Safe Haskell	None
Language	Haskell2010

FP.Prelude.Lattice

Documentation

data PartialOrdering Source

Constructors

PLT
PEQ
PGT
PUN

class POrd a where Source

Methods

(⊑⊒) :: a -> a -> PartialOrdering infix 6 Source

Instances

POrd 𝔹 Source
POrd 𝔻 Source
POrd 𝕀 Source
POrd ℕᵀ Source
POrd ℕ Source
POrd ℤ Source
POrd Exp Source
POrd Time Source
POrd VarAddr Source
POrd ExpAddr Source
OrdFunctorial POrd ID Source
Bifunctorial * POrd (,) Source
Functorial * POrd ID Source
OrdFunctorial POrd ς => OrdPolymorphic * POrd (NondetJoinΠ ς) Source
(OrdPolymorphic * POrd ς, Ord s) => OrdPolymorphic * POrd (PolyStateΠ s ς) Source
(OrdFunctorial POrd ς, POrd s, JoinLattice s, Ord s) => OrdPolymorphic * POrd (FlowJoinΠ s ς) Source
(OrdFunctorial POrd ς, POrd s, Ord s) => OrdFunctorial POrd (StateΠ s ς) Source
POrd a => POrd (Maybe a) Source
Ord a => POrd (𝒫ᵇ a) Source
POrd (𝒫 a) Source
POrd a => POrd (ID a) Source
POrd val => POrd (LamIfState val) Source
POrd val => POrd (LamIfStores val) Source
(POrd a, POrd b) => POrd (a, b) Source
(Ord k, POrd v, JoinLattice v) => POrd ((⇰♭⊔) k v) Source
POrd v => POrd ((⇰) k v) Source
(OrdFunctorial POrd ς, Ord a) => POrd (NondetJoinΠ ς a) Source
POrd (PathSensitiveΣ val a) Source
POrd val => POrd (FlowInsensitiveΣ val a) Source
POrd val => POrd (FlowSensitiveΣ val a) Source
(OrdFunctorial POrd ς, POrd s, POrd a, Ord s, Ord a) => POrd (StateΠ s ς a) Source
(OrdPolymorphic * POrd ς, Ord s, Ord a) => POrd (PolyStateΠ s ς a) Source
(OrdFunctorial POrd ς, POrd s, JoinLattice s, Ord s, Ord a) => POrd (FlowJoinΠ s ς a) Source
POrd (t (u a)) => POrd ((⊡) k k t u a) Source

fromOrdering :: Ordering -> PartialOrdering Source

(⊑) :: POrd a => a -> a -> 𝔹 infix 6 Source

(⊒) :: POrd a => a -> a -> 𝔹 infix 6 Source

(><) :: POrd a => a -> a -> 𝔹 infix 6 Source

poCompareFromLte :: (a -> a -> 𝔹) -> a -> a -> PartialOrdering Source

discretePO :: Eq a => a -> a -> PartialOrdering Source

class Bot a where Source

Methods

bot :: a Source

Instances

Bot 𝔹 Source
Bot 𝔻 Source
Bot 𝕀 Source
Bot ℕᵀ Source
Bot ℕ Source
Bot Loc Source
Bot SourceErrorTrace Source
Functorial * Bot ID Source
Bot (𝒫ᵇ a) Source
Bot (𝒫 a) Source
Bot a => Bot (AddTop a) Source
Bot (AddBot a) Source
Bot a => Bot (ID a) Source
Bot (LamIfStores val) Source
Bot b => Bot (a -> b) Source
(Bot a, Bot b) => Bot (a, b) Source
Bot ((⇰♭⊔) k v) Source
Bot ((⇰) k v) Source
Functorial * JoinLattice ς => Bot (NondetJoinΠ ς a) Source
Bot (PathSensitiveΣ val a) Source
Bot (FlowInsensitiveΣ val a) Source
Bot (FlowSensitiveΣ val a) Source
(Functorial * JoinLattice ς, JoinLattice s, JoinLattice a) => Bot (StateΠ s ς a) Source
Polymorphic * * JoinLattice m => Bot (PolyStateT s m a) Source
Polymorphic * * JoinLattice ς => Bot (PolyStateΠ s ς a) Source
Functorial * JoinLattice ς => Bot (FlowJoinΠ s ς a) Source
Bot (t (u a)) => Bot ((⊡) k k t u a) Source

class Join a where Source

Methods

(⊔) :: a -> a -> a infixr 4 Source

Instances

Join 𝔹 Source
Join 𝔻 Source
Join 𝕀 Source
Join ℕᵀ Source
Join ℕ Source
Join ℤ Source
Join Loc Source
Join LocRange Source
Join SourceErrorTrace Source
Functorial * Join ID Source
Join (𝒫ᵇ a) Source
Join (𝒫 a) Source
Join a => Join (AddTop a) Source
Join a => Join (AddBot a) Source
Join a => Join (ID a) Source
Join val => Join (LamIfStores val) Source
Join b => Join (a -> b) Source
(Join a, Join b) => Join (a, b) Source
Join ((⇰♭⊔) k v) Source
Join v => Join ((⇰) k v) Source
Functorial * JoinLattice ς => Join (NondetJoinΠ ς a) Source
Join (PathSensitiveΣ val a) Source
Join val => Join (FlowInsensitiveΣ val a) Source
Join val => Join (FlowSensitiveΣ val a) Source
(Functorial * JoinLattice ς, JoinLattice s, JoinLattice a) => Join (StateΠ s ς a) Source
Polymorphic * * JoinLattice m => Join (PolyStateT s m a) Source
Polymorphic * * JoinLattice ς => Join (PolyStateΠ s ς a) Source
Functorial * JoinLattice ς => Join (FlowJoinΠ s ς a) Source
Join (t (u a)) => Join ((⊡) k k t u a) Source

class Top a where Source

Methods

top :: a Source

Instances

Top 𝔹 Source
Top 𝔻 Source
Top 𝕀 Source
Top ℕᵀ Source
Functorial * Top ID Source
Top (AddTop a) Source
Top a => Top (AddBot a) Source
Top a => Top (ID a) Source
Top b => Top (a -> b) Source
(Top a, Top b) => Top (a, b) Source
Top (t (u a)) => Top ((⊡) k k t u a) Source

class Meet a where Source

Methods

(⊓) :: a -> a -> a infixr 5 Source

Instances

Meet 𝔹 Source
Meet 𝔻 Source
Meet 𝕀 Source
Meet ℕᵀ Source
Meet ℕ Source
Meet ℤ Source
Meet Loc Source
Functorial * Meet ID Source
Meet (𝒫 a) Source
Meet a => Meet (AddTop a) Source
Meet a => Meet (AddBot a) Source
Meet a => Meet (ID a) Source
Meet b => Meet (a -> b) Source
(Meet a, Meet b) => Meet (a, b) Source
Meet v => Meet ((⇰) k v) Source
Meet (t (u a)) => Meet ((⊡) k k t u a) Source

class Dual a where Source

Methods

dual :: a -> a Source

Instances

Dual 𝔹 Source
Dual b => Dual (a -> b) Source
(Dual a, Dual b) => Dual (a, b) Source

class Difference a where Source

Methods

(⊟) :: a -> a -> a infix 4 Source

Instances

Ord a => Difference (𝒫ᵇ a) Source
Difference (𝒫 a) Source
Difference a => Difference (ID a) Source
Difference val => Difference (LamIfStores val) Source
Difference b => Difference (a -> b) Source
(Difference a, Difference b) => Difference (a, b) Source
(Ord k, JoinLattice v, Difference v) => Difference ((⇰♭⊔) k v) Source
Difference v => Difference ((⇰) k v) Source
Difference (ς (𝒫ᵇ a)) => Difference (NondetJoinΠ ς a) Source
Difference (PathSensitiveΣ val a) Source
Difference val => Difference (FlowInsensitiveΣ val a) Source
Difference val => Difference (FlowSensitiveΣ val a) Source
Difference (ς (a, s)) => Difference (StateΠ s ς a) Source
Difference (ς (a, s)) => Difference (PolyStateΠ s ς a) Source
Difference (ς ((⇰♭⊔) a s)) => Difference (FlowJoinΠ s ς a) Source

class (Bot a, Join a) => JoinLattice a Source

Instances

JoinLattice 𝔹 Source
JoinLattice 𝔻 Source
JoinLattice 𝕀 Source
JoinLattice ℕᵀ Source
JoinLattice ℕ Source
JoinLattice SourceErrorTrace Source
Functorial * JoinLattice ID Source
Functorial * JoinLattice ς => Polymorphic * * JoinLattice (NondetJoinΠ ς) Source
Polymorphic * * JoinLattice m => Polymorphic * * JoinLattice (PolyStateT s m) Source
Polymorphic * * JoinLattice ς => Polymorphic * * JoinLattice (PolyStateΠ s ς) Source
Functorial * JoinLattice ς => Polymorphic * * JoinLattice (FlowJoinΠ s ς) Source
JoinLattice a => Functorial * JoinLattice ((,) a) Source
(Functorial * JoinLattice ς, JoinLattice s) => Functorial * JoinLattice (StateΠ s ς) Source
JoinLattice (𝒫ᵇ a) Source
JoinLattice (𝒫 a) Source
JoinLattice a => JoinLattice (AddTop a) Source
Join a => JoinLattice (AddBot a) Source
JoinLattice a => JoinLattice (ID a) Source
Join val => JoinLattice (LamIfStores val) Source
JoinLattice b => JoinLattice (a -> b) Source
(JoinLattice a, JoinLattice b) => JoinLattice (a, b) Source
JoinLattice ((⇰♭⊔) k v) Source
Join v => JoinLattice ((⇰) k v) Source
Functorial * JoinLattice ς => JoinLattice (NondetJoinΠ ς a) Source
JoinLattice (PathSensitiveΣ val a) Source
Join val => JoinLattice (FlowInsensitiveΣ val a) Source
Join val => JoinLattice (FlowSensitiveΣ val a) Source
(Functorial * JoinLattice ς, JoinLattice s, JoinLattice a) => JoinLattice (StateΠ s ς a) Source
Polymorphic * * JoinLattice m => JoinLattice (PolyStateT s m a) Source
Polymorphic * * JoinLattice ς => JoinLattice (PolyStateΠ s ς a) Source
Functorial * JoinLattice ς => JoinLattice (FlowJoinΠ s ς a) Source
JoinLattice (t (u a)) => JoinLattice ((⊡) k k t u a) Source

class (Top a, Meet a) => MeetLattice a Source

Instances

MeetLattice 𝔹 Source
MeetLattice 𝔻 Source
MeetLattice 𝕀 Source
MeetLattice ℕᵀ Source
Meet a => MeetLattice (AddTop a) Source
MeetLattice a => MeetLattice (AddBot a) Source
MeetLattice b => MeetLattice (a -> b) Source
(MeetLattice a, MeetLattice b) => MeetLattice (a, b) Source
MeetLattice (t (u a)) => MeetLattice ((⊡) k k t u a) Source

class (JoinLattice a, MeetLattice a) => Lattice a Source

Instances

Lattice 𝔹 Source
Lattice 𝔻 Source
Lattice 𝕀 Source
Lattice ℕᵀ Source
(JoinLattice a, Meet a) => Lattice (AddTop a) Source
(Join a, MeetLattice a) => Lattice (AddBot a) Source
Lattice b => Lattice (a -> b) Source
(Lattice a, Lattice b) => Lattice (a, b) Source
Lattice (t (u a)) => Lattice ((⊡) k k t u a) Source

class (Lattice a, Dual a) => DualLattice a Source

Instances

DualLattice 𝔹 Source
DualLattice b => DualLattice (a -> b) Source
(DualLattice a, DualLattice b) => DualLattice (a, b) Source

dictJoin :: (ToFold (k, v) t, Ord k, JoinLattice v) => t -> k ⇰ v Source

mapKeyJoin :: (Ord k₂, JoinLattice v) => (k₁ -> k₂) -> (k₁ ⇰ v) -> k₂ ⇰ v Source

concretizeDictJoin :: (Ord k, JoinLattice v) => (k ⇰♭⊔ v) -> k ⇰ v Source

lazyDictJoin :: ToFold (k, v) t => t -> k ⇰♭⊔ v Source

mapKeyLazyDictJoin :: (k -> k') -> (k ⇰♭⊔ v) -> k' ⇰♭⊔ v Source

joins :: (JoinLattice a, ToFold a t) => t -> a Source

meets :: (MeetLattice a, ToFold a t) => t -> a Source

data AddBot a Source

Constructors

Bot
AddBot a

Instances

Functor AddBot Source
Eq a => Eq (AddBot a) Source
Ord a => Ord (AddBot a) Source
(Join a, MeetLattice a) => Lattice (AddBot a) Source
MeetLattice a => MeetLattice (AddBot a) Source
Join a => JoinLattice (AddBot a) Source
Meet a => Meet (AddBot a) Source
Top a => Top (AddBot a) Source
Join a => Join (AddBot a) Source
Bot (AddBot a) Source

elimAddBot :: b -> (a -> b) -> AddBot a -> b Source

data AddTop a Source

Constructors

AddTop a
Top

Instances

Functor AddTop Source
Eq a => Eq (AddTop a) Source
Ord a => Ord (AddTop a) Source
(JoinLattice a, Meet a) => Lattice (AddTop a) Source
Meet a => MeetLattice (AddTop a) Source
JoinLattice a => JoinLattice (AddTop a) Source
Meet a => Meet (AddTop a) Source
Top (AddTop a) Source
Join a => Join (AddTop a) Source
Bot a => Bot (AddTop a) Source