fixedHeight; ≈-equiv; ≈-refl; ≈-sym; ≈-trans; ≼-refl; ≼-cong; ≺-cong; ≈-⊔-cong; absorb-⊔-⊓; ⊔-comm; ⊓-comm)
open IsDecidable decA using () renaming (R-dec to ≈-dec)
open NatProps using (+-comm; m+1+n≰m)
module ChainA = Chain _≈_ ≈-equiv _≺_ ≺-cong
A-BoundedBelow : Σ A (λ ⊥ᴬ (a : A) ⊥ᴬ a)
A-BoundedBelow = (⊥ᴬ , ⊥ᴬ≼)
where
⊥ᴬ : A
⊥ᴬ = proj₁ (proj₁ (proj₁ fixedHeight))
⊥ᴬ≼ : (a : A) ⊥ᴬ a
⊥ᴬ≼ a
with ≈-dec a ⊥ᴬ
... | yes a≈⊥ᴬ = ≼-cong a≈⊥ᴬ ≈-refl (≼-refl a)
... | no a̷≈⊥ᴬ with ≈-dec ⊥ᴬ (a ⊥ᴬ)
... | yes ⊥ᴬ≈a⊓⊥ᴬ = (a , ≈-trans (⊔-comm ⊥ᴬ a) (≈-trans (≈-⊔-cong (≈-refl {a}) ⊥ᴬ≈a⊓⊥ᴬ) (absorb-⊔-⊓ a ⊥ᴬ)))
... | no ⊥ᴬ̷≈a⊓⊥ᴬ = absurd (m+1+n≰m h h+1≤h)
where
⊥ᴬ⊓a̷≈⊥ᴬ : ¬ (⊥ᴬ a) ⊥ᴬ
⊥ᴬ⊓a̷≈⊥ᴬ = λ ⊥ᴬ⊓a≈⊥ᴬ ⊥ᴬ̷≈a⊓⊥ᴬ (≈-trans (≈-sym ⊥ᴬ⊓a≈⊥ᴬ) (⊓-comm _ _))
x≺⊥ᴬ : (⊥ᴬ a) ⊥ᴬ
x≺⊥ᴬ = ((⊥ᴬ , ≈-trans (⊔-comm _ _) (≈-trans (≈-refl {⊥ᴬ (⊥ᴬ a)}) (absorb-⊔-⊓ ⊥ᴬ a))) , ⊥ᴬ⊓a̷≈⊥ᴬ)
h+1≤h : h + 1 h
h+1≤h rewrite (+-comm h 1) = proj₂ fixedHeight (ChainA.step x≺⊥ᴬ ≈-refl (proj₂ (proj₁ fixedHeight)))
module FixedPoint {a} {A : Set a}
(h : )
(_≈_ : A A Set a)
(_⊔_ : A A A) (_⊓_ : A A A)
(isFiniteHeightLattice : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_)
(f : A A) (Monotonicᶠ : Monotonic (IsFiniteHeightLattice._≼_ isFiniteHeightLattice)
(IsFiniteHeightLattice._≼_ isFiniteHeightLattice) f) where
open IsFiniteHeightLattice isFiniteHeightLattice