Make partiality more explicit in library functions of Lem shallow embedding

Some functions are partial, e.g. converting a bitvector to an integer, which
might fail for the bit list representation due to undefined bits.  Undefined
cases can be handled in different ways:

- call Lem's failwith, which maps to undefined/ARB in Isabelle and HOL (the
  default so far),
- return an option type,
- raise a failure  in the monad, or
- use a bitstream oracle to resolve undefined bits.

This patch adds different versions of partial functions corresponding to those
options.  The desired behaviour can be selected by choosing a binding in the
Sail prelude.  The naming scheme is that the failwith version is the default,
while the other versions have the suffixes _maybe, _fail, and _oracle,
respectively.

1 files changed, 42 insertions, 119 deletions

diff --git a/lib/isabelle/State_monad_lemmas.thy b/lib/isabelle/State_monad_lemmas.thy
index 291157f5..e7286fcf 100644
--- a/lib/isabelle/State_monad_lemmas.thy
+++ b/lib/isabelle/State_monad_lemmas.thy
@@ -4,23 +4,15 @@ theory State_monad_lemmas
     Sail_values_lemmas
 begin
 
-context
+(*context
   notes returnS_def[simp] and failS_def[simp] and throwS_def[simp] and readS_def[simp] and updateS_def[simp]
-begin
-
-abbreviation "bindS_aux f \<equiv> (\<lambda>r. case r of (Value a, s') \<Rightarrow> f a s' | (Ex e, s') \<Rightarrow> {(Ex e, s')})"
-abbreviation "bindS_app ms f \<equiv> \<Union>(bindS_aux f ` ms)"
+begin*)
 
 lemma bindS_ext_cong[fundef_cong]:
   assumes m: "m1 s = m2 s"
     and f: "\<And>a s'. (Value a, s') \<in> (m2 s) \<Longrightarrow> f1 a s' = f2 a s'"
   shows "bindS m1 f1 s = bindS m2 f2 s"
   using assms unfolding bindS_def by (auto split: result.splits)
-(* proof -
-  have "bindS_app (m2 s) f1 = bindS_app (m2 s) f2"
-    using f by (auto split: result.splits)
-  then show ?thesis using m by (auto simp: bindS_def)
-qed *)
 
 lemma bindS_cong[fundef_cong]:
   assumes m: "m1 = m2"
@@ -29,21 +21,16 @@ lemma bindS_cong[fundef_cong]:
   using assms by (intro ext bindS_ext_cong; blast)
 
 lemma bindS_returnS_left[simp]: "bindS (returnS x) f = f x"
-  by (auto simp add: bindS_def)
+  by (auto simp add: bindS_def returnS_def)
 
 lemma bindS_returnS_right[simp]: "bindS m returnS = (m :: ('regs, 'a, 'e) monadS)"
-  by (intro ext) (auto simp: bindS_def split: result.splits)
-(* proof -
-  have "List.concat (map (bindS_aux returnS) ms) = ms" for ms :: "(('a, 'e) result \<times> 'regs sequential_state) list"
-    by (induction ms) (auto split: result.splits)
-  then show ?thesis unfolding bindS_def by blast
-qed *)
+  by (intro ext) (auto simp: bindS_def returnS_def split: result.splits)
 
 lemma bindS_readS: "bindS (readS f) m = (\<lambda>s. m (f s) s)"
-  by (auto simp: bindS_def)
+  by (auto simp: bindS_def readS_def returnS_def)
 
 lemma bindS_updateS: "bindS (updateS f) m = (\<lambda>s. m () (f s))"
-  by (auto simp: bindS_def)
+  by (auto simp: bindS_def updateS_def returnS_def)
 
 
 lemma result_cases:
@@ -84,17 +71,6 @@ lemma monadS_eqI:
   shows "m = m'"
   using assms by (intro ext monadS_ext_eqI)
 
-lemma
-  assumes "(Value a, s') \<in> bindS m f s"
-  obtains a' s'' where "(Value a', s'') \<in> m s" and "(Value a, s') \<in> f a' s''"
-  using assms by (auto simp: bindS_def split: result.splits)
-
-lemma
-  assumes "(Ex e, s') \<in> bindS m f s"
-  obtains (Left) "(Ex e, s') \<in> m s"
-  | (Right) a s'' where "(Value a, s'') \<in> m s" and "(Ex e, s') \<in> f a s''"
-  using assms by (auto simp: bindS_def split: result.splits)
-
 lemma bindS_cases:
   assumes "(r, s') \<in> bindS m f s"
   obtains (Value) a a' s'' where "r = Value a" and "(Value a', s'') \<in> m s" and "(Value a, s') \<in> f a' s''"
@@ -110,15 +86,9 @@ lemma bindS_intros:
 
 lemma bindS_assoc[simp]: "bindS (bindS m f) g = bindS m (\<lambda>x. bindS (f x) g)"
   by (auto elim!: bindS_cases intro: bindS_intros monadS_eqI)
-(*proof -
-  have "List.concat (map (bindS_aux g) (List.concat (map (bindS_aux f) xs))) =
-        List.concat (map (bindS_aux (\<lambda>x s. List.concat (map (bindS_aux g) (f x s)))) xs)" for xs
-    by (induction xs) (auto split: result.splits)
-  then show ?thesis unfolding bindS_def by auto
-qed*)
-
-lemma bindS_failS[simp]: "bindS (failS msg) f = failS msg" by (auto simp: bindS_def)
-lemma bindS_throwS[simp]: "bindS (throwS e) f = throwS e" by (auto simp: bindS_def)
+
+lemma bindS_failS[simp]: "bindS (failS msg) f = failS msg" by (auto simp: bindS_def failS_def)
+lemma bindS_throwS[simp]: "bindS (throwS e) f = throwS e" by (auto simp: bindS_def throwS_def)
 declare seqS_def[simp]
 
 lemma Value_bindS_elim:
@@ -132,18 +102,10 @@ lemma Ex_bindS_elim:
   | (Right) s'' a' where "(Value a', s'') \<in> m s" and "(Ex e, s') \<in> f a' s''"
   using assms by (auto elim: bindS_cases)
 
-abbreviation
-  "try_catchS_aux h r \<equiv>
-    (case r of
-       (Value a, s') => returnS a s'
-     | (Ex (Throw e), s') => h e s'
-     | (Ex (Failure msg), s') => {(Ex (Failure msg), s')})"
-abbreviation "try_catchS_app ms h \<equiv> \<Union>(try_catchS_aux h ` ms)"
-
 lemma try_catchS_returnS[simp]: "try_catchS (returnS a) h = returnS a"
   and try_catchS_failS[simp]: "try_catchS (failS msg) h = failS msg"
   and try_catchS_throwS[simp]: "try_catchS (throwS e) h = h e"
-  by (auto simp: try_catchS_def)
+  by (auto simp: try_catchS_def returnS_def failS_def throwS_def)
 
 lemma try_catchS_cong[cong]:
   assumes "\<And>s. m1 s = m2 s" and "\<And>e s. h1 e s = h2 e s"
@@ -155,13 +117,14 @@ lemma try_catchS_cases:
   obtains (Value) a where "r = Value a" and "(Value a, s') \<in> m s"
   | (Fail) msg where "r = Ex (Failure msg)" and "(Ex (Failure msg), s') \<in> m s"
   | (h) e s'' where "(Ex (Throw e), s'') \<in> m s" and "(r, s') \<in> h e s''"
-  using assms by (cases r rule: result_cases) (auto simp: try_catchS_def split: result.splits ex.splits)
+  using assms
+  by (cases r rule: result_cases) (auto simp: try_catchS_def returnS_def split: result.splits ex.splits)
 
 lemma try_catchS_intros:
   "\<And>m h s a s'. (Value a, s') \<in> m s \<Longrightarrow> (Value a, s') \<in> try_catchS m h s"
   "\<And>m h s msg s'. (Ex (Failure msg), s') \<in> m s \<Longrightarrow> (Ex (Failure msg), s') \<in> try_catchS m h s"
   "\<And>m h s e s'' r s'. (Ex (Throw e), s'') \<in> m s \<Longrightarrow> (r, s') \<in> h e s'' \<Longrightarrow> (r, s') \<in> try_catchS m h s"
-  by (auto simp: try_catchS_def intro: bexI[rotated])
+  by (auto simp: try_catchS_def returnS_def intro: bexI[rotated])
 
 fun ignore_throw_aux :: "(('a, 'e1) result \<times> 's) \<Rightarrow> (('a, 'e2) result \<times> 's) set" where
   "ignore_throw_aux (Value a, s') = {(Value a, s')}"
@@ -169,42 +132,23 @@ fun ignore_throw_aux :: "(('a, 'e1) result \<times> 's) \<Rightarrow> (('a, 'e2)
 | "ignore_throw_aux (Ex (Failure msg), s') = {(Ex (Failure msg), s')}"
 definition "ignore_throw m s \<equiv> \<Union>(ignore_throw_aux ` m s)"
 
-(*fun ignore_throw_aux :: "(('a, 'e1) result \<times> 's) \<Rightarrow> (('a, 'e2) result \<times> 's) list" where
-  "ignore_throw_aux r \<equiv>
-    (case r of
-       (Value a, s') => returnS a s'
-     | (Ex (Throw e), s') => h e s'
-     | (Ex (Failure msg), s') => {(Ex (Failure msg), s')})"
-fun ignore_throw_app :: "(('a, 'e1) result \<times> 's) list \<Rightarrow> (('a, 'e2) result \<times> 's) list" where
-  "ignore_throw_app [] = []"
-| "ignore_throw_app ((Value a, s) # ms) = (Value a, s) # ignore_throw_app ms"
-| "ignore_throw_app ((Ex (Failure msg), s) # ms) = (Ex (Failure msg), s) # ignore_throw_app ms"
-| "ignore_throw_app ((Ex (Throw e), s) # ms) = ignore_throw_app ms"
-abbreviation ignore_throw :: "('r, 'a, 'e1) monadS \<Rightarrow> ('r, 'a, 'e2) monadS" where
-  "ignore_throw m \<equiv> \<lambda>s. ignore_throw_app (m s)"
-
-lemma [simp]: "ignore_throw_app ms = (Ex (Throw e), s) # ms' \<longleftrightarrow> False"
-  by (induction ms rule: ignore_throw_app.induct) auto
-
-lemma ignore_throw_app_append[simp]:
-  "ignore_throw_app (ms1 @ ms2) = ignore_throw_app ms1 @ ignore_throw_app ms2"
-  by (induction ms1 rule: ignore_throw_app.induct) auto
-
-lemma ignore_throw_app_bindS_app[simp]:
-  "ignore_throw_app (bindS_app ms f) = bindS_app (ignore_throw_app ms) (ignore_throw \<circ> f)"
-  by (induction ms rule: ignore_throw_app.induct) (auto split: result.splits)*)
-
-lemma [simp]:
+lemma ignore_throw_cong:
+  assumes "\<And>s. m1 s = m2 s"
+  shows "ignore_throw m1 = ignore_throw m2"
+  using assms by (auto simp: ignore_throw_def)
+
+lemma ignore_throw_aux_member_simps[simp]:
   "(Value a, s') \<in> ignore_throw_aux ms \<longleftrightarrow> ms = (Value a, s')"
   "(Ex (Throw e), s') \<in> ignore_throw_aux ms \<longleftrightarrow> False"
   "(Ex (Failure msg), s') \<in> ignore_throw_aux ms \<longleftrightarrow> ms = (Ex (Failure msg), s')"
   by (cases ms rule: result_state_cases; auto)+
 
-(*lemma [simp]: "(Value a, s') \<in> ignore_throw m s \<longleftrightarrow> (Value a, s') \<in> m s"
+lemma ignore_throw_member_simps[simp]:
+  "(Value a, s') \<in> ignore_throw m s \<longleftrightarrow> (Value a, s') \<in> m s"
   "(Value a, s') \<in> ignore_throw m s \<longleftrightarrow> (Value a, s') \<in> m s"
   "(Ex (Throw e), s') \<in> ignore_throw m s \<longleftrightarrow> False"
-  "(Ex (Failure msg), s') \<in> ignore_throw_aux ms \<longleftrightarrow> ms = (Ex (Failure msg), s')"
-  by (auto simp: ignore_throw_def)*)
+  "(Ex (Failure msg), s') \<in> ignore_throw m s \<longleftrightarrow> (Ex (Failure msg), s') \<in> m s"
+  by (auto simp: ignore_throw_def)
 
 lemma ignore_throw_cases:
   assumes no_throw: "ignore_throw m s = m s"
@@ -232,43 +176,21 @@ proof
   also have "\<dots> = bindS m2 (\<lambda>a. try_catchS (f a) h) s" using m2 by (intro bindS_ext_cong) auto
   finally show "try_catchS (bindS m1 f) h s = bindS m2 (\<lambda>a. try_catchS (f a) h) s" .
 qed
-(*proof
-  fix s
-  have 1: "try_catchS_app (bindS_app ms f) h =
-        bindS_app (ignore_throw_app ms) (\<lambda>a s'. try_catchS_app (f a s') h)"
-    if "ignore_throw_app ms = ms" for ms
-    using that by (induction ms rule: ignore_throw_app.induct) auto
-  then show "try_catchS (bindS m1 f) h s = bindS m2 (\<lambda>a. try_catchS (f a) h) s"
-    using m1 unfolding try_catchS_def bindS_def m2[symmetric] by blast
-qed*)
-
-(*lemma no_throwI:
-  fixes m1 :: "('regs, 'a, 'e1) monadS" and m2 :: "('regs, 'a, 'e2) monadS"
-  assumes "\<And>a s'. (Value a, s') \<in> m1 s \<longleftrightarrow> (Value a, s') \<in> m2 s"
-    and "\<And>msg s'. (Ex (Failure msg), s') \<in> m1 s \<longleftrightarrow> (Ex (Failure msg), s') \<in> m2 s"
-    and "\<And>e s'. (Ex (Throw e), s') \<notin> m1 s"
-    and "\<And>e s'. (Ex (Throw e), s') \<notin> m2 s"
-  shows "ignore_throw m1 s = m2 s"
-  using assms by (intro monadS_ext_eqI) (auto simp: ignore_throw_def)*)
-
-(*lemma no_throw_bindSI:
-  assumes "ignore_throw m1 s = m2 s"
-and "\<And>a s'. (Value a, s') \<in> m2 s \<Longrightarrow> ignore_throw (f1 a) s' = f2 a s'"
-shows "ignore_throw (bindS m1 f1) s = bindS m2 f2 s"
-  using assms unfolding ignore_throw_bindS apply (intro monadS_ext_eqI) apply auto apply (auto elim!: bindS_cases intro: bindS_intros)
-   defer thm bindS_intros
-   apply (intro bindS_intros(3)) apply auto
-  apply (intro bindS_intros(3)) apply auto
-  done
-
-lemma
-  "\<And>BC rk a sz s. ignore_throw (read_mem_bytesS BC rk a sz) s = read_mem_bytesS BC rk a sz s"
-  unfolding read_mem_bytesS_def Let_def seqS_def
-  apply (intro no_throw_bindSI)
-  oops*)
+
+lemma no_throw_basic_builtins[simp]:
+  "ignore_throw (returnS a) = returnS a"
+  "\<And>f. ignore_throw (readS f) = readS f"
+  "\<And>f. ignore_throw (updateS f) = updateS f"
+  "ignore_throw (chooseS xs) = chooseS xs"
+  "ignore_throw (failS msg) = failS msg"
+  "ignore_throw (maybe_failS msg x) = maybe_failS msg x"
+  unfolding ignore_throw_def returnS_def chooseS_def maybe_failS_def failS_def readS_def updateS_def
+  by (intro ext; auto split: option.splits)+
+
+lemmas ignore_throw_option_case_distrib =
+  option.case_distrib[where h = "\<lambda>c. ignore_throw c s" and option = "c s" for c s]
 
 lemma no_throw_mem_builtins:
-  "\<And>a. ignore_throw (returnS a) = returnS a"
   "\<And>BC rk a sz s. ignore_throw (read_mem_bytesS BC rk a sz) s = read_mem_bytesS BC rk a sz s"
   "\<And>BC a s. ignore_throw (read_tagS BC a) s = read_tagS BC a s"
   "\<And>BC wk a sz s. ignore_throw (write_mem_eaS BC wk a sz) s = write_mem_eaS BC wk a sz s"
@@ -280,20 +202,21 @@ lemma no_throw_mem_builtins:
   unfolding read_mem_bytesS_def read_memS_def read_tagS_def write_mem_eaS_def
   unfolding write_mem_valS_def write_mem_bytesS_def write_tagS_def
   unfolding excl_resultS_def undefined_boolS_def
-  by (auto simp: ignore_throw_def bindS_def chooseS_def Let_def split: option.splits prod.splits)
+  by (auto cong: bindS_cong bindS_ext_cong ignore_throw_cong option.case_cong
+           simp: option.case_distrib prod.case_distrib ignore_throw_option_case_distrib comp_def)
 
 lemma no_throw_read_memS: "ignore_throw (read_memS BCa BCb rk a sz) s = read_memS BCa BCb rk a sz s"
   by (auto simp: read_memS_def no_throw_mem_builtins cong: bindS_ext_cong)
 
 lemma no_throw_read_regvalS: "ignore_throw (read_regvalS r reg_name) s = read_regvalS r reg_name s"
-  by (cases r) (auto simp: ignore_throw_def bindS_def split: option.splits)
+  by (cases r) (auto simp: option.case_distrib cong: bindS_cong option.case_cong)
 
 lemma no_throw_write_regvalS: "ignore_throw (write_regvalS r reg_name v) s = write_regvalS r reg_name v s"
-  by (cases r) (auto simp: ignore_throw_def bindS_def split: option.splits)
+  by (cases r) (auto simp: option.case_distrib cong: bindS_cong option.case_cong)
 
-lemmas no_throw_builtins[simp, intro] =
+lemmas no_throw_builtins[simp] =
   no_throw_mem_builtins no_throw_read_regvalS no_throw_write_regvalS no_throw_read_memS
 
-end
+(* end *)
 
 end