Equational Reasoning with Applicative Functors. Institute of Information Security

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

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

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

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

5 Equational Reasoning with Applicative Functors Andreas Lochbihler Joshua Schneider Institute of Information Security pure f (g ) pure h (k ) g k f h don t care model effects state probabilities error non-determinism streams

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

7 Task: Label a binary tree with distinct numbers! c b q a d datatype α tree 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 / 35

8 Task: Label a binary tree with distinct numbers! c b q a d datatype α tree L α N (α tree) (α tree) :: 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 / 35

9 Task: Label a binary tree with distinct numbers! c b q a d datatype α tree L α N (α tree) (α tree) state :: α tree nat tree state where α state nat α nat monadic α M α state return :: α α M ( >) :: α M (α β M) β M (L ) fresh > λx. return (L x ) (N l r) l > λl. r > λr. return (N l r ) Example suggested by G. Hutton, D. Fulger: Reasoning about effects: seeing the wood through the trees. TFP 2008 A. Lochbihler (ETH Zurich) ITP / 35

10 Task: Label a binary tree with distinct numbers! c b q a d datatype α tree L α N (α tree) (α tree) state :: α tree nat tree state where α state nat α nat monadic α M α state applicative 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 (L ) fresh > λx. return (L x ) (N l r) l > λl. r > λr. return (N l r ) (L ) pure L fresh (N l r) pure N l r Example suggested by G. Hutton, D. Fulger: Reasoning about effects: seeing the wood through the trees. TFP 2008 A. Lochbihler (ETH Zurich) ITP / 35

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

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

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

14 Labelling trees and lists c a q leaves :: α tree α list leaves (L x) x [ ] leaves (N l r) leaves l ++ leaves r leaves pure leaves [c, a, q] [0, 1, 2] :: α list nat list state [ ] pure [ ] ( xs) pure ( ) fresh xs Lemma: pure leaves t (leaves t) Proof by induction on t. Case L x: pure leaves (L x) (leaves (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 / 35

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

16 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 / 35

17 Lifting equations over applicative functors [Hinze 2010] pure leaves (pure L fresh) pure ( ) fresh pure [ ] λ β Isabelle kernel α ML syntactic formalisation λ β Isabelle HOL α follows x. leaves (L x) x [ ] A. Lochbihler (ETH Zurich) ITP / 35

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

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

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

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

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

23 Lifting equations over applicative functors pure function opaque arguments Canonical form applicative expression pure f x 1 x 2... x n [Hinze 2010] [McBride, Paterson] pure leaves (pure L fresh) pure ( ) fresh pure [ ] pure (λx. leaves (L x)) fresh 2. Generalise opaque arguments X. pure (λx. leaves (L x)) X 3. Equality is a congruence X. pure (λx. leaves (L x)) X 4. Use extensionality 1. Convert to canonical form pure (λx. x [ ]) fresh pure (λx. x [ ]) X pure (λx. x [ ]) X x. leaves (L x) x [ ] A. Lochbihler (ETH Zurich) ITP / 35

24 Lifting equations over applicative functors pure function opaque arguments Canonical form applicative expression pure f x 1 x 2... x n [Hinze 2010] [McBride, Paterson] pure leaves (pure L fresh) pure ( ) fresh pure [ ] pure (λx. leaves (L x)) fresh 2. Generalise opaque arguments X. pure (λx. leaves (L x)) X 3. Equality is a congruence X. pure (λx. leaves (L x)) X 4. Use extensionality 1. Convert to canonical form pure (λx. x [ ]) fresh pure (λx. x [ ]) X pure (λx. x [ ]) X x. leaves (L x) x [ ] Same opaque args. on both sides! A. Lochbihler (ETH Zurich) ITP / 35

25 Tree mirroring c a q mirror :: α tree α tree mirror (L x) L x mirror (N l r) N (mirror r) (mirror l) mirror? pure mirror q a c :: α tree nat tree state (L ) pure L fresh (N l r) pure N l r Lemma: Proof by induction on t. Case N l r: (mirror t) pure mirror t? pure (λr l. N (mirror r ) (mirror l )) r l pure (λl r. mirror (N l r )) l r A. Lochbihler (ETH Zurich) ITP / 35

26 Tree mirroring c a q mirror :: α tree α tree mirror (L x) L x mirror (N l r) N (mirror r) (mirror l) mirror pure mirror q a c :: α tree nat tree state (L ) pure L fresh (N l r) pure N l r Lemma: Proof by induction on t. Case N l r: (mirror t) pure mirror t? pure (λr l. N (mirror r ) (mirror l )) r l pure (λl r. mirror (N l r )) l r A. Lochbihler (ETH Zurich) ITP / 35

27 Tree mirroring and random labels c a q mirror :: α tree α tree mirror (L x) L x mirror (N l r) N (mirror r) (mirror l) mirror pure mirror q a c :: α tree nat tree probability (L ) pure L fresh (N l r) pure N l r Lemma: (mirror t) pure mirror t if effects commute Proof by induction on t. Case N l r: pure (λr l. N (mirror r ) (mirror l )) r l pure (λl r. mirror (N l r )) l r A. Lochbihler (ETH Zurich) ITP / 35

28 Tree mirroring and random labels c a q mirror :: α tree α tree mirror (L x) L x mirror (N l r) N (mirror r) (mirror l) mirror pure mirror q a c Criterion for commutative effects: :: α tree nat tree probability pure (λf (L x y. ) f y x) pure f L x fresh y f y x (N Cl r) puref N x y l f r y x Lemma: (mirror t) pure mirror t if effects commute Proof by induction on t. Case N l r: pure (λr l. N (mirror r ) (mirror l )) r l pure (λl r. mirror (N l r )) l r A. Lochbihler (ETH Zurich) ITP / 35

29 Subtrees c a q mirror pure mirror q a c Lemma: (right t) pure right t Proof by case analysis on t. right pure right Case N l r: pure (λr. r ) r? pure (λ r. r ) l r a c A. Lochbihler (ETH Zurich) ITP / 35

30 Subtrees c a q Criterion for omissible effects: pure (λx y. x) x y x mirror pure mirror K x y x q a c Lemma: if effects are omissible (right t) pure right t Proof by case analysis on t. right pure right Case N l r: pure (λr. r ) r pure (λ r. r ) l r a c A. Lochbihler (ETH Zurich) ITP / 35

31 Combinatorial basis BCKW B K omissible W idempotent commutative Declarative characterisation of liftable equations Modular implementation via bracket abstraction C A. Lochbihler (ETH Zurich) ITP / 35

32 Combinatorial basis BCKW B K omissible state exception probability reader W idempotent commutative C Declarative characterisation of liftable equations Modular implementation via bracket abstraction User declares and proves combinator properties at registration A. Lochbihler (ETH Zurich) ITP / 35

33 Combinatorial basis BCKW B ordered non-determinism K omissible nondeterminism probability non-standard numbers reader stream zip list state exception idempotent monoid monoid W idempotent parser non-determinism with failure subprobability commutative commutative monoid C maybe list Declarative characterisation of liftable equations Modular implementation via bracket abstraction User declares and proves combinator properties at registration A. Lochbihler (ETH Zurich) ITP / 35

34 Summary Lifting.shtml x B K functor registration x proof tactic C W classify effects guides formalisation of the meta theory A. Lochbihler (ETH Zurich) ITP / 35

35 Summary Lifting.shtml x B K W x C functor registration proof tactic classify effects monads? generate? guides beyond equality? formalisation of the meta theory A. Lochbihler (ETH Zurich) ITP / 35