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21
Lattice/AboveBelow.agda
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@ -10,14 +10,10 @@ module Lattice.AboveBelow {a} (A : Set a)
open import Data.Empty using (⊥-elim)
open import Data.Product using (_,_)
open import Data.Nat using (_≤_; ℕ; z≤n; s≤s; suc)
open import Function using (_∘_)
open import Relation.Binary.PropositionalEquality as Eq
using (_≡_; sym; subst; refl)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; subst; refl)
import Chain
open IsEquivalence ≈₁-equiv using ()
renaming (≈-refl to ≈₁-refl; ≈-sym to ≈₁-sym; ≈-trans to ≈₁-trans)
open IsEquivalence ≈₁-equiv using () renaming (≈-refl to ≈₁-refl; ≈-sym to ≈₁-sym; ≈-trans to ≈₁-trans)
data AboveBelow : Set a where
⊥ : AboveBelow
@ -65,9 +61,6 @@ data _≈_ : AboveBelow → AboveBelow → Set a where
≈-dec [ x ] ⊥ = no λ ()
≈-dec [ x ] ⊤ = no λ ()
-- Any object can be wrapped in an 'above below' to make it a lattice,
-- since ⊤ and ⊥ are the largest and least elements, and the rest are left
-- unordered. That's what this module does.
module Plain where
_⊔_ : AboveBelow → AboveBelow → AboveBelow
⊥ ⊔ x = x
@ -106,8 +99,7 @@ module Plain where
... | yes x≈₁y = ⊥-elim (x̷≈₁y x≈₁y)
... | no x̷≈₁y = refl
≈-⊔-cong : ∀ {ab₁ ab₂ ab₃ ab₄} → ab₁ ≈ ab₂ → ab₃ ≈ ab₄ →
(ab₁ ⊔ ab₃) ≈ (ab₂ ⊔ ab₄)
≈-⊔-cong : ∀ {ab₁ ab₂ ab₃ ab₄} → ab₁ ≈ ab₂ → ab₃ ≈ ab₄ → (ab₁ ⊔ ab₃) ≈ (ab₂ ⊔ ab₄)
≈-⊔-cong ≈-⊤-⊤ ≈-⊤-⊤ = ≈-⊤-⊤
≈-⊔-cong ≈-⊤-⊤ ≈-⊥-⊥ = ≈-⊤-⊤
≈-⊔-cong ≈-⊥-⊥ ≈-⊤-⊤ = ≈-⊤-⊤
@ -123,8 +115,7 @@ module Plain where
... | no a₁̷≈a₃ | yes a₂≈a₄ = ⊥-elim (a₁̷≈a₃ (≈₁-trans a₁≈a₂ (≈₁-trans a₂≈a₄ (≈₁-sym a₃≈a₄))))
... | no _ | no _ = ≈-⊤-⊤
⊔-assoc : ∀ (ab₁ ab₂ ab₃ : AboveBelow) →
((ab₁ ⊔ ab₂) ⊔ ab₃) ≈ (ab₁ ⊔ (ab₂ ⊔ ab₃))
⊔-assoc : ∀ (ab₁ ab₂ ab₃ : AboveBelow) → ((ab₁ ⊔ ab₂) ⊔ ab₃) ≈ (ab₁ ⊔ (ab₂ ⊔ ab₃))
⊔-assoc ⊤ ab₂ ab₃ = ≈-⊤-⊤
⊔-assoc ⊥ ab₂ ab₃ = ≈-refl
⊔-assoc [ x₁ ] ⊤ ab₃ = ≈-⊤-⊤
@ -135,7 +126,7 @@ module Plain where
with ≈₁-dec x₂ x₃ | ≈₁-dec x₁ x₂
... | no x₂̷≈x₃ | no _ rewrite x̷≈y⇒[x]⊔[y]≡⊤ x₂̷≈x₃ = ≈-⊤-⊤
... | no x₂̷≈x₃ | yes x₁≈x₂ rewrite x̷≈y⇒[x]⊔[y]≡⊤ x₂̷≈x₃
rewrite x̷≈y⇒[x]⊔[y]≡⊤ (x₂̷≈x₃ ∘ (≈₁-trans (≈₁-sym x₁≈x₂))) = ≈-⊤-⊤
rewrite x̷≈y⇒[x]⊔[y]≡⊤ λ x₁≈x₃ → x₂̷≈x₃ (≈₁-trans (≈₁-sym x₁≈x₂) x₁≈x₃) = ≈-⊤-⊤
... | yes x₂≈x₃ | yes x₁≈x₂ rewrite x≈y⇒[x]⊔[y]≡[x] x₂≈x₃
rewrite x≈y⇒[x]⊔[y]≡[x] x₁≈x₂
rewrite x≈y⇒[x]⊔[y]≡[x] (≈₁-trans x₁≈x₂ x₂≈x₃) = ≈-refl
@ -148,7 +139,7 @@ module Plain where
⊔-comm x ⊥ rewrite x⊔⊥≡x x = ≈-refl
⊔-comm [ x₁ ] [ x₂ ] with ≈₁-dec x₁ x₂
... | yes x₁≈x₂ rewrite x≈y⇒[x]⊔[y]≡[x] (≈₁-sym x₁≈x₂) = ≈-lift x₁≈x₂
... | no x₁̷≈x₂ rewrite x̷≈y⇒[x]⊔[y]≡⊤ (x₁̷≈x₂ ∘ ≈₁-sym) = ≈-⊤-⊤
... | no x₁̷≈x₂ rewrite x̷≈y⇒[x]⊔[y]≡⊤ λ x₂≈x₁ → (x₁̷≈x₂ (≈₁-sym x₂≈x₁)) = ≈-⊤-⊤
⊔-idemp : ∀ ab → (ab ⊔ ab) ≈ ab
⊔-idemp ⊤ = ≈-⊤-⊤

36
Lattice/FiniteMap.agda
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@ -1,13 +1,13 @@
open import Lattice
open import Relation.Binary.PropositionalEquality as Eq
using (_≡_;refl; sym; trans; cong; subst)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
open import Relation.Binary.Definitions using (Decidable)
open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)
open import Data.List using (List)
module Lattice.FiniteMap {a b : Level} (A : Set a) (B : Set b)
(_≈₂_ : B → B → Set b)
(_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
(≡-dec-A : IsDecidable (_≡_ {a} {A}))
(≡-dec-A : Decidable (_≡_ {a} {A}))
(lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A lB as Map
@ -43,36 +43,22 @@ module _ (ks : List A) where
≈-dec ≈₂-dec fm₁ fm₂ = ≈ᵐ-dec ≈₂-dec (proj₁ fm₁) (proj₁ fm₂)
_⊔_ : FiniteMap → FiniteMap → FiniteMap
_⊔_ (m₁ , km₁≡ks) (m₂ , km₂≡ks) =
( m₁ ⊔ᵐ m₂
, trans (sym (⊔-equal-keys {m₁} {m₂} (trans (km₁≡ks) (sym km₂≡ks))))
km₁≡ks
)
_⊔_ (m₁ , km₁≡ks) (m₂ , km₂≡ks) = (m₁ ⊔ᵐ m₂ , trans (sym (⊔-equal-keys {m₁} {m₂} (trans (km₁≡ks) (sym km₂≡ks)))) km₁≡ks)
_⊓_ : FiniteMap → FiniteMap → FiniteMap
_⊓_ (m₁ , km₁≡ks) (m₂ , km₂≡ks) =
( m₁ ⊓ᵐ m₂
, trans (sym (⊓-equal-keys {m₁} {m₂} (trans (km₁≡ks) (sym km₂≡ks))))
km₁≡ks
)
_⊓_ (m₁ , km₁≡ks) (m₂ , km₂≡ks) = (m₁ ⊓ᵐ m₂ , trans (sym (⊓-equal-keys {m₁} {m₂} (trans (km₁≡ks) (sym km₂≡ks)))) km₁≡ks)
≈-equiv : IsEquivalence FiniteMap _≈_
≈-equiv = record
{ ≈-refl =
λ {(m , _)} → IsEquivalence.≈-refl ≈ᵐ-equiv {m}
; ≈-sym =
λ {(m₁ , _)} {(m₂ , _)} → IsEquivalence.≈-sym ≈ᵐ-equiv {m₁} {m₂}
; ≈-trans =
λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} →
IsEquivalence.≈-trans ≈ᵐ-equiv {m₁} {m₂} {m₃}
{ ≈-refl = λ {(m , _)} → IsEquivalence.≈-refl ≈ᵐ-equiv {m}
; ≈-sym = λ {(m₁ , _)} {(m₂ , _)} → IsEquivalence.≈-sym ≈ᵐ-equiv {m₁} {m₂}
; ≈-trans = λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} → IsEquivalence.≈-trans ≈ᵐ-equiv {m₁} {m₂} {m₃}
}
isUnionSemilattice : IsSemilattice FiniteMap _≈_ _⊔_
isUnionSemilattice = record
{ ≈-equiv = ≈-equiv
; ≈-⊔-cong =
λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} {(m₄ , _)} m₁≈m₂ m₃≈m₄ →
≈ᵐ-⊔ᵐ-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄
; ≈-⊔-cong = λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} {(m₄ , _)} m₁≈m₂ m₃≈m₄ → ≈ᵐ-⊔ᵐ-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄
; ⊔-assoc = λ (m₁ , _) (m₂ , _) (m₃ , _) → ⊔ᵐ-assoc m₁ m₂ m₃
; ⊔-comm = λ (m₁ , _) (m₂ , _) → ⊔ᵐ-comm m₁ m₂
; ⊔-idemp = λ (m , _) → ⊔ᵐ-idemp m
@ -81,9 +67,7 @@ module _ (ks : List A) where
isIntersectSemilattice : IsSemilattice FiniteMap _≈_ _⊓_
isIntersectSemilattice = record
{ ≈-equiv = ≈-equiv
; ≈-⊔-cong =
λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} {(m₄ , _)} m₁≈m₂ m₃≈m₄ →
≈ᵐ-⊓ᵐ-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄
; ≈-⊔-cong = λ {(m₁ , _)} {(m₂ , _)} {(m₃ , _)} {(m₄ , _)} m₁≈m₂ m₃≈m₄ → ≈ᵐ-⊓ᵐ-cong {m₁} {m₂} {m₃} {m₄} m₁≈m₂ m₃≈m₄
; ⊔-assoc = λ (m₁ , _) (m₂ , _) (m₃ , _) → ⊓ᵐ-assoc m₁ m₂ m₃
; ⊔-comm = λ (m₁ , _) (m₂ , _) → ⊓ᵐ-comm m₁ m₂
; ⊔-idemp = λ (m , _) → ⊓ᵐ-idemp m

289
Lattice/FiniteValueMap.agda
View File

@ -4,8 +4,7 @@
open import Lattice
open import Equivalence
open import Relation.Binary.PropositionalEquality as Eq
using (_≡_; refl; sym; trans; cong; subst)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
open import Relation.Binary.Definitions using (Decidable)
open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)
open import Function.Definitions using (Inverseˡ; Inverseʳ)
@ -28,45 +27,18 @@ open import Data.List.Relation.Unary.All using (All)
open import Data.List.Relation.Unary.Any using (Any; here; there)
open import Relation.Nullary using (¬_)
open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A lB
using
( subset-impl
; locate; forget
; _∈_
; Map-functional
; Expr-Provenance
; _∩_; _∪_; `_
; in₁; in₂; bothᵘ; single
; ⊔-combines
)
open import Lattice.Map A B _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A lB using (subset-impl; locate; forget; _∈_; Map-functional; Expr-Provenance; _∩_; _∪_; `_; in₁; in₂; bothᵘ; single; ⊔-combines)
open import Lattice.FiniteMap A B _≈₂_ _⊔₂_ _⊓₂_ ≡-dec-A lB public
module IterProdIsomorphism where
open import Data.Unit using (⊤; tt)
open import Lattice.Unit using ()
renaming
( _≈_ to _≈ᵘ_
; _⊔_ to _⊔ᵘ_
; _⊓_ to _⊓ᵘ_
; ≈-dec to ≈ᵘ-dec
; isLattice to isLatticeᵘ
; ≈-equiv to ≈ᵘ-equiv
; fixedHeight to fixedHeightᵘ
)
open import Lattice.IterProd _≈₂_ _≈ᵘ_ _⊔₂_ _⊔ᵘ_ _⊓₂_ _⊓ᵘ_ lB isLatticeᵘ
as IP
using (IterProd)
open IsLattice lB using ()
renaming
( ≈-trans to ≈₂-trans
; ≈-sym to ≈₂-sym
; FixedHeight to FixedHeight₂
)
open import Lattice.Unit using () renaming (_≈_ to _≈ᵘ_; _⊔_ to _⊔ᵘ_; _⊓_ to _⊓ᵘ_; ≈-dec to ≈ᵘ-dec; isLattice to isLatticeᵘ; ≈-equiv to ≈ᵘ-equiv; fixedHeight to fixedHeightᵘ)
open import Lattice.IterProd _≈₂_ _≈ᵘ_ _⊔₂_ _⊔ᵘ_ _⊓₂_ _⊓ᵘ_ lB isLatticeᵘ as IP using (IterProd)
open IsLattice lB using () renaming (≈-trans to ≈₂-trans; ≈-sym to ≈₂-sym; FixedHeight to FixedHeight₂)
from : ∀ {ks : List A} → FiniteMap ks → IterProd (length ks)
from {[]} (([] , _) , _) = tt
from {k ∷ ks'} (((k' , v) ∷ fm' , push _ uks') , refl) =
(v , from ((fm' , uks'), refl))
from {k ∷ ks'} (((k' , v) ∷ fm' , push _ uks') , refl) = (v , from ((fm' , uks'), refl))
to : ∀ {ks : List A} → Unique ks → IterProd (length ks) → FiniteMap ks
to {[]} _ ⊤ = (([] , empty) , refl)
@ -93,20 +65,17 @@ module IterProdIsomorphism where
_⊆ᵐ_ : ∀ {ks₁ ks₂ : List A} → FiniteMap ks₁ → FiniteMap ks₂ → Set
_⊆ᵐ_ fm₁ fm₂ = subset-impl (proj₁ (proj₁ fm₁)) (proj₁ (proj₁ fm₂))
_≈ⁱᵖ_ : ∀ {n : ℕ} → IterProd n → IterProd n → Set
_≈ⁱᵖ_ {n} = IP._≈_ n
_≈ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → Set
_≈ⁱᵖ_ {ks} = IP._≈_ (length ks)
_⊔ⁱᵖ_ : ∀ {ks : List A} →
IterProd (length ks) → IterProd (length ks) → IterProd (length ks)
_⊔ⁱᵖ_ : ∀ {ks : List A} → IterProd (length ks) → IterProd (length ks) → IterProd (length ks)
_⊔ⁱᵖ_ {ks} = IP._⊔_ (length ks)
_∈ᵐ_ : ∀ {ks : List A} → A × B → FiniteMap ks → Set
_∈ᵐ_ {ks} k,v fm = k,v ∈ proj₁ fm
-- The left inverse is: from (to x) = x
from-to-inverseˡ : ∀ {ks : List A} (uks : Unique ks) →
Inverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {length ks})
(from {ks}) (to {ks} uks)
Inverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {ks}) (from {ks}) (to {ks} uks) -- from (to x) = x
from-to-inverseˡ {[]} _ _ = IsEquivalence.≈-refl (IP.≈-equiv 0)
from-to-inverseˡ {k ∷ ks'} (push k≢ks' uks') (v , rest)
with ((fm' , ufm') , refl) ← to uks' rest in p rewrite sym p =
@ -118,47 +87,33 @@ module IterProdIsomorphism where
-- The map has its own uniqueness proof, but the call to 'to' needs a standalone
-- uniqueness proof too. Work with both proofs as needed to thread things through.
--
-- The right inverse is: to (from x) = x
from-to-inverseʳ : ∀ {ks : List A} (uks : Unique ks) →
Inverseʳ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {length ks})
(from {ks}) (to {ks} uks)
from-to-inverseʳ {[]} _ (([] , empty) , kvs≡ks) rewrite kvs≡ks =
( (λ k v ())
, (λ k v ())
)
from-to-inverseʳ {k ∷ ks'} uks@(push _ uks'₁) fm₁@(((k , v) ∷ fm'₁ , push _ uks'₂) , refl)
with to uks'₁ (from ((fm'₁ , uks'₂) , refl))
| from-to-inverseʳ {ks'} uks'₁ ((fm'₁ , uks'₂) , refl)
... | ((fm'₂ , ufm'₂) , _)
| (fm'₂⊆fm'₁ , fm'₁⊆fm'₂) = (m₂⊆m₁ , m₁⊆m₂)
Inverseʳ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {ks}) (from {ks}) (to {ks} uks) -- to (from x) = x
from-to-inverseʳ {[]} _ (([] , empty) , kvs≡ks) rewrite kvs≡ks = ((λ k v ()) , (λ k v ()))
from-to-inverseʳ {k ∷ ks'} uks@(push k≢ks'₁ uks'₁) fm₁@(m₁@((k , v) ∷ fm'₁ , push k≢ks'₂ uks'₂) , refl)
with to uks'₁ (from ((fm'₁ , uks'₂) , refl)) | from-to-inverseʳ {ks'} uks'₁ ((fm'₁ , uks'₂) , refl)
... | ((fm'₂ , ufm'₂) , _) | (fm'₂⊆fm'₁ , fm'₁⊆fm'₂) = (m₂⊆m₁ , m₁⊆m₂)
where
kvs₁ = (k , v) ∷ fm'₁
kvs₂ = (k , v) ∷ fm'₂
m₁⊆m₂ : subset-impl kvs₁ kvs₂
m₁⊆m₂ k' v' (here refl) =
(v' , (IsLattice.≈-refl lB , here refl))
m₁⊆m₂ k' v' (here refl) = (v' , (IsLattice.≈-refl lB , here refl))
m₁⊆m₂ k' v' (there k',v'∈fm'₁) =
let (v'' , (v'≈v'' , k',v''∈fm'₂)) =
fm'₁⊆fm'₂ k' v' k',v'∈fm'₁
let (v'' , (v'≈v'' , k',v''∈fm'₂)) = fm'₁⊆fm'₂ k' v' k',v'∈fm'₁
in (v'' , (v'≈v'' , there k',v''∈fm'₂))
m₂⊆m₁ : subset-impl kvs₂ kvs₁
m₂⊆m₁ k' v' (here refl) =
(v' , (IsLattice.≈-refl lB , here refl))
m₂⊆m₁ k' v' (here refl) = (v' , (IsLattice.≈-refl lB , here refl))
m₂⊆m₁ k' v' (there k',v'∈fm'₂) =
let (v'' , (v'≈v'' , k',v''∈fm'₁)) =
fm'₂⊆fm'₁ k' v' k',v'∈fm'₂
let (v'' , (v'≈v'' , k',v''∈fm'₁)) = fm'₂⊆fm'₁ k' v' k',v'∈fm'₂
in (v'' , (v'≈v'' , there k',v''∈fm'₁))
private
first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
Σ B (λ v → (k , v) ∈ proj₁ fm)
first-key-in-map : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → Σ B (λ v → (k , v) ∈ proj₁ fm)
first-key-in-map (((k , v) ∷ _ , _) , refl) = (v , here refl)
from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
proj₁ (from fm) ≡ proj₁ (first-key-in-map fm)
from-first-value : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₁ (from fm) ≡ proj₁ (first-key-in-map fm)
from-first-value {k} {ks} (((k , v) ∷ _ , push _ _) , refl) = refl
-- We need pop because reasoning about two distinct 'refl' pattern
@ -168,183 +123,109 @@ module IterProdIsomorphism where
pop : ∀ {k : A} {ks : List A} → FiniteMap (k ∷ ks) → FiniteMap ks
pop (((_ ∷ fm') , push _ ufm') , refl) = ((fm' , ufm') , refl)
pop-≈ : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) →
fm₁ ≈ᵐ fm₂ → pop fm₁ ≈ᵐ pop fm₂
pop-≈ {k} {ks} fm₁ fm₂ (fm₁⊆fm₂ , fm₂⊆fm₁) =
(narrow fm₁⊆fm₂ , narrow fm₂⊆fm₁)
pop-≈ : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) → fm₁ ≈ᵐ fm₂ → pop fm₁ ≈ᵐ pop fm₂
pop-≈ {k} {ks} fm₁ fm₂ (fm₁⊆fm₂ , fm₂⊆fm₁) = (narrow fm₁⊆fm₂ , narrow fm₂⊆fm₁)
where
narrow₁ : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} →
fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ fm₂
narrow₁ {(_ ∷ _ , push _ _) , refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁ =
kvs₁⊆kvs₂ k' v' (there k',v'∈fm'₁)
narrow₁ : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ fm₂
narrow₁ {(_ ∷ _ , push _ _) , refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁ = kvs₁⊆kvs₂ k' v' (there k',v'∈fm'₁)
narrow₂ : ∀ {fm₁ : FiniteMap ks} {fm₂ : FiniteMap (k ∷ ks)} →
fm₁ ⊆ᵐ fm₂ → fm₁ ⊆ᵐ pop fm₂
narrow₂ : ∀ {fm₁ : FiniteMap ks} {fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂ → fm₁ ⊆ᵐ pop fm₂
narrow₂ {fm₁} {fm₂ = (_ ∷ fm'₂ , push k≢ks _) , kvs≡ks@refl} kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
with kvs₁⊆kvs₂ k' v' k',v'∈fm'₁
... | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) =
⊥-elim (All¬-¬Any k≢ks (forget {m = proj₁ fm₁} k',v'∈fm'₁))
... | (v'' , (v'≈v'' , there k',v'∈fm'₂)) =
(v'' , (v'≈v'' , k',v'∈fm'₂))
... | (v'' , (v'≈v'' , here refl)) rewrite sym (proj₂ fm₁) = ⊥-elim (All¬-¬Any k≢ks (forget {m = proj₁ fm₁} k',v'∈fm'₁))
... | (v'' , (v'≈v'' , there k',v'∈fm'₂)) = (v'' , (v'≈v'' , k',v'∈fm'₂))
narrow : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} →
fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ pop fm₂
narrow : ∀ {fm₁ fm₂ : FiniteMap (k ∷ ks)} → fm₁ ⊆ᵐ fm₂ → pop fm₁ ⊆ᵐ pop fm₂
narrow {fm₁} {fm₂} x = narrow₂ {pop fm₁} (narrow₁ {fm₂ = fm₂} x)
k,v∈pop⇒k,v∈ : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) →
(k' , v) ∈ᵐ pop fm → (¬ k ≡ k' × ((k' , v) ∈ᵐ fm))
k,v∈pop⇒k,v∈ : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) → (k' , v) ∈ᵐ pop fm → (¬ k ≡ k' × ((k' , v) ∈ᵐ fm))
k,v∈pop⇒k,v∈ {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k',v∈fm =
( (λ { refl → All¬-¬Any k≢ks (forget {m = (fm' , uks')} k',v∈fm) })
, there k',v∈fm
)
((λ { refl → All¬-¬Any k≢ks (forget {m = (fm' , uks')} k',v∈fm) }), there k',v∈fm)
k,v∈⇒k,v∈pop : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) →
¬ k ≡ k' → (k' , v) ∈ᵐ fm → (k' , v) ∈ᵐ pop fm
k,v∈⇒k,v∈pop (m@(_ ∷ _ , push k≢ks _) , refl) k≢k' (here refl) = ⊥-elim (k≢k' refl)
k,v∈⇒k,v∈pop (m@(_ ∷ _ , push k≢ks _) , refl) k≢k' (there k,v'∈fm') = k,v'∈fm'
k,v∈⇒k,v∈pop : ∀ {k : A} {ks : List A} {k' : A} {v : B} (fm : FiniteMap (k ∷ ks)) → ¬ k ≡ k' → (k' , v) ∈ᵐ fm → (k' , v) ∈ᵐ pop fm
k,v∈⇒k,v∈pop {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k≢k' (here refl) = ⊥-elim (k≢k' refl)
k,v∈⇒k,v∈pop {k} {ks} {k'} {v} (m@((k , _) ∷ fm' , push k≢ks uks') , refl) k≢k' (there k,v'∈fm') = k,v'∈fm'
FromBothMaps : ∀ (k : A) (v : B) {ks : List A} (fm₁ fm₂ : FiniteMap ks) → Set
FromBothMaps k v fm₁ fm₂ =
Σ (B × B)
(λ (v₁ , v₂) → ( (v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : FiniteMap ks) {k : A} {v : B} →
(k , v) ∈ᵐ (fm₁ ⊔ᵐ fm₂) → FromBothMaps k v fm₁ fm₂
Provenance-union : ∀ {ks : List A} (fm₁ fm₂ : FiniteMap ks) {k : A} {v : B} → (k , v) ∈ᵐ (fm₁ ⊔ᵐ fm₂) → Σ (B × B) (λ (v₁ , v₂) → ((v ≡ v₁ ⊔₂ v₂) × ((k , v₁) ∈ᵐ fm₁ × (k , v₂) ∈ᵐ fm₂)))
Provenance-union fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂)
... | (_ , (in₁ (single k,v∈m₁) k∉km₂ , _))
with k∈km₁ ← (forget {m = m₁} k,v∈m₁)
rewrite trans ks₁≡ks (sym ks₂≡ks) =
⊥-elim (k∉km₂ k∈km₁)
... | (_ , (in₂ k∉km₁ (single k,v∈m₂) , _))
with k∈km₂ ← (forget {m = m₂} k,v∈m₂)
rewrite trans ks₁≡ks (sym ks₂≡ks) =
⊥-elim (k∉km₁ k∈km₂)
... | (_ , (in₁ (single k,v∈m₁) k∉km₂ , _)) with k∈km₁ ← (forget {m = m₁} k,v∈m₁) rewrite trans ks₁≡ks (sym ks₂≡ks) = ⊥-elim (k∉km₂ k∈km₁)
... | (_ , (in₂ k∉km₁ (single k,v∈m₂) , _)) with k∈km₂ ← (forget {m = m₂} k,v∈m₂) rewrite trans ks₁≡ks (sym ks₂≡ks) = ⊥-elim (k∉km₁ k∈km₂)
... | (v₁⊔v₂ , (bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) , k,v₁⊔v₂∈m₁m₂))
rewrite Map-functional {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂ =
((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
rewrite Map-functional {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂ = ((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
pop-⊔-distr : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) →
pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) =
(pfm₁fm₂⊆pfm₁pfm₂ , pfm₁pfm₂⊆pfm₁fm₂)
pop-⊔-distr : ∀ {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ∷ ks)) → pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) = (pfm₁fm₂⊆pfm₁pfm₂ , pfm₁pfm₂⊆pfm₁fm₂)
where
-- pfm₁fm₂⊆pfm₁pfm₂ = {!!}
pfm₁fm₂⊆pfm₁pfm₂ : pop (fm₁ ⊔ᵐ fm₂) ⊆ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
pfm₁fm₂⊆pfm₁pfm₂ k' v' k',v'∈pfm₁fm₂
with (k≢k' , k',v'∈fm₁fm₂) ← k,v∈pop⇒k,v∈ (fm₁ ⊔ᵐ fm₂) k',v'∈pfm₁fm₂
with ((v₁ , v₂) , (refl , (k,v₁∈fm₁ , k,v₂∈fm₂)))
← Provenance-union fm₁ fm₂ k',v'∈fm₁fm₂
with ((v₁ , v₂) , (refl , (k,v₁∈fm₁ , k,v₂∈fm₂))) ← Provenance-union fm₁ fm₂ k',v'∈fm₁fm₂
with k',v₁∈pfm₁ ← k,v∈⇒k,v∈pop fm₁ k≢k' k,v₁∈fm₁
with k',v₂∈pfm₂ ← k,v∈⇒k,v∈pop fm₂ k≢k' k,v₂∈fm₂
=
( v₁ ⊔₂ v₂
, (IsLattice.≈-refl lB
, ⊔-combines {m₁ = proj₁ (pop fm₁)}
{m₂ = proj₁ (pop fm₂)}
k',v₁∈pfm₁ k',v₂∈pfm₂
)
)
= (v₁ ⊔₂ v₂ , (IsLattice.≈-refl lB , ⊔-combines {m₁ = proj₁ (pop fm₁)} {m₂ = proj₁ (pop fm₂)} k',v₁∈pfm₁ k',v₂∈pfm₂))
pfm₁pfm₂⊆pfm₁fm₂ : (pop fm₁ ⊔ᵐ pop fm₂) ⊆ᵐ pop (fm₁ ⊔ᵐ fm₂)
pfm₁pfm₂⊆pfm₁fm₂ k' v' k',v'∈pfm₁pfm₂
with ((v₁ , v₂) , (refl , (k,v₁∈pfm₁ , k,v₂∈pfm₂)))
← Provenance-union (pop fm₁) (pop fm₂) k',v'∈pfm₁pfm₂
with ((v₁ , v₂) , (refl , (k,v₁∈pfm₁ , k,v₂∈pfm₂))) ← Provenance-union (pop fm₁) (pop fm₂) k',v'∈pfm₁pfm₂
with (k≢k' , k',v₁∈fm₁) ← k,v∈pop⇒k,v∈ fm₁ k,v₁∈pfm₁
with (_ , k',v₂∈fm₂) ← k,v∈pop⇒k,v∈ fm₂ k,v₂∈pfm₂
=
( v₁ ⊔₂ v₂
, ( IsLattice.≈-refl lB
, k,v∈⇒k,v∈pop (fm₁ ⊔ᵐ fm₂) k≢k'
(⊔-combines {m₁ = m₁} {m₂ = m₂}
k',v₁∈fm₁ k',v₂∈fm₂)
)
)
= (v₁ ⊔₂ v₂ , (IsLattice.≈-refl lB , k,v∈⇒k,v∈pop (fm₁ ⊔ᵐ fm₂) k≢k' (⊔-combines {m₁ = m₁} {m₂ = m₂} k',v₁∈fm₁ k',v₂∈fm₂)))
from-rest : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) →
proj₂ (from fm) ≡ from (pop fm)
from-rest : ∀ {k : A} {ks : List A} (fm : FiniteMap (k ∷ ks)) → proj₂ (from fm) ≡ from (pop fm)
from-rest (((_ ∷ fm') , push _ ufm') , refl) = refl
from-preserves-≈ : ∀ {ks : List A} → {fm₁ fm₂ : FiniteMap ks} →
fm₁ ≈ᵐ fm₂ → (_≈ⁱᵖ_ {length ks}) (from fm₁) (from fm₂)
from-preserves-≈ {[]} {_} {_} _ = IsEquivalence.≈-refl ≈ᵘ-equiv
from-preserves-≈ : ∀ {ks : List A} → {fm₁ fm₂ : FiniteMap ks} → fm₁ ≈ᵐ fm₂ → (_≈ⁱᵖ_ {ks}) (from fm₁) (from fm₂)
from-preserves-≈ {[]} {([] , _) , _} {([] , _) , _} _ = IsEquivalence.≈-refl ≈ᵘ-equiv
from-preserves-≈ {k ∷ ks'} {fm₁@(m₁ , _)} {fm₂@(m₂ , _)} fm₁≈fm₂@(kvs₁⊆kvs₂ , kvs₂⊆kvs₁)
with first-key-in-map fm₁
| first-key-in-map fm₂
| from-first-value fm₁
| from-first-value fm₂
with first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value fm₁ | from-first-value fm₂
... | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl
with kvs₁⊆kvs₂ _ _ k,v₁∈fm₁
... | (v₁' , (v₁≈v₁' , k,v₁'∈fm₂))
rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₁'∈fm₂
rewrite from-rest fm₁ rewrite from-rest fm₂
=
( v₁≈v₁'
, from-preserves-≈ {ks'} {pop fm₁} {pop fm₂}
(pop-≈ fm₁ fm₂ fm₁≈fm₂)
)
= (v₁≈v₁' , from-preserves-≈ {ks'} {pop fm₁} {pop fm₂} (pop-≈ fm₁ fm₂ fm₁≈fm₂))
to-preserves-≈ : ∀ {ks : List A} (uks : Unique ks) {ip₁ ip₂ : IterProd (length ks)} →
_≈ⁱᵖ_ {length ks} ip₁ ip₂ → to uks ip₁ ≈ᵐ to uks ip₂
to-preserves-≈ : ∀ {ks : List A} (uks : Unique ks) {ip₁ ip₂ : IterProd (length ks)} → _≈ⁱᵖ_ {ks} ip₁ ip₂ → to uks ip₁ ≈ᵐ to uks ip₂
to-preserves-≈ {[]} empty {tt} {tt} _ = ((λ k v ()), (λ k v ()))
to-preserves-≈ {k ∷ ks'} uks@(push k≢ks' uks') {ip₁@(v₁ , rest₁)} {ip₂@(v₂ , rest₂)} (v₁≈v₂ , rest₁≈rest₂) = (fm₁⊆fm₂ , fm₂⊆fm₁)
where
inductive-step : ∀ {v₁ v₂ : B} {rest₁ rest₂ : IterProd (length ks')} →
v₁ ≈₂ v₂ → _≈ⁱᵖ_ {length ks'} rest₁ rest₂ →
to uks (v₁ , rest₁) ⊆ᵐ to uks (v₂ , rest₂)
inductive-step {v₁} {v₂} {rest₁} {rest₂} v₁≈v₂ rest₁≈rest₂ k v k,v∈kvs₁
fm₁⊆fm₂ : to uks ip₁ ⊆ᵐ to uks ip₂
fm₁⊆fm₂ k v k,v∈kvs₁
with ((fm'₁ , ufm'₁) , fm'₁≡ks') ← to uks' rest₁ in p₁
with ((fm'₂ , ufm'₂) , fm'₂≡ks') ← to uks' rest₂ in p₂
with k,v∈kvs₁
... | here refl = (v₂ , (v₁≈v₂ , here refl))
... | there k,v∈fm'₁ with refl ← p₁ with refl ← p₂ =
let
(fm'₁⊆fm'₂ , _) = to-preserves-≈ uks' {rest₁} {rest₂}
rest₁≈rest₂
(v' , (v≈v' , k,v'∈kvs₁)) = fm'₁⊆fm'₂ k v k,v∈fm'₁
in
(v' , (v≈v' , there k,v'∈kvs₁))
fm₁⊆fm₂ : to uks ip₁ ⊆ᵐ to uks ip₂
fm₁⊆fm₂ = inductive-step v₁≈v₂ rest₁≈rest₂
... | there k,v∈fm'₁ with refl ← p₁ with refl ← p₂ = let (v' , (v≈v' , k,v'∈kvs₁)) = proj₁ (to-preserves-≈ uks' {rest₁} {rest₂} rest₁≈rest₂) k v k,v∈fm'₁ in (v' , (v≈v' , there k,v'∈kvs₁))
fm₂⊆fm₁ : to uks ip₂ ⊆ᵐ to uks ip₁
fm₂⊆fm₁ = inductive-step (≈₂-sym v₁≈v₂)
(IP.≈-sym (length ks') rest₁≈rest₂)
fm₂⊆fm₁ k v k,v∈kvs₂
with ((fm'₁ , ufm'₁) , fm'₁≡ks') ← to uks' rest₁ in p₁
with ((fm'₂ , ufm'₂) , fm'₂≡ks') ← to uks' rest₂ in p₂
with k,v∈kvs₂
... | here refl = (v₁ , (IsLattice.≈-sym lB v₁≈v₂ , here refl))
... | there k,v∈fm'₂ with refl ← p₁ with refl ← p₂ = let (v' , (v≈v' , k,v'∈kvs₂)) = proj₂ (to-preserves-≈ uks' {rest₁} {rest₂} rest₁≈rest₂) k v k,v∈fm'₂ in (v' , (v≈v' , there k,v'∈kvs₂))
from-⊔-distr : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) →
_≈ⁱᵖ_ {length ks} (from (fm₁ ⊔ᵐ fm₂))
(_⊔ⁱᵖ_ {ks} (from fm₁) (from fm₂))
from-⊔-distr : ∀ {ks : List A} → (fm₁ fm₂ : FiniteMap ks) → _≈ⁱᵖ_ {ks} (from (fm₁ ⊔ᵐ fm₂)) (_⊔ⁱᵖ_ {ks} (from fm₁) (from fm₂))
from-⊔-distr {[]} fm₁ fm₂ = IsEquivalence.≈-refl ≈ᵘ-equiv
from-⊔-distr {k ∷ ks} fm₁@(m₁ , _) fm₂@(m₂ , _)
with first-key-in-map (fm₁ ⊔ᵐ fm₂)
| first-key-in-map fm₁
| first-key-in-map fm₂
| from-first-value (fm₁ ⊔ᵐ fm₂)
| from-first-value fm₁ | from-first-value fm₂
with first-key-in-map (fm₁ ⊔ᵐ fm₂) | first-key-in-map fm₁ | first-key-in-map fm₂ | from-first-value (fm₁ ⊔ᵐ fm₂) | from-first-value fm₁ | from-first-value fm₂
... | (v , k,v∈fm₁fm₂) | (v₁ , k,v₁∈fm₁) | (v₂ , k,v₂∈fm₂) | refl | refl | refl
with Expr-Provenance k ((` m₁) ∪ (` m₂)) (forget {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂)
... | (_ , (in₁ _ k∉km₂ , _)) = ⊥-elim (k∉km₂ (forget {m = m₂}
k,v₂∈fm₂))
... | (_ , (in₂ k∉km₁ _ , _)) = ⊥-elim (k∉km₁ (forget {m = m₁}
k,v₁∈fm₁))
... | (_ , (in₁ _ k∉km₂ , _)) = ⊥-elim (k∉km₂ (forget {m = m₂} k,v₂∈fm₂))
... | (_ , (in₂ k∉km₁ _ , _)) = ⊥-elim (k∉km₁ (forget {m = m₁} k,v₁∈fm₁))
... | (v₁⊔v₂ , (bothᵘ {v₁'} {v₂'} (single k,v₁'∈m₁) (single k,v₂'∈m₂) , k,v₁⊔v₂∈m₁m₂))
rewrite Map-functional {m = m₁} k,v₁∈fm₁ k,v₁'∈m₁
rewrite Map-functional {m = m₂} k,v₂∈fm₂ k,v₂'∈m₂
rewrite Map-functional {m = proj₁ (fm₁ ⊔ᵐ fm₂)} k,v∈fm₁fm₂ k,v₁⊔v₂∈m₁m₂
rewrite from-rest (fm₁ ⊔ᵐ fm₂) rewrite from-rest fm₁ rewrite from-rest fm₂
= ( IsLattice.≈-refl lB
, IsEquivalence.≈-trans
(IP.≈-equiv (length ks))
(from-preserves-≈ {_} {pop (fm₁ ⊔ᵐ fm₂)}
{pop fm₁ ⊔ᵐ pop fm₂}
(pop-⊔-distr fm₁ fm₂))
((from-⊔-distr (pop fm₁) (pop fm₂)))
, IsEquivalence.≈-trans (IP.≈-equiv (length ks)) (from-preserves-≈ {_} {pop (fm₁ ⊔ᵐ fm₂)} {pop fm₁ ⊔ᵐ pop fm₂} (pop-⊔-distr fm₁ fm₂)) ((from-⊔-distr (pop fm₁) (pop fm₂)))
)
to-⊔-distr : ∀ {ks : List A} (uks : Unique ks) → (ip₁ ip₂ : IterProd (length ks)) →
to uks (_⊔ⁱᵖ_ {ks} ip₁ ip₂) ≈ᵐ (to uks ip₁ ⊔ᵐ to uks ip₂)
to-⊔-distr : ∀ {ks : List A} (uks : Unique ks) → (ip₁ ip₂ : IterProd (length ks)) → to uks (_⊔ⁱᵖ_ {ks} ip₁ ip₂) ≈ᵐ (to uks ip₁ ⊔ᵐ to uks ip₂)
to-⊔-distr {[]} empty tt tt = ((λ k v ()), (λ k v ()))
to-⊔-distr {ks@(k ∷ ks')} uks@(push k≢ks' uks') ip₁@(v₁ , rest₁) ip₂@(v₂ , rest₂) = (fm⊆fm₁fm₂ , fm₁fm₂⊆fm)
where
@ -358,45 +239,27 @@ module IterProdIsomorphism where
fm⊆fm₁fm₂ k v (here refl) =
(v₁ ⊔₂ v₂
, (IsLattice.≈-refl lB
, ⊔-combines {k} {v₁} {v₂} {proj₁ fm₁} {proj₁ fm₂}
(here refl) (here refl)
, ⊔-combines {k} {v₁} {v₂} {proj₁ fm₁} {proj₁ fm₂} (here refl) (here refl)
)
)
fm⊆fm₁fm₂ k' v (there k',v∈fm')
with (fm'⊆fm'₁fm'₂ , _) ← to-⊔-distr uks' rest₁ rest₂
with (v' , (v₁⊔v₂≈v' , k',v'∈fm'₁fm'₂))
← fm'⊆fm'₁fm'₂ k' v k',v∈fm'
with (_ , (refl , (v₁∈fm'₁ , v₂∈fm'₂)))
← Provenance-union fm₁' fm₂' k',v'∈fm'₁fm'₂ =
( v'
, ( v₁⊔v₂≈v'
, ⊔-combines {m₁ = proj₁ fm₁} {m₂ = proj₁ fm₂}
(there v₁∈fm'₁) (there v₂∈fm'₂)
)
)
with (v' , (v₁⊔v₂≈v' , k',v'∈fm'₁fm'₂)) ← fm'⊆fm'₁fm'₂ k' v k',v∈fm'
with (_ , (refl , (v₁∈fm'₁ , v₂∈fm'₂))) ← Provenance-union fm₁' fm₂' k',v'∈fm'₁fm'₂ =
(v' , (v₁⊔v₂≈v' , ⊔-combines {m₁ = proj₁ fm₁} {m₂ = proj₁ fm₂} (there v₁∈fm'₁) (there v₂∈fm'₂)))
fm₁fm₂⊆fm : (fm₁ ⊔ᵐ fm₂) ⊆ᵐ fm
fm₁fm₂⊆fm k' v k',v∈fm₁fm₂
with (_ , fm'₁fm'₂⊆fm')
← to-⊔-distr uks' rest₁ rest₂
with (_ , (refl , (v₁∈fm₁ , v₂∈fm₂)))
← Provenance-union fm₁ fm₂ k',v∈fm₁fm₂
with (_ , fm'₁fm'₂⊆fm') ← to-⊔-distr uks' rest₁ rest₂
with (_ , (refl , (v₁∈fm₁ , v₂∈fm₂))) ← Provenance-union fm₁ fm₂ k',v∈fm₁fm₂
with v₁∈fm₁ | v₂∈fm₂
... | here refl | here refl =
(v , (IsLattice.≈-refl lB , here refl))
... | here refl | there k',v₂∈fm₂' =
⊥-elim (All¬-¬Any k≢ks' (subst (k' ∈ˡ_) (proj₂ fm₂')
(forget {m = proj₁ fm₂'} k',v₂∈fm₂')))
... | there k',v₁∈fm₁' | here refl =
⊥-elim (All¬-¬Any k≢ks' (subst (k' ∈ˡ_) (proj₂ fm₁')
(forget {m = proj₁ fm₁'} k',v₁∈fm₁')))
... | here refl | here refl = (v , (IsLattice.≈-refl lB , here refl))
... | here refl | there k',v₂∈fm₂' = ⊥-elim (All¬-¬Any k≢ks' (subst (λ list → k' ∈ˡ list) (proj₂ fm₂') (forget {m = proj₁ fm₂'} k',v₂∈fm₂')))
... | there k',v₁∈fm₁' | here refl = ⊥-elim (All¬-¬Any k≢ks' (subst (λ list → k' ∈ˡ list) (proj₂ fm₁') (forget {m = proj₁ fm₁'} k',v₁∈fm₁')))
... | there k',v₁∈fm₁' | there k',v₂∈fm₂' =
let
k',v₁v₂∈fm₁'fm₂' =
⊔-combines {m₁ = proj₁ fm₁'} {m₂ = proj₁ fm₂'}
k',v₁∈fm₁' k',v₂∈fm₂'
(v' , (v₁⊔v₂≈v' , v'∈fm')) =
fm'₁fm'₂⊆fm' _ _ k',v₁v₂∈fm₁'fm₂'
k',v₁v₂∈fm₁'fm₂' = ⊔-combines {m₁ = proj₁ fm₁'} {m₂ = proj₁ fm₂'} k',v₁∈fm₁' k',v₂∈fm₂'
(v' , (v₁⊔v₂≈v' , v'∈fm')) = fm'₁fm'₂⊆fm' _ _ k',v₁v₂∈fm₁'fm₂'
in
(v' , (v₁⊔v₂≈v' , there v'∈fm'))

10
Main.agda
View File

@ -56,4 +56,14 @@ dumbFunction-Monotonic {m₁} {m₂} m₁≼m₂ = ⊔-Monotonicˡ dumb {m₁} {
open import Fixedpoint {0ℓ} {FiniteHeightMap xyzw} {8} {_≈_} {_⊔_} {_⊓_} (≈-dec fhlᵘ xyzw-Unique ≈ᵘ-dec) (FiniteHeightLattice.isFiniteHeightLattice fhlⁱᵖ) dumbFunction (λ {m₁} {m₂} m₁≼m₂ → dumbFunction-Monotonic {m₁} {m₂} m₁≼m₂)
-- module Fixedpoint {a} {A : Set a}
-- {h : ℕ}
-- {_≈_ : A → A → Set a}
-- {_⊔_ : A → A → A} {_⊓_ : A → A → A}
-- (≈-dec : IsDecidable _≈_)
-- (flA : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_)
-- (f : A → A)
-- (Monotonicᶠ : Monotonic (IsFiniteHeightLattice._≼_ flA)
-- (IsFiniteHeightLattice._≼_ flA) f) where
main = run {0ℓ} (putStrLn (showMap aᶠ))