src/HOL/Datatype.thy
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(*  Title:      HOL/Datatype.thy
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    ID:         $Id$
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    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* Datatypes *}
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theory Datatype = Datatype_Universe:
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subsection {* Representing primitive types *}
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rep_datatype bool
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  distinct True_not_False False_not_True
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  induction bool_induct
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declare case_split [cases type: bool]
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  -- "prefer plain propositional version"
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rep_datatype unit
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  induction unit_induct
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rep_datatype prod
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  inject Pair_eq
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  induction prod_induct
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rep_datatype sum
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  distinct Inl_not_Inr Inr_not_Inl
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  inject Inl_eq Inr_eq
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  induction sum_induct
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ML {*
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  val [sum_case_Inl, sum_case_Inr] = thms "sum.cases";
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  bind_thm ("sum_case_Inl", sum_case_Inl);
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  bind_thm ("sum_case_Inr", sum_case_Inr);
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*} -- {* compatibility *}
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lemma surjective_sum: "sum_case (%x::'a. f (Inl x)) (%y::'b. f (Inr y)) s = f(s)"
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  apply (rule_tac s = s in sumE)
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   apply (erule ssubst)
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   apply (rule sum_case_Inl)
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  apply (erule ssubst)
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  apply (rule sum_case_Inr)
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  done
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lemma sum_case_weak_cong: "s = t ==> sum_case f g s = sum_case f g t"
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  -- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
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  by (erule arg_cong)
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lemma sum_case_inject:
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  "sum_case f1 f2 = sum_case g1 g2 ==> (f1 = g1 ==> f2 = g2 ==> P) ==> P"
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proof -
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  assume a: "sum_case f1 f2 = sum_case g1 g2"
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  assume r: "f1 = g1 ==> f2 = g2 ==> P"
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  show P
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    apply (rule r)
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     apply (rule ext)
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     apply (cut_tac x = "Inl x" in a [THEN fun_cong], simp)
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    apply (rule ext)
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    apply (cut_tac x = "Inr x" in a [THEN fun_cong], simp)
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    done
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qed
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constdefs
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  Suml :: "('a => 'c) => 'a + 'b => 'c"
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  "Suml == (%f. sum_case f arbitrary)"
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  Sumr :: "('b => 'c) => 'a + 'b => 'c"
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  "Sumr == sum_case arbitrary"
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lemma Suml_inject: "Suml f = Suml g ==> f = g"
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  by (unfold Suml_def) (erule sum_case_inject)
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lemma Sumr_inject: "Sumr f = Sumr g ==> f = g"
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  by (unfold Sumr_def) (erule sum_case_inject)
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subsection {* Finishing the datatype package setup *}
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text {* Belongs to theory @{text Datatype_Universe}; hides popular names. *}
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hide const Push Node Atom Leaf Numb Lim Split Case Suml Sumr
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hide type node item
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subsection {* Further cases/induct rules for tuples *}
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lemma prod_cases3 [case_names fields, cases type]:
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    "(!!a b c. y = (a, b, c) ==> P) ==> P"
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  apply (cases y)
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  apply (case_tac b, blast)
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  done
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lemma prod_induct3 [case_names fields, induct type]:
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    "(!!a b c. P (a, b, c)) ==> P x"
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  by (cases x) blast
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lemma prod_cases4 [case_names fields, cases type]:
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    "(!!a b c d. y = (a, b, c, d) ==> P) ==> P"
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  apply (cases y)
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  apply (case_tac c, blast)
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  done
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lemma prod_induct4 [case_names fields, induct type]:
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    "(!!a b c d. P (a, b, c, d)) ==> P x"
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  by (cases x) blast
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lemma prod_cases5 [case_names fields, cases type]:
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    "(!!a b c d e. y = (a, b, c, d, e) ==> P) ==> P"
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  apply (cases y)
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  apply (case_tac d, blast)
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  done
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lemma prod_induct5 [case_names fields, induct type]:
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    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
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  by (cases x) blast
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lemma prod_cases6 [case_names fields, cases type]:
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    "(!!a b c d e f. y = (a, b, c, d, e, f) ==> P) ==> P"
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  apply (cases y)
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  apply (case_tac e, blast)
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  done
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lemma prod_induct6 [case_names fields, induct type]:
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    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
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  by (cases x) blast
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lemma prod_cases7 [case_names fields, cases type]:
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    "(!!a b c d e f g. y = (a, b, c, d, e, f, g) ==> P) ==> P"
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  apply (cases y)
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  apply (case_tac f, blast)
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  done
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lemma prod_induct7 [case_names fields, induct type]:
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    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
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  by (cases x) blast
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subsection {* The option type *}
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datatype 'a option = None | Some 'a
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lemma not_None_eq [iff]: "(x ~= None) = (EX y. x = Some y)"
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  by (induct x) auto
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lemma not_Some_eq [iff]: "(ALL y. x ~= Some y) = (x = None)"
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  by (induct x) auto
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lemma option_caseE:
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  "(case x of None => P | Some y => Q y) ==>
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    (x = None ==> P ==> R) ==>
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    (!!y. x = Some y ==> Q y ==> R) ==> R"
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  by (cases x) simp_all
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subsubsection {* Operations *}
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consts
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  the :: "'a option => 'a"
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primrec
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  "the (Some x) = x"
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consts
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  o2s :: "'a option => 'a set"
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primrec
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  "o2s None = {}"
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  "o2s (Some x) = {x}"
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lemma ospec [dest]: "(ALL x:o2s A. P x) ==> A = Some x ==> P x"
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  by simp
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ML_setup {* claset_ref() := claset() addSD2 ("ospec", thm "ospec") *}
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lemma elem_o2s [iff]: "(x : o2s xo) = (xo = Some x)"
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  by (cases xo) auto
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lemma o2s_empty_eq [simp]: "(o2s xo = {}) = (xo = None)"
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  by (cases xo) auto
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constdefs
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  option_map :: "('a => 'b) => ('a option => 'b option)"
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  "option_map == %f y. case y of None => None | Some x => Some (f x)"
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lemma option_map_None [simp]: "option_map f None = None"
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  by (simp add: option_map_def)
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lemma option_map_Some [simp]: "option_map f (Some x) = Some (f x)"
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  by (simp add: option_map_def)
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lemma option_map_is_None[iff]:
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 "(option_map f opt = None) = (opt = None)"
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by (simp add: option_map_def split add: option.split)
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lemma option_map_eq_Some [iff]:
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    "(option_map f xo = Some y) = (EX z. xo = Some z & f z = y)"
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by (simp add: option_map_def split add: option.split)
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lemma option_map_comp:
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 "option_map f (option_map g opt) = option_map (f o g) opt"
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by (simp add: option_map_def split add: option.split)
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lemma option_map_o_sum_case [simp]:
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    "option_map f o sum_case g h = sum_case (option_map f o g) (option_map f o h)"
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  apply (rule ext)
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  apply (simp split add: sum.split)
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  done
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end