Equational Reasoning with Applicative Functors Andreas Lochbihler - PowerPoint PPT Presentation

Equational Reasoning with Applicative Functors Andreas Lochbihler Joshua Schneider Institute of Information Security

Equational Reasoning with Applicative Functors Andreas Lochbihler Joshua Schneider Institute of Information Security model effects state probabilities � error non-determinism streams 1 2 3 4 · · ·

Equational Reasoning with Applicative Functors Andreas Lochbihler Joshua Schneider Institute of Information Security f ( g � ) = h ( k � ) model effects k state probabilities g � error h non-determinism f streams 1 2 3 4 · · ·

Equational Reasoning with Applicative Functors Andreas Lochbihler Joshua Schneider Institute of Information Security pure f ⋄ ( g � ) = pure h ⋄ ( k � ) model effects k state probabilities g � error h non-determinism f streams 1 2 3 4 · · ·

Equational Reasoning with Applicative Functors Andreas Lochbihler Joshua Schneider Institute of Information Security pure f ⋄ ( g � ) = pure h ⋄ ( k � ) model effects k state probabilities g � error h don’t non-determinism care f streams 1 2 3 4 · · ·

Contributions ◮ Isabelle/HOL package for reasoning about applicative effects ⋄ ⋄ x = x = functor registration proof tactic classify effects ◮ Meta theory formalised and algorithms verified ◮ Used in several examples and case studies A. Lochbihler (ETH Zurich) ITP 2016 6 / 35

Task: Label a binary tree with distinct numbers! lbl q a d 2 3 4 c b datatype α tree = 0 1 L α | N ( α tree ) ( α tree ) Example suggested by G. Hutton, D. Fulger: Reasoning about effects: seeing the wood through the trees. TFP 2008 A. Lochbihler (ETH Zurich) ITP 2016 7 / 35

Task: Label a binary tree with distinct numbers! lbl q a d 2 3 4 c b datatype α tree = 0 1 L α | N ( α tree ) ( α tree ) lbl :: α tree ⇒ nat tree Example suggested by G. Hutton, D. Fulger: Reasoning about effects: seeing the wood through the trees. TFP 2008 A. Lochbihler (ETH Zurich) ITP 2016 8 / 35

Task: Label a binary tree with distinct numbers! lbl q a d 2 3 4 c b datatype α tree = 0 1 L α | N ( α tree ) ( α tree ) state lbl :: α tree ⇒ nat tree state α state = nat ⇒ α × nat where monadic α M = α state return :: α ⇒ α M ( > =) :: α M ⇒ ( α ⇒ β M ) ⇒ β M > lbl (L ) = fresh > = λ x ′ . return (L x ′ ) > lbl (N l r ) = = λ r ′ . return (N l ′ r ′ ) lbl l > = λ l ′ . lbl r > > > Example suggested by G. Hutton, D. Fulger: Reasoning about effects: seeing the wood through the trees. TFP 2008 A. Lochbihler (ETH Zurich) ITP 2016 9 / 35

Task: Label a binary tree with distinct numbers! lbl q a d 2 3 4 c b datatype α tree = 0 1 L α | N ( α tree ) ( α tree ) state lbl :: α tree ⇒ nat tree state α state = nat ⇒ α × nat where monadic applicative α M = α state α F = α state return :: α ⇒ α M pure :: α ⇒ α F ( > =) :: α M ⇒ ( α ⇒ β M ) ⇒ β M ( ⋄ ) :: ( α ⇒ β ) F ⇒ α F ⇒ β F > lbl (L ) = fresh > = λ x ′ . return (L x ′ ) lbl (L ) = pure L ⋄ fresh > lbl (N l r ) = lbl (N l r ) = pure N ⋄ lbl l ⋄ lbl r = λ r ′ . return (N l ′ r ′ ) lbl l > = λ l ′ . lbl r > > > Example suggested by G. Hutton, D. Fulger: Reasoning about effects: seeing the wood through the trees. TFP 2008 A. Lochbihler (ETH Zurich) ITP 2016 10 / 35

Labelling trees and lists leaves :: α tree ⇒ α list lbl leaves (L x ) = x · [ ] q 2 leaves (N l r ) = leaves l + + leaves r c a 0 1 leaves pure leaves lbl ′ :: α list ⇒ nat list state lbl ′ [ ] lbl ′ = pure [ ] [c, a, q] [0, 1, 2] lbl ′ ( · xs ) = pure ( · ) ⋄ fresh ⋄ lbl ′ xs A. Lochbihler (ETH Zurich) ITP 2016 11 / 35

Labelling trees and lists leaves :: α tree ⇒ α list lbl leaves (L x ) = x · [ ] q 2 leaves (N l r ) = leaves l + + leaves r c a 0 1 leaves pure leaves lbl ′ :: α list ⇒ nat list state lbl ′ [ ] lbl ′ = pure [ ] [c, a, q] [0, 1, 2] lbl ′ ( · xs ) = pure ( · ) ⋄ fresh ⋄ lbl ′ xs pure leaves ⋄ lbl t = lbl ′ (leaves t ) Lemma: Proof by induction on t . = lbl ′ (leaves (L x )) Case L x : pure leaves ⋄ lbl (L x )

Labelling trees and lists leaves :: α tree ⇒ α list lbl leaves (L x ) = x · [ ] q 2 leaves (N l r ) = leaves l + + leaves r c a 0 1 leaves pure leaves lbl ′ :: α list ⇒ nat list state lbl ′ [ ] lbl ′ = pure [ ] [c, a, q] [0, 1, 2] lbl ′ ( · xs ) = pure ( · ) ⋄ fresh ⋄ lbl ′ xs pure leaves ⋄ lbl t = lbl ′ (leaves t ) Lemma: Proof by induction on t . = lbl ′ (leaves (L x )) Case L x : pure leaves ⋄ lbl (L x ) pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] ∀ x . leaves ( L x ) = ( · ) x [ ] A. Lochbihler (ETH Zurich) ITP 2016 13 / 35

Labelling trees and lists leaves :: α tree ⇒ α list lbl leaves (L x ) = x · [ ] q 2 leaves (N l r ) = leaves l + + leaves r c a 0 1 leaves pure leaves lbl ′ :: α list ⇒ nat list state lbl ′ [ ] lbl ′ = pure [ ] [c, a, q] [0, 1, 2] lbl ′ ( · xs ) = pure ( · ) ⋄ fresh ⋄ lbl ′ xs pure leaves ⋄ lbl t = lbl ′ (leaves t ) Lemma: Proof by induction on t . = lbl ′ (leaves (L x )) Case L x : pure leaves ⋄ lbl (L x ) pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] ⇑ holds by the applicative laws ∀ x . leaves ( L x ) = ( · ) x [ ] A. Lochbihler (ETH Zurich) ITP 2016 14 / 35

Labelling trees and lists leaves :: α tree ⇒ α list lbl leaves (L x ) = x · [ ] q 2 leaves (N l r ) = leaves l + + leaves r c a 0 1 leaves pure leaves lbl ′ :: α list ⇒ nat list state lbl ′ [ ] lbl ′ = pure [ ] [c, a, q] [0, 1, 2] lbl ′ ( · xs ) = pure ( · ) ⋄ fresh ⋄ lbl ′ xs pure leaves ⋄ lbl t = lbl ′ (leaves t ) Lemma: Proof by induction on t . = lbl ′ (leaves (L x )) Case L x : pure leaves ⋄ lbl (L x ) pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] ⇑ holds by the applicative laws apply applicative lifting ∀ x . leaves ( L x ) = ( · ) x [ ] A. Lochbihler (ETH Zurich) ITP 2016 15 / 35

Lifting equations over applicative functors [Hinze 2010] pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] ∀ x . leaves (L x ) = x · [ ] A. Lochbihler (ETH Zurich) ITP 2016 16 / 35

Lifting equations over applicative functors [Hinze 2010] pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] ML e e kernel HOL l l l syntactic l e e b b a a ∀ ∀ s s I I = = formalisation α α λ λ β → β → f o l l o w s ∀ x . leaves (L x ) = x · [ ] A. Lochbihler (ETH Zurich) ITP 2016 17 / 35

Lifting equations over applicative functors [Hinze 2010] pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] 1. Convert to canonical form pure ( λ x . leaves (L x )) ⋄ fresh = pure ( λ x . x · [ ]) ⋄ fresh ∀ x . leaves (L x ) = x · [ ] A. Lochbihler (ETH Zurich) ITP 2016 18 / 35

Lifting equations over applicative functors [Hinze 2010] Canonical form [McBride, Paterson] applicative expression �→ pure f ⋄ x 1 ⋄ x 2 ⋄ . . . ⋄ x n pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] 1. Convert to canonical form pure ( λ x . leaves (L x )) ⋄ fresh = pure ( λ x . x · [ ]) ⋄ fresh ∀ x . leaves (L x ) = x · [ ] A. Lochbihler (ETH Zurich) ITP 2016 19 / 35

Lifting equations over applicative functors [Hinze 2010] pure function opaque arguments opaque arguments opaque arguments opaque arguments opaque arguments Canonical form [McBride, Paterson] applicative expression �→ pure f ⋄ x 1 ⋄ x 2 ⋄ . . . ⋄ x n pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] 1. Convert to canonical form pure ( λ x . leaves (L x )) ⋄ fresh = pure ( λ x . x · [ ]) ⋄ fresh ∀ x . leaves (L x ) = x · [ ] A. Lochbihler (ETH Zurich) ITP 2016 20 / 35

Lifting equations over applicative functors [Hinze 2010] pure function opaque arguments opaque arguments opaque arguments opaque arguments opaque arguments Canonical form [McBride, Paterson] applicative expression �→ pure f ⋄ x 1 ⋄ x 2 ⋄ . . . ⋄ x n pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] 1. Convert to canonical form pure ( λ x . leaves (L x )) ⋄ fresh = pure ( λ x . x · [ ]) ⋄ fresh 2. Generalise opaque arguments ∀ X . pure ( λ x . leaves (L x )) ⋄ X = pure ( λ x . x · [ ]) ⋄ X ∀ x . leaves (L x ) = x · [ ] A. Lochbihler (ETH Zurich) ITP 2016 21 / 35

Lifting equations over applicative functors [Hinze 2010] pure function opaque arguments opaque arguments opaque arguments opaque arguments opaque arguments Canonical form [McBride, Paterson] applicative expression �→ pure f ⋄ x 1 ⋄ x 2 ⋄ . . . ⋄ x n pure leaves ⋄ (pure L ⋄ fresh) = pure ( · ) ⋄ fresh ⋄ pure [ ] 1. Convert to canonical form pure ( λ x . leaves (L x )) ⋄ fresh = pure ( λ x . x · [ ]) ⋄ fresh 2. Generalise opaque arguments ∀ X . pure ( λ x . leaves (L x )) ⋄ X = pure ( λ x . x · [ ]) ⋄ X 3. Equality is a congruence ∀ X . pure ( λ x . leaves (L x )) ⋄ X = pure ( λ x . x · [ ]) ⋄ X ∀ x . leaves (L x ) = x · [ ] A. Lochbihler (ETH Zurich) ITP 2016 22 / 35