(* Title: HOL/Corec_Examples/Tests/GPV_Bare_Bones.thy
Author: Andreas Lochbihler, ETH Zuerich
Author: Jasmin Blanchette, Inria, LORIA, MPII
Copyright 2016
A bare bones version of generative probabilistic values (GPVs).
*)
section \<open>A Bare Bones Version of Generative Probabilistic Values (GPVs)\<close>
theory GPV_Bare_Bones
imports "~~/src/HOL/Library/BNF_Corec"
begin
datatype 'a pmf = return_pmf 'a
type_synonym 'a spmf = "'a option pmf"
abbreviation map_spmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a spmf \<Rightarrow> 'b spmf"
where "map_spmf f \<equiv> map_pmf (map_option f)"
abbreviation return_spmf :: "'a \<Rightarrow> 'a spmf"
where "return_spmf x \<equiv> return_pmf (Some x)"
datatype (generat_pures: 'a, generat_outs: 'b, generat_conts: 'c) generat =
Pure (result: 'a)
| IO ("output": 'b) (continuation: "'c")
codatatype (results'_gpv: 'a, outs'_gpv: 'out, 'in) gpv =
GPV (the_gpv: "('a, 'out, 'in \<Rightarrow> ('a, 'out, 'in) gpv) generat spmf")
codatatype ('a, 'call, 'ret, 'x) gpv' =
GPV' (the_gpv': "('a, 'call, 'ret \<Rightarrow> ('a, 'call, 'ret, 'x) gpv') generat spmf")
consts bind_gpv' :: "('a, 'call, 'ret) gpv \<Rightarrow> ('a \<Rightarrow> ('b, 'call, 'ret, 'a) gpv') \<Rightarrow> ('b, 'call, 'ret, 'a) gpv'"
definition bind_spmf :: "'a spmf \<Rightarrow> ('a \<Rightarrow> 'b spmf) \<Rightarrow> 'b spmf"
where "bind_spmf x f = undefined x (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
friend_of_corec bind_gpv'
where "bind_gpv' r f =
GPV' (map_spmf (map_generat id id (op \<circ> (case_sum id (\<lambda>r. bind_gpv' r f))))
(bind_spmf (the_gpv r)
(case_generat (\<lambda>x. map_spmf (map_generat id id (op \<circ> Inl)) (the_gpv' (f x)))
(\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))))"
sorry
end