1(* Title: HOL/Corec_Examples/Tests/GPV_Bare_Bones.thy 2 Author: Andreas Lochbihler, ETH Zuerich 3 Author: Jasmin Blanchette, Inria, LORIA, MPII 4 Copyright 2016 5 6A bare bones version of generative probabilistic values (GPVs). 7*) 8 9section \<open>A Bare Bones Version of Generative Probabilistic Values (GPVs)\<close> 10 11theory GPV_Bare_Bones 12imports "HOL-Library.BNF_Corec" 13begin 14 15datatype 'a pmf = return_pmf 'a 16 17type_synonym 'a spmf = "'a option pmf" 18 19abbreviation map_spmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a spmf \<Rightarrow> 'b spmf" 20where "map_spmf f \<equiv> map_pmf (map_option f)" 21 22abbreviation return_spmf :: "'a \<Rightarrow> 'a spmf" 23where "return_spmf x \<equiv> return_pmf (Some x)" 24 25datatype (generat_pures: 'a, generat_outs: 'b, generat_conts: 'c) generat = 26 Pure (result: 'a) 27| IO ("output": 'b) (continuation: "'c") 28 29codatatype (results'_gpv: 'a, outs'_gpv: 'out, 'in) gpv = 30 GPV (the_gpv: "('a, 'out, 'in \<Rightarrow> ('a, 'out, 'in) gpv) generat spmf") 31 32codatatype ('a, 'call, 'ret, 'x) gpv' = 33 GPV' (the_gpv': "('a, 'call, 'ret \<Rightarrow> ('a, 'call, 'ret, 'x) gpv') generat spmf") 34 35consts bind_gpv' :: "('a, 'call, 'ret) gpv \<Rightarrow> ('a \<Rightarrow> ('b, 'call, 'ret, 'a) gpv') \<Rightarrow> ('b, 'call, 'ret, 'a) gpv'" 36 37definition bind_spmf :: "'a spmf \<Rightarrow> ('a \<Rightarrow> 'b spmf) \<Rightarrow> 'b spmf" 38where "bind_spmf x f = undefined x (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')" 39 40friend_of_corec bind_gpv' 41where "bind_gpv' r f = 42GPV' (map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. bind_gpv' r f)))) 43 (bind_spmf (the_gpv r) 44 (case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv' (f x))) 45 (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))))" 46 sorry 47 48end 49