(* Title: HOL/Library/Predicate_Compile_Alternative_Defs.thy
Author: Lukas Bulwahn, TU Muenchen
*)
theory Predicate_Compile_Alternative_Defs
imports Main
begin
section \<open>Common constants\<close>
declare HOL.if_bool_eq_disj[code_pred_inline]
declare bool_diff_def[code_pred_inline]
declare inf_bool_def[abs_def, code_pred_inline]
declare less_bool_def[abs_def, code_pred_inline]
declare le_bool_def[abs_def, code_pred_inline]
lemma min_bool_eq [code_pred_inline]: "(min :: bool => bool => bool) == (\)"
by (rule eq_reflection) (auto simp add: fun_eq_iff min_def)
lemma [code_pred_inline]:
"((A::bool) \ (B::bool)) = ((A \ \ B) \ (B \ \ A))"
by fast
setup \<open>Predicate_Compile_Data.ignore_consts [\<^const_name>\<open>Let\<close>]\<close>
section \<open>Pairs\<close>
setup \<open>Predicate_Compile_Data.ignore_consts [\<^const_name>\<open>fst\<close>, \<^const_name>\<open>snd\<close>, \<^const_name>\<open>case_prod\<close>]\<close>
section \<open>Filters\<close>
(*TODO: shouldn't this be done by typedef? *)
setup \<open>Predicate_Compile_Data.ignore_consts [\<^const_name>\<open>Abs_filter\<close>, \<^const_name>\<open>Rep_filter\<close>]\<close>
section \<open>Bounded quantifiers\<close>
declare Ball_def[code_pred_inline]
declare Bex_def[code_pred_inline]
section \<open>Operations on Predicates\<close>
lemma Diff[code_pred_inline]:
"(A - B) = (%x. A x \ \ B x)"
by (simp add: fun_eq_iff)
lemma subset_eq[code_pred_inline]:
"(P :: 'a \ bool) < (Q :: 'a \ bool) \ ((\x. Q x \ (\ P x)) \ (\x. P x \ Q x))"
by (rule eq_reflection) (auto simp add: less_fun_def le_fun_def)
lemma set_equality[code_pred_inline]:
"A = B \ (\x. A x \ B x) \ (\x. B x \ A x)"
by (auto simp add: fun_eq_iff)
section \<open>Setup for Numerals\<close>
setup \<open>Predicate_Compile_Data.ignore_consts [\<^const_name>\<open>numeral\<close>]\<close>
setup \<open>Predicate_Compile_Data.keep_functions [\<^const_name>\<open>numeral\<close>]\<close>
setup \<open>Predicate_Compile_Data.ignore_consts [\<^const_name>\<open>Char\<close>]\<close>
setup \<open>Predicate_Compile_Data.keep_functions [\<^const_name>\<open>Char\<close>]\<close>
setup \<open>Predicate_Compile_Data.ignore_consts [\<^const_name>\<open>divide\<close>, \<^const_name>\<open>modulo\<close>, \<^const_name>\<open>times\<close>]\<close>
section \<open>Arithmetic operations\<close>
subsection \<open>Arithmetic on naturals and integers\<close>
definition plus_eq_nat :: "nat => nat => nat => bool"
where
"plus_eq_nat x y z = (x + y = z)"
definition minus_eq_nat :: "nat => nat => nat => bool"
where
"minus_eq_nat x y z = (x - y = z)"
definition plus_eq_int :: "int => int => int => bool"
where
"plus_eq_int x y z = (x + y = z)"
definition minus_eq_int :: "int => int => int => bool"
where
"minus_eq_int x y z = (x - y = z)"
definition subtract
where
[code_unfold]: "subtract x y = y - x"
setup \<open>
let
val Fun = Predicate_Compile_Aux.Fun
val Input = Predicate_Compile_Aux.Input
val Output = Predicate_Compile_Aux.Output
val Bool = Predicate_Compile_Aux.Bool
val iio = Fun (Input, Fun (Input, Fun (Output, Bool)))
val ioi = Fun (Input, Fun (Output, Fun (Input, Bool)))
val oii = Fun (Output, Fun (Input, Fun (Input, Bool)))
val ooi = Fun (Output, Fun (Output, Fun (Input, Bool)))
val plus_nat = Core_Data.functional_compilation \<^const_name>\<open>plus\<close> iio
val minus_nat = Core_Data.functional_compilation \<^const_name>\<open>minus\<close> iio
fun subtract_nat compfuns (_ : typ) =
let
val T = Predicate_Compile_Aux.mk_monadT compfuns \<^typ>\<open>nat\<close>
in
absdummy \<^typ>\<open>nat\<close> (absdummy \<^typ>\<open>nat\<close>
(Const (\<^const_name>\<open>If\<close>, \<^typ>\<open>bool\<close> --> T --> T --> T) $
(\<^term>\<open>(>) :: nat => nat => bool\<close> $ Bound 1 $ Bound 0) $
Predicate_Compile_Aux.mk_empty compfuns \<^typ>\<open>nat\<close> $
Predicate_Compile_Aux.mk_single compfuns
(\<^term>\<open>(-) :: nat => nat => nat\<close> $ Bound 0 $ Bound 1)))
end
fun enumerate_addups_nat compfuns (_ : typ) =
absdummy \<^typ>\<open>nat\<close> (Predicate_Compile_Aux.mk_iterate_upto compfuns \<^typ>\<open>nat * nat\<close>
(absdummy \<^typ>\<open>natural\<close> (\<^term>\<open>Pair :: nat => nat => nat * nat\<close> $
(\<^term>\<open>nat_of_natural\<close> $ Bound 0) $
(\<^term>\<open>(-) :: nat => nat => nat\<close> $ Bound 1 $ (\<^term>\<open>nat_of_natural\<close> $ Bound 0))),
\<^term>\<open>0 :: natural\<close>, \<^term>\<open>natural_of_nat\<close> $ Bound 0))
fun enumerate_nats compfuns (_ : typ) =
let
val (single_const, _) = strip_comb (Predicate_Compile_Aux.mk_single compfuns \<^term>\<open>0 :: nat\<close>)
val T = Predicate_Compile_Aux.mk_monadT compfuns \<^typ>\<open>nat\<close>
in
absdummy \<^typ>\<open>nat\<close> (absdummy \<^typ>\<open>nat\<close>
(Const (\<^const_name>\<open>If\<close>, \<^typ>\<open>bool\<close> --> T --> T --> T) $
(\<^term>\<open>(=) :: nat => nat => bool\<close> $ Bound 0 $ \<^term>\<open>0::nat\<close>) $
(Predicate_Compile_Aux.mk_iterate_upto compfuns \<^typ>\<open>nat\<close> (\<^term>\<open>nat_of_natural\<close>,
\<^term>\<open>0::natural\<close>, \<^term>\<open>natural_of_nat\<close> $ Bound 1)) $
(single_const $ (\<^term>\<open>(+) :: nat => nat => nat\<close> $ Bound 1 $ Bound 0))))
end
in
Core_Data.force_modes_and_compilations \<^const_name>\<open>plus_eq_nat\<close>
[(iio, (plus_nat, false)), (oii, (subtract_nat, false)), (ioi, (subtract_nat, false)),
(ooi, (enumerate_addups_nat, false))]
#> Predicate_Compile_Fun.add_function_predicate_translation
(\<^term>\<open>plus :: nat => nat => nat\<close>, \<^term>\<open>plus_eq_nat\<close>)
#> Core_Data.force_modes_and_compilations \<^const_name>\<open>minus_eq_nat\<close>
[(iio, (minus_nat, false)), (oii, (enumerate_nats, false))]
#> Predicate_Compile_Fun.add_function_predicate_translation
(\<^term>\<open>minus :: nat => nat => nat\<close>, \<^term>\<open>minus_eq_nat\<close>)
#> Core_Data.force_modes_and_functions \<^const_name>\<open>plus_eq_int\<close>
[(iio, (\<^const_name>\<open>plus\<close>, false)), (ioi, (\<^const_name>\<open>subtract\<close>, false)),
(oii, (\<^const_name>\<open>subtract\<close>, false))]
#> Predicate_Compile_Fun.add_function_predicate_translation
(\<^term>\<open>plus :: int => int => int\<close>, \<^term>\<open>plus_eq_int\<close>)
#> Core_Data.force_modes_and_functions \<^const_name>\<open>minus_eq_int\<close>
[(iio, (\<^const_name>\<open>minus\<close>, false)), (oii, (\<^const_name>\<open>plus\<close>, false)),
(ioi, (\<^const_name>\<open>minus\<close>, false))]
#> Predicate_Compile_Fun.add_function_predicate_translation
(\<^term>\<open>minus :: int => int => int\<close>, \<^term>\<open>minus_eq_int\<close>)
end
\<close>
subsection \<open>Inductive definitions for ordering on naturals\<close>
inductive less_nat
where
"less_nat 0 (Suc y)"
| "less_nat x y ==> less_nat (Suc x) (Suc y)"
lemma less_nat[code_pred_inline]:
"x < y = less_nat x y"
apply (rule iffI)
apply (induct x arbitrary: y)
apply (case_tac y) apply (auto intro: less_nat.intros)
apply (case_tac y)
apply (auto intro: less_nat.intros)
apply (induct rule: less_nat.induct)
apply auto
done
inductive less_eq_nat
where
"less_eq_nat 0 y"
| "less_eq_nat x y ==> less_eq_nat (Suc x) (Suc y)"
lemma [code_pred_inline]:
"x <= y = less_eq_nat x y"
apply (rule iffI)
apply (induct x arbitrary: y)
apply (auto intro: less_eq_nat.intros)
apply (case_tac y) apply (auto intro: less_eq_nat.intros)
apply (induct rule: less_eq_nat.induct)
apply auto done
section \<open>Alternative list definitions\<close>
subsection \<open>Alternative rules for \<open>length\<close>\<close>
definition size_list' :: "'a list => nat"
where "size_list' = size"
lemma size_list'_simps:
"size_list' [] = 0"
"size_list' (x # xs) = Suc (size_list' xs)"
by (auto simp add: size_list'_def)
declare size_list'_simps[code_pred_def]
declare size_list'_def[symmetric, code_pred_inline]
subsection \<open>Alternative rules for \<open>list_all2\<close>\<close>
lemma list_all2_NilI [code_pred_intro]: "list_all2 P [] []"
by auto
lemma list_all2_ConsI [code_pred_intro]: "list_all2 P xs ys ==> P x y ==> list_all2 P (x#xs) (y#ys)"
by auto
code_pred [skip_proof] list_all2
proof -
case list_all2
from this show thesis
apply -
apply (case_tac xb)
apply (case_tac xc)
apply auto
apply (case_tac xc)
apply auto
done
qed
subsection \<open>Alternative rules for membership in lists\<close>
declare in_set_member[code_pred_inline]
lemma member_intros [code_pred_intro]:
"List.member (x#xs) x"
"List.member xs x \ List.member (y#xs) x"
by(simp_all add: List.member_def)
code_pred List.member
by(auto simp add: List.member_def elim: list.set_cases)
code_identifier constant member_i_i
\<rightharpoonup> (SML) "List.member_i_i"
and (OCaml) "List.member_i_i"
and (Haskell) "List.member_i_i"
and (Scala) "List.member_i_i"
code_identifier constant member_i_o
\<rightharpoonup> (SML) "List.member_i_o"
and (OCaml) "List.member_i_o"
and (Haskell) "List.member_i_o"
and (Scala) "List.member_i_o"
section \<open>Setup for String.literal\<close>
setup \<open>Predicate_Compile_Data.ignore_consts [\<^const_name>\<open>String.Literal\<close>]\<close>
section \<open>Simplification rules for optimisation\<close>
lemma [code_pred_simp]: "\ False == True"
by auto
lemma [code_pred_simp]: "\ True == False"
by auto
lemma less_nat_k_0 [code_pred_simp]: "less_nat k 0 == False"
unfolding less_nat[symmetric] by auto
end
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