Adjust behavior of eval to not require constant 'k in vars' threading

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-03-22 17:15:40 -07:00
parent f0da9a9020
commit 1a7b2a1736
2 changed files with 54 additions and 54 deletions

View File

@ -3,6 +3,7 @@ module Analysis.Sign where
open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
open import Data.Nat using (suc)
open import Data.Product using (_×_; proj₁; _,_)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.List using (List; _∷_; []; foldr; cartesianProduct; cartesianProductWith)
open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst)
@ -61,8 +62,7 @@ open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = s
-- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
open AB.Plain 0ˢ using ()
renaming
( finiteHeightLattice to finiteHeightLatticeᵍ
; isLattice to isLatticeᵍ
( isLattice to isLatticeᵍ
; fixedHeight to fixedHeightᵍ
; _≼_ to _≼ᵍ_
; _⊔_ to _⊔ᵍ_
@ -130,6 +130,8 @@ module WithProg (prog : Program) where
; _updating_via_ to _updatingᵛ_via_
; locate to locateᵛ
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
; all-equal-keys to all-equal-keysᵛ
; ∈k-dec to ∈k-decᵛ
)
open IsLattice isLatticeᵛ
using ()
@ -206,57 +208,57 @@ module WithProg (prog : Program) where
states-in-Map : (s : State) (sv : StateVariables) s ∈kᵐ sv
states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
eval : (e : Expr) ( k k ∈ᵉ e k ∈ˡ vars) VariableSigns SignLattice
eval (e₁ + e₂) k∈e⇒k∈vars vs =
plus (eval e₁ (λ k k∈e₁ k∈e⇒k∈vars k (in⁺₁ k∈e₁)) vs)
(eval e₂ (λ k k∈e₂ k∈e⇒k∈vars k (in⁺₂ k∈e₂)) vs)
eval (e₁ - e₂) k∈e⇒k∈vars vs =
minus (eval e₁ (λ k k∈e₁ k∈e⇒k∈vars k (in⁻₁ k∈e₁)) vs)
(eval e₂ (λ k k∈e₂ k∈e⇒k∈vars k (in⁻₂ k∈e₂)) vs)
eval (` k) k∈e⇒k∈vars vs = proj₁ (locateᵛ {k} {vs} (vars-in-Map k vs (k∈e⇒k∈vars k here)))
eval (# 0) _ _ = [ 0ˢ ]ᵍ
eval (# (suc n')) _ _ = [ + ]ᵍ
eval : (e : Expr) VariableSigns SignLattice
eval (e₁ + e₂) vs = plus (eval e₁ vs) (eval e₂ vs)
eval (e₁ - e₂) vs = minus (eval e₁ vs) (eval e₂ vs)
eval (` k) vs
with ∈k-decᵛ k (proj₁ (proj₁ vs))
... | yes k∈vs = proj₁ (locateᵛ {k} {vs} k∈vs)
... | no _ = ⊤ᵍ
eval (# 0) _ = [ 0ˢ ]ᵍ
eval (# (suc n')) _ = [ + ]ᵍ
eval-Mono : (e : Expr) (k∈e⇒k∈vars : k k ∈ᵉ e k ∈ˡ vars) Monotonic _≼ᵛ_ _≼ᵍ_ (eval e k∈e⇒k∈vars)
eval-Mono (e₁ + e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
eval-Mono : (e : Expr) Monotonic _≼ᵛ_ _≼ᵍ_ (eval e)
eval-Mono (e₁ + e₂) {vs₁} {vs₂} vs₁≼vs₂ =
let
-- TODO: can this be done with less boilerplate?
k∈e₁⇒k∈vars = λ k k∈e₁ k∈e⇒k∈vars k (in⁺₁ k∈e₁)
k∈e₂⇒k∈vars = λ k k∈e₂ k∈e⇒k∈vars k (in⁺₂ k∈e₂)
g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
g₁vs₁ = eval e₁ vs₁
g₂vs₁ = eval e₂ vs₁
g₁vs₂ = eval e₁ vs₂
g₂vs₂ = eval e₂ vs₂
in
≼ᵍ-trans
{plus g₁vs₁ g₂vs₁} {plus g₁vs₂ g₂vs₁} {plus g₁vs₂ g₂vs₂}
(plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
(plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
eval-Mono (e₁ - e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
(plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
(plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
eval-Mono (e₁ - e₂) {vs₁} {vs₂} vs₁≼vs₂ =
let
-- TODO: here too -- can this be done with less boilerplate?
k∈e₁⇒k∈vars = λ k k∈e₁ k∈e⇒k∈vars k (in⁻₁ k∈e₁)
k∈e₂⇒k∈vars = λ k k∈e₂ k∈e⇒k∈vars k (in⁻₂ k∈e₂)
g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
g₁vs₁ = eval e₁ vs₁
g₂vs₁ = eval e₂ vs₁
g₁vs₂ = eval e₁ vs₂
g₂vs₂ = eval e₂ vs₂
in
≼ᵍ-trans
{minus g₁vs₁ g₂vs₁} {minus g₁vs₂ g₂vs₁} {minus g₁vs₂ g₂vs₂}
(minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
(minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
eval-Mono (` k) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
let
(v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} (vars-in-Map k vs₁ (k∈e⇒k∈vars k here))
(v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} (vars-in-Map k vs₂ (k∈e⇒k∈vars k here))
in
m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
eval-Mono (# 0) _ _ = ≈ᵍ-refl
eval-Mono (# (suc n')) _ _ = ≈ᵍ-refl
(minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ {vs₁} {vs₂} vs₁≼vs₂))
(minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ {vs₁} {vs₂} vs₁≼vs₂))
eval-Mono (` k) {vs₁@((kvs₁ , _) , _)} {vs₂@((kvs₂ , _), _)} vs₁≼vs₂
with ∈k-decᵛ k kvs₁ | ∈k-decᵛ k kvs₂
... | yes k∈kvs₁ | yes k∈kvs₂ =
let
(v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} k∈kvs₁
(v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} k∈kvs₂
in
m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
... | yes k∈kvs₁ | no k∉kvs₂ = ⊥-elim (k∉kvs₂ (subst (λ l k ∈ˡ l) (all-equal-keysᵛ vs₁ vs₂) k∈kvs₁))
... | no k∉kvs₁ | yes k∈kvs₂ = ⊥-elim (k∉kvs₁ (subst (λ l k ∈ˡ l) (all-equal-keysᵛ vs₂ vs₁) k∈kvs₂))
... | no k∉kvs₁ | no k∉kvs₂ = IsLattice.≈-refl isLatticeᵍ
eval-Mono (# 0) _ = ≈ᵍ-refl
eval-Mono (# (suc n')) _ = ≈ᵍ-refl
private module _ (k : String) (e : Expr) (k∈e⇒k∈vars : k k ∈ᵉ e k ∈ˡ vars) where
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 [])
private module _ (k : String) (e : Expr) where
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 [])
renaming
( f' to updateVariablesFromExpression
; f'-Monotonic to updateVariablesFromExpression-Mono
@ -265,28 +267,23 @@ module WithProg (prog : Program) where
updateVariablesForState : State StateVariables VariableSigns
updateVariablesForState s sv
-- More weirdness here. Apparently, capturing the with-equality proof
-- using 'in p' makes code that reasons about this function (below)
-- throw ill-typed with-abstraction errors. Instead, make use of the
-- fact that later with-clauses are generalized over earlier ones to
-- construct a specialization of vars-complete for (code s).
with code s | (λ k vars-complete {k} s)
... | k e | k∈codes⇒k∈vars =
with code s
... | k e =
let
(vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
in
updateVariablesFromExpression k e (λ k k∈e k∈codes⇒k∈vars k (in←₂ k∈e)) vs
updateVariablesFromExpression k e vs
updateVariablesForState-Monoʳ : (s : State) Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂
with code s | (λ k vars-complete {k} s)
... | k e | k∈codes⇒k∈vars =
with code s
... | k e =
let
(vs₁ , s,vs₁∈sv₁) = locateᵐ {s} {sv₁} (states-in-Map s sv₁)
(vs₂ , s,vs₂∈sv₂) = locateᵐ {s} {sv₂} (states-in-Map s sv₂)
vs₁≼vs₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ sv₁ sv₂ sv₁≼sv₂ s,vs₁∈sv₁ s,vs₂∈sv₂
in
updateVariablesFromExpression-Mono k e (λ k k∈e k∈codes⇒k∈vars k (in←₂ k∈e)) {vs₁} {vs₂} vs₁≼vs₂
updateVariablesFromExpression-Mono k e {vs₁} {vs₂} vs₁≼vs₂
open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x x) (λ a₁≼a₂ a₁≼a₂) updateVariablesForState updateVariablesForState-Monoʳ states
renaming

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@ -12,7 +12,7 @@ module Lattice.FiniteMap {a b : Level} {A : Set a} {B : Set b}
open IsLattice lB using () renaming (_≼_ to _≼₂_)
open import Lattice.Map ≡-dec-A lB as Map
using (Map; ⊔-equal-keys; ⊓-equal-keys; ∈k-dec)
using (Map; ⊔-equal-keys; ⊓-equal-keys)
renaming
( _≈_ to _≈ᵐ_
; _⊔_ to _⊔ᵐ_
@ -37,6 +37,7 @@ open import Lattice.Map ≡-dec-A lB as Map
; updating-via-keys-≡ to updatingᵐ-via-keys-≡
; f'-Monotonic to f'-Monotonicᵐ
; _≼_ to _≼ᵐ_
; ∈k-dec to ∈k-decᵐ
)
open import Data.List.Membership.Propositional using () renaming (_∈_ to _∈ˡ_)
open import Data.Product using (_×_; _,_; Σ; proj₁ ; proj₂)
@ -82,6 +83,8 @@ module WithKeys (ks : List A) where
_∈k_ : A FiniteMap Set a
_∈k_ k (m₁ , _) = k ∈ˡ (keysᵐ m₁)
∈k-dec = ∈k-decᵐ
locate : {k : A} {fm : FiniteMap} k ∈k fm Σ B (λ v (k , v) fm)
locate {k} {fm = (m , _)} k∈kfm = locateᵐ {k} {m} k∈kfm
@ -182,7 +185,7 @@ module WithKeys (ks : List A) where
fm₁ fm₂ Pairwise _≼₂_ (fm₁ [ ks' ]) (fm₂ [ ks' ])
m₁≼m₂⇒m₁[ks]≼m₂[ks] _ _ [] _ = []
m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁@(m₁ , km₁≡ks) fm₂@(m₂ , km₂≡ks) (k ks'') m₁≼m₂
with ∈k-dec k (proj₁ m₁) | ∈k-dec k (proj₁ m₂)
with ∈k-dec k (proj₁ m₁) | ∈k-dec k (proj₁ m₂)
... | yes k∈km₁ | yes k∈km₂ =
let
(v₁ , k,v₁∈m₁) = locateᵐ {m = m₁} k∈km₁