src/HOL/ex/RPred.thy

no longer declare .psimps rules as [simp].

This regularly caused confusion (e.g., they show up in simp traces
when the regular simp rules are disabled). In the few places where the
rules are used, explicitly mentioning them actually clarifies the
proof text.

theory RPred
imports Quickcheck Random Predicate
begin

types 'a "rpred" = "Random.seed \<Rightarrow> ('a Predicate.pred \<times> Random.seed)"

section {* The RandomPred Monad *}

text {* A monad to combine the random state monad and the predicate monad *}

definition bot :: "'a rpred"
  where "bot = Pair (bot_class.bot)"

definition return :: "'a => 'a rpred"
  where "return x = Pair (Predicate.single x)"

definition bind :: "'a rpred \<Rightarrow> ('a \<Rightarrow> 'b rpred) \<Rightarrow> 'b rpred"
(* (infixl "\<guillemotright>=" 60) *)
  where "bind RP f =
  (\<lambda>s. let (P, s') = RP s;
        (s1, s2) = Random.split_seed s'
    in (Predicate.bind P (%a. fst (f a s1)), s2))"

definition supp :: "'a rpred \<Rightarrow> 'a rpred \<Rightarrow> 'a rpred"
(* (infixl "\<squnion>" 80) *)
where
  "supp RP1 RP2 = (\<lambda>s. let (P1, s') = RP1 s; (P2, s'') = RP2 s'
  in (semilattice_sup_class.sup P1 P2, s''))"

definition if_rpred :: "bool \<Rightarrow> unit rpred"
where
  "if_rpred b = (if b then return () else bot)"

(* Missing a good definition for negation: not_rpred *)

definition not_rpred :: "unit rpred \<Rightarrow> unit rpred"
where
  "not_rpred P = (\<lambda>s. let (P', s') = P s in if Predicate.eval P' () then (Orderings.bot, s') else (Predicate.single (), s'))"

definition lift_pred :: "'a Predicate.pred \<Rightarrow> 'a rpred"
  where
  "lift_pred = Pair"

definition lift_random :: "(Random.seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed) \<Rightarrow> 'a rpred"
  where "lift_random g = scomp g (Pair o (Predicate.single o fst))"

definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a rpred \<Rightarrow> 'b rpred)"
  where "map f P = bind P (return o f)"

hide_const (open) return bind supp map
  
end