Adjust behavior of eval to not require constant 'k in vars' threading
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -3,6 +3,7 @@ module Analysis.Sign where
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open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
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open import Data.Nat using (suc)
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open import Data.Product using (_×_; proj₁; _,_)
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open import Data.Empty using (⊥; ⊥-elim)
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open import Data.List using (List; _∷_; []; foldr; cartesianProduct; cartesianProductWith)
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
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open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst)
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@ -61,8 +62,7 @@ open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = s
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-- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
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open AB.Plain 0ˢ using ()
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renaming
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( finiteHeightLattice to finiteHeightLatticeᵍ
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; isLattice to isLatticeᵍ
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( isLattice to isLatticeᵍ
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; fixedHeight to fixedHeightᵍ
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; _≼_ to _≼ᵍ_
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; _⊔_ to _⊔ᵍ_
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@ -130,6 +130,8 @@ module WithProg (prog : Program) where
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; _updating_via_ to _updatingᵛ_via_
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; locate to locateᵛ
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; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
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; all-equal-keys to all-equal-keysᵛ
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; ∈k-dec to ∈k-decᵛ
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)
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open IsLattice isLatticeᵛ
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using ()
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@ -206,57 +208,57 @@ module WithProg (prog : Program) where
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states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
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states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
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eval : ∀ (e : Expr) → (∀ k → k ∈ᵉ e → k ∈ˡ vars) → VariableSigns → SignLattice
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eval (e₁ + e₂) k∈e⇒k∈vars vs =
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plus (eval e₁ (λ k k∈e₁ → k∈e⇒k∈vars k (in⁺₁ k∈e₁)) vs)
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(eval e₂ (λ k k∈e₂ → k∈e⇒k∈vars k (in⁺₂ k∈e₂)) vs)
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eval (e₁ - e₂) k∈e⇒k∈vars vs =
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minus (eval e₁ (λ k k∈e₁ → k∈e⇒k∈vars k (in⁻₁ k∈e₁)) vs)
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(eval e₂ (λ k k∈e₂ → k∈e⇒k∈vars k (in⁻₂ k∈e₂)) vs)
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eval (` k) k∈e⇒k∈vars vs = proj₁ (locateᵛ {k} {vs} (vars-in-Map k vs (k∈e⇒k∈vars k here)))
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eval (# 0) _ _ = [ 0ˢ ]ᵍ
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eval (# (suc n')) _ _ = [ + ]ᵍ
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eval : ∀ (e : Expr) → VariableSigns → SignLattice
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eval (e₁ + e₂) vs = plus (eval e₁ vs) (eval e₂ vs)
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eval (e₁ - e₂) vs = minus (eval e₁ vs) (eval e₂ vs)
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eval (` k) vs
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with ∈k-decᵛ k (proj₁ (proj₁ vs))
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... | yes k∈vs = proj₁ (locateᵛ {k} {vs} k∈vs)
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... | no _ = ⊤ᵍ
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eval (# 0) _ = [ 0ˢ ]ᵍ
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eval (# (suc n')) _ = [ + ]ᵍ
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eval-Mono : ∀ (e : Expr) (k∈e⇒k∈vars : ∀ k → k ∈ᵉ e → k ∈ˡ vars) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e k∈e⇒k∈vars)
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eval-Mono (e₁ + e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
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eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e)
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eval-Mono (e₁ + e₂) {vs₁} {vs₂} vs₁≼vs₂ =
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let
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-- TODO: can this be done with less boilerplate?
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k∈e₁⇒k∈vars = λ k k∈e₁ → k∈e⇒k∈vars k (in⁺₁ k∈e₁)
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k∈e₂⇒k∈vars = λ k k∈e₂ → k∈e⇒k∈vars k (in⁺₂ k∈e₂)
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g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
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g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
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g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
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g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
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g₁vs₁ = eval e₁ vs₁
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g₂vs₁ = eval e₂ vs₁
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g₁vs₂ = eval e₁ vs₂
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g₂vs₂ = eval e₂ vs₂
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in
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≼ᵍ-trans
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{plus g₁vs₁ g₂vs₁} {plus g₁vs₂ g₂vs₁} {plus g₁vs₂ g₂vs₂}
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(plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
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(plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
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eval-Mono (e₁ - e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
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(plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
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(plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
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eval-Mono (e₁ - e₂) {vs₁} {vs₂} vs₁≼vs₂ =
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let
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-- TODO: here too -- can this be done with less boilerplate?
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k∈e₁⇒k∈vars = λ k k∈e₁ → k∈e⇒k∈vars k (in⁻₁ k∈e₁)
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k∈e₂⇒k∈vars = λ k k∈e₂ → k∈e⇒k∈vars k (in⁻₂ k∈e₂)
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g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
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g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
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g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
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g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
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g₁vs₁ = eval e₁ vs₁
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g₂vs₁ = eval e₂ vs₁
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g₁vs₂ = eval e₁ vs₂
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g₂vs₂ = eval e₂ vs₂
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in
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≼ᵍ-trans
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{minus g₁vs₁ g₂vs₁} {minus g₁vs₂ g₂vs₁} {minus g₁vs₂ g₂vs₂}
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(minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
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(minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
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eval-Mono (` k) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
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(minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
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(minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
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eval-Mono (` k) {vs₁@((kvs₁ , _) , _)} {vs₂@((kvs₂ , _), _)} vs₁≼vs₂
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with ∈k-decᵛ k kvs₁ | ∈k-decᵛ k kvs₂
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... | yes k∈kvs₁ | yes k∈kvs₂ =
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let
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(v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} (vars-in-Map k vs₁ (k∈e⇒k∈vars k here))
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(v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} (vars-in-Map k vs₂ (k∈e⇒k∈vars k here))
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(v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} k∈kvs₁
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(v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} k∈kvs₂
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in
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m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
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eval-Mono (# 0) _ _ = ≈ᵍ-refl
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eval-Mono (# (suc n')) _ _ = ≈ᵍ-refl
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... | yes k∈kvs₁ | no k∉kvs₂ = ⊥-elim (k∉kvs₂ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₁ vs₂) k∈kvs₁))
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... | no k∉kvs₁ | yes k∈kvs₂ = ⊥-elim (k∉kvs₁ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₂ vs₁) k∈kvs₂))
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... | no k∉kvs₁ | no k∉kvs₂ = IsLattice.≈-refl isLatticeᵍ
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eval-Mono (# 0) _ = ≈ᵍ-refl
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eval-Mono (# (suc n')) _ = ≈ᵍ-refl
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private module _ (k : String) (e : Expr) (k∈e⇒k∈vars : ∀ k → k ∈ᵉ e → k ∈ˡ vars) where
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open VariableSignsFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e k∈e⇒k∈vars) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
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private module _ (k : String) (e : Expr) where
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open VariableSignsFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
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renaming
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( f' to updateVariablesFromExpression
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; f'-Monotonic to updateVariablesFromExpression-Mono
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@ -265,28 +267,23 @@ module WithProg (prog : Program) where
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updateVariablesForState : State → StateVariables → VariableSigns
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updateVariablesForState s sv
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-- More weirdness here. Apparently, capturing the with-equality proof
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-- using 'in p' makes code that reasons about this function (below)
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-- throw ill-typed with-abstraction errors. Instead, make use of the
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-- fact that later with-clauses are generalized over earlier ones to
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-- construct a specialization of vars-complete for (code s).
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with code s | (λ k → vars-complete {k} s)
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... | k ← e | k∈codes⇒k∈vars =
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with code s
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... | k ← e =
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let
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(vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
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in
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updateVariablesFromExpression k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) vs
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updateVariablesFromExpression k e vs
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updateVariablesForState-Monoʳ : ∀ (s : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
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updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂
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with code s | (λ k → vars-complete {k} s)
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... | k ← e | k∈codes⇒k∈vars =
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with code s
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... | k ← e =
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let
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(vs₁ , s,vs₁∈sv₁) = locateᵐ {s} {sv₁} (states-in-Map s sv₁)
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(vs₂ , s,vs₂∈sv₂) = locateᵐ {s} {sv₂} (states-in-Map s sv₂)
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vs₁≼vs₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ sv₁ sv₂ sv₁≼sv₂ s,vs₁∈sv₁ s,vs₂∈sv₂
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in
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updateVariablesFromExpression-Mono k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) {vs₁} {vs₂} vs₁≼vs₂
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updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
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open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) updateVariablesForState updateVariablesForState-Monoʳ states
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renaming
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@ -12,7 +12,7 @@ module Lattice.FiniteMap {a b : Level} {A : Set a} {B : Set b}
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open IsLattice lB using () renaming (_≼_ to _≼₂_)
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open import Lattice.Map ≡-dec-A lB as Map
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using (Map; ⊔-equal-keys; ⊓-equal-keys; ∈k-dec)
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using (Map; ⊔-equal-keys; ⊓-equal-keys)
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renaming
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( _≈_ to _≈ᵐ_
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; _⊔_ to _⊔ᵐ_
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@ -37,6 +37,7 @@ open import Lattice.Map ≡-dec-A lB as Map
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; updating-via-keys-≡ to updatingᵐ-via-keys-≡
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; f'-Monotonic to f'-Monotonicᵐ
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; _≼_ to _≼ᵐ_
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; ∈k-dec to ∈k-decᵐ
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)
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open import Data.List.Membership.Propositional using () renaming (_∈_ to _∈ˡ_)
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open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
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@ -82,6 +83,8 @@ module WithKeys (ks : List A) where
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_∈k_ : A → FiniteMap → Set a
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_∈k_ k (m₁ , _) = k ∈ˡ (keysᵐ m₁)
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∈k-dec = ∈k-decᵐ
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locate : ∀ {k : A} {fm : FiniteMap} → k ∈k fm → Σ B (λ v → (k , v) ∈ fm)
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locate {k} {fm = (m , _)} k∈kfm = locateᵐ {k} {m} k∈kfm
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@ -182,7 +185,7 @@ module WithKeys (ks : List A) where
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fm₁ ≼ fm₂ → Pairwise _≼₂_ (fm₁ [ ks' ]) (fm₂ [ ks' ])
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m₁≼m₂⇒m₁[ks]≼m₂[ks] _ _ [] _ = []
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m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁@(m₁ , km₁≡ks) fm₂@(m₂ , km₂≡ks) (k ∷ ks'') m₁≼m₂
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with ∈k-dec k (proj₁ m₁) | ∈k-dec k (proj₁ m₂)
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with ∈k-decᵐ k (proj₁ m₁) | ∈k-decᵐ k (proj₁ m₂)
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... | yes k∈km₁ | yes k∈km₂ =
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let
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(v₁ , k,v₁∈m₁) = locateᵐ {m = m₁} k∈km₁
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