Add decidability proofs for properties

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Analysis/Forward.agda

@@ -21,7 +21,7 @@ open IsFiniteHeightLattice isFiniteHeightLatticeˡ
     using () renaming (isLattice to isLatticeˡ)
 
 module WithProg (prog : Program) where
-    open import Analysis.Forward.Lattices L prog hiding (≈ᵛ-Decidable) -- to disambiguate instance resolution
+    open import Analysis.Forward.Lattices L prog hiding (≈ᵛ-Decidable; ≈ᵐ-Decidable) -- to disambiguate instance resolution
     open import Analysis.Forward.Evaluation L prog
     open Program prog
 
@@ -66,7 +66,7 @@ module WithProg (prog : Program) where
                            (joinAll-Mono {sv₁} {sv₂} sv₁≼sv₂)
 
         -- The fixed point of the 'analyze' function is our final goal.
-        open import Fixedpoint ≈ᵐ-Decidable isFiniteHeightLatticeᵐ analyze (λ {m₁} {m₂} m₁≼m₂ → analyze-Mono {m₁} {m₂} m₁≼m₂)
+        open import Fixedpoint analyze (λ {m₁} {m₂} m₁≼m₂ → analyze-Mono {m₁} {m₂} m₁≼m₂)
             using ()
             renaming (aᶠ to result; aᶠ≈faᶠ to result≈analyze-result)
             public

Fixedpoint.agda

@@ -5,8 +5,8 @@ module Fixedpoint {a} {A : Set a}
                   {h : ℕ}
                   {_≈_ : A → A → Set a}
                   {_⊔_ : A → A → A} {_⊓_ : A → A → A}
-                  (≈-Decidable : IsDecidable _≈_)
-                  (flA : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_)
+                  {{ ≈-Decidable : IsDecidable _≈_ }}
+                  {{flA : IsFiniteHeightLattice A h _≈_ _⊔_ _⊓_}}
                   (f : A → A)
                   (Monotonicᶠ : Monotonic (IsFiniteHeightLattice._≼_ flA)
                                           (IsFiniteHeightLattice._≼_ flA) f) where
@@ -28,24 +28,9 @@ private
         using ()
         renaming
             ( ⊥ to ⊥ᴬ
-            ; longestChain to longestChainᴬ
             ; bounded to boundedᴬ
             )
 
-
-    ⊥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 = ≈-trans (⊔-comm ⊥ᴬ a) (≈-trans (≈-⊔-cong (≈-refl {a}) ⊥ᴬ≈a⊓⊥ᴬ) (absorb-⊔-⊓ a ⊥ᴬ))
-    ...         | no ⊥ᴬ̷≈a⊓⊥ᴬ = ⊥-elim (ChainA.Bounded-suc-n boundedᴬ (ChainA.step x≺⊥ᴬ ≈-refl longestChainᴬ))
-                    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 = (≈-trans (⊔-comm _ _) (≈-trans (≈-refl {⊥ᴬ ⊔ (⊥ᴬ ⊓ a)}) (absorb-⊔-⊓ ⊥ᴬ a)) , ⊥ᴬ⊓a̷≈⊥ᴬ)
-
     -- using 'g', for gas, here helps make sure the function terminates.
     -- since A forms a fixed-height lattice, we must find a solution after
     -- 'h' steps at most. Gas is set up such that as soon as it runs
@@ -65,7 +50,7 @@ private
                     c' rewrite +-comm 1 hᶜ = ChainA.concat c (ChainA.step a₂≺fa₂ ≈-refl (ChainA.done (≈-refl {f a₂})))
 
     fix : Σ A (λ a → a ≈ f a)
-    fix = doStep (suc h) 0 ⊥ᴬ ⊥ᴬ (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))
+    fix = doStep (suc h) 0 ⊥ᴬ ⊥ᴬ (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥≼ (f ⊥ᴬ))
 
 aᶠ : A
 aᶠ = proj₁ fix
@@ -85,4 +70,4 @@ private
     ...   | no  _ = stepPreservesLess g' _ _ _ b b≈fb (≼-cong ≈-refl (≈-sym b≈fb) (Monotonicᶠ a₂≼b)) _ _ _
 
 aᶠ≼ : ∀ (a : A) → a ≈ f a → aᶠ ≼ a
-aᶠ≼ a a≈fa = stepPreservesLess (suc h) 0 ⊥ᴬ ⊥ᴬ a a≈fa (⊥ᴬ≼ a) (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))
+aᶠ≼ a a≈fa = stepPreservesLess (suc h) 0 ⊥ᴬ ⊥ᴬ a a≈fa (⊥≼ a) (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥≼ (f ⊥ᴬ))

Isomorphism.agda

@@ -68,6 +68,7 @@ module TransportFiniteHeight
         renaming (⊥ to ⊥₁; ⊤ to ⊤₁; bounded to bounded₁; longestChain to c)
 
     instance
+        fixedHeight : IsLattice.FixedHeight lB height
         fixedHeight = record
             { ⊥ = f ⊥₁
             ; ⊤ = f ⊤₁

Language/Graphs.agda

@@ -124,28 +124,6 @@ buildCfg (s₁ then s₂) = buildCfg s₁ ↦ buildCfg s₂
 buildCfg (if _ then s₁ else s₂) = buildCfg s₁ ∙ buildCfg s₂
 buildCfg (while _ repeat s) = loop (buildCfg s)
 
-private
-    z≢sf : ∀ {n : ℕ} (f : Fin n) → ¬ (zero ≡ suc f)
-    z≢sf f ()
-
-    z≢mapsfs : ∀ {n : ℕ} (fs : List (Fin n)) → All (λ sf → ¬ zero ≡ sf) (List.map suc fs)
-    z≢mapsfs [] = []
-    z≢mapsfs (f ∷ fs') = z≢sf f ∷ z≢mapsfs fs'
-
-    finValues : ∀ (n : ℕ) → Σ (List (Fin n)) Unique
-    finValues 0 = ([] , Utils.empty)
-    finValues (suc n') =
-        let
-            (inds' , unids') = finValues n'
-        in
-            ( zero ∷ List.map suc inds'
-            , push (z≢mapsfs inds') (Unique-map suc suc-injective unids')
-            )
-
-    finValues-complete : ∀ (n : ℕ) (f : Fin n) → f ListMem.∈ (proj₁ (finValues n))
-    finValues-complete (suc n') zero = RelAny.here refl
-    finValues-complete (suc n') (suc f') = RelAny.there (x∈xs⇒fx∈fxs suc (finValues-complete n' f'))
-
 module _ (g : Graph) where
     open import Data.Product.Properties as ProdProp using ()
     private _≟_ = ProdProp.≡-dec (FinProp._≟_ {Graph.size g})
@@ -154,13 +132,13 @@ module _ (g : Graph) where
     open import Data.List.Membership.DecPropositional (_≟_) using (_∈?_)
 
     indices : List (Graph.Index g)
-    indices = proj₁ (finValues (Graph.size g))
+    indices = proj₁ (fins (Graph.size g))
 
     indices-complete : ∀ (idx : (Graph.Index g)) → idx ListMem.∈ indices
-    indices-complete = finValues-complete (Graph.size g)
+    indices-complete = fins-complete (Graph.size g)
 
     indices-Unique : Unique indices
-    indices-Unique = proj₂ (finValues (Graph.size g))
+    indices-Unique = proj₂ (fins (Graph.size g))
 
     predecessors : (Graph.Index g) → List (Graph.Index g)
     predecessors idx = List.filter (λ idx' → (idx' , idx) ∈? (Graph.edges g)) indices

Lattice.agda

@@ -4,7 +4,8 @@ open import Equivalence
 import Chain
 
 open import Relation.Binary.Core using (_Preserves_⟶_ )
-open import Relation.Nullary using (Dec; ¬_)
+open import Relation.Nullary using (Dec; ¬_; yes; no)
+open import Data.Empty using (⊥-elim)
 open import Data.Nat as Nat using (ℕ)
 open import Data.Product using (_×_; Σ; _,_)
 open import Data.Sum using (_⊎_; inj₁; inj₂)
@@ -236,6 +237,28 @@ record IsFiniteHeightLattice {a} (A : Set a)
     field
         {{fixedHeight}} : FixedHeight h
 
+    -- If the equality is decidable, we can prove that the top and bottom
+    -- elements of the chain are least and greatest elements of the lattice
+    module _ {{≈-Decidable : IsDecidable _≈_}} where
+        open IsDecidable ≈-Decidable using () renaming (R-dec to ≈-dec)
+
+        module MyChain = Chain _≈_ ≈-equiv _≺_ ≺-cong
+        open MyChain.Height fixedHeight using (⊥; ⊤) public
+        open MyChain.Height fixedHeight using (bounded; longestChain)
+
+        ⊥≼ : ∀ (a : A) → ⊥ ≼ a
+        ⊥≼ a with ≈-dec a ⊥
+        ...   | yes a≈⊥ = ≼-cong a≈⊥ ≈-refl (≼-refl a)
+        ...   | no a̷≈⊥ with ≈-dec ⊥ (a ⊓ ⊥)
+        ...         | yes ⊥≈a⊓⊥ = ≈-trans (⊔-comm ⊥ a) (≈-trans (≈-⊔-cong (≈-refl {a}) ⊥≈a⊓⊥) (absorb-⊔-⊓ a ⊥))
+        ...         | no ⊥ᴬ̷≈a⊓⊥ = ⊥-elim (MyChain.Bounded-suc-n bounded (MyChain.step x≺⊥ ≈-refl longestChain))
+                        where
+                            ⊥⊓a̷≈⊥ : ¬ (⊥ ⊓ a) ≈ ⊥
+                            ⊥⊓a̷≈⊥ = λ ⊥⊓a≈⊥ → ⊥ᴬ̷≈a⊓⊥ (≈-trans (≈-sym ⊥⊓a≈⊥) (⊓-comm _ _))
+
+                            x≺⊥ : (⊥ ⊓ a) ≺ ⊥
+                            x≺⊥ = (≈-trans (⊔-comm _ _) (≈-trans (≈-refl {⊥ ⊔ (⊥ ⊓ a)}) (absorb-⊔-⊓ ⊥ a)) , ⊥⊓a̷≈⊥)
+
 module ChainMapping {a b} {A : Set a} {B : Set b}
     {_≈₁_ : A → A → Set a} {_≈₂_ : B → B → Set b}
     {_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B}

Utils.agda

@@ -1,19 +1,31 @@
 module Utils where
 
 open import Agda.Primitive using () renaming (_⊔_ to _⊔ℓ_)
-open import Data.Product as Prod using (_×_)
+open import Data.Product as Prod using (Σ; _×_; _,_; proj₁; proj₂)
 open import Data.Nat using (ℕ; suc)
+open import Data.Fin as Fin using (Fin; suc; zero)
+open import Data.Fin.Properties using (suc-injective)
 open import Data.List using (List; cartesianProduct; []; _∷_; _++_; foldr; filter) renaming (map to mapˡ)
-open import Data.List.Membership.Propositional using (_∈_)
+open import Data.List.Membership.Propositional using (_∈_; lose)
 open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
-open import Data.List.Relation.Unary.All using (All; []; _∷_; map)
-open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
+open import Data.List.Relation.Unary.All using (All; []; _∷_; map; all?; lookup)
+open import Data.List.Relation.Unary.All.Properties using (++⁻ˡ; ++⁻ʳ)
+open import Data.List.Relation.Unary.Any as Any using (Any; here; there; any?) -- TODO: re-export these with nicer names from map
 open import Data.Sum using (_⊎_)
 open import Function.Definitions using (Injective)
+open import Relation.Binary using (Antisymmetric) renaming (Decidable to Decidable²)
 open import Relation.Binary.PropositionalEquality using (_≡_; sym; refl; cong)
-open import Relation.Nullary using (¬_; yes; no)
+open import Relation.Nullary using (¬_; yes; no; Dec)
+open import Relation.Nullary.Decidable using (¬?)
 open import Relation.Unary using (Decidable)
 
+All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
+All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
+All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs
+
+Decidable-¬ : ∀ {p c} {C : Set c} {P : C → Set p} → Decidable P → Decidable (λ x → ¬ P x)
+Decidable-¬ Decidable-P x = ¬? (Decidable-P x)
+
 data Unique {c} {C : Set c} : List C → Set c where
     empty : Unique []
     push : ∀ {x : C} {xs : List C}
@@ -34,6 +46,24 @@ Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') =
         help {[]} _ = x'≢x ∷ []
         help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
 
+Unique-++⁻ˡ : ∀ {c} {C : Set c} (xs : List C) {ys : List C} → Unique (xs ++ ys) → Unique xs
+Unique-++⁻ˡ [] Unique-ys = empty
+Unique-++⁻ˡ (x ∷ xs) {ys} (push x≢xs++ys Unique-xs++ys) = push (++⁻ˡ xs {ys = ys} x≢xs++ys) (Unique-++⁻ˡ xs Unique-xs++ys)
+
+Unique-++⁻ʳ : ∀ {c} {C : Set c} (xs : List C) {ys : List C} → Unique (xs ++ ys) → Unique ys
+Unique-++⁻ʳ [] Unique-ys = Unique-ys
+Unique-++⁻ʳ (x ∷ xs) {ys} (push x≢xs++ys Unique-xs++ys) = Unique-++⁻ʳ xs Unique-xs++ys
+
+Unique-∈-++ˡ : ∀ {c} {C : Set c} {x : C} (xs : List C) {ys : List C} →  Unique (xs ++ ys) → x ∈ xs → ¬ x ∈ ys
+Unique-∈-++ˡ [] _ ()
+Unique-∈-++ˡ {x = x} (x' ∷ xs) (push x≢xs++ys _) (here refl) = All¬-¬Any (++⁻ʳ xs x≢xs++ys)
+Unique-∈-++ˡ {x = x} (x' ∷ xs) (push _ Unique-xs++ys) (there x̷∈xs) = Unique-∈-++ˡ xs Unique-xs++ys x̷∈xs
+
+Unique-narrow : ∀ {c} {C : Set c} {x : C} (xs : List C) {ys : List C} →  Unique (xs ++ ys) → x ∈ xs →  Unique (x ∷ ys)
+Unique-narrow [] _ ()
+Unique-narrow {x = x} (x' ∷ xs) (push x≢xs++ys Unique-xs++ys) (here refl) = push (++⁻ʳ xs x≢xs++ys) (Unique-++⁻ʳ xs Unique-xs++ys)
+Unique-narrow {x = x} (x' ∷ xs) (push _ Unique-xs++ys) (there x̷∈xs) = Unique-narrow xs Unique-xs++ys x̷∈xs
+
 All-≢-map : ∀ {c d} {C : Set c} {D : Set d} (x : C) {xs : List C} (f : C → D) →
             Injective (_≡_ {_} {C}) (_≡_ {_} {D}) f →
             All (λ x' → ¬ x ≡ x') xs → All (λ y' → ¬ (f x) ≡ y') (mapˡ f xs)
@@ -46,9 +76,8 @@ Unique-map : ∀ {c d} {C : Set c} {D : Set d} {l : List C} (f : C → D) →
 Unique-map {l = []} _ _ _ = empty
 Unique-map {l = x ∷ xs} f f-Injecitve (push x≢xs uxs) = push (All-≢-map x f f-Injecitve x≢xs) (Unique-map f f-Injecitve uxs)
 
-All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
-All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
-All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs
+¬Any-map : ∀ {p₁ p₂ c} {C : Set c} {P₁ : C → Set p₁} {P₂ : C → Set p₂} {l : List C} → (∀ {x} → P₁ x → P₂ x) → ¬ Any P₂ l → ¬ Any P₁ l
+¬Any-map f ¬Any-P₂ Any-P₁ = ¬Any-P₂ (Any.map f Any-P₁)
 
 All-single : ∀ {p c} {C : Set c} {P : C → Set p} {c : C} {l : List C} → All P l → c ∈ l → P c
 All-single {c = c} {l = x ∷ xs} (p ∷ ps) (here refl) = p
@@ -106,3 +135,34 @@ _∧_ P Q a = P a × Q a
 
 it : ∀ {a} {A : Set a} {{_ : A}} → A
 it {{x}} = x
+
+z≢sf : ∀ {n : ℕ} (f : Fin n) → ¬ (Fin.zero ≡ Fin.suc f)
+z≢sf f ()
+
+z≢mapsfs : ∀ {n : ℕ} (fs : List (Fin n)) → All (λ sf → ¬ zero ≡ sf) (mapˡ suc fs)
+z≢mapsfs [] = []
+z≢mapsfs (f ∷ fs') = z≢sf f ∷ z≢mapsfs fs'
+
+fins : ∀ (n : ℕ) → Σ (List (Fin n)) Unique
+fins 0 = ([] , empty)
+fins (suc n') =
+    let
+        (inds' , unids') = fins n'
+    in
+        ( zero ∷ mapˡ suc inds'
+        , push (z≢mapsfs inds') (Unique-map suc suc-injective unids')
+        )
+
+fins-complete : ∀ (n : ℕ) (f : Fin n) → f ∈ (proj₁ (fins n))
+fins-complete (suc n') zero = here refl
+fins-complete (suc n') (suc f') = there (x∈xs⇒fx∈fxs suc (fins-complete n' f'))
+
+findUniversal : ∀ {p c} {C : Set c} {R : C → C → Set p} (l : List C) →  Decidable² R →
+                Dec (Any (λ x → All (R x) l) l)
+findUniversal l Rdec = any? (λ x → all? (Rdec x) l) l
+
+findUniversal-unique : ∀ {p c} {C : Set c} (R : C → C → Set p) (l : List C) →
+                       Antisymmetric _≡_ R →
+                       ∀ x₁ x₂ → x₁ ∈ l → x₂ ∈ l → All (R x₁) l → All (R x₂) l →
+                       x₁ ≡ x₂
+findUniversal-unique R l Rantisym x₁ x₂ x₁∈l x₂∈l Allx₁ Allx₂ = Rantisym (lookup Allx₁ x₂∈l) (lookup Allx₂ x₁∈l)