Seeing the wood through the trees
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- Stanley Cross
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1 Seeing the wood through the trees A DIY guide to reasoning about effects Graham Hutton and Diana Fulger University of Nottingham December 16, 2007 Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
2 Background Monads Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
3 Background Monads make writing programmes easier Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
4 Background Monads make writing programmes easier should make reasoning about them easier too? Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
5 Background Monads make writing programmes easier should make reasoning about them easier too? But Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
6 Background Monads make writing programmes easier should make reasoning about them easier too? But existing techniques have not gone mainstream Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
7 Background Monads make writing programmes easier should make reasoning about them easier too? But existing techniques have not gone mainstream Let s try a simple example! Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
8 Reasoning on trees data Tree a = Leaf a Node (Tree a) (Tree a) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
9 Reasoning on trees data Tree a = Leaf a Node (Tree a) (Tree a) label :: Tree a Int (Tree Int, Int) label (Leaf x) n = (Leaf n, n+1) label (Node l r) n = (Node l r, n") where (l, n ) = label l n (r, n") = label r n raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
10 Reasoning on trees data Tree a = Leaf a Node (Tree a) (Tree a) label :: Tree a Int (Tree Int, Int) label (Leaf x) n = (Leaf n, n+1) label (Node l r) n = (Node l r, n") where (l, n ) = label l n (r, n") = label r n Prove label correct? raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
11 Reasoning on trees data Tree a = Leaf a Node (Tree a) (Tree a) label :: Tree a Int (Tree Int, Int) label (Leaf x) n = (Leaf n, n+1) label (Node l r) n = (Node l r, n") where (l, n ) = label l n (r, n") = label r n Prove label correct? equationally! raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
12 Define correct? Specification preserves shape raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
13 Define correct? Specification preserves shape distinct labels Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
14 Simple correctness labels :: Tree a [a] labels (Leaf x) = [x] labels (Node l r) = labels l ++ labels r raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
15 Simple correctness labels :: Tree a [a] labels (Leaf x) = [x] labels (Node l r) = labels l ++ labels r Theorem (Correctness of label) For all finite trees t: nodups (labels (fst (label t 0))) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
16 Simple correctness labels :: Tree a [a] labels (Leaf x) = [x] labels (Node l r) = labels l ++ labels r Theorem (Correctness of label) For all finite trees t: nodups (labels (fst (label t 0))) nodups :: Eq a [a] Bool nodups [] = True nodups (x:xs) = not(elem x xs) && nodups xs Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
17 Proof steps identify generated labels raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
18 Proof steps identify generated labels Lemma (Behaviour of label) For every finite tree t and initial integer label n, label t n = (t, n ) n < n labels t = [n.. n 1] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
19 Proof steps identify generated labels Lemma (Behaviour of label) For every finite tree t and initial integer label n, label t n = (t, n ) n < n labels t = [n.. n 1] labels are unique raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
20 Proof steps identify generated labels Lemma (Behaviour of label) For every finite tree t and initial integer label n, label t n = (t, n ) n < n labels t = [n.. n 1] labels are unique Lemma (Haskell) Intervals have no duplicates: for all integers a b: nodups [a.. b] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
21 State monad label :: Tree a Int (Tree Int, Int) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
22 State monad label :: Tree a Int (Tree Int, Int) type ST s a = s (a,s) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
23 State monad label :: Tree a Int (Tree Int, Int) type ST s a = s (a,s) instance Monad (ST s) where -- return :: a ST s a return v = λs (v,s) -- (>>=) :: ST s a (a ST s b) ST s b st >>= f = λs let (v,s ) = st s in (f v) s Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
24 Monadic labelling label :: Tree a Int (Tree Int, Int) label (Leaf x) n = (Leaf n, n+1) label (Node l r) n = (Node l r, n") where (l, n ) = label l n (r, n") = label r n raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
25 Monadic labelling label :: Tree a Int (Tree Int, Int) label (Leaf x) n = (Leaf n, n+1) label (Node l r) n = (Node l r, n") where (l, n ) = label l n (r, n") = label r n label :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
26 Monadic labelling label :: Tree a Int (Tree Int, Int) label (Leaf x) n = (Leaf n, n+1) label (Node l r) n = (Node l r, n") where (l, n ) = label l n (r, n") = label r n label :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) fresh :: ST Int Int fresh = λn (n, n+1) Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
27 Monadic labelling label :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
28 Monadic labelling label :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) single-assignment variables Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
29 Monadic labelling label :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) single-assignment variables refreshing fresh Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
30 Monadic labelling label :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) single-assignment variables refreshing fresh reasoning any easier? Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
31 What we would like to see Leaf x Leaf fresh Node l r Node (label l) (label r) Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
32 Applicative functors Effectful generalisation of application Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
33 Applicative functors Effectful generalisation of application class Functor f Applicative f where pure :: a f a ( ) :: f (a b) f a f b Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
34 Applicative functors Effectful generalisation of application class Functor f Applicative f where pure :: a f a ( ) :: f (a b) f a f b Monad Applicative return pure ap Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
35 Applicative functors class Functor f Applicative f where pure :: a f a ( ) :: f (a b) f a f b Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
36 Applicative functors class Functor f Applicative f where pure :: a f a ( ) :: f (a b) f a f b pure id v =v pure ( ) u v w =u (v w) pure f pure x =pure (f x) u pure y =pure (λf f y) u identity composition homomorphism interchange raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
37 Applicative functors class Functor f Applicative f where pure :: a f a ( ) :: f (a b) f a f b pure id v =v pure ( ) u v w =u (v w) pure f pure x =pure (f x) u pure y =pure (λf f y) u identity composition homomorphism interchange pure f x1... xn = [[f x1 x2... xn]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
38 State applicative functor instance Applicative (ST s) where -- pure :: a ST s a pure v = λs (v,s) --( ) :: ST s (a b) ST s a ST s b mf mx = λs let (f, s ) = mf s (x, s") = mx s in (f x, s") Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
39 Applicative labelling label :: Tree a ST Int (Tree Int) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
40 Applicative labelling label monadic expression :: Tree a ST Int (Tree Int) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
41 Applicative labelling label monadic expression :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
42 Applicative labelling label monadic expression :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) applicative expression raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
43 Applicative labelling label monadic expression :: Tree a ST Int (Tree Int) label (Leaf x) = do n fresh return (Leaf n) label (Node l r) = do l label l r label r return (Node l r ) applicative expression label (Leaf _) = [[ Leaf fresh ]] label (Node l r) = [[ Node (label l) (label r) ]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
44 Applicative approach Theorem (Correctness of label) For all finite trees t: nodups (append (labels from) label t) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
45 Applicative approach Theorem (Correctness of label) For all finite trees t: nodups (append (labels from) label t) Lemma (Behaviour of label) For all finite trees t: append (labels from) label t = from Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
46 Applicative labelling label :: Tree a ST Int (Tree Int) label (Leaf _) = [[ Leaf fresh ]] label (Node l r) = [[ Node (label l) (label r) ]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
47 Applicative labelling label :: Tree a ST Int (Tree Int) label (Leaf _) = [[ Leaf fresh ]] label (Node l r) = [[ Node (label l) (label r) ]] state threading Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
48 Applicative labelling label :: Tree a ST Int (Tree Int) label (Leaf _) = [[ Leaf fresh ]] label (Node l r) = [[ Node (label l) (label r) ]] state threading fresh label generation raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
49 Applicative labelling label :: Tree a ST Int (Tree Int) label (Leaf _) = [[ Leaf fresh ]] label (Node l r) = [[ Node (label l) (label r) ]] state threading fresh label generation label :: Tree a ST [b] (Tree b) label (Leaf _) = [[ Leaf fetch ]] label (Node l r) = [[ Node (label l) (label r) ]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
50 Applicative labelling label :: Tree a ST Int (Tree Int) label (Leaf _) = [[ Leaf fresh ]] label (Node l r) = [[ Node (label l) (label r) ]] state threading fresh label generation label :: Tree a ST [b] (Tree b) label (Leaf _) = [[ Leaf fetch ]] label (Node l r) = [[ Node (label l) (label r) ]] fetch :: [b] (b, [b]) fetch (x:xs) = (x, xs) Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
51 Factorisation Theorem (Factorising label) For all finite trees t: (id from) label t = (label t) from Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
52 Factorisation Theorem (Factorising label) For all finite trees t: (id from) label t = (label t) from Proof by induction, using equational reasoning Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
53 Base case (id from) label (Leaf ) Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
54 Base case (id from) label (Leaf ) = (id from) [[Leaf fresh]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
55 Base case (id from) label (Leaf ) = (id from) [[Leaf fresh]] = (id from) (Leaf id) fresh Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
56 Base case (id from) label (Leaf ) = (id from) [[Leaf fresh]] = (id from) (Leaf id) fresh = (Leaf id) (id from) fresh Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
57 Base case (id from) label (Leaf ) = (id from) [[Leaf fresh]] = (id from) (Leaf id) fresh = (Leaf id) (id from) fresh = (Leaf id) fetch from Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
58 Base case (id from) label (Leaf ) = (id from) [[Leaf fresh]] = (id from) (Leaf id) fresh = (Leaf id) (id from) fresh = (Leaf id) fetch from = [[Leaf fetch]] from Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
59 Base case (id from) label (Leaf ) = (id from) [[Leaf fresh]] = (id from) (Leaf id) fresh = (Leaf id) (id from) fresh = (Leaf id) fetch from = [[Leaf fetch]] from = label (Leaf ) from raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
60 Base case (id from) label (Leaf ) = (id from) [[Leaf fresh]] = (id from) (Leaf id) fresh = (Leaf id) (id from) fresh = (Leaf id) fetch from = [[Leaf fetch]] from = label (Leaf ) from Did we just expand ST out raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
61 Base case (id from) label (Leaf ) = (id from) [[Leaf fresh]] = (id from) (Leaf id) fresh = (Leaf id) (id from) fresh = (Leaf id) fetch from = [[Leaf fetch]] from = label (Leaf ) from Did we just expand ST out (only to rebuild back again)?? raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
62 What ST does... Lemma (Bracket notation) For all f : a b and x : ST s a: [[f x]] = (f id) x Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
63 Base case revisited (id from) label (Leaf ) Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
64 Base case revisited (id from) label (Leaf ) = (id from) [[Leaf fresh]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
65 Base case revisited (id from) label (Leaf ) = (id from) [[Leaf fresh]] = [[Leaf ((id from) fresh)]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
66 Base case revisited (id from) label (Leaf ) = (id from) [[Leaf fresh]] = [[Leaf ((id from) fresh)]] = [[Leaf (fetch from)]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
67 Base case revisited (id from) label (Leaf ) = (id from) [[Leaf fresh]] = [[Leaf ((id from) fresh)]] = [[Leaf (fetch from)]] = [[Leaf fetch]] from Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
68 Base case revisited (id from) label (Leaf ) = (id from) [[Leaf fresh]] = [[Leaf ((id from) fresh)]] = [[Leaf (fetch from)]] = [[Leaf fetch]] from = label (Leaf ) from Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
69 State fusions Lemma (Input state fusion) For all f : a b, x : ST s a and g : s s : [[f x]] g = [[f (x g)]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
70 State fusions Lemma (Input state fusion) For all f : a b, x : ST s a and g : s s : [[f x]] g = [[f (x g)]] Typing issue [[f (x g)]] :: s (s, a), raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
71 State fusions Lemma (Input state fusion) For all f : a b, x : ST s a and g : s s : [[f x]] g = [[f (x g)]] Typing issue [[f (x g)]] :: s (s, a), not a valid ST applicative expression! raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
72 State fusions Lemma (Input state fusion) For all f : a b, x : ST s a and g : s s : [[f x]] g = [[f (x g)]] Typing issue [[f (x g)]] :: s (s, a), not a valid ST applicative expression! Expand out the ST types and the problem goes away Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
73 State fusions Lemma (Input state fusion) For all f : a b, x : ST s a and g : s s : [[f x]] g = [[f (x g)]] Typing issue [[f (x g)]] :: s (s, a), not a valid ST applicative expression! Expand out the ST types and the problem goes away Lemma (Output state fusion) For all f : s s, g : a b and x : ST s a: (id f ) [[g x]] = [[g ((id f ) x)]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
74 What ST does... for functions with 2 arguments raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
75 What ST does... for functions with 2 arguments Lemma For all f : a b c, x : ST s a and y : ST s b: [[f x y]] = (uncurry f id) assoc (id y) x Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
76 Inductive step (id from) label (Node l r) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
77 Inductive step (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
78 Inductive step (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] = (id from) (uncurry Node id) assoc (id label r) label l raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
79 Inductive step (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] = (id from) (uncurry Node id) assoc (id label r) label l = (uncurry Node id) (id from) assoc (id label r) label l = (uncurry Node id) assoc (id ((id from) label r)) label l = (uncurry Node id) assoc (id (label r from)) label l = (uncurry Node id) assoc (id label r) (id from) label l = (uncurry Node id) assoc (id label r) label l from = [[Node (label l) (label r)]] from = label (Node l r) from raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
80 State fusions Lemma (Input state fusion) [[f x y]] g = [[f (x g) y]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
81 State fusions Lemma (Input state fusion) [[f x y]] g = [[f (x g) y]] Lemma (State shifting) [[f ((id g) x) y]] = [[f x (y g)]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
82 State fusions Lemma (Input state fusion) [[f x y]] g = [[f (x g) y]] Lemma (State shifting) [[f ((id g) x) y]] = [[f x (y g)]] Lemma (Output state fusion) (id f ) [[g x y]] = [[g x ((id f ) y)]] Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
83 Inductive step revisited (id from) label (Node l r) raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
84 Inductive step revisited (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
85 Inductive step revisited (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] = [[Node (label l) ((id from) label r)]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
86 Inductive step revisited (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] = [[Node (label l) ((id from) label r)]] = [[Node (label l) (label r from)]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
87 Inductive step revisited (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] = [[Node (label l) ((id from) label r)]] = [[Node (label l) (label r from)]] = [[Node ((id from) label l) (label r)]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
88 Inductive step revisited (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] = [[Node (label l) ((id from) label r)]] = [[Node (label l) (label r from)]] = [[Node ((id from) label l) (label r)]] = [[Node (label l from) (label r)]] raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
89 Inductive step revisited (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] = [[Node (label l) ((id from) label r)]] = [[Node (label l) (label r from)]] = [[Node ((id from) label l) (label r)]] = [[Node (label l from) (label r)]] = [[Node (label l) (label r)]] from raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
90 Inductive step revisited (id from) label (Node l r) = (id from) [[Node (label l) (label r)]] = [[Node (label l) ((id from) label r)]] = [[Node (label l) (label r from)]] = [[Node ((id from) label l) (label r)]] = [[Node (label l from) (label r)]] = [[Node (label l) (label r)]] from = label (Node l r) from raham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
91 Conclusions Equational proof of correctness on monadic/applicative expression Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
92 Conclusions Equational proof of correctness on monadic/applicative expression Factorisation Loosening the type system Applicative properties of the ST state transformer Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
93 Conclusions Equational proof of correctness on monadic/applicative expression Factorisation Loosening the type system Applicative properties of the ST state transformer Similar proof for the prefix-based labelling with Reader monad, without typing problems Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
94 Conclusions Equational proof of correctness on monadic/applicative expression Factorisation Loosening the type system Applicative properties of the ST state transformer Similar proof for the prefix-based labelling with Reader monad, without typing problems... and further work Alternative labellings Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
95 Conclusions Equational proof of correctness on monadic/applicative expression Factorisation Loosening the type system Applicative properties of the ST state transformer Similar proof for the prefix-based labelling with Reader monad, without typing problems... and further work Alternative labellings Traversable... and further along Compiler correctness Graham Hutton and Diana Fulger (University of Nottingham) Seeing the wood through the trees December 16, / 1
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