1(* Title: HOL/Corec_Examples/Paper_Examples.thy 2 Author: Andreas Lochbihler, ETH Zuerich 3 Author: Andrei Popescu, TU Muenchen 4 Copyright 2016 5 6Small examples from the paper "Friends with Benefits". 7*) 8 9section \<open>Small Examples from the Paper ``Friends with Benefits''\<close> 10 11theory Paper_Examples 12imports "HOL-Library.BNF_Corec" "HOL-Library.FSet" Complex_Main 13begin 14 15section \<open>Examples from the introduction\<close> 16 17codatatype 'a stream = SCons (shd: 'a) (stl: "'a stream") (infixr "\<lhd>" 65) 18 19corec "natsFrom" :: "nat \<Rightarrow> nat stream" where 20 "natsFrom n = n \<lhd> natsFrom (n + 1)" 21 22corec (friend) add1 :: "nat stream \<Rightarrow> nat stream" 23where "add1 ns = (shd ns + 1) \<lhd> add1 (stl ns)" 24 25corec natsFrom' :: "nat \<Rightarrow> nat stream" where 26 "natsFrom' n = n \<lhd> add1 (natsFrom' n)" 27 28section \<open>Examples from section 3\<close> 29 30text \<open>We curry the example functions in this section because infix syntax works only for curried functions.\<close> 31 32corec (friend) Plus :: "nat stream \<Rightarrow> nat stream \<Rightarrow> nat stream" (infix "\<oplus>" 67) where 33 "x\<^sub>1 \<oplus> x\<^sub>2 = (shd x\<^sub>1 + shd x\<^sub>2) \<lhd> (stl x\<^sub>1 \<oplus> stl x\<^sub>2)" 34 35section \<open>Examples from section 4\<close> 36 37codatatype 'a llist = LNil | LCons 'a "'a llist" 38 39corec collatz :: "nat \<Rightarrow> nat llist" where 40 "collatz n = (if n \<le> 1 then LNil 41 else if even n then collatz (n div 2) 42 else LCons n (collatz (3 * n + 1)))" 43 44datatype 'a nelist = NEList (hd:'a) (tl:"'a list") 45 46primrec (transfer) snoc :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a nelist" (infix "\<rhd>" 64) where 47 "[] \<rhd> a = NEList a []" 48|"(b # bs) \<rhd> a = NEList b (bs @ [a])" 49 50corec (friend) inter :: "nat stream nelist \<Rightarrow> nat stream" where 51"inter xss = shd (hd xss) \<lhd> inter (tl xss \<rhd> stl (hd xss))" 52 53corec (friend) inter' :: "nat stream nelist \<Rightarrow> nat stream" where 54"inter' xss = (case hd xss of x \<lhd> xs \<Rightarrow> x \<lhd> inter' (tl xss \<rhd> xs))" 55 56corec zero :: "nat stream" where "zero = 0 \<lhd> zero" 57 58section \<open>Examples from Blanchette et al. (ICFP 2015)\<close> 59 60corec oneTwos :: "nat stream" where "oneTwos = 1 \<lhd> 2 \<lhd> oneTwos" 61 62corec everyOther :: "'a stream \<Rightarrow> 'a stream" 63where "everyOther xs = shd xs \<lhd> everyOther (stl (stl xs))" 64 65corec fibA :: "nat stream" 66where "fibA = 0 \<lhd> (1 \<lhd> fibA \<oplus> fibA)" 67 68corec fibB :: "nat stream" 69where "fibB = (0 \<lhd> 1 \<lhd> fibB) \<oplus> (0 \<lhd> fibB)" 70 71corec (friend) times :: "nat stream \<Rightarrow> nat stream \<Rightarrow> nat stream" (infix "\<otimes>" 69) 72where "xs \<otimes> ys = (shd xs * shd ys) \<lhd> xs \<otimes> stl ys \<oplus> stl xs \<otimes> ys" 73 74corec (friend) exp :: "nat stream \<Rightarrow> nat stream" 75where "exp xs = 2 ^ shd xs \<lhd> (stl xs \<otimes> exp xs)" 76 77corec facA :: "nat stream" 78where "facA = (1 \<lhd> facA) \<otimes> (1 \<lhd> facA)" 79 80corec facB :: "nat stream" 81where "facB = exp (0 \<lhd> facB)" 82 83corec (friend) sfsup :: "nat stream fset \<Rightarrow> nat stream" 84where "sfsup X = Sup (fset (fimage shd X)) \<lhd> sfsup (fimage stl X)" 85 86codatatype tree = Node (val: nat) (sub: "tree list") 87 88corec (friend) tplus :: "tree \<Rightarrow> tree \<Rightarrow> tree" 89where "tplus t u = Node (val t + val u) (map (\<lambda>(t', u'). tplus t' u') (zip (sub t) (sub u)))" 90 91corec (friend) ttimes :: "tree \<Rightarrow> tree \<Rightarrow> tree" 92where "ttimes t u = Node (val t * val u) 93 (map (\<lambda>(t, u). tplus (ttimes t u) (ttimes t u)) (zip (sub t) (sub u)))" 94 95corecursive primes :: "nat \<Rightarrow> nat \<Rightarrow> nat stream" 96where "primes m n = 97 (if (m = 0 \<and> n > 1) \<or> coprime m n then n \<lhd> primes (m * n) (n + 1) else primes m (n + 1))" 98apply (relation "measure (\<lambda>(m, n). if n = 0 then 1 else if coprime m n then 0 else m - n mod m)") 99 apply (auto simp: mod_Suc diff_less_mono2 intro: Suc_lessI elim!: not_coprimeE) 100 apply (metis dvd_1_iff_1 dvd_eq_mod_eq_0 mod_0 mod_Suc mod_Suc_eq mod_mod_cancel) 101 done 102 103corec facC :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat stream" 104where "facC n a i = (if i = 0 then a \<lhd> facC (n + 1) 1 (n + 1) else facC n (a * i) (i - 1))" 105 106corec catalan :: "nat \<Rightarrow> nat stream" 107where "catalan n = (if n > 0 then catalan (n - 1) \<oplus> (0 \<lhd> catalan (n + 1)) else 1 \<lhd> catalan 1)" 108 109corec (friend) heart :: "nat stream \<Rightarrow> nat stream \<Rightarrow> nat stream" (infix "\<heartsuit>" 65) 110where "xs \<heartsuit> ys = SCons (shd xs * shd ys) ((((xs \<heartsuit> stl ys) \<oplus> (stl xs \<otimes> ys)) \<heartsuit> ys) \<otimes> ys)" 111 112corec (friend) g :: "'a stream \<Rightarrow> 'a stream" 113where "g xs = shd xs \<lhd> g (g (stl xs))" 114 115end 116