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The Lifting Lemma WG2.8

The Lifting Lemma

Ralf Hinze

Computing Laboratory, University of Oxford Wolfson Building, Parks Road, Oxford, OX1 3QD, England ralf.hinze@comlab.ox.ac.uk http://www.comlab.ox.ac.uk/ralf.hinze/

June 2009

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The Lifting Lemma WG2.8

Mathematicians do it . . .

(f + g)(x) = f(x) + g(x)

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The Lifting Lemma WG2.8

. . . over and . . .

A + B = { a + b | a ∈ A, b ∈ B}

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The Lifting Lemma WG2.8

. . . and over again.

(x1, x2, x3) + (y1, y2, y3) = (x1 + y1, x2 + y2, x3 + y3)

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The Lifting Lemma WG2.8

Haskell programmers do it . . .

data Maybe α = Nothing | Just α (+) :: Maybe N → Maybe N → Maybe N Nothing + n = Nothing m + Nothing = Nothing Just a + Just b = Just (a + b)

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The Lifting Lemma WG2.8

. . . over and over again.

(+) :: IO N → IO N → IO N m + n = do {a ← m; b ← n; return (a + b)}

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The Lifting Lemma WG2.8

I do it: lifting

data Stream α = Cons {head :: α, tail :: Stream α} (+) :: Stream N → Stream N → Stream N s + t = Cons (head s + head t) (tail s + tail t)

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The Lifting Lemma WG2.8

Since the arithmetic operations are defined point-wise, the familiar arithmetic laws also hold for streams.

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The Lifting Lemma WG2.8

More general, given a point-level identity, does the lifted version hold as well?

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The Lifting Lemma WG2.8

(x + y) + z = x + (y + z)

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The Lifting Lemma WG2.8

x + y = y + x

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The Lifting Lemma WG2.8

x ∗ 0 = 0

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The Lifting Lemma WG2.8

Idioms

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The Lifting Lemma WG2.8

Idioms aka applicative functors

class Idiom ι where pure :: α → ι α (⋄) :: ι (α → β) → (ι α → ι β)

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The Lifting Lemma WG2.8

instance Idiom (τ→) where pure a = λx → a f ⋄ g = λx → (f x) (g x)

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The Lifting Lemma WG2.8

instance Idiom (τ→) where pure a = λx → a f ⋄ g = λx → (f x) (g x) So, pure is the K combinator and ⋄ is the S combinator.

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The Lifting Lemma WG2.8

instance Idiom Stream where pure a = s where s = a ≺ s s ⋄ t = Cons ((head s) (head t)) (tail s ⋄ tail t)

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The Lifting Lemma WG2.8

Lifting, generically

(+) :: (Idiom ι) ⇒ ι N → ι N → ι N u + v = pure (+) ⋄ u ⋄ v (⋆) :: (Idiom ι) ⇒ ι α → ι β → ι (α, β) u ⋆ v = pure (, ) ⋄ u ⋄ v

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The Lifting Lemma WG2.8

Idiom laws

pure id ⋄ u = u (identity) pure (·) ⋄ u ⋄ v ⋄ w = u ⋄ (v ⋄ w) (composition) pure f ⋄ pure x = pure (f x) (homomorphism) u ⋄ pure x = pure (⋄x) ⋄ u (interchange)

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The Lifting Lemma WG2.8

pure f ⋄ (u ⋆ v) = { definition of ⋆ } pure f ⋄ (pure (, ) ⋄ u ⋄ v) = { idiom composition } pure (·) ⋄ pure f ⋄ (pure (, ) ⋄ u) ⋄ v = { idiom homomorphism } pure (f ·) ⋄ (pure (, ) ⋄ u) ⋄ v = { idiom composition } pure (·) ⋄ pure (f ·) ⋄ pure (, ) ⋄ u ⋄ v = { idiom homomorphism } pure ((f ·) · (, )) ⋄ u ⋄ v = { ((f ·) · (, )) x y = (f · (, ) x) y = f ((, ) x y) = curry f x y } pure (curry f) ⋄ u ⋄ v

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The Lifting Lemma WG2.8

The Idiomatic Calculus

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The Lifting Lemma WG2.8

Syntax: variables

data Ix :: ∗ → ∗ → ∗ where Zero :: Ix (ρ, α) α Succ :: Ix ρ β → Ix (ρ, α) β

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The Lifting Lemma WG2.8

Syntax: terms

data Term :: ∗ → ∗ → ∗ where Con :: α → Term ρ α Var :: Ix ρ α → Term ρ α App :: Term ρ (α → β) → Term ρ α → Term ρ β

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The Lifting Lemma WG2.8

Semantics: variables

data Env :: (∗ → ∗) → (∗ → ∗) where Empty :: Env ι () Push :: Env ι ρ → ι α → Env ι (ρ, α) We write Empty as and Push η u as η, u. acc :: Ix ρ α → Env ι ρ → ι α acc Zero η, v = v acc (Succ n) η, v = acc n η

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The Lifting Lemma WG2.8

Semantics: terms

I :: (Idiom ι) ⇒ Term ρ α → Env ι ρ → ι α ICon vη = pure v IVar nη = acc n η IApp e1 e2η = Ie1η ⋄ Ie2η

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The Lifting Lemma WG2.8

What about abstraction?

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The Lifting Lemma WG2.8

Syntax

data Term :: ∗ → ∗ → ∗where Con :: α → Term ρ α Var :: Ix ρ α → Term ρ α App :: Term ρ (α → β) → Term ρ α → Term ρ β Abs :: Term (ρ, α) β → Term ρ (α → β)

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The Lifting Lemma WG2.8

An idiom is very similar to an applicative structure.

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The Lifting Lemma WG2.8

Semantics

I :: (Idiom ι) ⇒ Term ρ α → Env ι ρ → ι α ICon vη = pure v IVar nη = acc n η IApp e1 e2η = Ie1η ⋄ Ie2η IAbs eη = the unique function f such that ∀v . f ⋄ v = Ieη, v

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The Lifting Lemma WG2.8

Uniqueness?

Extensionality: (∀u . f ⋄ u = g ⋄ u) = ⇒ f = g

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The Lifting Lemma WG2.8

Existence?

Combinatory model condition: pure K ⋄ u ⋄ v = u pure S ⋄ u ⋄ v ⋄ w = (u ⋄ w) ⋄ (v ⋄ w) Ensures that ι has enough points.

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The Lifting Lemma WG2.8

Idiomatic interpretation specialised to the identity idiom:

• :: Term ρ α → Env Id ρ → α

Con vη = v Var nη = acc n η App e1 e2η = (e1η) (e2η) Abs eη = λv → eη, v

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The Lifting Lemma WG2.8

. . . written in a pointfree style:

• :: Term ρ α → Env Id ρ → α

Con v = K v Var n = acc n App e1 e2 = S e1 e2 Abs e = curry e

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The Lifting Lemma WG2.8

The Lifting Lemma

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The Lifting Lemma WG2.8

The lifting lemma

Ie = pure e

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The Lifting Lemma WG2.8

pure (+) ⋄ (pure (∗) ⋄ u ⋄ w) ⋄ (pure (∗) ⋄ v ⋄ w) = { definition of I } IAbs (Abs (Abs (2 ∗ 0 + 1 ∗ 0))) ⋄ u ⋄ v ⋄ w = { lifting lemma } pure Abs (Abs (Abs (2 ∗ 0 + 1 ∗ 0))) ⋄ u ⋄ v ⋄ w = { definition of } pure (λx y z → x ∗ z + y ∗ z) ⋄ u ⋄ v ⋄ w = { arithmetic } pure (λx y z → (x + y) ∗ z) ⋄ u ⋄ v ⋄ w = { definition of } pure Abs (Abs (Abs ((2 + 1) ∗ 0))) ⋄ u ⋄ v ⋄ w = { lifting lemma } IAbs (Abs (Abs ((2 + 1) ∗ 0))) ⋄ u ⋄ v ⋄ w = { definition of I } pure (∗) ⋄ (pure (+) u v) ⋄ w

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The Lifting Lemma WG2.8

The lifting lemma, general form

Ieη = pure e ⋄ zip η zip :: (Idiom ι) ⇒ Env ι ρ → ι (Env Id ρ) zip = pure zip η, v = pure , ⋄ η ⋄ v

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The Lifting Lemma WG2.8

Proof: Case e = Con v:

pure Con v ⋄ zip η = { definition of } pure (K v) ⋄ zip η = { idiom homomorphism } pure K ⋄ pure v ⋄ zip η = { combinatory model condition I } pure v = { definition of I } ICon vη

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The Lifting Lemma WG2.8

Proof: Case e = Var n:

pure Var n ⋄ zip η = { definition of } pure (acc n) ⋄ zip η = { Lemma: pure (acc n) ⋄ zip η = acc n η } acc n η = { definition of I } IVar nη

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The Lifting Lemma WG2.8

Proof: Case e = App e1 e2:

pure App e1 e2 ⋄ zip η = { definition of } pure (S e1 e2) ⋄ zip η = { idiom homomorphism } pure S ⋄ pure e1 ⋄ pure e2 ⋄ zip η = { combinatory model condition II } (pure e1 ⋄ zip η) ⋄ (pure e2 ⋄ zip η) = { ex hypothesi } Ie1η ⋄ Ie2η = { definition of I } IApp e1 e2η

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The Lifting Lemma WG2.8

Proof: Case e = Abs e:

pure Abs e ⋄ zip η = { definition of } pure (curry e) ⋄ zip η = { proof obligation (see next slide) } f = { definition of I } IAbs eη

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The Lifting Lemma WG2.8

Proof obligation

pure (curry e) ⋄ zip η = f ⇐ ⇒ { extensionality } pure (curry e) ⋄ zip η ⋄ v = f ⋄ v ⇐ ⇒ { definition of f } pure (curry e) ⋄ zip η ⋄ v = Ieη, v ⇐ ⇒ { curry-lemma (see above) } pure e ⋄ (zip η ⋆ v) = Ieη, v ⇐ ⇒ { definition of zip } pure e ⋄ (zip η, v) = Ieη, v ⇐ ⇒ { ex hypothesi } True

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The Lifting Lemma WG2.8

What about . . .

• τ→: √;
• Set: , but λI-calculus;
• Vector: √;
• Maybe: , but λI-calculus;
• IO: , but NF Lemma;
• Stream: √.

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