Theory Refine_Imperative_HOL.Sepref_Monadify
section ‹Monadify›
theory Sepref_Monadify
imports Sepref_Basic Sepref_Id_Op
begin
text ‹
In this phase, a monadic program is converted to complete monadic form,
that is, computation of compound expressions are made visible as top-level
operations in the monad.
The monadify process is separated into 2 steps.
\begin{enumerate}
\item In a first step, eta-expansion is used to add missing operands
to operations and combinators. This way, operators and combinators
always occur with the same arity, which simplifies further processing.
\item In a second step, computation of compound operands is flattened,
introducing new bindings for the intermediate values.
\end{enumerate}
›
definition SP
where [simp]: "SP x ≡ x"
lemma SP_cong[cong]: "SP x ≡ SP x" by simp
lemma PR_CONST_cong[cong]: "PR_CONST x ≡ PR_CONST x" by simp
definition RCALL
where [simp]: "RCALL D ≡ D"
definition EVAL
where [simp]: "EVAL x ≡ RETURN x"
text ‹
Internally, the package first applies rewriting rules from
‹sepref_monadify_arity›, which use eta-expansion to ensure that
every combinator has enough actual parameters. Moreover, this phase will
mark recursive calls by the tag @{const RCALL}.
Next, rewriting rules from ‹sepref_monadify_comb› are used to
add @{const EVAL}-tags to plain expressions that should be evaluated
in the monad. The @{const EVAL} tags are flattened using a default simproc
that generates left-to-right argument order.
›
lemma monadify_simps:
"Refine_Basic.bind$(RETURN$x)$(λ⇩2x. f x) = f x"
"EVAL$x ≡ RETURN$x"
by simp_all
definition [simp]: "PASS ≡ RETURN"
lemma remove_pass_simps:
"Refine_Basic.bind$(PASS$x)$(λ⇩2x. f x) ≡ f x"
"Refine_Basic.bind$m$(λ⇩2x. PASS$x) ≡ m"
by simp_all
definition COPY :: "'a ⇒ 'a"
where [simp]: "COPY x ≡ x"
lemma RET_COPY_PASS_eq: "RETURN$(COPY$p) = PASS$p" by simp
named_theorems_rev sepref_monadify_arity "Sepref.Monadify: Arity alignment equations"
named_theorems_rev sepref_monadify_comb "Sepref.Monadify: Combinator equations"
ML ‹
structure Sepref_Monadify = struct
local
fun cr_var (i,T) = ("v"^string_of_int i, Free ("__v"^string_of_int i,T))
fun lambda2_name n t = let
val t = @{mk_term "PROTECT2 ?t DUMMY"}
in
Term.lambda_name n t
end
fun
bind_args exp0 [] = exp0
| bind_args exp0 ((x,m)::xms) = let
val lr = bind_args exp0 xms
|> incr_boundvars 1
|> lambda2_name x
in @{mk_term "Refine_Basic.bind$?m$?lr"} end
fun monadify t = let
val (f,args) = Autoref_Tagging.strip_app t
val _ = not (is_Abs f) orelse
raise TERM ("monadify: higher-order",[t])
val argTs = map fastype_of args
val args = map (fn a => @{mk_term "EVAL$?a"}) args
val argVs = tag_list 0 argTs
|> map cr_var
val res0 = let
val x = Autoref_Tagging.list_APP (f,map #2 argVs)
in
@{mk_term "SP (RETURN$?x)"}
end
val res = bind_args res0 (argVs ~~ args)
in
res
end
fun monadify_conv_aux ctxt ct = case Thm.term_of ct of
@{mpat "EVAL$_"} => let
fun tac goal_ctxt =
simp_tac (put_simpset HOL_basic_ss goal_ctxt addsimps @{thms monadify_simps SP_def}) 1
in (
Refine_Util.f_tac_conv ctxt (dest_comb #> #2 #> monadify) tac) ct
end
| t => raise TERM ("monadify_conv",[t])
in
val monadify_simproc =
\<^simproc_setup>‹passive monadify ("EVAL$a") = ‹K (try o monadify_conv_aux)››;
end
local
open Sepref_Basic
fun mark_params t = let
val (P,c,Q,R,a) = dest_hn_refine t
val pps = strip_star P |> map_filter (dest_hn_ctxt_opt #> map_option #2)
fun tr env (t as @{mpat "RETURN$?x"}) =
if is_Bound x orelse member (aconv) pps x then
@{mk_term env: "PASS$?x"}
else t
| tr env (t1$t2) = tr env t1 $ tr env t2
| tr env (Abs (x,T,t)) = Abs (x,T,tr (T::env) t)
| tr _ t = t
val a = tr [] a
in
mk_hn_refine (P,c,Q,R,a)
end
in
fun mark_params_conv ctxt = Refine_Util.f_tac_conv ctxt
(mark_params)
(fn goal_ctxt => simp_tac (put_simpset HOL_basic_ss goal_ctxt addsimps @{thms PASS_def}) 1)
end
local
open Sepref_Basic
fun dp ctxt (@{mpat "Refine_Basic.bind$(PASS$?p)$(?t' AS⇩p (λ_. PROTECT2 _ DUMMY))"}) =
let
val (t',ps) = let
val ((t',rc),ctxt) = dest_lambda_rc ctxt t'
val f = case t' of @{mpat "PROTECT2 ?f _"} => f | _ => raise Match
val (f,ps) = dp ctxt f
val t' = @{mk_term "PROTECT2 ?f DUMMY"}
val t' = rc t'
in
(t',ps)
end
val dup = member (aconv) ps p
val t = if dup then
@{mk_term "Refine_Basic.bind$(RETURN$(COPY$?p))$?t'"}
else
@{mk_term "Refine_Basic.bind$(PASS$?p)$?t'"}
in
(t,p::ps)
end
| dp ctxt (t1$t2) = (#1 (dp ctxt t1) $ #1 (dp ctxt t2),[])
| dp ctxt (t as (Abs _)) = (apply_under_lambda (#1 oo dp) ctxt t,[])
| dp _ t = (t,[])
fun dp_conv ctxt = Refine_Util.f_tac_conv ctxt
(#1 o dp ctxt)
(fn goal_ctxt =>
ALLGOALS (simp_tac (put_simpset HOL_basic_ss goal_ctxt addsimps @{thms RET_COPY_PASS_eq})))
in
fun dup_tac ctxt = CONVERSION (Sepref_Basic.hn_refine_concl_conv_a dp_conv ctxt)
end
fun arity_tac ctxt = let
val arity1_ss = put_simpset HOL_basic_ss ctxt
addsimps ((Named_Theorems_Rev.get ctxt @{named_theorems_rev sepref_monadify_arity}))
|> Simplifier.add_cong @{thm SP_cong}
|> Simplifier.add_cong @{thm PR_CONST_cong}
val arity2_ss = put_simpset HOL_basic_ss ctxt
addsimps @{thms beta SP_def}
in
simp_tac arity1_ss THEN' simp_tac arity2_ss
end
fun comb_tac ctxt = let
val comb1_ss = put_simpset HOL_basic_ss ctxt
addsimps (Named_Theorems_Rev.get ctxt @{named_theorems_rev sepref_monadify_comb})
addsimprocs [monadify_simproc]
|> Simplifier.add_cong @{thm SP_cong}
|> Simplifier.add_cong @{thm PR_CONST_cong}
val comb2_ss = put_simpset HOL_basic_ss ctxt
addsimps @{thms SP_def}
in
simp_tac comb1_ss THEN' simp_tac comb2_ss
end
fun mark_params_tac ctxt = CONVERSION (
Refine_Util.HOL_concl_conv mark_params_conv ctxt)
fun contains_eval @{mpat "Trueprop (hn_refine _ _ _ _ ?a)"} =
Term.exists_subterm (fn @{mpat EVAL} => true | _ => false) a
| contains_eval t = raise TERM("contains_eval",[t]);
fun remove_pass_tac ctxt =
simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms remove_pass_simps})
fun monadify_tac dbg ctxt = let
open Sepref_Basic
in
PHASES' [
("arity", arity_tac, 0),
("comb", comb_tac, 0),
("check_EVAL", K (CONCL_COND' (not o contains_eval)), 0),
("mark_params", mark_params_tac, 0),
("dup", dup_tac, 0),
("remove_pass", remove_pass_tac, 0)
] (flag_phases_ctrl dbg) ctxt
end
end
›
lemma dflt_arity[sepref_monadify_arity]:
"RETURN ≡ λ⇩2x. SP RETURN$x"
"RECT ≡ λ⇩2B x. SP RECT$(λ⇩2D x. B$(λ⇩2x. RCALL$D$x)$x)$x"
"case_list ≡ λ⇩2fn fc l. SP case_list$fn$(λ⇩2x xs. fc$x$xs)$l"
"case_prod ≡ λ⇩2fp p. SP case_prod$(λ⇩2a b. fp$a$b)$p"
"case_option ≡ λ⇩2fn fs ov. SP case_option$fn$(λ⇩2x. fs$x)$ov"
"If ≡ λ⇩2b t e. SP If$b$t$e"
"Let ≡ λ⇩2x f. SP Let$x$(λ⇩2x. f$x)"
by (simp_all only: SP_def APP_def PROTECT2_def RCALL_def)
lemma dflt_comb[sepref_monadify_comb]:
"⋀B x. RECT$B$x ≡ Refine_Basic.bind$(EVAL$x)$(λ⇩2x. SP (RECT$B$x))"
"⋀D x. RCALL$D$x ≡ Refine_Basic.bind$(EVAL$x)$(λ⇩2x. SP (RCALL$D$x))"
"⋀fn fc l. case_list$fn$fc$l ≡ Refine_Basic.bind$(EVAL$l)$(λ⇩2l. (SP case_list$fn$fc$l))"
"⋀fp p. case_prod$fp$p ≡ Refine_Basic.bind$(EVAL$p)$(λ⇩2p. (SP case_prod$fp$p))"
"⋀fn fs ov. case_option$fn$fs$ov
≡ Refine_Basic.bind$(EVAL$ov)$(λ⇩2ov. (SP case_option$fn$fs$ov))"
"⋀b t e. If$b$t$e ≡ Refine_Basic.bind$(EVAL$b)$(λ⇩2b. (SP If$b$t$e))"
"⋀x. RETURN$x ≡ Refine_Basic.bind$(EVAL$x)$(λ⇩2x. SP (RETURN$x))"
"⋀x f. Let$x$f ≡ Refine_Basic.bind$(EVAL$x)$(λ⇩2x. (SP Let$x$f))"
by (simp_all)
lemma dflt_plain_comb[sepref_monadify_comb]:
"EVAL$(If$b$t$e) ≡ Refine_Basic.bind$(EVAL$b)$(λ⇩2b. If$b$(EVAL$t)$(EVAL$e))"
"EVAL$(case_list$fn$(λ⇩2x xs. fc x xs)$l) ≡
Refine_Basic.bind$(EVAL$l)$(λ⇩2l. case_list$(EVAL$fn)$(λ⇩2x xs. EVAL$(fc x xs))$l)"
"EVAL$(case_prod$(λ⇩2a b. fp a b)$p) ≡
Refine_Basic.bind$(EVAL$p)$(λ⇩2p. case_prod$(λ⇩2a b. EVAL$(fp a b))$p)"
"EVAL$(case_option$fn$(λ⇩2x. fs x)$ov) ≡
Refine_Basic.bind$(EVAL$ov)$(λ⇩2ov. case_option$(EVAL$fn)$(λ⇩2x. EVAL$(fs x))$ov)"
"EVAL $ (Let $ v $ (λ⇩2x. f x)) ≡ (⤜) $ (EVAL $ v) $ (λ⇩2x. EVAL $ (f x))"
apply (rule eq_reflection, simp split: list.split prod.split option.split)+
done
lemma evalcomb_PR_CONST[sepref_monadify_comb]:
"EVAL$(PR_CONST x) ≡ SP (RETURN$(PR_CONST x))"
by simp
end