Theory Predicate_Compile_Alternative_Defs

theory Predicate_Compile_Alternative_Defs
imports Main
theory Predicate_Compile_Alternative_Defs
imports Main
begin

section {* Common constants *}

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) == (op &)"
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 {* Predicate_Compile_Data.ignore_consts [@{const_name Let}] *}

section {* Pairs *}

setup {* Predicate_Compile_Data.ignore_consts [@{const_name fst}, @{const_name snd}, @{const_name prod_case}] *}

section {* Bounded quantifiers *}

declare Ball_def[code_pred_inline]
declare Bex_def[code_pred_inline]

section {* Operations on Predicates *}

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 {* Setup for Numerals *}

setup {* Predicate_Compile_Data.ignore_consts [@{const_name numeral}, @{const_name neg_numeral}] *}
setup {* Predicate_Compile_Data.keep_functions [@{const_name numeral}, @{const_name neg_numeral}] *}

setup {* Predicate_Compile_Data.ignore_consts [@{const_name div}, @{const_name mod}, @{const_name times}] *}

section {* Arithmetic operations *}

subsection {* Arithmetic on naturals and integers *}

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 {*
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 plus} iio
val minus_nat = Core_Data.functional_compilation @{const_name "minus"} iio
fun subtract_nat compfuns (_ : typ) =
let
val T = Predicate_Compile_Aux.mk_monadT compfuns @{typ nat}
in
absdummy @{typ nat} (absdummy @{typ nat}
(Const (@{const_name "If"}, @{typ bool} --> T --> T --> T) $
(@{term "op > :: nat => nat => bool"} $ Bound 1 $ Bound 0) $
Predicate_Compile_Aux.mk_empty compfuns @{typ nat} $
Predicate_Compile_Aux.mk_single compfuns
(@{term "op - :: nat => nat => nat"} $ Bound 0 $ Bound 1)))
end
fun enumerate_addups_nat compfuns (_ : typ) =
absdummy @{typ nat} (Predicate_Compile_Aux.mk_iterate_upto compfuns @{typ "nat * nat"}
(absdummy @{typ natural} (@{term "Pair :: nat => nat => nat * nat"} $
(@{term "nat_of_natural"} $ Bound 0) $
(@{term "op - :: nat => nat => nat"} $ Bound 1 $ (@{term "nat_of_natural"} $ Bound 0))),
@{term "0 :: natural"}, @{term "natural_of_nat"} $ Bound 0))
fun enumerate_nats compfuns (_ : typ) =
let
val (single_const, _) = strip_comb (Predicate_Compile_Aux.mk_single compfuns @{term "0 :: nat"})
val T = Predicate_Compile_Aux.mk_monadT compfuns @{typ nat}
in
absdummy @{typ nat} (absdummy @{typ nat}
(Const (@{const_name If}, @{typ bool} --> T --> T --> T) $
(@{term "op = :: nat => nat => bool"} $ Bound 0 $ @{term "0::nat"}) $
(Predicate_Compile_Aux.mk_iterate_upto compfuns @{typ nat} (@{term "nat_of_natural"},
@{term "0::natural"}, @{term "natural_of_nat"} $ Bound 1)) $
(single_const $ (@{term "op + :: nat => nat => nat"} $ Bound 1 $ Bound 0))))
end
in
Core_Data.force_modes_and_compilations @{const_name plus_eq_nat}
[(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 "plus :: nat => nat => nat"}, @{term "plus_eq_nat"})
#> Core_Data.force_modes_and_compilations @{const_name minus_eq_nat}
[(iio, (minus_nat, false)), (oii, (enumerate_nats, false))]
#> Predicate_Compile_Fun.add_function_predicate_translation
(@{term "minus :: nat => nat => nat"}, @{term "minus_eq_nat"})
#> Core_Data.force_modes_and_functions @{const_name plus_eq_int}
[(iio, (@{const_name plus}, false)), (ioi, (@{const_name subtract}, false)),
(oii, (@{const_name subtract}, false))]
#> Predicate_Compile_Fun.add_function_predicate_translation
(@{term "plus :: int => int => int"}, @{term "plus_eq_int"})
#> Core_Data.force_modes_and_functions @{const_name minus_eq_int}
[(iio, (@{const_name minus}, false)), (oii, (@{const_name plus}, false)),
(ioi, (@{const_name minus}, false))]
#> Predicate_Compile_Fun.add_function_predicate_translation
(@{term "minus :: int => int => int"}, @{term "minus_eq_int"})
end
*}


subsection {* Inductive definitions for ordering on naturals *}

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 {* Alternative list definitions *}

subsection {* Alternative rules for @{text length} *}

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 {* Alternative rules for @{text list_all2} *}

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
apply fastforce
done
qed

section {* Setup for String.literal *}

setup {* Predicate_Compile_Data.ignore_consts [@{const_name "STR"}] *}

section {* Simplification rules for optimisation *}

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