src/HOL/Product_Type.thy
author haftmann
Thu Jun 10 12:24:01 2010 +0200 (2010-06-10)
changeset 37389 09467cdfa198
parent 37387 3581483cca6c
child 37411 c88c44156083
permissions -rw-r--r--
qualified type "*"; qualified constants Pair, fst, snd, split
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(*  Title:      HOL/Product_Type.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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header {* Cartesian products *}
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theory Product_Type
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imports Typedef Inductive Fun
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uses
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  ("Tools/split_rule.ML")
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  ("Tools/inductive_codegen.ML")
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  ("Tools/inductive_set.ML")
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begin
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subsection {* @{typ bool} is a datatype *}
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rep_datatype True False by (auto intro: bool_induct)
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declare case_split [cases type: bool]
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  -- "prefer plain propositional version"
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lemma
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  shows [code]: "eq_class.eq False P \<longleftrightarrow> \<not> P"
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    and [code]: "eq_class.eq True P \<longleftrightarrow> P" 
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    and [code]: "eq_class.eq P False \<longleftrightarrow> \<not> P" 
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    and [code]: "eq_class.eq P True \<longleftrightarrow> P"
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    and [code nbe]: "eq_class.eq P P \<longleftrightarrow> True"
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  by (simp_all add: eq)
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code_const "eq_class.eq \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
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  (Haskell infixl 4 "==")
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code_instance bool :: eq
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  (Haskell -)
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subsection {* The @{text unit} type *}
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typedef unit = "{True}"
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proof
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  show "True : ?unit" ..
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qed
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definition
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  Unity :: unit    ("'(')")
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where
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  "() = Abs_unit True"
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lemma unit_eq [no_atp]: "u = ()"
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  by (induct u) (simp add: unit_def Unity_def)
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text {*
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  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
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  this rule directly --- it loops!
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*}
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ML {*
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  val unit_eq_proc =
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    let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in
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      Simplifier.simproc @{theory} "unit_eq" ["x::unit"]
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      (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)
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    end;
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  Addsimprocs [unit_eq_proc];
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*}
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rep_datatype "()" by simp
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
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  by simp
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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
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  by (rule triv_forall_equality)
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text {*
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  This rewrite counters the effect of @{text unit_eq_proc} on @{term
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  [source] "%u::unit. f u"}, replacing it by @{term [source]
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  f} rather than by @{term [source] "%u. f ()"}.
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*}
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lemma unit_abs_eta_conv [simp,no_atp]: "(%u::unit. f ()) = f"
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  by (rule ext) simp
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instantiation unit :: default
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begin
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definition "default = ()"
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instance ..
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end
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lemma [code]:
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  "eq_class.eq (u\<Colon>unit) v \<longleftrightarrow> True" unfolding eq unit_eq [of u] unit_eq [of v] by rule+
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code_type unit
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  (SML "unit")
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  (OCaml "unit")
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  (Haskell "()")
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  (Scala "Unit")
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code_const Unity
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  (SML "()")
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  (OCaml "()")
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  (Haskell "()")
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  (Scala "()")
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code_instance unit :: eq
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  (Haskell -)
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code_const "eq_class.eq \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
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  (Haskell infixl 4 "==")
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code_reserved SML
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  unit
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code_reserved OCaml
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  unit
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code_reserved Scala
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  Unit
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subsection {* The product type *}
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subsubsection {* Type definition *}
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definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
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  "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
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typedef (prod) ('a, 'b) "*" (infixr "*" 20)
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  = "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
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proof
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  fix a b show "Pair_Rep a b \<in> ?prod"
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    by rule+
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qed
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type_notation (xsymbols)
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  "*"  ("(_ \<times>/ _)" [21, 20] 20)
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type_notation (HTML output)
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  "*"  ("(_ \<times>/ _)" [21, 20] 20)
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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where
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  "Pair a b = Abs_prod (Pair_Rep a b)"
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rep_datatype (prod) Pair proof -
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  fix P :: "'a \<times> 'b \<Rightarrow> bool" and p
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  assume "\<And>a b. P (Pair a b)"
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  then show "P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
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next
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  fix a c :: 'a and b d :: 'b
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  have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d"
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    by (auto simp add: Pair_Rep_def expand_fun_eq)
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  moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod"
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    by (auto simp add: prod_def)
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  ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"
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    by (simp add: Pair_def Abs_prod_inject)
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qed
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subsubsection {* Tuple syntax *}
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definition split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
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  split_prod_case: "split == prod_case"
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text {*
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  Patterns -- extends pre-defined type @{typ pttrn} used in
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  abstractions.
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*}
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nonterminals
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  tuple_args patterns
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syntax
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  "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
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  "_tuple_arg"  :: "'a => tuple_args"                   ("_")
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  "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
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  "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
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  ""            :: "pttrn => patterns"                  ("_")
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  "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
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translations
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  "(x, y)" == "CONST Pair x y"
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  "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
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  "%(x, y, zs). b" == "CONST split (%x (y, zs). b)"
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  "%(x, y). b" == "CONST split (%x y. b)"
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  "_abs (CONST Pair x y) t" => "%(x, y). t"
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  -- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
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     The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *}
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(*reconstruct pattern from (nested) splits, avoiding eta-contraction of body;
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  works best with enclosing "let", if "let" does not avoid eta-contraction*)
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print_translation {*
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let
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  fun split_tr' [Abs (x, T, t as (Abs abs))] =
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        (* split (%x y. t) => %(x,y) t *)
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        let
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          val (y, t') = atomic_abs_tr' abs;
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          val (x', t'') = atomic_abs_tr' (x, T, t');
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        in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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        end
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    | split_tr' [Abs (x, T, (s as Const (@{const_syntax split}, _) $ t))] =
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        (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
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        let
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          val Const (@{syntax_const "_abs"}, _) $
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            (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t];
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          val (x', t'') = atomic_abs_tr' (x, T, t');
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        in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x' $
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              (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''
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        end
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    | split_tr' [Const (@{const_syntax split}, _) $ t] =
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        (* split (split (%x y z. t)) => %((x, y), z). t *)
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        split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
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    | split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =
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        (* split (%pttrn z. t) => %(pttrn,z). t *)
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        let val (z, t) = atomic_abs_tr' abs in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t
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        end
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    | split_tr' _ = raise Match;
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in [(@{const_syntax split}, split_tr')] end
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*}
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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
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typed_print_translation {*
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let
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  fun split_guess_names_tr' _ T [Abs (x, _, Abs _)] = raise Match
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    | split_guess_names_tr' _ T [Abs (x, xT, t)] =
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        (case (head_of t) of
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          Const (@{const_syntax split}, _) => raise Match
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        | _ =>
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          let 
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            val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
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            val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
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            val (x', t'') = atomic_abs_tr' (x, xT, t');
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          in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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          end)
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    | split_guess_names_tr' _ T [t] =
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        (case head_of t of
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          Const (@{const_syntax split}, _) => raise Match
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        | _ =>
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          let
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            val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
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            val (y, t') = atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
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            val (x', t'') = atomic_abs_tr' ("x", xT, t');
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          in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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          end)
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    | split_guess_names_tr' _ _ _ = raise Match;
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in [(@{const_syntax split}, split_guess_names_tr')] end
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*}
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subsubsection {* Code generator setup *}
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lemma split_case_cert:
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  assumes "CASE \<equiv> split f"
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  shows "CASE (a, b) \<equiv> f a b"
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  using assms by (simp add: split_prod_case)
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setup {*
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  Code.add_case @{thm split_case_cert}
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*}
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code_type *
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  (SML infix 2 "*")
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  (OCaml infix 2 "*")
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  (Haskell "!((_),/ (_))")
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  (Scala "((_),/ (_))")
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code_const Pair
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  (SML "!((_),/ (_))")
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  (OCaml "!((_),/ (_))")
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  (Haskell "!((_),/ (_))")
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  (Scala "!((_),/ (_))")
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code_instance * :: eq
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  (Haskell -)
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code_const "eq_class.eq \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
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  (Haskell infixl 4 "==")
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types_code
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  "*"     ("(_ */ _)")
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attach (term_of) {*
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fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
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*}
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attach (test) {*
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fun gen_id_42 aG aT bG bT i =
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  let
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    val (x, t) = aG i;
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    val (y, u) = bG i
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  in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
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*}
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consts_code
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  "Pair"    ("(_,/ _)")
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setup {*
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let
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fun strip_abs_split 0 t = ([], t)
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  | strip_abs_split i (Abs (s, T, t)) =
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      let
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        val s' = Codegen.new_name t s;
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        val v = Free (s', T)
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      in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
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  | strip_abs_split i (u as Const (@{const_name split}, _) $ t) =
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      (case strip_abs_split (i+1) t of
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        (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
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      | _ => ([], u))
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  | strip_abs_split i t =
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      strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0));
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fun let_codegen thy defs dep thyname brack t gr =
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  (case strip_comb t of
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    (t1 as Const (@{const_name Let}, _), t2 :: t3 :: ts) =>
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    let
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      fun dest_let (l as Const (@{const_name Let}, _) $ t $ u) =
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          (case strip_abs_split 1 u of
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             ([p], u') => apfst (cons (p, t)) (dest_let u')
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           | _ => ([], l))
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        | dest_let t = ([], t);
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      fun mk_code (l, r) gr =
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        let
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          val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr;
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          val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1;
haftmann@37166
   336
        in ((pl, pr), gr2) end
haftmann@37166
   337
    in case dest_let (t1 $ t2 $ t3) of
haftmann@37166
   338
        ([], _) => NONE
haftmann@37166
   339
      | (ps, u) =>
haftmann@37166
   340
          let
haftmann@37166
   341
            val (qs, gr1) = fold_map mk_code ps gr;
haftmann@37166
   342
            val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
haftmann@37166
   343
            val (pargs, gr3) = fold_map
haftmann@37166
   344
              (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
haftmann@37166
   345
          in
haftmann@37166
   346
            SOME (Codegen.mk_app brack
haftmann@37166
   347
              (Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, flat
haftmann@37166
   348
                  (separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
haftmann@37166
   349
                    [Pretty.block [Codegen.str "val ", pl, Codegen.str " =",
haftmann@37166
   350
                       Pretty.brk 1, pr]]) qs))),
haftmann@37166
   351
                Pretty.brk 1, Codegen.str "in ", pu,
haftmann@37166
   352
                Pretty.brk 1, Codegen.str "end"])) pargs, gr3)
haftmann@37166
   353
          end
haftmann@37166
   354
    end
haftmann@37166
   355
  | _ => NONE);
haftmann@37166
   356
haftmann@37166
   357
fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of
haftmann@37166
   358
    (t1 as Const (@{const_name split}, _), t2 :: ts) =>
haftmann@37166
   359
      let
haftmann@37166
   360
        val ([p], u) = strip_abs_split 1 (t1 $ t2);
haftmann@37166
   361
        val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr;
haftmann@37166
   362
        val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
haftmann@37166
   363
        val (pargs, gr3) = fold_map
haftmann@37166
   364
          (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
haftmann@37166
   365
      in
haftmann@37166
   366
        SOME (Codegen.mk_app brack
haftmann@37166
   367
          (Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>",
haftmann@37166
   368
            Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2)
haftmann@37166
   369
      end
haftmann@37166
   370
  | _ => NONE);
haftmann@37166
   371
haftmann@37166
   372
in
haftmann@37166
   373
haftmann@37166
   374
  Codegen.add_codegen "let_codegen" let_codegen
haftmann@37166
   375
  #> Codegen.add_codegen "split_codegen" split_codegen
haftmann@37166
   376
haftmann@37166
   377
end
haftmann@37166
   378
*}
haftmann@37166
   379
haftmann@37166
   380
haftmann@37166
   381
subsubsection {* Fundamental operations and properties *}
wenzelm@11838
   382
haftmann@26358
   383
lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@37166
   384
  by (cases p) simp
nipkow@10213
   385
haftmann@37389
   386
definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where
haftmann@37389
   387
  "fst p = (case p of (a, b) \<Rightarrow> a)"
wenzelm@11838
   388
haftmann@37389
   389
definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where
haftmann@37389
   390
  "snd p = (case p of (a, b) \<Rightarrow> b)"
wenzelm@11838
   391
haftmann@22886
   392
lemma fst_conv [simp, code]: "fst (a, b) = a"
haftmann@37166
   393
  unfolding fst_def by simp
wenzelm@11838
   394
haftmann@22886
   395
lemma snd_conv [simp, code]: "snd (a, b) = b"
haftmann@37166
   396
  unfolding snd_def by simp
oheimb@11025
   397
haftmann@37166
   398
code_const fst and snd
haftmann@37166
   399
  (Haskell "fst" and "snd")
haftmann@26358
   400
haftmann@37166
   401
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
haftmann@37166
   402
  by (simp add: expand_fun_eq split: prod.split)
haftmann@26358
   403
wenzelm@11838
   404
lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838
   405
  by simp
wenzelm@11838
   406
wenzelm@11838
   407
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   408
  by simp
wenzelm@11838
   409
haftmann@26358
   410
lemma pair_collapse [simp]: "(fst p, snd p) = p"
wenzelm@11838
   411
  by (cases p) simp
wenzelm@11838
   412
haftmann@26358
   413
lemmas surjective_pairing = pair_collapse [symmetric]
wenzelm@11838
   414
haftmann@37166
   415
lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166
   416
  by (cases s, cases t) simp
haftmann@37166
   417
haftmann@37166
   418
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
haftmann@37166
   419
  by (simp add: Pair_fst_snd_eq)
haftmann@37166
   420
haftmann@37166
   421
lemma split_conv [simp, code]: "split f (a, b) = f a b"
haftmann@37166
   422
  by (simp add: split_prod_case)
haftmann@37166
   423
haftmann@37166
   424
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@37166
   425
  by (rule split_conv [THEN iffD2])
haftmann@37166
   426
haftmann@37166
   427
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@37166
   428
  by (rule split_conv [THEN iffD1])
haftmann@37166
   429
haftmann@37166
   430
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
haftmann@37166
   431
  by (simp add: split_prod_case expand_fun_eq split: prod.split)
haftmann@37166
   432
haftmann@37166
   433
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@37166
   434
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
haftmann@37166
   435
  by (simp add: split_prod_case expand_fun_eq split: prod.split)
haftmann@37166
   436
haftmann@37166
   437
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@37166
   438
  by (cases x) simp
haftmann@37166
   439
haftmann@37166
   440
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@37166
   441
  by (cases p) simp
haftmann@37166
   442
haftmann@37166
   443
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
haftmann@37166
   444
  by (simp add: split_prod_case prod_case_unfold)
haftmann@37166
   445
haftmann@37166
   446
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
haftmann@37166
   447
  -- {* Prevents simplification of @{term c}: much faster *}
haftmann@37166
   448
  by (erule arg_cong)
haftmann@37166
   449
haftmann@37166
   450
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@37166
   451
  by (simp add: split_eta)
haftmann@37166
   452
wenzelm@11838
   453
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   454
proof
wenzelm@11820
   455
  fix a b
wenzelm@11820
   456
  assume "!!x. PROP P x"
wenzelm@19535
   457
  then show "PROP P (a, b)" .
wenzelm@11820
   458
next
wenzelm@11820
   459
  fix x
wenzelm@11820
   460
  assume "!!a b. PROP P (a, b)"
wenzelm@19535
   461
  from `PROP P (fst x, snd x)` show "PROP P x" by simp
wenzelm@11820
   462
qed
wenzelm@11820
   463
wenzelm@11838
   464
text {*
wenzelm@11838
   465
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   466
  Simplifier because it also affects premises in congrence rules,
wenzelm@11838
   467
  where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838
   468
  ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@11838
   469
*}
wenzelm@11838
   470
haftmann@26358
   471
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   472
wenzelm@26480
   473
ML {*
wenzelm@11838
   474
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   475
  val safe_full_simp_tac = generic_simp_tac true (true, false, false);
wenzelm@11838
   476
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@16121
   477
    fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
wenzelm@11838
   478
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   479
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   480
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   481
      | exists_paired_all _ = false;
wenzelm@11838
   482
    val ss = HOL_basic_ss
wenzelm@26340
   483
      addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@11838
   484
      addsimprocs [unit_eq_proc];
wenzelm@11838
   485
  in
wenzelm@11838
   486
    val split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   487
      if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   488
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   489
      if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838
   490
    fun split_all th =
wenzelm@26340
   491
   if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
wenzelm@11838
   492
  end;
wenzelm@26340
   493
*}
wenzelm@11838
   494
wenzelm@26340
   495
declaration {* fn _ =>
wenzelm@26340
   496
  Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
wenzelm@16121
   497
*}
wenzelm@11838
   498
wenzelm@11838
   499
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   500
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   501
  by fast
wenzelm@11838
   502
haftmann@26358
   503
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   504
  by fast
haftmann@26358
   505
wenzelm@11838
   506
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   507
  -- {* Can't be added to simpset: loops! *}
haftmann@26358
   508
  by (simp add: split_eta)
wenzelm@11838
   509
wenzelm@11838
   510
text {*
wenzelm@11838
   511
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   512
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   513
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   514
  existing proofs very inefficient; similarly for @{text
haftmann@26358
   515
  split_beta}.
haftmann@26358
   516
*}
wenzelm@11838
   517
wenzelm@26480
   518
ML {*
wenzelm@11838
   519
local
wenzelm@35364
   520
  val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta};
wenzelm@35364
   521
  fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364
   522
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364
   523
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364
   524
    | Pair_pat _ _ _ = false;
wenzelm@35364
   525
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364
   526
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364
   527
    | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364
   528
    | no_args _ _ _ = true;
wenzelm@35364
   529
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
wenzelm@35364
   530
    | split_pat tp i (Const (@{const_name split}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364
   531
    | split_pat tp i _ = NONE;
wenzelm@20044
   532
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@35364
   533
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
wenzelm@18328
   534
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
wenzelm@11838
   535
wenzelm@35364
   536
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364
   537
    | beta_term_pat k i (t $ u) =
wenzelm@35364
   538
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364
   539
    | beta_term_pat k i t = no_args k i t;
wenzelm@35364
   540
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364
   541
    | eta_term_pat _ _ _ = false;
wenzelm@11838
   542
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364
   543
    | subst arg k i (t $ u) =
wenzelm@35364
   544
        if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364
   545
        else (subst arg k i t $ subst arg k i u)
wenzelm@35364
   546
    | subst arg k i t = t;
wenzelm@35364
   547
  fun beta_proc ss (s as Const (@{const_name split}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   548
        (case split_pat beta_term_pat 1 t of
wenzelm@35364
   549
          SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531
   550
        | NONE => NONE)
wenzelm@35364
   551
    | beta_proc _ _ = NONE;
wenzelm@35364
   552
  fun eta_proc ss (s as Const (@{const_name split}, _) $ Abs (_, _, t)) =
wenzelm@11838
   553
        (case split_pat eta_term_pat 1 t of
wenzelm@35364
   554
          SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531
   555
        | NONE => NONE)
wenzelm@35364
   556
    | eta_proc _ _ = NONE;
wenzelm@11838
   557
in
wenzelm@32010
   558
  val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc);
wenzelm@32010
   559
  val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc);
wenzelm@11838
   560
end;
wenzelm@11838
   561
wenzelm@11838
   562
Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838
   563
*}
wenzelm@11838
   564
berghofe@26798
   565
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   566
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   567
blanchet@35828
   568
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838
   569
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   570
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   571
wenzelm@11838
   572
text {*
wenzelm@11838
   573
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   574
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   575
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   576
  current goal contains one of those constants.
wenzelm@11838
   577
*}
wenzelm@11838
   578
blanchet@35828
   579
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   580
by (subst split_split, simp)
wenzelm@11838
   581
wenzelm@11838
   582
text {*
wenzelm@11838
   583
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   584
wenzelm@11838
   585
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@11838
   586
  call @{text simp} using @{thm [source] split} as rewrite. *}
wenzelm@11838
   587
wenzelm@11838
   588
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   589
  apply (simp only: split_tupled_all)
wenzelm@11838
   590
  apply (simp (no_asm_simp))
wenzelm@11838
   591
  done
wenzelm@11838
   592
wenzelm@11838
   593
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   594
  apply (simp only: split_tupled_all)
wenzelm@11838
   595
  apply (simp (no_asm_simp))
wenzelm@11838
   596
  done
wenzelm@11838
   597
wenzelm@11838
   598
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37166
   599
  by (induct p) (auto simp add: split_prod_case)
wenzelm@11838
   600
wenzelm@11838
   601
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37166
   602
  by (induct p) (auto simp add: split_prod_case)
wenzelm@11838
   603
wenzelm@11838
   604
lemma splitE2:
wenzelm@11838
   605
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   606
proof -
wenzelm@11838
   607
  assume q: "Q (split P z)"
wenzelm@11838
   608
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   609
  show R
wenzelm@11838
   610
    apply (rule r surjective_pairing)+
wenzelm@11838
   611
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   612
    done
wenzelm@11838
   613
qed
wenzelm@11838
   614
wenzelm@11838
   615
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   616
  by simp
wenzelm@11838
   617
wenzelm@11838
   618
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   619
  by simp
wenzelm@11838
   620
wenzelm@11838
   621
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   622
by (simp only: split_tupled_all, simp)
wenzelm@11838
   623
wenzelm@18372
   624
lemma mem_splitE:
haftmann@37166
   625
  assumes major: "z \<in> split c p"
haftmann@37166
   626
    and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"
wenzelm@18372
   627
  shows Q
haftmann@37166
   628
  by (rule major [unfolded split_prod_case prod_case_unfold] cases surjective_pairing)+
wenzelm@11838
   629
wenzelm@11838
   630
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   631
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   632
wenzelm@26340
   633
ML {*
wenzelm@11838
   634
local (* filtering with exists_p_split is an essential optimization *)
wenzelm@35364
   635
  fun exists_p_split (Const (@{const_name split},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838
   636
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   637
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   638
    | exists_p_split _ = false;
wenzelm@35364
   639
  val ss = HOL_basic_ss addsimps @{thms split_conv};
wenzelm@11838
   640
in
wenzelm@11838
   641
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   642
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   643
end;
wenzelm@26340
   644
*}
wenzelm@26340
   645
wenzelm@11838
   646
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   647
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@26340
   648
declaration {* fn _ =>
wenzelm@26340
   649
  Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
wenzelm@16121
   650
*}
wenzelm@11838
   651
blanchet@35828
   652
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   653
  by (rule ext) fast
wenzelm@11838
   654
blanchet@35828
   655
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   656
  by (rule ext) fast
wenzelm@11838
   657
wenzelm@11838
   658
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   659
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   660
  by (rule ext) blast
wenzelm@11838
   661
nipkow@14337
   662
(* Do NOT make this a simp rule as it
nipkow@14337
   663
   a) only helps in special situations
nipkow@14337
   664
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   665
*)
nipkow@14337
   666
lemma split_comp_eq: 
paulson@20415
   667
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   668
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   669
  by (rule ext) auto
oheimb@14101
   670
haftmann@26358
   671
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   672
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   673
   apply auto
haftmann@26358
   674
  done
haftmann@26358
   675
wenzelm@11838
   676
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   677
  by blast
wenzelm@11838
   678
wenzelm@11838
   679
(*
wenzelm@11838
   680
the following  would be slightly more general,
wenzelm@11838
   681
but cannot be used as rewrite rule:
wenzelm@11838
   682
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   683
### ?y = .x
wenzelm@11838
   684
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   685
by (rtac some_equality 1)
paulson@14208
   686
by ( Simp_tac 1)
paulson@14208
   687
by (split_all_tac 1)
paulson@14208
   688
by (Asm_full_simp_tac 1)
wenzelm@11838
   689
qed "The_split_eq";
wenzelm@11838
   690
*)
wenzelm@11838
   691
wenzelm@11838
   692
text {*
wenzelm@11838
   693
  Setup of internal @{text split_rule}.
wenzelm@11838
   694
*}
wenzelm@11838
   695
haftmann@24699
   696
lemmas prod_caseI = prod.cases [THEN iffD2, standard]
haftmann@24699
   697
haftmann@24699
   698
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@24699
   699
  by auto
haftmann@24699
   700
haftmann@24699
   701
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@24699
   702
  by (auto simp: split_tupled_all)
haftmann@24699
   703
haftmann@24699
   704
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@24699
   705
  by (induct p) auto
haftmann@24699
   706
haftmann@24699
   707
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@24699
   708
  by (induct p) auto
haftmann@24699
   709
haftmann@24699
   710
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
haftmann@24699
   711
declare prod_caseE' [elim!] prod_caseE [elim!]
haftmann@24699
   712
haftmann@24844
   713
lemma prod_case_split:
haftmann@24699
   714
  "prod_case = split"
haftmann@24699
   715
  by (auto simp add: expand_fun_eq)
haftmann@24699
   716
bulwahn@26143
   717
lemma prod_case_beta:
bulwahn@26143
   718
  "prod_case f p = f (fst p) (snd p)"
bulwahn@26143
   719
  unfolding prod_case_split split_beta ..
bulwahn@26143
   720
haftmann@24699
   721
lemma prod_cases3 [cases type]:
haftmann@24699
   722
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   723
  by (cases y, case_tac b) blast
haftmann@24699
   724
haftmann@24699
   725
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   726
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   727
  by (cases x) blast
haftmann@24699
   728
haftmann@24699
   729
lemma prod_cases4 [cases type]:
haftmann@24699
   730
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   731
  by (cases y, case_tac c) blast
haftmann@24699
   732
haftmann@24699
   733
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   734
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   735
  by (cases x) blast
haftmann@24699
   736
haftmann@24699
   737
lemma prod_cases5 [cases type]:
haftmann@24699
   738
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   739
  by (cases y, case_tac d) blast
haftmann@24699
   740
haftmann@24699
   741
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   742
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   743
  by (cases x) blast
haftmann@24699
   744
haftmann@24699
   745
lemma prod_cases6 [cases type]:
haftmann@24699
   746
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   747
  by (cases y, case_tac e) blast
haftmann@24699
   748
haftmann@24699
   749
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   750
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   751
  by (cases x) blast
haftmann@24699
   752
haftmann@24699
   753
lemma prod_cases7 [cases type]:
haftmann@24699
   754
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   755
  by (cases y, case_tac f) blast
haftmann@24699
   756
haftmann@24699
   757
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   758
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   759
  by (cases x) blast
haftmann@24699
   760
haftmann@37166
   761
lemma split_def:
haftmann@37166
   762
  "split = (\<lambda>c p. c (fst p) (snd p))"
haftmann@37166
   763
  unfolding split_prod_case prod_case_unfold ..
haftmann@37166
   764
haftmann@37166
   765
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@37166
   766
  "internal_split == split"
haftmann@37166
   767
haftmann@37166
   768
lemma internal_split_conv: "internal_split c (a, b) = c a b"
haftmann@37166
   769
  by (simp only: internal_split_def split_conv)
haftmann@37166
   770
haftmann@37166
   771
use "Tools/split_rule.ML"
haftmann@37166
   772
setup Split_Rule.setup
haftmann@37166
   773
haftmann@37166
   774
hide_const internal_split
haftmann@37166
   775
haftmann@24699
   776
haftmann@26358
   777
subsubsection {* Derived operations *}
haftmann@26358
   778
haftmann@37387
   779
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387
   780
  "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166
   781
haftmann@37166
   782
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166
   783
  by (simp add: curry_def)
haftmann@37166
   784
haftmann@37166
   785
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166
   786
  by (simp add: curry_def)
haftmann@37166
   787
haftmann@37166
   788
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166
   789
  by (simp add: curry_def)
haftmann@37166
   790
haftmann@37166
   791
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166
   792
  by (simp add: curry_def)
haftmann@37166
   793
haftmann@37166
   794
lemma curry_split [simp]: "curry (split f) = f"
haftmann@37166
   795
  by (simp add: curry_def split_def)
haftmann@37166
   796
haftmann@37166
   797
lemma split_curry [simp]: "split (curry f) = f"
haftmann@37166
   798
  by (simp add: curry_def split_def)
haftmann@37166
   799
haftmann@26358
   800
text {*
haftmann@26358
   801
  The composition-uncurry combinator.
haftmann@26358
   802
*}
haftmann@26358
   803
haftmann@26588
   804
notation fcomp (infixl "o>" 60)
haftmann@26358
   805
haftmann@37166
   806
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60) where
haftmann@26588
   807
  "f o\<rightarrow> g = (\<lambda>x. split g (f x))"
haftmann@26358
   808
haftmann@26588
   809
lemma scomp_apply:  "(f o\<rightarrow> g) x = split g (f x)"
haftmann@26588
   810
  by (simp add: scomp_def)
haftmann@26358
   811
haftmann@26588
   812
lemma Pair_scomp: "Pair x o\<rightarrow> f = f x"
haftmann@26588
   813
  by (simp add: expand_fun_eq scomp_apply)
haftmann@26358
   814
haftmann@26588
   815
lemma scomp_Pair: "x o\<rightarrow> Pair = x"
haftmann@26588
   816
  by (simp add: expand_fun_eq scomp_apply)
haftmann@26358
   817
haftmann@26588
   818
lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)"
haftmann@26588
   819
  by (simp add: expand_fun_eq split_twice scomp_def)
haftmann@26358
   820
haftmann@26588
   821
lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)"
haftmann@26588
   822
  by (simp add: expand_fun_eq scomp_apply fcomp_def split_def)
haftmann@26358
   823
haftmann@26588
   824
lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)"
haftmann@26588
   825
  by (simp add: expand_fun_eq scomp_apply fcomp_apply)
haftmann@26358
   826
haftmann@31202
   827
code_const scomp
haftmann@31202
   828
  (Eval infixl 3 "#->")
haftmann@31202
   829
haftmann@26588
   830
no_notation fcomp (infixl "o>" 60)
haftmann@26588
   831
no_notation scomp (infixl "o\<rightarrow>" 60)
haftmann@26358
   832
haftmann@26358
   833
text {*
haftmann@26358
   834
  @{term prod_fun} --- action of the product functor upon
krauss@36664
   835
  functions.
haftmann@26358
   836
*}
haftmann@21195
   837
haftmann@26358
   838
definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
haftmann@28562
   839
  [code del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   840
haftmann@28562
   841
lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)"
haftmann@26358
   842
  by (simp add: prod_fun_def)
haftmann@26358
   843
nipkow@37278
   844
lemma fst_prod_fun[simp]: "fst (prod_fun f g x) = f (fst x)"
nipkow@37278
   845
by (cases x, auto)
nipkow@37278
   846
nipkow@37278
   847
lemma snd_prod_fun[simp]: "snd (prod_fun f g x) = g (snd x)"
nipkow@37278
   848
by (cases x, auto)
nipkow@37278
   849
nipkow@37278
   850
lemma fst_comp_prod_fun[simp]: "fst \<circ> prod_fun f g = f \<circ> fst"
nipkow@37278
   851
by (rule ext) auto
nipkow@37278
   852
nipkow@37278
   853
lemma snd_comp_prod_fun[simp]: "snd \<circ> prod_fun f g = g \<circ> snd"
nipkow@37278
   854
by (rule ext) auto
nipkow@37278
   855
nipkow@37278
   856
nipkow@37278
   857
lemma prod_fun_compose:
nipkow@37278
   858
  "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
nipkow@37278
   859
by (rule ext) auto
haftmann@26358
   860
haftmann@26358
   861
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
haftmann@26358
   862
  by (rule ext) auto
haftmann@26358
   863
haftmann@26358
   864
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
haftmann@26358
   865
  apply (rule image_eqI)
haftmann@26358
   866
  apply (rule prod_fun [symmetric], assumption)
haftmann@26358
   867
  done
haftmann@21195
   868
haftmann@26358
   869
lemma prod_fun_imageE [elim!]:
haftmann@26358
   870
  assumes major: "c: (prod_fun f g)`r"
haftmann@26358
   871
    and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
haftmann@26358
   872
  shows P
haftmann@26358
   873
  apply (rule major [THEN imageE])
haftmann@37166
   874
  apply (case_tac x)
haftmann@26358
   875
  apply (rule cases)
haftmann@26358
   876
   apply (blast intro: prod_fun)
haftmann@26358
   877
  apply blast
haftmann@26358
   878
  done
haftmann@26358
   879
nipkow@37278
   880
haftmann@37166
   881
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
haftmann@37166
   882
  "apfst f = prod_fun f id"
haftmann@26358
   883
haftmann@37166
   884
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
haftmann@37166
   885
  "apsnd f = prod_fun id f"
haftmann@26358
   886
haftmann@26358
   887
lemma apfst_conv [simp, code]:
haftmann@26358
   888
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   889
  by (simp add: apfst_def)
haftmann@26358
   890
hoelzl@33638
   891
lemma apsnd_conv [simp, code]:
haftmann@26358
   892
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   893
  by (simp add: apsnd_def)
haftmann@21195
   894
haftmann@33594
   895
lemma fst_apfst [simp]:
haftmann@33594
   896
  "fst (apfst f x) = f (fst x)"
haftmann@33594
   897
  by (cases x) simp
haftmann@33594
   898
haftmann@33594
   899
lemma fst_apsnd [simp]:
haftmann@33594
   900
  "fst (apsnd f x) = fst x"
haftmann@33594
   901
  by (cases x) simp
haftmann@33594
   902
haftmann@33594
   903
lemma snd_apfst [simp]:
haftmann@33594
   904
  "snd (apfst f x) = snd x"
haftmann@33594
   905
  by (cases x) simp
haftmann@33594
   906
haftmann@33594
   907
lemma snd_apsnd [simp]:
haftmann@33594
   908
  "snd (apsnd f x) = f (snd x)"
haftmann@33594
   909
  by (cases x) simp
haftmann@33594
   910
haftmann@33594
   911
lemma apfst_compose:
haftmann@33594
   912
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594
   913
  by (cases x) simp
haftmann@33594
   914
haftmann@33594
   915
lemma apsnd_compose:
haftmann@33594
   916
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594
   917
  by (cases x) simp
haftmann@33594
   918
haftmann@33594
   919
lemma apfst_apsnd [simp]:
haftmann@33594
   920
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594
   921
  by (cases x) simp
haftmann@33594
   922
haftmann@33594
   923
lemma apsnd_apfst [simp]:
haftmann@33594
   924
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594
   925
  by (cases x) simp
haftmann@33594
   926
haftmann@33594
   927
lemma apfst_id [simp] :
haftmann@33594
   928
  "apfst id = id"
haftmann@33594
   929
  by (simp add: expand_fun_eq)
haftmann@33594
   930
haftmann@33594
   931
lemma apsnd_id [simp] :
haftmann@33594
   932
  "apsnd id = id"
haftmann@33594
   933
  by (simp add: expand_fun_eq)
haftmann@33594
   934
haftmann@33594
   935
lemma apfst_eq_conv [simp]:
haftmann@33594
   936
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594
   937
  by (cases x) simp
haftmann@33594
   938
haftmann@33594
   939
lemma apsnd_eq_conv [simp]:
haftmann@33594
   940
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594
   941
  by (cases x) simp
haftmann@33594
   942
hoelzl@33638
   943
lemma apsnd_apfst_commute:
hoelzl@33638
   944
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638
   945
  by simp
haftmann@21195
   946
haftmann@26358
   947
text {*
haftmann@26358
   948
  Disjoint union of a family of sets -- Sigma.
haftmann@26358
   949
*}
haftmann@26358
   950
haftmann@26358
   951
definition  Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where
haftmann@26358
   952
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
   953
haftmann@26358
   954
abbreviation
haftmann@26358
   955
  Times :: "['a set, 'b set] => ('a * 'b) set"
haftmann@26358
   956
    (infixr "<*>" 80) where
haftmann@26358
   957
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
   958
haftmann@26358
   959
notation (xsymbols)
haftmann@26358
   960
  Times  (infixr "\<times>" 80)
berghofe@15394
   961
haftmann@26358
   962
notation (HTML output)
haftmann@26358
   963
  Times  (infixr "\<times>" 80)
haftmann@26358
   964
haftmann@26358
   965
syntax
wenzelm@35115
   966
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
   967
translations
wenzelm@35115
   968
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358
   969
haftmann@26358
   970
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
   971
  by (unfold Sigma_def) blast
haftmann@26358
   972
haftmann@26358
   973
lemma SigmaE [elim!]:
haftmann@26358
   974
    "[| c: Sigma A B;
haftmann@26358
   975
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
   976
     |] ==> P"
haftmann@26358
   977
  -- {* The general elimination rule. *}
haftmann@26358
   978
  by (unfold Sigma_def) blast
haftmann@20588
   979
haftmann@26358
   980
text {*
haftmann@26358
   981
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
   982
  eigenvariables.
haftmann@26358
   983
*}
haftmann@26358
   984
haftmann@26358
   985
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
   986
  by blast
haftmann@26358
   987
haftmann@26358
   988
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
   989
  by blast
haftmann@26358
   990
haftmann@26358
   991
lemma SigmaE2:
haftmann@26358
   992
    "[| (a, b) : Sigma A B;
haftmann@26358
   993
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
   994
     |] ==> P"
haftmann@26358
   995
  by blast
haftmann@20588
   996
haftmann@26358
   997
lemma Sigma_cong:
haftmann@26358
   998
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
   999
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
  1000
  by auto
haftmann@26358
  1001
haftmann@26358
  1002
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
  1003
  by blast
haftmann@26358
  1004
haftmann@26358
  1005
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
  1006
  by blast
haftmann@26358
  1007
haftmann@26358
  1008
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
  1009
  by blast
haftmann@26358
  1010
haftmann@26358
  1011
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
  1012
  by auto
haftmann@21908
  1013
haftmann@26358
  1014
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
  1015
  by auto
haftmann@26358
  1016
haftmann@26358
  1017
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
  1018
  by auto
haftmann@26358
  1019
haftmann@26358
  1020
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
  1021
  by blast
haftmann@26358
  1022
haftmann@26358
  1023
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
  1024
  by blast
haftmann@26358
  1025
haftmann@26358
  1026
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
  1027
  by (blast elim: equalityE)
haftmann@20588
  1028
haftmann@26358
  1029
lemma SetCompr_Sigma_eq:
haftmann@26358
  1030
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
  1031
  by blast
haftmann@26358
  1032
haftmann@26358
  1033
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358
  1034
  by blast
haftmann@26358
  1035
haftmann@26358
  1036
lemma UN_Times_distrib:
haftmann@26358
  1037
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358
  1038
  -- {* Suggested by Pierre Chartier *}
haftmann@26358
  1039
  by blast
haftmann@26358
  1040
blanchet@35828
  1041
lemma split_paired_Ball_Sigma [simp,no_atp]:
haftmann@26358
  1042
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358
  1043
  by blast
haftmann@26358
  1044
blanchet@35828
  1045
lemma split_paired_Bex_Sigma [simp,no_atp]:
haftmann@26358
  1046
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358
  1047
  by blast
haftmann@21908
  1048
haftmann@26358
  1049
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358
  1050
  by blast
haftmann@26358
  1051
haftmann@26358
  1052
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
haftmann@26358
  1053
  by blast
haftmann@26358
  1054
haftmann@26358
  1055
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358
  1056
  by blast
haftmann@26358
  1057
haftmann@26358
  1058
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
haftmann@26358
  1059
  by blast
haftmann@26358
  1060
haftmann@26358
  1061
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358
  1062
  by blast
haftmann@26358
  1063
haftmann@26358
  1064
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358
  1065
  by blast
haftmann@21908
  1066
haftmann@26358
  1067
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358
  1068
  by blast
haftmann@26358
  1069
haftmann@26358
  1070
text {*
haftmann@26358
  1071
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
  1072
  matching, especially when the rules are re-oriented.
haftmann@26358
  1073
*}
haftmann@21908
  1074
haftmann@26358
  1075
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
nipkow@28719
  1076
by blast
haftmann@26358
  1077
haftmann@26358
  1078
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
nipkow@28719
  1079
by blast
haftmann@26358
  1080
haftmann@26358
  1081
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
nipkow@28719
  1082
by blast
haftmann@26358
  1083
hoelzl@36622
  1084
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622
  1085
  by auto
hoelzl@36622
  1086
hoelzl@36622
  1087
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
hoelzl@36622
  1088
  by (auto intro!: image_eqI)
hoelzl@36622
  1089
hoelzl@36622
  1090
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
hoelzl@36622
  1091
  by (auto intro!: image_eqI)
hoelzl@36622
  1092
nipkow@28719
  1093
lemma insert_times_insert[simp]:
nipkow@28719
  1094
  "insert a A \<times> insert b B =
nipkow@28719
  1095
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
nipkow@28719
  1096
by blast
haftmann@26358
  1097
paulson@33271
  1098
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
haftmann@37166
  1099
  by (auto, case_tac "f x", auto)
paulson@33271
  1100
nipkow@37278
  1101
text{* The following @{const prod_fun} lemmas are due to Joachim Breitner: *}
nipkow@37278
  1102
nipkow@37278
  1103
lemma prod_fun_inj_on:
nipkow@37278
  1104
  assumes "inj_on f A" and "inj_on g B"
nipkow@37278
  1105
  shows "inj_on (prod_fun f g) (A \<times> B)"
nipkow@37278
  1106
proof (rule inj_onI)
nipkow@37278
  1107
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
nipkow@37278
  1108
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
nipkow@37278
  1109
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
nipkow@37278
  1110
  assume "prod_fun f g x = prod_fun f g y"
nipkow@37278
  1111
  hence "fst (prod_fun f g x) = fst (prod_fun f g y)" by (auto)
nipkow@37278
  1112
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
nipkow@37278
  1113
  with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
nipkow@37278
  1114
  have "fst x = fst y" by (auto dest:dest:inj_onD)
nipkow@37278
  1115
  moreover from `prod_fun f g x = prod_fun f g y`
nipkow@37278
  1116
  have "snd (prod_fun f g x) = snd (prod_fun f g y)" by (auto)
nipkow@37278
  1117
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
nipkow@37278
  1118
  with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
nipkow@37278
  1119
  have "snd x = snd y" by (auto dest:dest:inj_onD)
nipkow@37278
  1120
  ultimately show "x = y" by(rule prod_eqI)
nipkow@37278
  1121
qed
nipkow@37278
  1122
nipkow@37278
  1123
lemma prod_fun_surj:
nipkow@37278
  1124
  assumes "surj f" and "surj g"
nipkow@37278
  1125
  shows "surj (prod_fun f g)"
nipkow@37278
  1126
unfolding surj_def
nipkow@37278
  1127
proof
nipkow@37278
  1128
  fix y :: "'b \<times> 'd"
nipkow@37278
  1129
  from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
nipkow@37278
  1130
  moreover
nipkow@37278
  1131
  from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)
nipkow@37278
  1132
  ultimately have "(fst y, snd y) = prod_fun f g (a,b)" by auto
nipkow@37278
  1133
  thus "\<exists>x. y = prod_fun f g x" by auto
nipkow@37278
  1134
qed
nipkow@37278
  1135
nipkow@37278
  1136
lemma prod_fun_surj_on:
nipkow@37278
  1137
  assumes "f ` A = A'" and "g ` B = B'"
nipkow@37278
  1138
  shows "prod_fun f g ` (A \<times> B) = A' \<times> B'"
nipkow@37278
  1139
unfolding image_def
nipkow@37278
  1140
proof(rule set_ext,rule iffI)
nipkow@37278
  1141
  fix x :: "'a \<times> 'c"
nipkow@37278
  1142
  assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = prod_fun f g x}"
nipkow@37278
  1143
  then obtain y where "y \<in> A \<times> B" and "x = prod_fun f g y" by blast
nipkow@37278
  1144
  from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
nipkow@37278
  1145
  moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto
nipkow@37278
  1146
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
nipkow@37278
  1147
  with `x = prod_fun f g y` show "x \<in> A' \<times> B'" by (cases y, auto)
nipkow@37278
  1148
next
nipkow@37278
  1149
  fix x :: "'a \<times> 'c"
nipkow@37278
  1150
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
nipkow@37278
  1151
  from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto
nipkow@37278
  1152
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
nipkow@37278
  1153
  moreover from `image g B = B'` and `snd x \<in> B'`
nipkow@37278
  1154
  obtain b where "b \<in> B" and "snd x = g b" by auto
nipkow@37278
  1155
  ultimately have "(fst x, snd x) = prod_fun f g (a,b)" by auto
nipkow@37278
  1156
  moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto
nipkow@37278
  1157
  ultimately have "\<exists>y \<in> A \<times> B. x = prod_fun f g y" by auto
nipkow@37278
  1158
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = prod_fun f g y}" by auto
nipkow@37278
  1159
qed
nipkow@37278
  1160
haftmann@35822
  1161
lemma swap_inj_on:
hoelzl@36622
  1162
  "inj_on (\<lambda>(i, j). (j, i)) A"
hoelzl@36622
  1163
  by (auto intro!: inj_onI)
haftmann@35822
  1164
haftmann@35822
  1165
lemma swap_product:
haftmann@35822
  1166
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@35822
  1167
  by (simp add: split_def image_def) blast
haftmann@35822
  1168
hoelzl@36622
  1169
lemma image_split_eq_Sigma:
hoelzl@36622
  1170
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
hoelzl@36622
  1171
proof (safe intro!: imageI vimageI)
hoelzl@36622
  1172
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622
  1173
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622
  1174
    using * eq[symmetric] by auto
hoelzl@36622
  1175
qed simp_all
haftmann@35822
  1176
haftmann@21908
  1177
haftmann@37166
  1178
subsection {* Inductively defined sets *}
berghofe@15394
  1179
haftmann@37389
  1180
use "Tools/inductive_codegen.ML"
haftmann@37389
  1181
setup Inductive_Codegen.setup
haftmann@37389
  1182
haftmann@31723
  1183
use "Tools/inductive_set.ML"
haftmann@31723
  1184
setup Inductive_Set.setup
haftmann@24699
  1185
haftmann@37166
  1186
haftmann@37166
  1187
subsection {* Legacy theorem bindings and duplicates *}
haftmann@37166
  1188
haftmann@37166
  1189
lemma PairE:
haftmann@37166
  1190
  obtains x y where "p = (x, y)"
haftmann@37166
  1191
  by (fact prod.exhaust)
haftmann@37166
  1192
haftmann@37166
  1193
lemma Pair_inject:
haftmann@37166
  1194
  assumes "(a, b) = (a', b')"
haftmann@37166
  1195
    and "a = a' ==> b = b' ==> R"
haftmann@37166
  1196
  shows R
haftmann@37166
  1197
  using assms by simp
haftmann@37166
  1198
haftmann@37166
  1199
lemmas Pair_eq = prod.inject
haftmann@37166
  1200
haftmann@37166
  1201
lemmas split = split_conv  -- {* for backwards compatibility *}
haftmann@37166
  1202
nipkow@10213
  1203
end