Unnesting of Copatterns RTA-TLCA, 15 July 2014 Anton Setzer - PowerPoint PPT Presentation

Unnesting of Copatterns RTA-TLCA, 15 July 2014 Anton Setzer Andreas Abel Brigitte Pientka David Thibodeau Swansea Gothenburg Montreal Montreal UK Sweden Canada Canada Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 1/ 30

Pattern and Copattern Matching Unnesting of Copatterns/Patterns Proof of Conservatity and Preservation of SN/WN Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 2/ 30

Pattern and Copattern Matching Pattern and Copattern Matching Unnesting of Copatterns/Patterns Proof of Conservatity and Preservation of SN/WN Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 3/ 30

Pattern and Copattern Matching Natural Numbers Syntax for Algebraic Data Types in Paper Nat := µ X . � zero 1 | suc X � Introduction Rule (All constructors will have exactly one argument) zero : 1 → Nat suc : Nat → Nat Elimination Rule (Pattern Matching) pred : Nat → Nat pred (zero x ) = ? pred (suc n ) = ? Full recursion allowed (normalisation for individual terms see later) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 4/ 30

Pattern and Copattern Matching Nested Patterns and CC-Pattern-Sets pred : Nat → Nat pred (zero x ) = ? pred (suc n ) = ? Pattern matching on 1 (containing ()) yields the nested pattern pred : Nat → Nat pred (zero ()) = ? pred (suc n ) = ? which formally is the coverage complete ( cc ) pattern set pred : Nat → Nat ⊳ | ( · ⊢ pred (zero ()) : Nat) ( n : Nat ⊢ pred (suc n ) : Nat) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 5/ 30

Pattern and Copattern Matching Coverage Complete Rule Sets (CC-Rule Sets) After full pattern derived we fill in the “?” pred : Nat → Nat pred (zero ()) = zero () pred (suc n ) = n corresponds to the coverage complete ( cc ) rule set pred : Nat → Nat ⊳ | ( · ⊢ pred (zero ()) − → zero () : Nat) ( n : Nat ⊢ pred (suc n ) − → n : Nat) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 6/ 30

Pattern and Copattern Matching Stream Stream := ν X . { head : Nat , tail : X } (In paper called StrN) . tail . tail . tail s 2 s 3 s 1 · · · . head . head . head n 2 n 3 n 1 Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 7/ 30

Pattern and Copattern Matching Stream Elimination Rule If s : Stream then s . head : Nat s . tail : Stream . head, . tail treated like application. Introduction Rule (Copattern Matching) inc : Nat → Stream inc n . head = n inc n . tail = inc ( n + 1) Informally inc n = n , n + 1 , n + 2 , . . . Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 8/ 30

Pattern and Copattern Matching CC-Rule/Pattern Set inc : Nat → Stream inc n . head = n inc n . tail = inc ( n + 1) This corresponds to the cc-pattern-set inc : Nat → Stream ⊳ | ( n : Nat ⊢ inc n . head : Nat) ( n : Nat ⊢ inc n . tail : Stream) and cc-rule-set inc : Nat → Stream ⊳ | ( n : Nat ⊢ inc n . head − → n : Nat) ( n : Nat ⊢ inc n . tail − → inc ( n + 1) : Stream) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 9/ 30

Pattern and Copattern Matching cyc n Let N be fixed . For n we define a stream which is informally given as cyc n = n , n − 1 , n − 2 , . . . , 0 , N , N − 1 , N − 2 , . . . , 0 , N , N − 1 , . . . . tail . tail . tail . tail cyc cyc N cyc 0 N − 1 · · · . head . head . head N N − 1 0 Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 10/ 30

Pattern and Copattern Matching Development of cyc (Paper contains rules for deriving cc-pattern/rule-sets .) The simplest pattern matching is by itself: cyc : Nat → Stream ⊳ | ( · ⊢ cyc : Nat → Stream) Copattern matching for functions is application : cyc : Nat → Stream ⊳ | ( n : Nat ⊢ cyc n : Stream) Copattern matching on Stream yields: cyc : Nat → Stream ⊳ | ( n : Nat ⊢ cyc n . head : Nat) ( n : Nat ⊢ cyc n . tail : Stream) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 11/ 30

Pattern and Copattern Matching Development of cyc (Cont.) Pattern matching on Nat yields: ( n : Nat ⊢ cyc n . head : Nat) cyc : Nat → Stream ⊳ | ( x : 1 ⊢ cyc (zero x ) . tail : Stream) ( n : Nat ⊢ cyc (suc n ) . tail : Stream) Pattern matching on 1 yields: ( n : Nat ⊢ cyc n . head : Nat) cyc : Nat → Stream ⊳ | ( · ⊢ cyc (zero ()) . tail : Stream) ( n : Nat ⊢ cyc (suc n ) . tail : Stream) By adding results we obtain a cc-rule-set: ( n : Nat ⊢ cyc n . head − → n :Nat) cyc : Nat → Stream ⊳ | ( · ⊢ cyc (zero ()) . tail − → cyc N :Stream) ( n : Nat ⊢ cyc (suc n ) . tail − → cyc n : Stream) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 12/ 30

Unnesting of Copatterns/Patterns Pattern and Copattern Matching Unnesting of Copatterns/Patterns Proof of Conservatity and Preservation of SN/WN Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 13/ 30

Unnesting of Copatterns/Patterns Unnesting of cyc ( n : Nat ⊢ cyc n . head − → n :Nat) ( · ⊢ cyc (zero ()) . tail − → cyc N :Stream) cyc : Nat → Stream ⊳ | ( n : Nat ⊢ cyc (suc n ) . tail − → cyc n : Stream) We unnest the last step (pattern matching on 1 ) and delegate it to a new function g 2 ( n : Nat ⊢ cyc n . head − → n :Nat) cyc : Nat → Stream ⊳ | ( x : 1 ⊢ cyc (zero x ) . tail − → g 2 x :Stream) ( n : Nat ⊢ cyc (suc n ) . tail − → cyc n : Stream) g 2 : 1 → Stream ⊳ | ( · ⊢ g 2 () − → cyc N : Stream) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 14/ 30

Unnesting of Copatterns/Patterns Unnesting of cyc (Cont) ( n : Nat ⊢ cyc n . head − → n :Nat) cyc : Nat → Stream ⊳ | ( x : 1 ⊢ cyc (zero x ) . tail − → g 2 x :Stream) ( n : Nat ⊢ cyc (suc n ) . tail − → cyc n : Stream) g 2 : 1 → Stream ⊳ | ( · ⊢ g 2 () − → cyc N : Stream) Now we unnest pattern matching on x : Nat and delegate it to a new function g 1 : cyc : Nat → Stream ⊳ | ( n : Nat ⊢ cyc n . head − → n : Nat) ( n : Nat ⊢ cyc n . tail − → g 1 n : Stream) g 1 : Nat → Stream ⊳ | ( x : 1 ⊢ g 1 (zero x ) − → g 2 x : Stream) ( n : Nat ⊢ g 1 (suc n ) − → cyc n : Stream) g 2 : 1 → Stream ⊳ | ( · ⊢ g 2 () − → cyc N : Stream) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 15/ 30

Unnesting of Copatterns/Patterns Simple Pattern ◮ End result is simple pattern: ◮ There is at most one proper pattern/copattern step which is the last one. Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 16/ 30

Proof of Conservatity and Preservation of SN/WN Pattern and Copattern Matching Unnesting of Copatterns/Patterns Proof of Conservatity and Preservation of SN/WN Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 17/ 30

Proof of Conservatity and Preservation of SN/WN Programs Definition (a) A program P is given by constants with their types and a cc-rule-set for each constant (referring to terms in the same language). (b) A program P ′ extends P if the it contains all the constants of P with the same types (but not necessarily the same cc-rule-sets). Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 18/ 30

Proof of Conservatity and Preservation of SN/WN Conservative Extensions, Preservation of SN, WN Definition Let P ′ be a program extending P . (a) P ′ is a conservative extension of P iff ∀ t , t ′ ∈ Term P . t − P t ′ ⇔ t − → ∗ → ∗ P ′ t ′ (b) P ′ preserves strong normalisation ( SN ) iff ∀ t ∈ Term P . t ∈ SN( P ) ⇔ t ∈ SN( P ′ ) (c) P ′ preserves weak normalisation ( WN ) iff ∀ t ∈ Term P . t ∈ WN( P ) ⇔ t ∈ WN( P ′ ) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 19/ 30

Proof of Conservatity and Preservation of SN/WN Conservative Extension P ′ P ′ P ′ · · · t n ∈ Term P′ t 1 ∈ Term P′ t 2 ∈ Term P′ P ′ P ′ t ′ ∈ Term P t ∈ Term P Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 20/ 30

Proof of Conservatity and Preservation of SN/WN Conservative Extension P ′ P ′ P ′ · · · t n ∈ Term P′ t 1 ∈ Term P′ t 2 ∈ Term P′ P ′ P ′ t ′ ∈ Term P t ∈ Term P P P ∃ t ′ ∃ t ′ 2 ∈ Term P ∃ t ′ 1 ∈ Term P m ∈ Term P · · · P P P Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 20/ 30

Proof of Conservatity and Preservation of SN/WN Preservation of ¬ SN t 1 ∈ Term P′ t 2 ∈ Term P′ t 3 ∈ Term P′ · · · t �∈ SN( P ′ ) Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 21/ 30

Proof of Conservatity and Preservation of SN/WN Preservation of ¬ SN t 1 ∈ Term P′ t 2 ∈ Term P′ t 3 ∈ Term P′ · · · t �∈ SN( P ′ ) t �∈ SN( P ) ∃ t ′ ∃ t ′ ∃ t ′ 1 ∈ Term P 2 ∈ Term P 2 ∈ Term P · · · Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 21/ 30

Proof of Conservatity and Preservation of SN/WN Main Theorem Theorem Let P be a program. There exist an extension of P which is ◮ conservative, ◮ preserves SN , ◮ preserves WN ◮ and has only simple patterns. Setzer, Abel, Pientka, Thibodeau Unnesting of Copatterns 22/ 30