Pattern and Copattern matching Anton Setzer Swansea University, - PowerPoint PPT Presentation

Pattern and Copattern matching Anton Setzer Swansea University, Swansea UK Leeds Logic Seminar, 13 May 2015 Anton Setzer (Swansea) Pattern and Copattern matching 1/ 46

Iteration, Recursion, Induction Coiteration, Corecursion Bisimilarity and Coinduction Proofs by Coinduction of Bisimilarity in Transition Systems Mixed Patterns and Copatterns Unnesting of Pattern/Copattern Matching Anton Setzer (Swansea) Pattern and Copattern matching 2/ 46

Iteration, Recursion, Induction Iteration, Recursion, Induction Coiteration, Corecursion Bisimilarity and Coinduction Proofs by Coinduction of Bisimilarity in Transition Systems Mixed Patterns and Copatterns Unnesting of Pattern/Copattern Matching Anton Setzer (Swansea) Pattern and Copattern matching 3/ 46

Iteration, Recursion, Induction N as an Initial Algebra ◮ N is initial algebra of the functor F ( X ) = 1 + X ◮ 0 + S ✲ N F ( N ) = 1 + N F ( g ) = 1 + g ∃ ! g f ′ ❄ ❄ ✲ A F ( A ) = 1 + A f ′ can be decomposed as f ′ = a + f Anton Setzer (Swansea) Pattern and Copattern matching 4/ 46

Iteration, Recursion, Induction Unique Iteration 1 + N 0 + S ✲ N 1 + g ∃ ! g ❄ a + f ✲ A ❄ 1 + A Unique existence of g means unique iteration : Given a : A and f : A → A there exists a unique g : N → A g 0 = a g ( S n ) = f ( g n ) i.e g ( S n 0) f n a = Anton Setzer (Swansea) Pattern and Copattern matching 5/ 46

Iteration, Recursion, Induction Unique Recursion ◮ From the principle of unique iteration we can prove the principle of unique (primitive) recursion: Given a : A and f : N → A → A there exists a unique g : N → A g 0 = a g ( S n ) = f n ( g n ) Anton Setzer (Swansea) Pattern and Copattern matching 6/ 46

Iteration, Recursion, Induction Induction ◮ From the principle of unique iteration we can prove the principle of induction: Assume A : N → Set , a : A 0 and f : ( n : N ) → A n → A ( S n ) There exists a unique g : ( n : N ) → A n g 0 = a g ( S n ) = f n ( g n ) ◮ Using induction we can prove that if we have two solutions for a iteration or recursion principle, they are pointwise equal, i.e. uniqueness of iteration and recursion. Anton Setzer (Swansea) Pattern and Copattern matching 7/ 46

Iteration, Recursion, Induction Pattern Matching ◮ The above means that we can define g : ( n : N ) → A n g 0 = for some a : A a for some a ′ : A depending on n a ′ g ( S n ) = where in the second case we can use the recursion hypothesis or induction hypothesis g n . ◮ This means we can define g n by pattern matching on n : N . Anton Setzer (Swansea) Pattern and Copattern matching 8/ 46

Iteration, Recursion, Induction Iteration, Recursion, Induction Theorem Assume N : Set , 0 : N , S : N → N . The following are equivalent ◮ The principle of unique iteration. ◮ The principle of unique recursion. ◮ The principle of unique induction. ◮ The principle of induction. Anton Setzer (Swansea) Pattern and Copattern matching 9/ 46

Coiteration, Corecursion Iteration, Recursion, Induction Coiteration, Corecursion Bisimilarity and Coinduction Proofs by Coinduction of Bisimilarity in Transition Systems Mixed Patterns and Copatterns Unnesting of Pattern/Copattern Matching Anton Setzer (Swansea) Pattern and Copattern matching 10/ 46

Coiteration, Corecursion Streams as a Final Coalgebra ◮ Dual of + is × , so we use for clarity a functor using product rather than disjoint union: ◮ Stream is the final coalgebra of F ( X ) = N × X f ✲ N × X = F ( X ) X ∃ ! g id × g = F ( g ) ❄ head × tail ✲ N × Stream ❄ Stream = F ( Stream ) ◮ We can decompose f as = f 0 × f 1 f Anton Setzer (Swansea) Pattern and Copattern matching 11/ 46

Coiteration, Corecursion Unique Coiteration f 0 × f 1 ✲ N × X X ∃ ! g id × g head × tail ❄ ❄ ✲ N × Stream Stream This corresponds to the principle of unique coiteration: There exists a unique g : A → Stream head ( g x ) = f 0 x ( g x ) = g ( f 1 x ) tail Anton Setzer (Swansea) Pattern and Copattern matching 12/ 46

Coiteration, Corecursion Unique Coiteration ◮ We had: head ( g x )) = f 0 x ( g x ) = g ( f 1 x ) tail ◮ By choosing f 0 , f 1 we can define g : X → Stream s.t. head ( g x ) = n for some n : N depending on x for some x ′ : X depending on x ( g x ) = g x ′ tail Anton Setzer (Swansea) Pattern and Copattern matching 13/ 46

Coiteration, Corecursion Unique Corecursion ◮ From unique coiteration we can derive unique corecursion : There exists a unique g : A → Stream ( g x ) = for some n : N depending on x head n for some x ′ : X depending on x ( g x ) = g x ′ tail or = for some s : Stream depending on x s ◮ This means we can define g x by copattern matching Anton Setzer (Swansea) Pattern and Copattern matching 14/ 46

Coiteration, Corecursion Examples ◮ We can define : ( N × Stream ) → Stream cons head ( cons ( n , s )) = n ( cons ( n , s )) = tail s Note: cons not primitive but defined by corecursion : N → Stream inc head ( inc n ) = n ( inc n ) = inc ( n + 1) tail Anton Setzer (Swansea) Pattern and Copattern matching 15/ 46

Coiteration, Corecursion Examples inc ′ : N → Stream ( inc ′ ( n )) head = n ( inc ′ ( n )) = inc ′′ ( n + 1) tail inc ′′ : N → Stream ( inc ′′ ( n )) head = n ( inc ′′ ( n )) = inc ′ ( n + 1) tail Anton Setzer (Swansea) Pattern and Copattern matching 16/ 46

Bisimilarity and Coinduction Iteration, Recursion, Induction Coiteration, Corecursion Bisimilarity and Coinduction Proofs by Coinduction of Bisimilarity in Transition Systems Mixed Patterns and Copatterns Unnesting of Pattern/Copattern Matching Anton Setzer (Swansea) Pattern and Copattern matching 17/ 46

Bisimilarity and Coinduction Bisimilarity ◮ Bisimilarity ∼ on Streams is an indexed final coalgebra . ◮ Consider the category Set Stream × Stream of binary relations ϕ : Stream × Stream → Set ◮ Let F ∼ : Set Stream × Stream → Set Stream × Stream F ∼ ( ϕ, ( s , s ′ )) = ( head s = head s ′ ) × ϕ ( tail s , tail s ′ ) Anton Setzer (Swansea) Pattern and Copattern matching 18/ 46

Bisimilarity and Coinduction Bisimilarity ◮ That ∼ is a F ∼ coalgebra means there exist elim ∼ : ( s , s ′ : Stream ) → s ∼ s ′ → ( head s = head s ′ ) × ( tail s ∼ tail s ′ ) i.e. s ∼ s ′ → ( head s = head s ′ ) ∧ (( tail s ) ∼ ( tail s ′ )) ◮ Let elim 0 ∼ and elim 1 ∼ the two components of elim ∼ , ( s , s ′ : Stream ) → s ∼ s ′ → head s = head s ′ elim 0 : ∼ ( s , s ′ : Stream ) → s ∼ s ′ → tail s ∼ tail s ′ elim 1 : ∼ and hide the first two arguments of elim i ∼ . Anton Setzer (Swansea) Pattern and Copattern matching 19/ 46

Bisimilarity and Coinduction Bisimilarity ◮ That ∼ is a final F ∼ -coalgebra means that it is the largest such relation: f ✲ head s = head s ′ ∧ ϕ ( tail s , tail s ′ ) ϕ ( s , s ′ ) ∃ ! g id ∧ g ❄ ❄ elim ∼ ✲ head s = head s ′ ∧ ( tail s ) ∼ ( tail s ′ ) s ∼ s ′ ◮ This means that ∀ s , s ′ .ϕ ( s , s ′ ) → head s = head s ′ ∧ ϕ ( tail s , tail s ′ ) then ∀ s , s ′ .ϕ ( s , s ′ ) → s ∼ s ′ Anton Setzer (Swansea) Pattern and Copattern matching 20/ 46

Bisimilarity and Coinduction Bisimilarity ◮ So we have s ∼ s ′ → head s = head s ′ ∧ ( tail s ) ∼ ( tail s ′ ) and if ∀ s , s ′ .ϕ ( s , s ′ ) → head s = head s ′ ∧ ϕ ( tail s , tail s ′ ) then ∀ s , s ′ .ϕ ( s , s ′ ) → s ∼ s ′ Anton Setzer (Swansea) Pattern and Copattern matching 21/ 46

Bisimilarity and Coinduction Corecursive Proof of Bisimilarity ◮ Because ∼ is a final coalgebra we can compute proofs of it by corecursion: ◮ We can define f : ( s , s ′ : Stream ) → ϕ s s ′ → s ∼ s ′ ∼ ( f s s ′ x ) elim 0 = an element of head s = head s ′ ∼ ( f s s ′ x ) elim 0 = an element of ( tail s ) ∼ ( tail s ′ ) where in the last line we can use ◮ either a proof of tail s ∼ tail s ′ defined before ◮ or use the corecursion hypothesis f ( tail s ) ( tail s ′ ) x ′ for some x ′ : ϕ ( tail s ) ( tail s ′ ) Anton Setzer (Swansea) Pattern and Copattern matching 22/ 46

Bisimilarity and Coinduction Coinduction Theorem Assume Stream : Set , head : Stream → N , tail : Stream → Stream . The following are equivalent ◮ The principle of unique coiteration. ◮ The principle of unique corecursion. ◮ The principle of iteration together with the principle that bisimilarity ∼ implies equality ∀ s , s ′ : Stream . s ∼ s ′ → s = s ′ Because of the possibility of defining elements of s ∼ s ′ the latter can be considered as a principle of coinduction . Anton Setzer (Swansea) Pattern and Copattern matching 23/ 46

Bisimilarity and Coinduction Principle of Coinduction ◮ Let ϕ : Stream → Stream → Set . ◮ We can prove ∀ s , s ′ : Stream .ϕ s s ′ → s = s ′ by showing ∀ s , s ′ : Stream .ϕ s s ′ → head s = head s ′ ∀ s , s ′ : Stream .ϕ s s ′ → tail s = tail s ′ where for proving tail s = tail s ′ we can use the coinduction hypothesis that ϕ ( tail s ) ( tail s ′ ) implies tail s = tail s ′ . Anton Setzer (Swansea) Pattern and Copattern matching 24/ 46