[PPT] - Dependently Typed Programming with Finite Sets Denis Firsov and PowerPoint Presentation

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Dependently Typed Programming with Finite Sets

Denis Firsov and Tarmo Uustalu

Institute of Cybernetics at TUT

November 13, 2015

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Outline

The problem and motivation Different notions of finiteness Pragmatic finite subsets Approaches to defining functions from finite sets Prover for quantified formulas over decidable properties on finite sets

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Motivating Example I

A finite set could be defined as an inductive enumeration type: data Pauli : Set where X : Pauli Y : Pauli Z : Pauli I : Pauli

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Motivating Example I

A finite set could be defined as an inductive enumeration type: data Pauli : Set where X : Pauli Y : Pauli Z : Pauli I : Pauli Next, we can list all the elements: listPauli : List Pauli listPauli = X :: Y :: Z :: I :: []

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Motivating Example I

A finite set could be defined as an inductive enumeration type: data Pauli : Set where X : Pauli Y : Pauli Z : Pauli I : Pauli Next, we can list all the elements: listPauli : List Pauli listPauli = X :: Y :: Z :: I :: [] Finally, we can prove that the list is indeed complete: allPauli : (x : Pauli) → x ∈ listPauli allPauli X = here allPauli Y = there here allPauli Z = there (there here) allPauli I = there (there (there here))

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Motivating Example II

Typically we also want decidable equality:

Dec : Set → Set Dec P = P ⊎ ¬ P

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Motivating Example II

Typically we also want decidable equality:

Dec : Set → Set Dec P = P ⊎ ¬ P _≡P?_ : (x1 x2 : Pauli) → Dec (x1 ≡ x2) X ≡P? X = inj1 refl X ≡P? Y = inj2 λ() X ≡P? Z = inj2 λ() X ≡P? I = inj2 λ() Y ≡P? X = inj2 λ() Y ≡P? Y = inj1 refl Y ≡P? Z = inj2 λ() Y ≡P? I = inj2 λ() Z ≡P? X = inj2 λ() Z ≡P? Y = inj2 λ() Z ≡P? Z = inj1 refl Z ≡P? I = inj2 λ() I ≡P? X = inj2 λ() I ≡P? Y = inj2 λ() I ≡P? Z = inj2 λ() I ≡P? I = inj1 refl

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Listable Finite Sets

All X xs says that list xs has all elements of type X. All : (X : Set) → List X → Set All X xs = (x : X) → x ∈ xs

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Listable Finite Sets

All X xs says that list xs has all elements of type X. All : (X : Set) → List X → Set All X xs = (x : X) → x ∈ xs A set X is listable if there is a list xs so that All X xs Listable : (X : Set) → Set Listable X = Σ[ xs ∈ List X ] All X xs

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Listable Finite Sets

All X xs says that list xs has all elements of type X. All : (X : Set) → List X → Set All X xs = (x : X) → x ∈ xs A set X is listable if there is a list xs so that All X xs Listable : (X : Set) → Set Listable X = Σ[ xs ∈ List X ] All X xs It is the same as asking for a surjection from an initial segment of N: FinSurj : (X : Set) → Set FinSurj X = Σ[ n ∈ N ] Σ[ fromFin ∈ (Fin n → X) ] Σ[ toFin ∈ (X → Fin n) ] ((x : X) → fromFin (toFin x) ≡ x)

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Listable Finite Sets

All X xs says that list xs has all elements of type X. All : (X : Set) → List X → Set All X xs = (x : X) → x ∈ xs A set X is listable if there is a list xs so that All X xs Listable : (X : Set) → Set Listable X = Σ[ xs ∈ List X ] All X xs It is the same as asking for a surjection from an initial segment of N: FinSurj : (X : Set) → Set FinSurj X = Σ[ n ∈ N ] Σ[ fromFin ∈ (Fin n → X) ] Σ[ toFin ∈ (X → Fin n) ] ((x : X) → fromFin (toFin x) ≡ x) surj2lstbl : {X : Set} → FinSurj X → Listable X lstbl2surj : {X : Set} → Listable X → FinSurj X

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Listable Finite Sets without Duplicates

NoDup xs says that the list xs has no duplicates: NoDup : {X : Set} → List X → Set NoDup {X} xs = (x : X) → (p1 p2 : x ∈ xs) → p1 ≡ p2

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Listable Finite Sets without Duplicates

NoDup xs says that the list xs has no duplicates: NoDup : {X : Set} → List X → Set NoDup {X} xs = (x : X) → (p1 p2 : x ∈ xs) → p1 ≡ p2 Now, we can define duplicate-free listability of a set: ListableNoDup : (X : Set) → Set ListableNoDup X = Σ[ xs ∈ List X ] All X xs × NoDup xs

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Listable Finite Sets without Duplicates

NoDup xs says that the list xs has no duplicates: NoDup : {X : Set} → List X → Set NoDup {X} xs = (x : X) → (p1 p2 : x ∈ xs) → p1 ≡ p2 Now, we can define duplicate-free listability of a set: ListableNoDup : (X : Set) → Set ListableNoDup X = Σ[ xs ∈ List X ] All X xs × NoDup xs This is equivalent to having a bijection from an initial segment of N: FinBij : (X : Set) → Set FinBij X = Σ[ n ∈ N ] Σ[ fromFin ∈ (Fin n → X) ] Σ[ toFin ∈ (X → Fin n) ] ((x : X) → fromFin (toFin x) ≡ x) × ((i : Fin n) → toFin (fromFin i) ≡ i)

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Listable Finite Sets without Duplicates

NoDup xs says that the list xs has no duplicates: NoDup : {X : Set} → List X → Set NoDup {X} xs = (x : X) → (p1 p2 : x ∈ xs) → p1 ≡ p2 Now, we can define duplicate-free listability of a set: ListableNoDup : (X : Set) → Set ListableNoDup X = Σ[ xs ∈ List X ] All X xs × NoDup xs This is equivalent to having a bijection from an initial segment of N: FinBij : (X : Set) → Set FinBij X = Σ[ n ∈ N ] Σ[ fromFin ∈ (Fin n → X) ] Σ[ toFin ∈ (X → Fin n) ] ((x : X) → fromFin (toFin x) ≡ x) × ((i : Fin n) → toFin (fromFin i) ≡ i) bij2lstblnd : {X : Set} → FinBij X → ListableNoDup X lstblnd2bij : {X : Set} → ListableNoDup X → FinBij X

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Listability and Decidable Equality

All four notions of finiteness (Listable, FinSurj, ListableNoDup, FinBij) are equivalent.

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Listability and Decidable Equality

All four notions of finiteness (Listable, FinSurj, ListableNoDup, FinBij) are equivalent. Indeed, equality on a listable set is decidable: lstbl2deq : {X : Set} → Listable X → DecEq X

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Listability and Decidable Equality

All four notions of finiteness (Listable, FinSurj, ListableNoDup, FinBij) are equivalent. Indeed, equality on a listable set is decidable: lstbl2deq : {X : Set} → Listable X → DecEq X Therefore, we can implement removal of duplicates and convert Listable to ListableNoDup: lstbl2nodup : {X : Set} → Listable X → ListableNoDup X

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Listability and Decidable Equality

All four notions of finiteness (Listable, FinSurj, ListableNoDup, FinBij) are equivalent. Indeed, equality on a listable set is decidable: lstbl2deq : {X : Set} → Listable X → DecEq X Therefore, we can implement removal of duplicates and convert Listable to ListableNoDup: lstbl2nodup : {X : Set} → Listable X → ListableNoDup X The other direction is trivial: nodup2lstbl : {X : Set} → ListableNoDup X → Listable X

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Listability: Example

The generic proof lstbl2deq also provides an alternative way for defining an equality decider for concrete listable types like Pauli: listablePauli : Listable Pauli listablePauli = listPauli , allPauli _≡P?_ : DecEq Pauli _≡P?_ = lstbl2deq listablePauli Recall that the direct implementation took |Pauli|2 lines of code.

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Finite Subsets

We can give a special definition of listability of a subset given by a predicate on some base set: ListableSub : (U : Set) → (U → Set) → Set ListableSub U P = Σ[ xs ∈ List U ] ((x : U) → P x → x ∈ xs) × ((x : U) → x ∈ xs → P x)

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Finite Subsets

We can give a special definition of listability of a subset given by a predicate on some base set: ListableSub : (U : Set) → (U → Set) → Set ListableSub U P = Σ[ xs ∈ List U ] ((x : U) → P x → x ∈ xs) × ((x : U) → x ∈ xs → P x) Listable sets are a special case of listable subsets: lstbl2lsub : {U : Set} → Listable U → ListableSub U (λ _ → ⊤) lsub2lstbl : {U : Set} → ListableSub U (λ _ → ⊤) → Listable U

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Decidable Equality for Finite Subsets

In general, equality on a finite subset is not decidable: deqLstblSub : {U : Set} → (P : U → Set) → ListableSub U P → (x1 x2 : U) → P x1 → P x2 → Dec (x1 ≡ x2) deqLstblSub = ???

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Decidable Equality for Finite Subsets

In general, equality on a finite subset is not decidable: deqLstblSub : {U : Set} → (P : U → Set) → ListableSub U P → (x1 x2 : U) → P x1 → P x2 → Dec (x1 ≡ x2) deqLstblSub = ??? If P is a mere proposition, then equality is decidable: deqLstblSub1 : {U : Set} → (P : U → Set) → ListableSub U P → ((x : U) → (p1 p2 : P x) → p1 ≡ p2) → (x1 x2 : U) → P x1 → P x2 → Dec (x1 ≡ x2)

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Decidable Equality for Finite Subsets

In general, equality on a finite subset is not decidable: deqLstblSub : {U : Set} → (P : U → Set) → ListableSub U P → (x1 x2 : U) → P x1 → P x2 → Dec (x1 ≡ x2) deqLstblSub = ??? If P is a mere proposition, then equality is decidable: deqLstblSub1 : {U : Set} → (P : U → Set) → ListableSub U P → ((x : U) → (p1 p2 : P x) → p1 ≡ p2) → (x1 x2 : U) → P x1 → P x2 → Dec (x1 ≡ x2) If the list contains no duplicates, then equality is decidable: deqLstblSub2 : {U : Set} → (P : U → Set) → (p : ListableSub U P) → NoDup (proj1 p) → (x1 x2 : U) → P x1 → P x2 → Dec (x1 ≡ x2)

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Pragmatic Finite Subsets

If a set P is decidable, we can effectively squash P:

_ : {P : Set} → Dec P → Set inj1 _ = ⊤ inj2 _ = ⊥

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Pragmatic Finite Subsets

If a set P is decidable, we can effectively squash P:

_ : {P : Set} → Dec P → Set inj1 _ = ⊤ inj2 _ = ⊥ We describe a finite subset by listing all its elements with or without duplicates: data FinSubDesc (U : Set) (eq : DecEq U) : Bool → Set where fsd-plain : List U → FinSubDesc U eq true fsd-nodup : (xs : List U) → { nodup? xs } → FinSubDesc U eq false

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Pragmatic Finite Subsets

If a set P is decidable, we can effectively squash P:

_ : {P : Set} → Dec P → Set inj1 _ = ⊤ inj2 _ = ⊥ We describe a finite subset by listing all its elements with or without duplicates: data FinSubDesc (U : Set) (eq : DecEq U) : Bool → Set where fsd-plain : List U → FinSubDesc U eq true fsd-nodup : (xs : List U) → { nodup? xs } → FinSubDesc U eq false An element of a FinSubDesc is a dependent pair of an element together with a squashed proof that it belongs to the subset: Elem : {U : Set}{eq : DecEq U}{b : Bool} → FinSubDesc U eq b → Set Elem {U} {eq} D = Σ[ x ∈ U ] x ∈? toList D

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