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Author SHA1 Message Date
3d2a507f2f Almost prove correctness
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 22:49:53 -07:00
82027ecd04 Move predecessor computation into Graphs
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 21:26:32 -07:00
734e82ff6d Wrap generated graphs to ensure entry and exit nodes have no extra edges
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 21:08:32 -07:00
69d1ecebae Prove that the bottom map's valyes are all bottoms
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 20:48:32 -07:00
b78cb91f2a Strengthen lemma about IterProd bottom to definition equality
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 20:20:11 -07:00
16fa4cd1d8 Use records rather than nested pairs to represent 'fixed height'
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 20:11:04 -07:00
95669b2c65 Prove that the iterated product is made from iterated bottom elements
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 19:45:15 -07:00
6857f60465 Rename the min/max elements top bottom and top in Prod
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 19:08:46 -07:00
f4392b32c0 Finish the last proof obligation for trace walking
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 19:01:36 -07:00
794c04eee9 Prove the foldr-implies lemma
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 18:37:50 -07:00
80069e76e6 Prove the recursive step of trace walking
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 17:56:47 -07:00
a22c0c9252 Prove a property of multi-key lookup
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 17:56:26 -07:00
20dc99ba1f Re-indent some code to take up less horizontal space
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 16:57:03 -07:00
b3a62da1fb Add a proof that edges lead to 'incoming' inclusion
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-05-09 16:56:45 -07:00
14 changed files with 333 additions and 136 deletions

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@ -10,16 +10,18 @@ module Analysis.Forward
open import Data.Empty using (⊥-elim) open import Data.Empty using (⊥-elim)
open import Data.String using (String) renaming (_≟_ to _≟ˢ_) open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
open import Data.Nat using (suc) open import Data.Nat using (suc)
open import Data.Product using (_×_; proj₁; _,_) open import Data.Product using (_×_; proj₁; proj₂; _,_)
open import Data.Sum using (inj₁; inj₂)
open import Data.List using (List; _∷_; []; foldr; foldl; cartesianProduct; cartesianProductWith) open import Data.List using (List; _∷_; []; foldr; foldl; cartesianProduct; cartesianProductWith)
open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_) open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
open import Data.List.Relation.Unary.Any as Any using () open import Data.List.Relation.Unary.Any as Any using ()
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst) open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; sym; trans; subst)
open import Relation.Nullary using (¬_; Dec; yes; no) open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Unit using () open import Data.Unit using ()
open import Function using (_∘_; flip) open import Function using (_∘_; flip)
import Chain
open import Utils using (Pairwise; _⇒_) open import Utils using (Pairwise; _⇒_; __)
import Lattice.FiniteValueMap import Lattice.FiniteValueMap
open IsFiniteHeightLattice isFiniteHeightLatticeˡ open IsFiniteHeightLattice isFiniteHeightLatticeˡ
@ -62,25 +64,32 @@ module WithProg (prog : Program) where
; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ ; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
; ⊔-idemp to ⊔ᵛ-idemp ; ⊔-idemp to ⊔ᵛ-idemp
) )
open Lattice.FiniteValueMap.IterProdIsomorphism _≟ˢ_ isLatticeˡ
using ()
renaming
( Provenance-union to Provenance-unionᵐ
)
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeˡ vars-Unique ≈ˡ-dec _ fixedHeightˡ open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeˡ vars-Unique ≈ˡ-dec _ fixedHeightˡ
using () using ()
renaming renaming
( isFiniteHeightLattice to isFiniteHeightLatticeᵛ ( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
; ⊥-contains-bottoms to ⊥ᵛ-contains-bottoms
) )
≈ᵛ-dec = ≈ˡ-dec⇒≈ᵛ-dec ≈ˡ-dec ≈ᵛ-dec = ≈ˡ-dec⇒≈ᵛ-dec ≈ˡ-dec
joinSemilatticeᵛ = IsFiniteHeightLattice.joinSemilattice isFiniteHeightLatticeᵛ joinSemilatticeᵛ = IsFiniteHeightLattice.joinSemilattice isFiniteHeightLatticeᵛ
fixedHeightᵛ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵛ fixedHeightᵛ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵛ
⊥ᵛ = proj₁ (proj₁ (proj₁ fixedHeightᵛ)) ⊥ᵛ = Chain.Height.⊥ fixedHeightᵛ
-- Finally, the map we care about is (state -> (variables -> value)). Bring that in. -- Finally, the map we care about is (state -> (variables -> value)). Bring that in.
module StateVariablesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟_ isLatticeᵛ states module StateVariablesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟_ isLatticeᵛ states
open StateVariablesFiniteMap open StateVariablesFiniteMap
using (_[_]; m₁≼m₂⇒m₁[ks]≼m₂[ks]) using (_[_]; []-∈; m₁≼m₂⇒m₁[ks]≼m₂[ks]; m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂)
renaming renaming
( FiniteMap to StateVariables ( FiniteMap to StateVariables
; isLattice to isLatticeᵐ ; isLattice to isLatticeᵐ
; _≈_ to _≈ᵐ_ ; _≈_ to _≈ᵐ_
; _∈_ to _∈ᵐ_
; _∈k_ to _∈kᵐ_ ; _∈k_ to _∈kᵐ_
; locate to locateᵐ ; locate to locateᵐ
; _≼_ to _≼ᵐ_ ; _≼_ to _≼ᵐ_
@ -113,8 +122,13 @@ module WithProg (prog : Program) where
variablesAt : State StateVariables VariableValues variablesAt : State StateVariables VariableValues
variablesAt s sv = proj₁ (locateᵐ {s} {sv} (states-in-Map s sv)) variablesAt s sv = proj₁ (locateᵐ {s} {sv} (states-in-Map s sv))
variablesAt-∈ : (s : State) (sv : StateVariables) (s , variablesAt s sv) ∈ᵐ sv
variablesAt-∈ s sv = proj₂ (locateᵐ {s} {sv} (states-in-Map s sv))
variablesAt-≈ : s sv₁ sv₂ sv₁ ≈ᵐ sv₂ variablesAt s sv₁ ≈ᵛ variablesAt s sv₂ variablesAt-≈ : s sv₁ sv₂ sv₁ ≈ᵐ sv₂ variablesAt s sv₁ ≈ᵛ variablesAt s sv₂
variablesAt-≈ = {!!} variablesAt-≈ s sv₁ sv₂ sv₁≈sv₂ =
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ sv₁ sv₂ sv₁≈sv₂
(states-in-Map s sv₁) (states-in-Map s sv₂)
-- build up the 'join' function, which follows from Exercise 4.26's -- build up the 'join' function, which follows from Exercise 4.26's
-- --
@ -232,11 +246,17 @@ module WithProg (prog : Program) where
module WithInterpretation (latticeInterpretationˡ : LatticeInterpretation isLatticeˡ) where module WithInterpretation (latticeInterpretationˡ : LatticeInterpretation isLatticeˡ) where
open LatticeInterpretation latticeInterpretationˡ open LatticeInterpretation latticeInterpretationˡ
using () using ()
renaming (⟦_⟧ to ⟦_⟧ˡ; ⟦⟧-respects-≈ to ⟦⟧ˡ-respects-≈ˡ) renaming
( ⟦_⟧ to ⟦_⟧ˡ
; ⟦⟧-respects-≈ to ⟦⟧ˡ-respects-≈ˡ
; ⟦⟧-⊔- to ⟦⟧ˡ-⊔ˡ-
)
⟦_⟧ᵛ : VariableValues Env Set ⟦_⟧ᵛ : VariableValues Env Set
⟦_⟧ᵛ vs ρ = {k l} (k , l) ∈ᵛ vs {v} (k , v) Language.∈ ρ l ⟧ˡ v ⟦_⟧ᵛ vs ρ = {k l} (k , l) ∈ᵛ vs {v} (k , v) Language.∈ ρ l ⟧ˡ v
⟦⊥ᵛ⟧ᵛ∅ : ⊥ᵛ ⟧ᵛ []
⟦⊥ᵛ⟧ᵛ∅ _ ()
⟦⟧ᵛ-respects-≈ᵛ : {vs₁ vs₂ : VariableValues} vs₁ ≈ᵛ vs₂ vs₁ ⟧ᵛ vs₂ ⟧ᵛ ⟦⟧ᵛ-respects-≈ᵛ : {vs₁ vs₂ : VariableValues} vs₁ ≈ᵛ vs₂ vs₁ ⟧ᵛ vs₂ ⟧ᵛ
⟦⟧ᵛ-respects-≈ᵛ {m₁ , _} {m₂ , _} ⟦⟧ᵛ-respects-≈ᵛ {m₁ , _} {m₂ , _}
@ -247,9 +267,21 @@ module WithProg (prog : Program) where
in in
⟦⟧ˡ-respects-≈ˡ (≈ˡ-sym l≈l') v ⟦l'⟧v ⟦⟧ˡ-respects-≈ˡ (≈ˡ-sym l≈l') v ⟦l'⟧v
⟦⟧ᵛ-⊔ᵛ- : {vs₁ vs₂ : VariableValues} ( vs₁ ⟧ᵛ vs₂ ⟧ᵛ) vs₁ ⊔ᵛ vs₂ ⟧ᵛ
⟦⟧ᵛ-⊔ᵛ- {vs₁} {vs₂} ρ ⟦vs₁⟧ρ⟦vs₂⟧ρ {k} {l} k,l∈vs₁₂ {v} k,v∈ρ
with ((l₁ , l₂) , (refl , (k,l₁∈vs₁ , k,l₂∈vs₂)))
Provenance-unionᵐ vs₁ vs₂ k,l∈vs₁₂
with ⟦vs₁⟧ρ⟦vs₂⟧ρ
... | inj₁ ⟦vs₁⟧ρ = ⟦⟧ˡ-⊔ˡ- {l₁} {l₂} v (inj₁ (⟦vs₁⟧ρ k,l₁∈vs₁ k,v∈ρ))
... | inj₂ ⟦vs₂⟧ρ = ⟦⟧ˡ-⊔ˡ- {l₁} {l₂} v (inj₂ (⟦vs₂⟧ρ k,l₂∈vs₂ k,v∈ρ))
⟦⟧ᵛ-foldr : {vs : VariableValues} {vss : List VariableValues} {ρ : Env} ⟦⟧ᵛ-foldr : {vs : VariableValues} {vss : List VariableValues} {ρ : Env}
vs ⟧ᵛ ρ vs ∈ˡ vss foldr _⊔ᵛ_ ⊥ᵛ vss ⟧ᵛ ρ vs ⟧ᵛ ρ vs ∈ˡ vss foldr _⊔ᵛ_ ⊥ᵛ vss ⟧ᵛ ρ
⟦⟧ᵛ-foldr = {!!} ⟦⟧ᵛ-foldr {vs} {vs vss'} {ρ = ρ} ⟦vs⟧ρ (Any.here refl) =
⟦⟧ᵛ-⊔ᵛ- {vs₁ = vs} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ (inj₁ ⟦vs⟧ρ)
⟦⟧ᵛ-foldr {vs} {vs' vss'} {ρ = ρ} ⟦vs⟧ρ (Any.there vs∈vss') =
⟦⟧ᵛ-⊔ᵛ- {vs₁ = vs'} {vs₂ = foldr _⊔ᵛ_ ⊥ᵛ vss'} ρ
(inj₂ (⟦⟧ᵛ-foldr ⟦vs⟧ρ vs∈vss'))
InterpretationValid : Set InterpretationValid : Set
InterpretationValid = {vs ρ e v} ρ , e ⇒ᵉ v vs ⟧ᵛ ρ eval e vs ⟧ˡ v InterpretationValid = {vs ρ e v} ρ , e ⇒ᵉ v vs ⟧ᵛ ρ eval e vs ⟧ˡ v
@ -275,13 +307,17 @@ module WithProg (prog : Program) where
updateVariablesFromStmt-fold-matches : {bss vs ρ₁ ρ₂} ρ₁ , bss ⇒ᵇˢ ρ₂ vs ⟧ᵛ ρ₁ foldl (flip updateVariablesFromStmt) vs bss ⟧ᵛ ρ₂ updateVariablesFromStmt-fold-matches : {bss vs ρ₁ ρ₂} ρ₁ , bss ⇒ᵇˢ ρ₂ vs ⟧ᵛ ρ₁ foldl (flip updateVariablesFromStmt) vs bss ⟧ᵛ ρ₂
updateVariablesFromStmt-fold-matches [] ⟦vs⟧ρ = ⟦vs⟧ρ updateVariablesFromStmt-fold-matches [] ⟦vs⟧ρ = ⟦vs⟧ρ
updateVariablesFromStmt-fold-matches {bs bss'} {vs} {ρ₁} {ρ₂} (ρ₁,bs⇒ρ ρ,bss'⇒ρ₂) ⟦vs⟧ρ = updateVariablesFromStmt-fold-matches {bs bss'} {vs} {ρ₁} {ρ₂} (ρ₁,bs⇒ρ ρ,bss'⇒ρ₂) ⟦vs⟧ρ =
updateVariablesFromStmt-fold-matches {bss'} {updateVariablesFromStmt bs vs} ρ,bss'⇒ρ₂ (updateVariablesFromStmt-matches ρ₁,bs⇒ρ ⟦vs⟧ρ) updateVariablesFromStmt-fold-matches
{bss'} {updateVariablesFromStmt bs vs} ρ,bss'⇒ρ₂
(updateVariablesFromStmt-matches ρ₁,bs⇒ρ ⟦vs⟧ρ)
updateVariablesForState-matches : {s sv ρ₁ ρ₂} ρ₁ , (code s) ⇒ᵇˢ ρ₂ variablesAt s sv ⟧ᵛ ρ₁ updateVariablesForState s sv ⟧ᵛ ρ₂ updateVariablesForState-matches : {s sv ρ₁ ρ₂} ρ₁ , (code s) ⇒ᵇˢ ρ₂ variablesAt s sv ⟧ᵛ ρ₁ updateVariablesForState s sv ⟧ᵛ ρ₂
updateVariablesForState-matches = updateVariablesFromStmt-fold-matches updateVariablesForState-matches =
updateVariablesFromStmt-fold-matches
updateAll-matches : {s sv ρ₁ ρ₂} ρ₁ , (code s) ⇒ᵇˢ ρ₂ variablesAt s sv ⟧ᵛ ρ₁ variablesAt s (updateAll sv) ⟧ᵛ ρ₂ updateAll-matches : {s sv ρ₁ ρ₂} ρ₁ , (code s) ⇒ᵇˢ ρ₂ variablesAt s sv ⟧ᵛ ρ₁ variablesAt s (updateAll sv) ⟧ᵛ ρ₂
updateAll-matches {s} {sv} ρ₁,bss⇒ρ ⟦vs⟧ρ rewrite variablesAt-updateAll s sv = updateAll-matches {s} {sv} ρ₁,bss⇒ρ ⟦vs⟧ρ
rewrite variablesAt-updateAll s sv =
updateVariablesForState-matches {s} {sv} ρ₁,bss⇒ρ ⟦vs⟧ρ updateVariablesForState-matches {s} {sv} ρ₁,bss⇒ρ ⟦vs⟧ρ
@ -289,12 +325,44 @@ module WithProg (prog : Program) where
stepTrace {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ ρ₁,bss⇒ρ = stepTrace {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ ρ₁,bss⇒ρ =
let let
-- I'd use rewrite, but Agda gets a memory overflow (?!). -- I'd use rewrite, but Agda gets a memory overflow (?!).
⟦joinAll-result⟧ρ = subst (λ vs vs ⟧ᵛ ρ₁) (sym (variablesAt-joinAll s₁ result)) ⟦joinForKey-s₁⟧ρ ⟦joinAll-result⟧ρ =
⟦analyze-result⟧ρ = updateAll-matches {sv = joinAll result} ρ₁,bss⇒ρ ⟦joinAll-result⟧ρ subst (λ vs vs ⟧ᵛ ρ₁)
analyze-result≈result = ≈ᵐ-sym {result} {updateAll (joinAll result)} result≈analyze-result (sym (variablesAt-joinAll s₁ result))
analyze-s₁≈s₁ = variablesAt-≈ s₁ (updateAll (joinAll result)) result (analyze-result≈result) ⟦joinForKey-s₁⟧ρ
⟦analyze-result⟧ρ =
updateAll-matches {sv = joinAll result}
ρ₁,bss⇒ρ ⟦joinAll-result⟧ρ
analyze-result≈result =
≈ᵐ-sym {result} {updateAll (joinAll result)}
result≈analyze-result
analyze-s₁≈s₁ =
variablesAt-≈ s₁ (updateAll (joinAll result))
result (analyze-result≈result)
in in
⟦⟧ᵛ-respects-≈ᵛ {variablesAt s₁ (updateAll (joinAll result))} {variablesAt s₁ result} (analyze-s₁≈s₁) ρ₂ ⟦analyze-result⟧ρ ⟦⟧ᵛ-respects-≈ᵛ {variablesAt s₁ (updateAll (joinAll result))} {variablesAt s₁ result} (analyze-s₁≈s₁) ρ₂ ⟦analyze-result⟧ρ
walkTrace : {s₁ s₂ ρ₁ ρ₂} joinForKey s₁ result ⟧ᵛ ρ₁ Trace {graph} s₁ s₂ ρ₁ ρ₂ variablesAt s₂ result ⟧ᵛ ρ₂ walkTrace : {s₁ s₂ ρ₁ ρ₂} joinForKey s₁ result ⟧ᵛ ρ₁ Trace {graph} s₁ s₂ ρ₁ ρ₂ variablesAt s₂ result ⟧ᵛ ρ₂
walkTrace {s₁} {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ (Trace-single ρ₁,bss⇒ρ) = stepTrace {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ ρ₁,bss⇒ρ walkTrace {s₁} {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ (Trace-single ρ₁,bss⇒ρ) =
stepTrace {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ ρ₁,bss⇒ρ
walkTrace {s₁} {s₂} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ (Trace-edge {ρ₂ = ρ} {idx₂ = s} ρ₁,bss⇒ρ s₁→s₂ tr) =
let
⟦result-s₁⟧ρ =
stepTrace {s₁} {ρ₁} {ρ} ⟦joinForKey-s₁⟧ρ ρ₁,bss⇒ρ
s₁∈incomingStates =
[]-∈ result (edge⇒incoming s₁→s₂)
(variablesAt-∈ s₁ result)
⟦joinForKey-s⟧ρ =
⟦⟧ᵛ-foldr ⟦result-s₁⟧ρ s₁∈incomingStates
in
walkTrace ⟦joinForKey-s⟧ρ tr
postulate initialState-pred-∅ : incoming initialState []
joinForKey-initialState-⊥ᵛ : joinForKey initialState result ⊥ᵛ
joinForKey-initialState-⊥ᵛ = cong (λ ins foldr _⊔ᵛ_ ⊥ᵛ (result [ ins ])) initialState-pred-∅
⟦joinAll-initialState⟧ᵛ∅ : joinForKey initialState result ⟧ᵛ []
⟦joinAll-initialState⟧ᵛ∅ = subst (λ vs vs ⟧ᵛ []) (sym joinForKey-initialState-⊥ᵛ) ⟦⊥ᵛ⟧ᵛ∅
analyze-correct : {ρ : Env} [] , rootStmt ⇒ˢ ρ variablesAt finalState result ⟧ᵛ ρ
analyze-correct {ρ} ∅,s⇒ρ = walkTrace {initialState} {finalState} {[]} {ρ} ⟦joinAll-initialState⟧ᵛ∅ (trace ∅,s⇒ρ)

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@ -10,7 +10,7 @@ open import Data.Nat using (; suc; _+_; _≤_)
open import Data.Nat.Properties using (+-comm; m+1+n≰m) open import Data.Nat.Properties using (+-comm; m+1+n≰m)
open import Data.Product using (_×_; Σ; _,_) open import Data.Product using (_×_; Σ; _,_)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl) open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
open import Data.Empty using () open import Data.Empty as Empty using ()
open IsEquivalence ≈-equiv open IsEquivalence ≈-equiv
@ -38,11 +38,16 @@ module _ where
Bounded : Set a Bounded : Set a
Bounded bound = {a₁ a₂ : A} {n : } Chain a₁ a₂ n n bound Bounded bound = {a₁ a₂ : A} {n : } Chain a₁ a₂ n n bound
Bounded-suc-n : {a₁ a₂ : A} {n : } Bounded n Chain a₁ a₂ (suc n) Bounded-suc-n : {a₁ a₂ : A} {n : } Bounded n Chain a₁ a₂ (suc n) Empty.
Bounded-suc-n {a₁} {a₂} {n} bounded c = (m+1+n≰m n n+1≤n) Bounded-suc-n {a₁} {a₂} {n} bounded c = (m+1+n≰m n n+1≤n)
where where
n+1≤n : n + 1 n n+1≤n : n + 1 n
n+1≤n rewrite (+-comm n 1) = bounded c n+1≤n rewrite (+-comm n 1) = bounded c
Height : Set a record Height (height : ) : Set a where
Height height = (Σ (A × A) (λ (a₁ , a₂) Chain a₁ a₂ height) × Bounded height) field
: A
: A
longestChain : Chain height
bounded : Bounded height

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@ -23,15 +23,21 @@ import Chain
module ChainA = Chain _≈_ ≈-equiv _≺_ ≺-cong module ChainA = Chain _≈_ ≈-equiv _≺_ ≺-cong
private private
⊥ᴬ : A open ChainA.Height fixedHeight
⊥ᴬ = proj₁ (proj₁ (proj₁ fixedHeight)) using ()
renaming
( to ⊥ᴬ
; longestChain to longestChainᴬ
; bounded to boundedᴬ
)
⊥ᴬ≼ : (a : A) ⊥ᴬ a ⊥ᴬ≼ : (a : A) ⊥ᴬ a
⊥ᴬ≼ a with ≈-dec a ⊥ᴬ ⊥ᴬ≼ a with ≈-dec a ⊥ᴬ
... | yes a≈⊥ᴬ = ≼-cong a≈⊥ᴬ ≈-refl (≼-refl a) ... | yes a≈⊥ᴬ = ≼-cong a≈⊥ᴬ ≈-refl (≼-refl a)
... | no a̷≈⊥ᴬ with ≈-dec ⊥ᴬ (a ⊥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)) ... | 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 (proj₂ fixedHeight) (ChainA.step x≺⊥ᴬ ≈-refl (proj₂ (proj₁ fixedHeight)))) ... | no ⊥ᴬ̷≈a⊓⊥ᴬ = ⊥-elim (ChainA.Bounded-suc-n boundedᴬ (ChainA.step x≺⊥ᴬ ≈-refl longestChainᴬ))
where where
⊥ᴬ⊓a̷≈⊥ᴬ : ¬ (⊥ᴬ a) ⊥ᴬ ⊥ᴬ⊓a̷≈⊥ᴬ : ¬ (⊥ᴬ a) ⊥ᴬ
⊥ᴬ⊓a̷≈⊥ᴬ = λ ⊥ᴬ⊓a≈⊥ᴬ ⊥ᴬ̷≈a⊓⊥ᴬ (≈-trans (≈-sym ⊥ᴬ⊓a≈⊥ᴬ) (⊓-comm _ _)) ⊥ᴬ⊓a̷≈⊥ᴬ = λ ⊥ᴬ⊓a≈⊥ᴬ ⊥ᴬ̷≈a⊓⊥ᴬ (≈-trans (≈-sym ⊥ᴬ⊓a≈⊥ᴬ) (⊓-comm _ _))
@ -45,7 +51,7 @@ private
-- out, we have exceeded h steps, which shouldn't be possible. -- out, we have exceeded h steps, which shouldn't be possible.
doStep : (g hᶜ : ) (a₁ a₂ : A) (c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ suc h) (a₂≼fa₂ : a₂ f a₂) Σ A (λ a a f a) doStep : (g hᶜ : ) (a₁ a₂ : A) (c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ suc h) (a₂≼fa₂ : a₂ f a₂) Σ A (λ a a f a)
doStep 0 hᶜ a₁ a₂ c g+hᶜ≡sh a₂≼fa₂ rewrite g+hᶜ≡sh = ⊥-elim (ChainA.Bounded-suc-n (proj₂ fixedHeight) c) doStep 0 hᶜ a₁ a₂ c g+hᶜ≡sh a₂≼fa₂ rewrite g+hᶜ≡sh = ⊥-elim (ChainA.Bounded-suc-n boundedᴬ c)
doStep (suc g') hᶜ a₁ a₂ c g+hᶜ≡sh a₂≼fa₂ rewrite sym (+-suc g' hᶜ) doStep (suc g') hᶜ a₁ a₂ c g+hᶜ≡sh a₂≼fa₂ rewrite sym (+-suc g' hᶜ)
with ≈-dec a₂ (f a₂) with ≈-dec a₂ (f a₂)
... | yes a₂≈fa₂ = (a₂ , a₂≈fa₂) ... | yes a₂≈fa₂ = (a₂ , a₂≈fa₂)
@ -71,7 +77,7 @@ private
(c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ suc h) (c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ suc h)
(a₂≼fa₂ : a₂ f a₂) (a₂≼fa₂ : a₂ f a₂)
proj₁ (doStep g hᶜ a₁ a₂ c g+hᶜ≡h a₂≼fa₂) a proj₁ (doStep g hᶜ a₁ a₂ c g+hᶜ≡h a₂≼fa₂) a
stepPreservesLess 0 _ _ _ _ _ _ c g+hᶜ≡sh _ rewrite g+hᶜ≡sh = ⊥-elim (ChainA.Bounded-suc-n (proj₂ fixedHeight) c) stepPreservesLess 0 _ _ _ _ _ _ c g+hᶜ≡sh _ rewrite g+hᶜ≡sh = ⊥-elim (ChainA.Bounded-suc-n boundedᴬ c)
stepPreservesLess (suc g') hᶜ a₁ a₂ a a≈fa a₂≼a c g+hᶜ≡sh a₂≼fa₂ rewrite sym (+-suc g' hᶜ) stepPreservesLess (suc g') hᶜ a₁ a₂ a a≈fa a₂≼a c g+hᶜ≡sh a₂≼fa₂ rewrite sym (+-suc g' hᶜ)
with ≈-dec a₂ (f a₂) with ≈-dec a₂ (f a₂)
... | yes _ = a₂≼a ... | yes _ = a₂≼a

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@ -38,8 +38,9 @@ module TransportFiniteHeight
open IsEquivalence ≈₁-equiv using () renaming (≈-sym to ≈₁-sym; ≈-trans to ≈₁-trans) open IsEquivalence ≈₁-equiv using () renaming (≈-sym to ≈₁-sym; ≈-trans to ≈₁-trans)
open IsEquivalence ≈₂-equiv using () renaming (≈-sym to ≈₂-sym; ≈-trans to ≈₂-trans) open IsEquivalence ≈₂-equiv using () renaming (≈-sym to ≈₂-sym; ≈-trans to ≈₂-trans)
open import Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong using () renaming (Chain to Chain₁; done to done₁; step to step₁) import Chain
open import Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong using () renaming (Chain to Chain₂; done to done₂; step to step₂) open Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong using () renaming (Chain to Chain₁; done to done₁; step to step₁)
open Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong using () renaming (Chain to Chain₂; done to done₂; step to step₂)
private private
f-Injective : Injective _≈₁_ _≈₂_ f f-Injective : Injective _≈₁_ _≈₂_ f
@ -65,10 +66,17 @@ module TransportFiniteHeight
isFiniteHeightLattice : IsFiniteHeightLattice B height _≈₂_ _⊔₂_ _⊓₂_ isFiniteHeightLattice : IsFiniteHeightLattice B height _≈₂_ _⊔₂_ _⊓₂_
isFiniteHeightLattice = isFiniteHeightLattice =
let let
(((a₁ , a₂) , c) , bounded₁) = IsFiniteHeightLattice.fixedHeight fhlA open Chain.Height (IsFiniteHeightLattice.fixedHeight fhlA)
using ()
renaming ( to ⊥₁; to ⊤₁; bounded to bounded₁; longestChain to c)
in record in record
{ isLattice = lB { isLattice = lB
; fixedHeight = (((f a₁ , f a₂), portChain₁ c) , λ c' bounded₁ (portChain₂ c')) ; fixedHeight = record
{ = f ⊥₁
; = f ⊤₁
; longestChain = portChain₁ c
; bounded = λ c' bounded₁ (portChain₂ c')
}
} }
finiteHeightLattice : FiniteHeightLattice B finiteHeightLattice : FiniteHeightLattice B

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@ -10,7 +10,7 @@ open import Data.Fin using (Fin; suc; zero)
open import Data.Fin.Properties as FinProp using (suc-injective) open import Data.Fin.Properties as FinProp using (suc-injective)
open import Data.List as List using (List; []; _∷_) open import Data.List as List using (List; []; _∷_)
open import Data.List.Membership.Propositional as ListMem using () open import Data.List.Membership.Propositional as ListMem using ()
open import Data.List.Relation.Unary.All using (All; []; _∷_) open import Data.List.Membership.Propositional.Properties as ListMemProp using (∈-filter⁺)
open import Data.List.Relation.Unary.Any as RelAny using () open import Data.List.Relation.Unary.Any as RelAny using ()
open import Data.Nat using (; suc) open import Data.Nat using (; suc)
open import Data.Product using (_,_; Σ; proj₁; proj₂) open import Data.Product using (_,_; Σ; proj₁; proj₂)
@ -27,43 +27,26 @@ open import Lattice.MapSet _≟ˢ_ using ()
; to-List to to-Listˢ ; to-List to to-Listˢ
) )
private
z≢sf : {n : } (f : Fin n) ¬ (zero suc f)
z≢sf f ()
z≢mapsfs : {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'
indices : (n : ) Σ (List (Fin n)) Unique
indices 0 = ([] , Utils.empty)
indices (suc n') =
let
(inds' , unids') = indices n'
in
( zero List.map suc inds'
, push (z≢mapsfs inds') (Unique-map suc suc-injective unids')
)
indices-complete : (n : ) (f : Fin n) f ListMem.∈ (proj₁ (indices n))
indices-complete (suc n') zero = RelAny.here refl
indices-complete (suc n') (suc f') = RelAny.there (x∈xs⇒fx∈fxs suc (indices-complete n' f'))
record Program : Set where record Program : Set where
field field
rootStmt : Stmt rootStmt : Stmt
graph : Graph graph : Graph
graph = buildCfg rootStmt graph = wrap (buildCfg rootStmt)
State : Set State : Set
State = Graph.Index graph State = Graph.Index graph
initialState : State initialState : State
initialState = proj₁ (buildCfg-input rootStmt) initialState = proj₁ (wrap-input (buildCfg rootStmt))
finalState : State finalState : State
finalState = proj₁ (buildCfg-output rootStmt) finalState = proj₁ (wrap-output (buildCfg rootStmt))
trace : {ρ : Env} [] , rootStmt ⇒ˢ ρ Trace {graph} initialState finalState [] ρ
trace {ρ} ∅,s⇒ρ
with MkEndToEndTrace idx₁ (RelAny.here refl) idx₂ (RelAny.here refl) tr
EndToEndTrace-wrap (buildCfg-sufficient ∅,s⇒ρ) = tr
private private
vars-Set : StringSet vars-Set : StringSet
@ -76,13 +59,13 @@ record Program : Set where
vars-Unique = proj₂ vars-Set vars-Unique = proj₂ vars-Set
states : List State states : List State
states = proj₁ (indices (Graph.size graph)) states = indices graph
states-complete : (s : State) s ListMem.∈ states states-complete : (s : State) s ListMem.∈ states
states-complete = indices-complete (Graph.size graph) states-complete = indices-complete graph
states-Unique : Unique states states-Unique : Unique states
states-Unique = proj₂ (indices (Graph.size graph)) states-Unique = indices-Unique graph
code : State List BasicStmt code : State List BasicStmt
code st = graph [ st ] code st = graph [ st ]
@ -99,4 +82,10 @@ record Program : Set where
open import Data.List.Membership.DecPropositional _≟ᵉ_ using (_∈?_) open import Data.List.Membership.DecPropositional _≟ᵉ_ using (_∈?_)
incoming : State List State incoming : State List State
incoming idx = List.filter (λ idx' (idx' , idx) ∈? (Graph.edges graph)) states incoming = predecessors graph
edge⇒incoming : {s₁ s₂ : State} (s₁ , s₂) ListMem.∈ (Graph.edges graph)
s₁ ListMem.∈ (incoming s₂)
edge⇒incoming {s₁} {s₂} s₁,s₂∈es =
∈-filter⁺ (λ s' (s' , s₂) ∈? (Graph.edges graph))
(states-complete s₁) s₁,s₂∈es

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@ -7,15 +7,19 @@ open import Data.Fin.Properties as FinProp using (suc-injective)
open import Data.List as List using (List; []; _∷_) open import Data.List as List using (List; []; _∷_)
open import Data.List.Membership.Propositional as ListMem using () open import Data.List.Membership.Propositional as ListMem using ()
open import Data.List.Membership.Propositional.Properties as ListMemProp using () open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
open import Data.List.Relation.Unary.All using (All; []; _∷_)
open import Data.List.Relation.Unary.Any as RelAny using ()
open import Data.Nat as Nat using (; suc) open import Data.Nat as Nat using (; suc)
open import Data.Nat.Properties using (+-assoc; +-comm) open import Data.Nat.Properties using (+-assoc; +-comm)
open import Data.Product using (_×_; Σ; _,_) open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
open import Data.Product.Properties as ProdProp using ()
open import Data.Vec using (Vec; []; _∷_; lookup; cast; _++_) open import Data.Vec using (Vec; []; _∷_; lookup; cast; _++_)
open import Data.Vec.Properties using (cast-is-id; ++-assoc; lookup-++ˡ; cast-sym; ++-identityʳ; lookup-++ʳ) open import Data.Vec.Properties using (cast-is-id; ++-assoc; lookup-++ˡ; cast-sym; ++-identityʳ; lookup-++ʳ)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; refl; subst; trans) open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; refl; subst; trans)
open import Relation.Nullary using (¬_)
open import Lattice open import Lattice
open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct) open import Utils using (Unique; push; Unique-map; x∈xs⇒fx∈fxs; ∈-cartesianProduct)
record Graph : Set where record Graph : Set where
constructor MkGraph constructor MkGraph
@ -112,8 +116,48 @@ singleton bss = record
; outputs = zero [] ; outputs = zero []
} }
wrap : Graph Graph
wrap g = singleton [] g singleton []
buildCfg : Stmt Graph buildCfg : Stmt Graph
buildCfg bs₁ = singleton (bs₁ []) buildCfg bs₁ = singleton (bs₁ [])
buildCfg (s₁ then s₂) = buildCfg s₁ buildCfg s₂ buildCfg (s₁ then s₂) = buildCfg s₁ buildCfg s₂
buildCfg (if _ then s₁ else s₂) = singleton [] (buildCfg s₁ buildCfg s₂) singleton [] buildCfg (if _ then s₁ else s₂) = singleton [] (buildCfg s₁ buildCfg s₂) singleton []
buildCfg (while _ repeat s) = loop (buildCfg s) buildCfg (while _ repeat s) = loop (buildCfg s)
private
z≢sf : {n : } (f : Fin n) ¬ (zero suc f)
z≢sf f ()
z≢mapsfs : {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 : ) 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 : ) (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.List.Membership.DecPropositional (ProdProp.≡-dec (FinProp._≟_ {Graph.size g}) (FinProp._≟_ {Graph.size g})) using (_∈?_)
indices : List (Graph.Index g)
indices = proj₁ (finValues (Graph.size g))
indices-complete : (idx : (Graph.Index g)) idx ListMem.∈ indices
indices-complete = finValues-complete (Graph.size g)
indices-Unique : Unique indices
indices-Unique = proj₂ (finValues (Graph.size g))
predecessors : (Graph.Index g) List (Graph.Index g)
predecessors idx = List.filter (λ idx' (idx' , idx) ∈? (Graph.edges g)) indices

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@ -18,22 +18,11 @@ open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym)
open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct; concat-∈) open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct; concat-∈)
wrap-input : (g : Graph) Σ (Graph.Index (wrap g)) (λ idx Graph.inputs (wrap g) idx [])
wrap-input g = (_ , refl)
buildCfg-input : (s : Stmt) let g = buildCfg s in Σ (Graph.Index g) (λ idx Graph.inputs g idx []) wrap-output : (g : Graph) Σ (Graph.Index (wrap g)) (λ idx Graph.outputs (wrap g) idx [])
buildCfg-input bs₁ = (zero , refl) wrap-output g = (_ , refl)
buildCfg-input (s₁ then s₂)
with (idx , p) buildCfg-input s₁ rewrite p = (_ , refl)
buildCfg-input (if _ then s₁ else s₂) = (zero , refl)
buildCfg-input (while _ repeat s)
with (idx , p) buildCfg-input s rewrite p = (_ , refl)
buildCfg-output : (s : Stmt) let g = buildCfg s in Σ (Graph.Index g) (λ idx Graph.outputs g idx [])
buildCfg-output bs₁ = (zero , refl)
buildCfg-output (s₁ then s₂)
with (idx , p) buildCfg-output s₂ rewrite p = (_ , refl)
buildCfg-output (if _ then s₁ else s₂) = (_ , refl)
buildCfg-output (while _ repeat s)
with (idx , p) buildCfg-output s rewrite p = (_ , refl)
Trace-∙ˡ : {g₁ g₂ : Graph} {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} Trace-∙ˡ : {g₁ g₂ : Graph} {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env}
Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ Trace {g₁} idx₁ idx₂ ρ₁ ρ₂
@ -224,6 +213,10 @@ EndToEndTrace-singleton ρ₁⇒ρ₂ = record
EndToEndTrace-singleton[] : (ρ : Env) EndToEndTrace {singleton []} ρ ρ EndToEndTrace-singleton[] : (ρ : Env) EndToEndTrace {singleton []} ρ ρ
EndToEndTrace-singleton[] env = EndToEndTrace-singleton [] EndToEndTrace-singleton[] env = EndToEndTrace-singleton []
EndToEndTrace-wrap : {g : Graph} {ρ₁ ρ₂ : Env}
EndToEndTrace {g} ρ₁ ρ₂ EndToEndTrace {wrap g} ρ₁ ρ₂
EndToEndTrace-wrap {g} {ρ₁} {ρ₂} etr = EndToEndTrace-singleton[] ρ₁ ++ etr ++ EndToEndTrace-singleton[] ρ₂
buildCfg-sufficient : {s : Stmt} {ρ₁ ρ₂ : Env} ρ₁ , s ⇒ˢ ρ₂ buildCfg-sufficient : {s : Stmt} {ρ₁ ρ₂ : Env} ρ₁ , s ⇒ˢ ρ₂
EndToEndTrace {buildCfg s} ρ₁ ρ₂ EndToEndTrace {buildCfg s} ρ₁ ρ₂
buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ) = buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ) =

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@ -355,7 +355,12 @@ module Plain (x : A) where
rewrite [x]≺y⇒y≡ _ _ [x]≺y with ≈-- y≈z = ⊥-elim (¬-Chain- c) rewrite [x]≺y⇒y≡ _ _ [x]≺y with ≈-- y≈z = ⊥-elim (¬-Chain- c)
fixedHeight : IsLattice.FixedHeight isLattice 2 fixedHeight : IsLattice.FixedHeight isLattice 2
fixedHeight = ((( , ) , longestChain) , isLongest) fixedHeight = record
{ =
; =
; longestChain = longestChain
; bounded = isLongest
}
isFiniteHeightLattice : IsFiniteHeightLattice AboveBelow 2 _≈_ _⊔_ _⊓_ isFiniteHeightLattice : IsFiniteHeightLattice AboveBelow 2 _≈_ _⊔_ _⊓_
isFiniteHeightLattice = record isFiniteHeightLattice = record

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@ -30,7 +30,9 @@ open import Lattice.Map ≡-dec-A lB as Map
; absorb-⊓-⊔ to absorb-⊓ᵐ-⊔ᵐ ; absorb-⊓-⊔ to absorb-⊓ᵐ-⊔ᵐ
; ≈-dec to ≈ᵐ-dec ; ≈-dec to ≈ᵐ-dec
; _[_] to _[_]ᵐ ; _[_] to _[_]ᵐ
; []-∈ to []ᵐ-∈
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ ; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
; m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ to m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ᵐ
; locate to locateᵐ ; locate to locateᵐ
; keys to keysᵐ ; keys to keysᵐ
; _updating_via_ to _updatingᵐ_via_ ; _updating_via_ to _updatingᵐ_via_
@ -104,6 +106,10 @@ module WithKeys (ks : List A) where
_[_] : FiniteMap List A List B _[_] : FiniteMap List A List B
_[_] (m₁ , _) ks = m₁ [ ks ]ᵐ _[_] (m₁ , _) ks = m₁ [ ks ]ᵐ
[]-∈ : {k : A} {v : B} {ks' : List A} (fm : FiniteMap)
k ∈ˡ ks' (k , v) fm v ∈ˡ (fm [ ks' ])
[]-∈ {k} {v} {ks'} (m , _) k∈ks' k,v∈fm = []ᵐ-∈ m k,v∈fm k∈ks'
≈-equiv : IsEquivalence FiniteMap _≈_ ≈-equiv : IsEquivalence FiniteMap _≈_
≈-equiv = record ≈-equiv = record
{ ≈-refl = { ≈-refl =
@ -159,6 +165,11 @@ module WithKeys (ks : List A) where
fm₁ fm₂ (k , v₁) fm₁ (k , v₂) fm₂ v₁ ≼₂ v₂ fm₁ fm₂ (k , v₁) fm₁ (k , v₂) fm₂ v₁ ≼₂ v₂
m₁≼m₂⇒m₁[k]≼m₂[k] (m₁ , _) (m₂ , _) m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂ m₁≼m₂⇒m₁[k]≼m₂[k] (m₁ , _) (m₂ , _) m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ : (fm₁ fm₂ : FiniteMap) {k : A}
fm₁ fm₂ (k∈kfm₁ : k ∈k fm₁) (k∈kfm₂ : k ∈k fm₂)
proj₁ (locate {fm = fm₁} k∈kfm₁) ≈₂ proj₁ (locate {fm = fm₂} k∈kfm₂)
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ (m₁ , _) (m₂ , _) = m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ᵐ m₁ m₂
module GeneralizedUpdate module GeneralizedUpdate
{l} {L : Set l} {l} {L : Set l}
{_≈ˡ_ : L L Set l} {_⊔ˡ_ : L L L} {_⊓ˡ_ : L L L} {_≈ˡ_ : L L Set l} {_⊔ˡ_ : L L L} {_⊓ˡ_ : L L L}

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@ -28,6 +28,7 @@ open import Data.List.Relation.Unary.All using (All)
open import Data.List.Relation.Unary.Any using (Any; here; there) open import Data.List.Relation.Unary.Any using (Any; here; there)
open import Relation.Nullary using (¬_) open import Relation.Nullary using (¬_)
open import Isomorphism using (IsInverseˡ; IsInverseʳ) open import Isomorphism using (IsInverseˡ; IsInverseʳ)
open import Chain using (Height)
open import Lattice.Map ≡-dec-A lB open import Lattice.Map ≡-dec-A lB
using using
@ -104,6 +105,14 @@ module IterProdIsomorphism where
_∈ᵐ_ : {ks : List A} A × B FiniteMap ks Set _∈ᵐ_ : {ks : List A} A × B FiniteMap ks Set
_∈ᵐ_ {ks} = _∈_ ks _∈ᵐ_ {ks} = _∈_ ks
to-build : {b : B} {ks : List A} (uks : Unique ks)
let fm = to uks (IP.build b tt (length ks))
in (k : A) (v : B) (k , v) ∈ᵐ fm v b
to-build {b} {k ks'} (push _ uks') k v (here refl) = refl
to-build {b} {k ks'} (push _ uks') k' v (there k',v∈m') =
to-build {ks = ks'} uks' k' v k',v∈m'
-- The left inverse is: from (to x) = x -- The left inverse is: from (to x) = x
from-to-inverseˡ : {ks : List A} (uks : Unique ks) from-to-inverseˡ : {ks : List A} (uks : Unique ks)
IsInverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {length ks}) IsInverseˡ (_≈ᵐ_ {ks}) (_≈ⁱᵖ_ {length ks})
@ -153,6 +162,26 @@ module IterProdIsomorphism where
fm'₂⊆fm'₁ k' v' k',v'∈fm'₂ fm'₂⊆fm'₁ k' v' k',v'∈fm'₂
in (v'' , (v'≈v'' , there k',v''∈fm'₁)) in (v'' , (v'≈v'' , there k',v''∈fm'₁))
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 fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
with Expr-Provenance-≡ ((` m₁) (` m₂)) k,v∈fm₁fm₂
... | in (single k,v∈m₁) k∉km₂
with k∈km₁ (forget k,v∈m₁)
rewrite trans ks₁≡ks (sym ks₂≡ks) =
⊥-elim (k∉km₂ k∈km₁)
... | in k∉km₁ (single k,v∈m₂)
with k∈km₂ (forget k,v∈m₂)
rewrite trans ks₁≡ks (sym ks₂≡ks) =
⊥-elim (k∉km₁ k∈km₂)
... | bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) =
((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
private private
first-key-in-map : {k : A} {ks : List A} (fm : FiniteMap (k ks)) first-key-in-map : {k : A} {ks : List A} (fm : FiniteMap (k ks))
Σ B (λ v (k , v) ∈ᵐ fm) Σ B (λ v (k , v) ∈ᵐ fm)
@ -204,26 +233,6 @@ module IterProdIsomorphism where
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' (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 (m@(_ _ , push k≢ks _) , 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 fm₁@(m₁ , ks₁≡ks) fm₂@(m₂ , ks₂≡ks) {k} {v} k,v∈fm₁fm₂
with Expr-Provenance-≡ ((` m₁) (` m₂)) k,v∈fm₁fm₂
... | in (single k,v∈m₁) k∉km₂
with k∈km₁ (forget k,v∈m₁)
rewrite trans ks₁≡ks (sym ks₂≡ks) =
⊥-elim (k∉km₂ k∈km₁)
... | in k∉km₁ (single k,v∈m₂)
with k∈km₂ (forget k,v∈m₂)
rewrite trans ks₁≡ks (sym ks₂≡ks) =
⊥-elim (k∉km₁ k∈km₂)
... | bothᵘ {v₁} {v₂} (single k,v₁∈m₁) (single k,v₂∈m₂) =
((v₁ , v₂) , (refl , (k,v₁∈m₁ , k,v₂∈m₂)))
pop-⊔-distr : {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ks)) pop-⊔-distr : {k : A} {ks : List A} (fm₁ fm₂ : FiniteMap (k ks))
pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂) pop (fm₁ ⊔ᵐ fm₂) ≈ᵐ (pop fm₁ ⊔ᵐ pop fm₂)
pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) = pop-⊔-distr {k} {ks} fm₁@(m₁ , _) fm₂@(m₂ , _) =
@ -407,3 +416,12 @@ module IterProdIsomorphism where
(to-⊔-distr uks) (from-⊔-distr {ks}) (to-⊔-distr uks) (from-⊔-distr {ks})
(from-to-inverseʳ uks) (from-to-inverseˡ uks) (from-to-inverseʳ uks) (from-to-inverseˡ uks)
using (isFiniteHeightLattice; finiteHeightLattice) public using (isFiniteHeightLattice; finiteHeightLattice) public
-- Helpful lemma: all entries of the 'bottom' map are assigned to bottom.
open Height (IsFiniteHeightLattice.fixedHeight isFiniteHeightLattice) using ()
⊥-contains-bottoms : {k : A} {v : B} (k , v) ∈ᵐ v (Height.⊥ fhB)
⊥-contains-bottoms {k} {v} k,v∈⊥
rewrite IP.⊥-built (length ks) ≈₂-dec ≈ᵘ-dec h₂ 0 fhB fixedHeightᵘ =
to-build uks k v k,v∈⊥

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@ -11,9 +11,11 @@ module Lattice.IterProd {a} {A B : Set a}
(lA : IsLattice A _≈₁_ _⊔₁_ _⊓₁_) (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where (lA : IsLattice A _≈₁_ _⊔₁_ _⊓₁_) (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_) where
open import Agda.Primitive using (lsuc) open import Agda.Primitive using (lsuc)
open import Data.Nat using (; suc; _+_) open import Data.Nat using (; zero; suc; _+_)
open import Data.Product using (_×_) open import Data.Product using (_×_; _,_; proj₁; proj₂)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; cong)
open import Utils using (iterate) open import Utils using (iterate)
open import Chain using (Height)
open IsLattice lA renaming (FixedHeight to FixedHeight₁) open IsLattice lA renaming (FixedHeight to FixedHeight₁)
open IsLattice lB renaming (FixedHeight to FixedHeight₂) open IsLattice lB renaming (FixedHeight to FixedHeight₂)
@ -30,6 +32,10 @@ IterProd k = iterate k (λ t → A × t) B
-- that are built up by the two iterations. So, do everything in one iteration. -- that are built up by the two iterations. So, do everything in one iteration.
-- This requires some odd code. -- This requires some odd code.
build : A B (k : ) IterProd k
build _ b zero = b
build a b (suc s) = (a , build a b s)
private private
record RequiredForFixedHeight : Set (lsuc a) where record RequiredForFixedHeight : Set (lsuc a) where
field field
@ -39,22 +45,37 @@ private
fhA : FixedHeight₁ h₁ fhA : FixedHeight₁ h₁
fhB : FixedHeight₂ h₂ fhB : FixedHeight₂ h₂
record IsFiniteHeightAndDecEq {A : Set a} {_≈_ : A A Set a} {_⊔_ : A A A} {_⊓_ : A A A} (isLattice : IsLattice A _≈_ _⊔_ _⊓_) : Set (lsuc a) where ⊥₁ : A
⊥₁ = Height.⊥ fhA
⊥₂ : B
⊥₂ = Height.⊥ fhB
⊥k : (k : ) IterProd k
⊥k = build ⊥₁ ⊥₂
record IsFiniteHeightWithBotAndDecEq {A : Set a} {_≈_ : A A Set a} {_⊔_ : A A A} {_⊓_ : A A A} (isLattice : IsLattice A _≈_ _⊔_ _⊓_) ( : A) : Set (lsuc a) where
field field
height : height :
fixedHeight : IsLattice.FixedHeight isLattice height fixedHeight : IsLattice.FixedHeight isLattice height
≈-dec : IsDecidable _≈_ ≈-dec : IsDecidable _≈_
record Everything (A : Set a) : Set (lsuc a) where ⊥-correct : Height.⊥ fixedHeight
record Everything (k : ) : Set (lsuc a) where
T = IterProd k
field field
_≈_ : A A Set a _≈_ : T T Set a
_⊔_ : A A A _⊔_ : T T T
_⊓_ : A A A _⊓_ : T T T
isLattice : IsLattice A _≈_ _⊔_ _⊓_ isLattice : IsLattice T _≈_ _⊔_ _⊓_
isFiniteHeightIfSupported : RequiredForFixedHeight IsFiniteHeightAndDecEq isLattice isFiniteHeightIfSupported :
(req : RequiredForFixedHeight)
IsFiniteHeightWithBotAndDecEq isLattice (RequiredForFixedHeight.⊥k req k)
everything : (k : ) Everything (IterProd k) everything : (k : ) Everything k
everything 0 = record everything 0 = record
{ _≈_ = _≈₂_ { _≈_ = _≈₂_
; _⊔_ = _⊔₂_ ; _⊔_ = _⊔₂_
@ -64,6 +85,7 @@ private
{ height = RequiredForFixedHeight.h₂ req { height = RequiredForFixedHeight.h₂ req
; fixedHeight = RequiredForFixedHeight.fhB req ; fixedHeight = RequiredForFixedHeight.fhB req
; ≈-dec = RequiredForFixedHeight.≈₂-dec req ; ≈-dec = RequiredForFixedHeight.≈₂-dec req
; ⊥-correct = refl
} }
} }
everything (suc k') = record everything (suc k') = record
@ -76,13 +98,16 @@ private
fhlRest = Everything.isFiniteHeightIfSupported everythingRest req fhlRest = Everything.isFiniteHeightIfSupported everythingRest req
in in
record record
{ height = (RequiredForFixedHeight.h₁ req) + IsFiniteHeightAndDecEq.height fhlRest { height = (RequiredForFixedHeight.h₁ req) + IsFiniteHeightWithBotAndDecEq.height fhlRest
; fixedHeight = ; fixedHeight =
P.fixedHeight P.fixedHeight
(RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightAndDecEq.≈-dec fhlRest) (RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightWithBotAndDecEq.≈-dec fhlRest)
(RequiredForFixedHeight.h₁ req) (IsFiniteHeightAndDecEq.height fhlRest) (RequiredForFixedHeight.h₁ req) (IsFiniteHeightWithBotAndDecEq.height fhlRest)
(RequiredForFixedHeight.fhA req) (IsFiniteHeightAndDecEq.fixedHeight fhlRest) (RequiredForFixedHeight.fhA req) (IsFiniteHeightWithBotAndDecEq.fixedHeight fhlRest)
; ≈-dec = P.≈-dec (RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightAndDecEq.≈-dec fhlRest) ; ≈-dec = P.≈-dec (RequiredForFixedHeight.≈₁-dec req) (IsFiniteHeightWithBotAndDecEq.≈-dec fhlRest)
; ⊥-correct =
cong ((Height.⊥ (RequiredForFixedHeight.fhA req)) ,_)
(IsFiniteHeightWithBotAndDecEq.⊥-correct fhlRest)
} }
} }
where where
@ -121,16 +146,22 @@ module _ (k : ) where
; fhB = fhB ; fhB = fhB
} }
fixedHeight = IsFiniteHeightWithBotAndDecEq.fixedHeight (Everything.isFiniteHeightIfSupported (everything k) required)
isFiniteHeightLattice = record isFiniteHeightLattice = record
{ isLattice = isLattice { isLattice = isLattice
; fixedHeight = IsFiniteHeightAndDecEq.fixedHeight (Everything.isFiniteHeightIfSupported (everything k) required) ; fixedHeight = fixedHeight
} }
finiteHeightLattice : FiniteHeightLattice (IterProd k) finiteHeightLattice : FiniteHeightLattice (IterProd k)
finiteHeightLattice = record finiteHeightLattice = record
{ height = IsFiniteHeightAndDecEq.height (Everything.isFiniteHeightIfSupported (everything k) required) { height = IsFiniteHeightWithBotAndDecEq.height (Everything.isFiniteHeightIfSupported (everything k) required)
; _≈_ = _≈_ ; _≈_ = _≈_
; _⊔_ = _⊔_ ; _⊔_ = _⊔_
; _⊓_ = _⊓_ ; _⊓_ = _⊓_
; isFiniteHeightLattice = isFiniteHeightLattice ; isFiniteHeightLattice = isFiniteHeightLattice
} }
⊥-built : Height.⊥ fixedHeight (build (Height.⊥ fhA) (Height.⊥ fhB) k)
⊥-built = IsFiniteHeightWithBotAndDecEq.⊥-correct (Everything.isFiniteHeightIfSupported (everything k) required)

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@ -1112,6 +1112,19 @@ _[_] m (k ∷ ks)
... | yes k∈km = proj₁ (locate {m = m} k∈km) (m [ ks ]) ... | yes k∈km = proj₁ (locate {m = m} k∈km) (m [ ks ])
... | no _ = m [ ks ] ... | no _ = m [ ks ]
[]-∈ : {k : A} {v : B} {ks : List A} (m : Map)
(k , v) m k ∈ˡ ks v ∈ˡ (m [ ks ])
[]-∈ {k} {v} {ks} m k,v∈m (here refl)
with ∈k-dec k (proj₁ m)
... | no k∉km = ⊥-elim (k∉km (forget k,v∈m))
... | yes k∈km
with (v' , k,v'∈m) locate {m = m} k∈km
rewrite Map-functional {m = m} k,v'∈m k,v∈m = here refl
[]-∈ {k} {v} {k' ks'} m k,v∈m (there k∈ks')
with ∈k-dec k' (proj₁ m)
... | no _ = []-∈ m k,v∈m k∈ks'
... | yes _ = there ([]-∈ m k,v∈m k∈ks')
m₁≼m₂⇒m₁[k]≼m₂[k] : (m₁ m₂ : Map) {k : A} {v₁ v₂ : B} m₁≼m₂⇒m₁[k]≼m₂[k] : (m₁ m₂ : Map) {k : A} {v₁ v₂ : B}
m₁ m₂ (k , v₁) m₁ (k , v₂) m₂ v₁ ≼₂ v₂ m₁ m₂ (k , v₁) m₁ (k , v₂) m₂ v₁ ≼₂ v₂
m₁≼m₂⇒m₁[k]≼m₂[k] m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂ m₁≼m₂⇒m₁[k]≼m₂[k] m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂
@ -1129,3 +1142,12 @@ m₁≼m₂⇒k∈km₁⇒k∈km₂ m₁ m₂ m₁≼m₂ k∈km₁ =
(v' , (v≈v' , k,v'∈m₂)) = (proj₁ m₁≼m₂) _ _ k,v∈m₁m₂ (v' , (v≈v' , k,v'∈m₂)) = (proj₁ m₁≼m₂) _ _ k,v∈m₁m₂
in in
forget k,v'∈m₂ forget k,v'∈m₂
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ : (m₁ m₂ : Map) {k : A}
m₁ m₂ (k∈km₁ : k ∈k m₁) (k∈km₂ : k ∈k m₂)
proj₁ (locate {m = m₁} k∈km₁) ≈₂ proj₁ (locate {m = m₂} k∈km₂)
m₁≈m₂⇒k∈m₁⇒k∈km₂⇒v₁≈v₂ m₁ m₂ {k} (m₁⊆m₂ , m₂⊆m₁) k∈km₁ k∈km₂
with (v₁ , k,v₁∈m₁) locate {m = m₁} k∈km₁
with (v₂ , k,v₂∈m₂) locate {m = m₂} k∈km₂
with (v₂' , (v₁≈v₂' , k,v₂'∈m₂)) m₁⊆m₂ k v₁ k,v₁∈m₁
rewrite Map-functional {m = m₂} k,v₂∈m₂ k,v₂'∈m₂ = v₁≈v₂'

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@ -143,17 +143,8 @@ module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
∙,b-Preserves-≈₁ : (b : B) (λ a (a , b)) Preserves _≈₁_ _≈_ ∙,b-Preserves-≈₁ : (b : B) (λ a (a , b)) Preserves _≈₁_ _≈_
∙,b-Preserves-≈₁ b {a₁} {a₂} a₁≈a₂ = (a₁≈a₂ , ≈₂-refl) ∙,b-Preserves-≈₁ b {a₁} {a₂} a₁≈a₂ = (a₁≈a₂ , ≈₂-refl)
amin : A open ChainA.Height fhA using () renaming ( to ⊥₁; to ⊤₁; longestChain to longestChain₁; bounded to bounded₁)
amin = proj₁ (proj₁ (proj₁ fhA)) open ChainB.Height fhB using () renaming ( to ⊥₂; to ⊤₂; longestChain to longestChain₂; bounded to bounded₂)
amax : A
amax = proj₂ (proj₁ (proj₁ fhA))
bmin : B
bmin = proj₁ (proj₁ (proj₁ fhB))
bmax : B
bmax = proj₂ (proj₁ (proj₁ fhB))
unzip : {a₁ a₂ : A} {b₁ b₂ : B} {n : } Chain (a₁ , b₁) (a₂ , b₂) n Σ ( × ) (λ (n₁ , n₂) ((Chain₁ a₁ a₂ n₁ × Chain₂ b₁ b₂ n₂) × (n n₁ + n₂))) unzip : {a₁ a₂ : A} {b₁ b₂ : B} {n : } Chain (a₁ , b₁) (a₂ , b₂) n Σ ( × ) (λ (n₁ , n₂) ((Chain₁ a₁ a₂ n₁ × Chain₂ b₁ b₂ n₂) × (n n₁ + n₂)))
unzip (done (a₁≈a₂ , b₁≈b₂)) = ((0 , 0) , ((done₁ a₁≈a₂ , done₂ b₁≈b₂) , ≤-refl)) unzip (done (a₁≈a₂ , b₁≈b₂)) = ((0 , 0) , ((done₁ a₁≈a₂ , done₂ b₁≈b₂) , ≤-refl))
@ -172,15 +163,16 @@ module _ (≈₁-dec : IsDecidable _≈₁_) (≈₂-dec : IsDecidable _≈₂_)
)) ))
fixedHeight : IsLattice.FixedHeight isLattice (h₁ + h₂) fixedHeight : IsLattice.FixedHeight isLattice (h₁ + h₂)
fixedHeight = fixedHeight = record
( ( ((amin , bmin) , (amax , bmax)) { = (⊥₁ , ⊥₂)
, concat ; = (⊤₁ , ⊤₂)
(ChainMapping₁.Chain-map (λ a (a , bmin)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) (proj₂ (proj₁ fhA))) ; longestChain = concat
(ChainMapping₂.Chain-map (λ b (amax , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) (proj₂ (proj₁ fhB))) (ChainMapping₁.Chain-map (λ a (a , ⊥₂)) (∙,b-Monotonic _) proj₁ (∙,b-Preserves-≈₁ _) longestChain₁)
) (ChainMapping₂.Chain-map (λ b (⊤₁ , b)) (a,∙-Monotonic _) proj₂ (a,∙-Preserves-≈₂ _) longestChain₂)
, λ a₁b₁a₂b₂ let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂ ; bounded = λ a₁b₁a₂b₂
in ≤-trans n≤n₁+n₂ (+-mono-≤ (proj₂ fhA a₁a₂) (proj₂ fhB b₁b₂)) let ((n₁ , n₂) , ((a₁a₂ , b₁b₂) , n≤n₁+n₂)) = unzip a₁b₁a₂b₂
) in ≤-trans n≤n₁+n₂ (+-mono-≤ (bounded₁ a₁a₂) (bounded₂ b₁b₂))
}
isFiniteHeightLattice : IsFiniteHeightLattice (A × B) (h₁ + h₂) _≈_ _⊔_ _⊓_ isFiniteHeightLattice : IsFiniteHeightLattice (A × B) (h₁ + h₂) _≈_ _⊔_ _⊓_
isFiniteHeightLattice = record isFiniteHeightLattice = record

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@ -108,7 +108,12 @@ private
isLongest (done _) = z≤n isLongest (done _) = z≤n
fixedHeight : IsLattice.FixedHeight isLattice 0 fixedHeight : IsLattice.FixedHeight isLattice 0
fixedHeight = (((tt , tt) , longestChain) , isLongest) fixedHeight = record
{ = tt
; = tt
; longestChain = longestChain
; bounded = isLongest
}
isFiniteHeightLattice : IsFiniteHeightLattice 0 _≈_ _⊔_ _⊓_ isFiniteHeightLattice : IsFiniteHeightLattice 0 _≈_ _⊔_ _⊓_
isFiniteHeightLattice = record isFiniteHeightLattice = record