Restructured algebra library, added ideals and quotient rings.
(* Title: HOL/Datatype.thy
ID: $Id$
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
*)
header {* Datatypes *}
theory Datatype
imports Datatype_Universe
begin
subsection {* Representing primitive types *}
rep_datatype bool
distinct True_not_False False_not_True
induction bool_induct
declare case_split [cases type: bool]
-- "prefer plain propositional version"
rep_datatype unit
induction unit_induct
rep_datatype prod
inject Pair_eq
induction prod_induct
rep_datatype sum
distinct Inl_not_Inr Inr_not_Inl
inject Inl_eq Inr_eq
induction sum_induct
ML {*
val [sum_case_Inl, sum_case_Inr] = thms "sum.cases";
bind_thm ("sum_case_Inl", sum_case_Inl);
bind_thm ("sum_case_Inr", sum_case_Inr);
*} -- {* compatibility *}
lemma surjective_sum: "sum_case (%x::'a. f (Inl x)) (%y::'b. f (Inr y)) s = f(s)"
apply (rule_tac s = s in sumE)
apply (erule ssubst)
apply (rule sum_case_Inl)
apply (erule ssubst)
apply (rule sum_case_Inr)
done
lemma sum_case_weak_cong: "s = t ==> sum_case f g s = sum_case f g t"
-- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
by (erule arg_cong)
lemma sum_case_inject:
"sum_case f1 f2 = sum_case g1 g2 ==> (f1 = g1 ==> f2 = g2 ==> P) ==> P"
proof -
assume a: "sum_case f1 f2 = sum_case g1 g2"
assume r: "f1 = g1 ==> f2 = g2 ==> P"
show P
apply (rule r)
apply (rule ext)
apply (cut_tac x = "Inl x" in a [THEN fun_cong], simp)
apply (rule ext)
apply (cut_tac x = "Inr x" in a [THEN fun_cong], simp)
done
qed
constdefs
Suml :: "('a => 'c) => 'a + 'b => 'c"
"Suml == (%f. sum_case f arbitrary)"
Sumr :: "('b => 'c) => 'a + 'b => 'c"
"Sumr == sum_case arbitrary"
lemma Suml_inject: "Suml f = Suml g ==> f = g"
by (unfold Suml_def) (erule sum_case_inject)
lemma Sumr_inject: "Sumr f = Sumr g ==> f = g"
by (unfold Sumr_def) (erule sum_case_inject)
subsection {* Finishing the datatype package setup *}
text {* Belongs to theory @{text Datatype_Universe}; hides popular names. *}
hide (open) const Push Node Atom Leaf Numb Lim Split Case Suml Sumr
hide (open) type node item
subsection {* Further cases/induct rules for tuples *}
lemma prod_cases3 [case_names fields, cases type]:
"(!!a b c. y = (a, b, c) ==> P) ==> P"
apply (cases y)
apply (case_tac b, blast)
done
lemma prod_induct3 [case_names fields, induct type]:
"(!!a b c. P (a, b, c)) ==> P x"
by (cases x) blast
lemma prod_cases4 [case_names fields, cases type]:
"(!!a b c d. y = (a, b, c, d) ==> P) ==> P"
apply (cases y)
apply (case_tac c, blast)
done
lemma prod_induct4 [case_names fields, induct type]:
"(!!a b c d. P (a, b, c, d)) ==> P x"
by (cases x) blast
lemma prod_cases5 [case_names fields, cases type]:
"(!!a b c d e. y = (a, b, c, d, e) ==> P) ==> P"
apply (cases y)
apply (case_tac d, blast)
done
lemma prod_induct5 [case_names fields, induct type]:
"(!!a b c d e. P (a, b, c, d, e)) ==> P x"
by (cases x) blast
lemma prod_cases6 [case_names fields, cases type]:
"(!!a b c d e f. y = (a, b, c, d, e, f) ==> P) ==> P"
apply (cases y)
apply (case_tac e, blast)
done
lemma prod_induct6 [case_names fields, induct type]:
"(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
by (cases x) blast
lemma prod_cases7 [case_names fields, cases type]:
"(!!a b c d e f g. y = (a, b, c, d, e, f, g) ==> P) ==> P"
apply (cases y)
apply (case_tac f, blast)
done
lemma prod_induct7 [case_names fields, induct type]:
"(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
by (cases x) blast
subsection {* The option type *}
datatype 'a option = None | Some 'a
lemma not_None_eq[iff]: "(x ~= None) = (EX y. x = Some y)"
by (induct x) auto
lemma not_Some_eq[iff]: "(ALL y. x ~= Some y) = (x = None)"
by (induct x) auto
text{*Although it may appear that both of these equalities are helpful
only when applied to assumptions, in practice it seems better to give
them the uniform iff attribute. *}
(*
lemmas not_None_eq_D = not_None_eq [THEN iffD1]
declare not_None_eq_D [dest!]
lemmas not_Some_eq_D = not_Some_eq [THEN iffD1]
declare not_Some_eq_D [dest!]
*)
lemma option_caseE:
"(case x of None => P | Some y => Q y) ==>
(x = None ==> P ==> R) ==>
(!!y. x = Some y ==> Q y ==> R) ==> R"
by (cases x) simp_all
subsubsection {* Operations *}
consts
the :: "'a option => 'a"
primrec
"the (Some x) = x"
consts
o2s :: "'a option => 'a set"
primrec
"o2s None = {}"
"o2s (Some x) = {x}"
lemma ospec [dest]: "(ALL x:o2s A. P x) ==> A = Some x ==> P x"
by simp
ML_setup {* change_claset (fn cs => cs addSD2 ("ospec", thm "ospec")) *}
lemma elem_o2s [iff]: "(x : o2s xo) = (xo = Some x)"
by (cases xo) auto
lemma o2s_empty_eq [simp]: "(o2s xo = {}) = (xo = None)"
by (cases xo) auto
constdefs
option_map :: "('a => 'b) => ('a option => 'b option)"
"option_map == %f y. case y of None => None | Some x => Some (f x)"
lemma option_map_None [simp]: "option_map f None = None"
by (simp add: option_map_def)
lemma option_map_Some [simp]: "option_map f (Some x) = Some (f x)"
by (simp add: option_map_def)
lemma option_map_is_None[iff]:
"(option_map f opt = None) = (opt = None)"
by (simp add: option_map_def split add: option.split)
lemma option_map_eq_Some [iff]:
"(option_map f xo = Some y) = (EX z. xo = Some z & f z = y)"
by (simp add: option_map_def split add: option.split)
lemma option_map_comp:
"option_map f (option_map g opt) = option_map (f o g) opt"
by (simp add: option_map_def split add: option.split)
lemma option_map_o_sum_case [simp]:
"option_map f o sum_case g h = sum_case (option_map f o g) (option_map f o h)"
apply (rule ext)
apply (simp split add: sum.split)
done
subsubsection {* Codegenerator setup *}
consts
is_none :: "'a option \<Rightarrow> bool"
primrec
"is_none None = True"
"is_none (Some x) = False"
lemma is_none_none [code inline]:
"(x = None) = (is_none x)"
by (cases x) simp_all
lemmas [code] = imp_conv_disj
lemma [code fun]:
"(\<not> True) = False" by (rule HOL.simp_thms)
lemma [code fun]:
"(\<not> False) = True" by (rule HOL.simp_thms)
lemma [code fun]:
shows "(False \<and> x) = False"
and "(True \<and> x) = x"
and "(x \<and> False) = False"
and "(x \<and> True) = x" by simp_all
lemma [code fun]:
shows "(False \<or> x) = x"
and "(True \<or> x) = True"
and "(x \<or> False) = x"
and "(x \<or> True) = True" by simp_all
declare
if_True [code fun]
if_False [code fun]
fst_conv [code]
snd_conv [code]
lemma split_is_prod_case [code inline]:
"split = prod_case"
by (simp add: expand_fun_eq split_def prod.cases)
code_typapp bool
ml (target_atom "bool")
haskell (target_atom "Bool")
code_constapp
True
ml (target_atom "true")
haskell (target_atom "True")
False
ml (target_atom "false")
haskell (target_atom "False")
Not
ml (target_atom "not")
haskell (target_atom "not")
"op &"
ml (infixl 1 "andalso")
haskell (infixl 3 "&&")
"op |"
ml (infixl 0 "orelse")
haskell (infixl 2 "||")
If
ml (target_atom "(if __/ then __/ else __)")
haskell (target_atom "(if __/ then __/ else __)")
code_typapp
*
ml (infix 2 "*")
haskell (target_atom "(__,/ __)")
code_constapp
Pair
ml (target_atom "(__,/ __)")
haskell (target_atom "(__,/ __)")
code_typapp
unit
ml (target_atom "unit")
haskell (target_atom "()")
code_constapp
Unity
ml (target_atom "()")
haskell (target_atom "()")
code_typapp
option
ml ("_ option")
haskell ("Maybe _")
code_constapp
None
ml (target_atom "NONE")
haskell (target_atom "Nothing")
Some
ml (target_atom "SOME")
haskell (target_atom "Just")
end