src/HOL/Product_Type.thy
author haftmann
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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_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 {* Unit *}
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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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text {* code generator setup *}
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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_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_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_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 {* Pairs *}
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subsubsection {* Product type, basic operations and concrete syntax *}
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definition
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  Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
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where
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  "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
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global
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typedef (Prod)
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  ('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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consts
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  Pair     :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
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  fst      :: "'a \<times> 'b \<Rightarrow> 'a"
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  snd      :: "'a \<times> 'b \<Rightarrow> 'b"
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  split    :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
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  curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"
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local
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defs
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  Pair_def:     "Pair a b == Abs_Prod (Pair_Rep a b)"
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  fst_def:      "fst p == THE a. EX b. p = Pair a b"
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  snd_def:      "snd p == THE b. EX a. p = Pair a b"
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  split_def:    "split == (%c p. c (fst p) (snd p))"
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  curry_def:    "curry == (%c x y. c (Pair x y))"
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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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text {* Towards a datatype declaration *}
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lemma surj_pair [simp]: "EX x y. p = (x, y)"
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  apply (unfold Pair_def)
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  apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
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  apply (erule exE, erule exE, rule exI, rule exI)
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  apply (rule Rep_Prod_inverse [symmetric, THEN trans])
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  apply (erule arg_cong)
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  done
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lemma PairE [cases type: *]:
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  obtains x y where "p = (x, y)"
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  using surj_pair [of p] by blast
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lemma ProdI: "Pair_Rep a b \<in> Prod"
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  unfolding Prod_def by rule+
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lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'"
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  unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast
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lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
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  apply (rule inj_on_inverseI)
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  apply (erule Abs_Prod_inverse)
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  done
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lemma Pair_inject:
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  assumes "(a, b) = (a', b')"
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    and "a = a' ==> b = b' ==> R"
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  shows R
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  apply (insert prems [unfolded Pair_def])
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  apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
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  apply (assumption | rule ProdI)+
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  done
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rep_datatype (prod) Pair
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proof -
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  fix P p
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  assume "\<And>x y. P (x, y)"
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  then show "P p" by (cases p) simp
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qed (auto elim: Pair_inject)
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lemmas Pair_eq = prod.inject
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lemma fst_conv [simp, code]: "fst (a, b) = a"
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  unfolding fst_def by blast
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lemma snd_conv [simp, code]: "snd (a, b) = b"
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  unfolding snd_def by blast
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subsubsection {* Basic rules and proof tools *}
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lemma fst_eqD: "fst (x, y) = a ==> x = a"
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  by simp
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02d75712061d got rid of ML proof scripts for Product_Type;
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lemma snd_eqD: "snd (x, y) = a ==> y = a"
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  by simp
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   318
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lemma pair_collapse [simp]: "(fst p, snd p) = p"
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  by (cases p) simp
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   321
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   322
lemmas surjective_pairing = pair_collapse [symmetric]
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   323
02d75712061d got rid of ML proof scripts for Product_Type;
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   324
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
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proof
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  fix a b
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  assume "!!x. PROP P x"
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  then show "PROP P (a, b)" .
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   329
next
015a82d4ee96 proper proof of split_paired_all (presently unused);
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   330
  fix x
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   331
  assume "!!a b. PROP P (a, b)"
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  from `PROP P (fst x, snd x)` show "PROP P x" by simp
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015a82d4ee96 proper proof of split_paired_all (presently unused);
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   333
qed
015a82d4ee96 proper proof of split_paired_all (presently unused);
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   334
11838
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   335
text {*
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   336
  The rule @{thm [source] split_paired_all} does not work with the
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   337
  Simplifier because it also affects premises in congrence rules,
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   338
  where this can lead to premises of the form @{text "!!a b. ... =
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   339
  ?P(a, b)"} which cannot be solved by reflexivity.
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   340
*}
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   341
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   342
lemmas split_tupled_all = split_paired_all unit_all_eq2
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   343
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ML {*
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  (* replace parameters of product type by individual component parameters *)
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  val safe_full_simp_tac = generic_simp_tac true (true, false, false);
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  local (* filtering with exists_paired_all is an essential optimization *)
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    fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
11838
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   349
          can HOLogic.dest_prodT T orelse exists_paired_all t
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   350
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
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      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
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   352
      | exists_paired_all _ = false;
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   353
    val ss = HOL_basic_ss
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   354
      addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
11838
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   355
      addsimprocs [unit_eq_proc];
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   356
  in
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   357
    val split_all_tac = SUBGOAL (fn (t, i) =>
02d75712061d got rid of ML proof scripts for Product_Type;
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   358
      if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
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   359
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
02d75712061d got rid of ML proof scripts for Product_Type;
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   360
      if exists_paired_all t then full_simp_tac ss i else no_tac);
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   361
    fun split_all th =
26340
a85fe32e7b2f more antiquotations;
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diff changeset
   362
   if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
11838
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diff changeset
   363
  end;
26340
a85fe32e7b2f more antiquotations;
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diff changeset
   364
*}
11838
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diff changeset
   365
26340
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   366
declaration {* fn _ =>
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diff changeset
   367
  Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
16121
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   368
*}
11838
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   369
02d75712061d got rid of ML proof scripts for Product_Type;
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   370
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
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   371
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
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   372
  by fast
02d75712061d got rid of ML proof scripts for Product_Type;
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diff changeset
   373
26358
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   374
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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   375
  by fast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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   376
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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   377
lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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diff changeset
   378
  by (cases s, cases t) simp
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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diff changeset
   379
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
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diff changeset
   380
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   381
  by (simp add: Pair_fst_snd_eq)
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diff changeset
   382
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   383
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   384
subsubsection {* @{text split} and @{text curry} *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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diff changeset
   385
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4e74209f113e `code func` now just `code`
haftmann
parents: 28537
diff changeset
   386
lemma split_conv [simp, code]: "split f (a, b) = f a b"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
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diff changeset
   387
  by (simp add: split_def)
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haftmann
parents: 26340
diff changeset
   388
28562
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diff changeset
   389
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   390
  by (simp add: curry_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   391
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   392
lemmas split = split_conv  -- {* for backwards compatibility *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   393
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   394
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   395
  by (rule split_conv [THEN iffD2])
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   396
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   397
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   398
  by (rule split_conv [THEN iffD1])
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   399
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   400
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   401
  by (simp add: curry_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   402
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   403
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   404
  by (simp add: curry_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   405
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   406
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   407
  by (simp add: curry_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   408
14189
de58f4d939e1 Added the constant "curry".
skalberg
parents: 14101
diff changeset
   409
lemma curry_split [simp]: "curry (split f) = f"
de58f4d939e1 Added the constant "curry".
skalberg
parents: 14101
diff changeset
   410
  by (simp add: curry_def split_def)
de58f4d939e1 Added the constant "curry".
skalberg
parents: 14101
diff changeset
   411
de58f4d939e1 Added the constant "curry".
skalberg
parents: 14101
diff changeset
   412
lemma split_curry [simp]: "split (curry f) = f"
de58f4d939e1 Added the constant "curry".
skalberg
parents: 14101
diff changeset
   413
  by (simp add: curry_def split_def)
de58f4d939e1 Added the constant "curry".
skalberg
parents: 14101
diff changeset
   414
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   415
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   416
  by (simp add: split_def id_def)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   417
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   418
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
36664
6302f9ad7047 repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents: 36622
diff changeset
   419
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   420
  by (rule ext) auto
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   421
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   422
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   423
  by (cases x) simp
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   424
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   425
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   426
  unfolding split_def ..
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   427
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   428
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   429
  -- {* Can't be added to simpset: loops! *}
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   430
  by (simp add: split_eta)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   431
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   432
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   433
  by (simp add: split_def)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   434
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   435
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   436
  -- {* Prevents simplification of @{term c}: much faster *}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   437
  by (erule arg_cong)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   438
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   439
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   440
  by (simp add: split_eta)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   441
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   442
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   443
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   444
  @{thm [source] split_eta} as a rewrite rule is not general enough,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   445
  and using @{thm [source] cond_split_eta} directly would render some
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   446
  existing proofs very inefficient; similarly for @{text
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   447
  split_beta}.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   448
*}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   449
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26358
diff changeset
   450
ML {*
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   451
local
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   452
  val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta};
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   453
  fun Pair_pat k 0 (Bound m) = (m = k)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   454
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   455
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   456
    | Pair_pat _ _ _ = false;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   457
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   458
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   459
    | no_args k i (Bound m) = m < k orelse m > k + i
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   460
    | no_args _ _ _ = true;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   461
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   462
    | split_pat tp i (Const (@{const_name split}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   463
    | split_pat tp i _ = NONE;
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19656
diff changeset
   464
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   465
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18220
diff changeset
   466
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   467
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   468
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   469
    | beta_term_pat k i (t $ u) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   470
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   471
    | beta_term_pat k i t = no_args k i t;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   472
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   473
    | eta_term_pat _ _ _ = false;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   474
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   475
    | subst arg k i (t $ u) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   476
        if Pair_pat k i (t $ u) then incr_boundvars k arg
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   477
        else (subst arg k i t $ subst arg k i u)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   478
    | subst arg k i t = t;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   479
  fun beta_proc ss (s as Const (@{const_name split}, _) $ Abs (_, _, t) $ arg) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   480
        (case split_pat beta_term_pat 1 t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   481
          SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f))
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
   482
        | NONE => NONE)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   483
    | beta_proc _ _ = NONE;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   484
  fun eta_proc ss (s as Const (@{const_name split}, _) $ Abs (_, _, t)) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   485
        (case split_pat eta_term_pat 1 t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   486
          SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
   487
        | NONE => NONE)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   488
    | eta_proc _ _ = NONE;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   489
in
32010
cb1a1c94b4cd more antiquotations;
wenzelm
parents: 31775
diff changeset
   490
  val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc);
cb1a1c94b4cd more antiquotations;
wenzelm
parents: 31775
diff changeset
   491
  val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc);
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   492
end;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   493
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   494
Addsimprocs [split_beta_proc, split_eta_proc];
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   495
*}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   496
26798
a9134a089106 split_beta is now declared as monotonicity rule, to allow bounded
berghofe
parents: 26588
diff changeset
   497
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   498
  by (subst surjective_pairing, rule split_conv)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   499
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   500
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   501
  -- {* For use with @{text split} and the Simplifier. *}
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15422
diff changeset
   502
  by (insert surj_pair [of p], clarify, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   503
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   504
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   505
  @{thm [source] split_split} could be declared as @{text "[split]"}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   506
  done after the Splitter has been speeded up significantly;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   507
  precompute the constants involved and don't do anything unless the
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   508
  current goal contains one of those constants.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   509
*}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   510
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   511
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   512
by (subst split_split, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   513
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   514
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   515
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   516
  \medskip @{term split} used as a logical connective or set former.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   517
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   518
  \medskip These rules are for use with @{text blast}; could instead
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   519
  call @{text simp} using @{thm [source] split} as rewrite. *}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   520
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   521
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   522
  apply (simp only: split_tupled_all)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   523
  apply (simp (no_asm_simp))
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   524
  done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   525
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   526
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   527
  apply (simp only: split_tupled_all)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   528
  apply (simp (no_asm_simp))
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   529
  done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   530
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   531
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   532
  by (induct p) (auto simp add: split_def)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   533
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   534
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   535
  by (induct p) (auto simp add: split_def)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   536
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   537
lemma splitE2:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   538
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   539
proof -
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   540
  assume q: "Q (split P z)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   541
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   542
  show R
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   543
    apply (rule r surjective_pairing)+
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   544
    apply (rule split_beta [THEN subst], rule q)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   545
    done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   546
qed
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   547
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   548
lemma splitD': "split R (a,b) c ==> R a b c"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   549
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   550
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   551
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   552
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   553
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   554
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   555
by (simp only: split_tupled_all, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   556
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   557
lemma mem_splitE:
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   558
  assumes major: "z: split c p"
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   559
    and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   560
  shows Q
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   561
  by (rule major [unfolded split_def] cases surjective_pairing)+
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   562
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   563
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   564
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   565
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   566
ML {*
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   567
local (* filtering with exists_p_split is an essential optimization *)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   568
  fun exists_p_split (Const (@{const_name split},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   569
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   570
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   571
    | exists_p_split _ = false;
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   572
  val ss = HOL_basic_ss addsimps @{thms split_conv};
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   573
in
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   574
val split_conv_tac = SUBGOAL (fn (t, i) =>
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   575
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   576
end;
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   577
*}
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   578
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   579
(* This prevents applications of splitE for already splitted arguments leading
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   580
   to quite time-consuming computations (in particular for nested tuples) *)
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   581
declaration {* fn _ =>
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   582
  Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
16121
wenzelm
parents: 15570
diff changeset
   583
*}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   584
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   585
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   586
  by (rule ext) fast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   587
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   588
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   589
  by (rule ext) fast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   590
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   591
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   592
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   593
  by (rule ext) blast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   594
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   595
(* Do NOT make this a simp rule as it
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   596
   a) only helps in special situations
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   597
   b) can lead to nontermination in the presence of split_def
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   598
*)
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   599
lemma split_comp_eq: 
20415
e3d2d7b01279 explicit type variables prevent empty sorts
paulson
parents: 20380
diff changeset
   600
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
e3d2d7b01279 explicit type variables prevent empty sorts
paulson
parents: 20380
diff changeset
   601
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   602
  by (rule ext) auto
14101
d25c23e46173 added upd_fst, upd_snd, some thms
oheimb
parents: 13480
diff changeset
   603
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   604
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   605
  apply (rule_tac x = "(a, b)" in image_eqI)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   606
   apply auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   607
  done
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   608
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   609
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   610
  by blast
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   611
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   612
(*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   613
the following  would be slightly more general,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   614
but cannot be used as rewrite rule:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   615
### Cannot add premise as rewrite rule because it contains (type) unknowns:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   616
### ?y = .x
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   617
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   618
by (rtac some_equality 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   619
by ( Simp_tac 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   620
by (split_all_tac 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   621
by (Asm_full_simp_tac 1)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   622
qed "The_split_eq";
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   623
*)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   624
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   625
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   626
  Setup of internal @{text split_rule}.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   627
*}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   628
25511
54db9b5080b8 more canonical attribute application
haftmann
parents: 24844
diff changeset
   629
definition
54db9b5080b8 more canonical attribute application
haftmann
parents: 24844
diff changeset
   630
  internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
54db9b5080b8 more canonical attribute application
haftmann
parents: 24844
diff changeset
   631
where
11032
83f723e86dac added hidden internal_split constant;
wenzelm
parents: 11025
diff changeset
   632
  "internal_split == split"
83f723e86dac added hidden internal_split constant;
wenzelm
parents: 11025
diff changeset
   633
83f723e86dac added hidden internal_split constant;
wenzelm
parents: 11025
diff changeset
   634
lemma internal_split_conv: "internal_split c (a, b) = c a b"
83f723e86dac added hidden internal_split constant;
wenzelm
parents: 11025
diff changeset
   635
  by (simp only: internal_split_def split_conv)
83f723e86dac added hidden internal_split constant;
wenzelm
parents: 11025
diff changeset
   636
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   637
use "Tools/split_rule.ML"
35365
2fcd08c62495 modernized structure Split_Rule;
wenzelm
parents: 35364
diff changeset
   638
setup Split_Rule.setup
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   639
36176
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 35831
diff changeset
   640
hide_const internal_split
11032
83f723e86dac added hidden internal_split constant;
wenzelm
parents: 11025
diff changeset
   641
10213
01c2744a3786 *** empty log message ***
nipkow
parents:
diff changeset
   642
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   643
lemmas prod_caseI = prod.cases [THEN iffD2, standard]
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   644
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   645
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   646
  by auto
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   647
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   648
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   649
  by (auto simp: split_tupled_all)
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   650
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   651
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   652
  by (induct p) auto
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   653
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   654
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   655
  by (induct p) auto
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   656
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   657
lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   658
  by (simp add: expand_fun_eq)
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   659
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   660
declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   661
declare prod_caseE' [elim!] prod_caseE [elim!]
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   662
24844
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
   663
lemma prod_case_split:
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   664
  "prod_case = split"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   665
  by (auto simp add: expand_fun_eq)
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   666
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   667
lemma prod_case_beta:
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   668
  "prod_case f p = f (fst p) (snd p)"
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   669
  unfolding prod_case_split split_beta ..
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   670
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   671
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   672
subsection {* Further cases/induct rules for tuples *}
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   673
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   674
lemma prod_cases3 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   675
  obtains (fields) a b c where "y = (a, b, c)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   676
  by (cases y, case_tac b) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   677
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   678
lemma prod_induct3 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   679
    "(!!a b c. P (a, b, c)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   680
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   681
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   682
lemma prod_cases4 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   683
  obtains (fields) a b c d where "y = (a, b, c, d)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   684
  by (cases y, case_tac c) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   685
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   686
lemma prod_induct4 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   687
    "(!!a b c d. P (a, b, c, d)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   688
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   689
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   690
lemma prod_cases5 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   691
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   692
  by (cases y, case_tac d) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   693
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   694
lemma prod_induct5 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   695
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   696
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   697
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   698
lemma prod_cases6 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   699
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   700
  by (cases y, case_tac e) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   701
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   702
lemma prod_induct6 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   703
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   704
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   705
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   706
lemma prod_cases7 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   707
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   708
  by (cases y, case_tac f) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   709
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   710
lemma prod_induct7 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   711
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   712
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   713
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   714
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   715
subsubsection {* Derived operations *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   716
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   717
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   718
  The composition-uncurry combinator.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   719
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   720
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   721
notation fcomp (infixl "o>" 60)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   722
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   723
definition
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   724
  scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o\<rightarrow>" 60)
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   725
where
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   726
  "f o\<rightarrow> g = (\<lambda>x. split g (f x))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   727
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   728
lemma scomp_apply:  "(f o\<rightarrow> g) x = split g (f x)"
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   729
  by (simp add: scomp_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   730
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   731
lemma Pair_scomp: "Pair x o\<rightarrow> f = f x"
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   732
  by (simp add: expand_fun_eq scomp_apply)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   733
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   734
lemma scomp_Pair: "x o\<rightarrow> Pair = x"
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   735
  by (simp add: expand_fun_eq scomp_apply)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   736
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   737
lemma scomp_scomp: "(f o\<rightarrow> g) o\<rightarrow> h = f o\<rightarrow> (\<lambda>x. g x o\<rightarrow> h)"
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   738
  by (simp add: expand_fun_eq split_twice scomp_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   739
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   740
lemma scomp_fcomp: "(f o\<rightarrow> g) o> h = f o\<rightarrow> (\<lambda>x. g x o> h)"
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   741
  by (simp add: expand_fun_eq scomp_apply fcomp_def split_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   742
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   743
lemma fcomp_scomp: "(f o> g) o\<rightarrow> h = f o> (g o\<rightarrow> h)"
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   744
  by (simp add: expand_fun_eq scomp_apply fcomp_apply)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   745
31202
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 30924
diff changeset
   746
code_const scomp
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 30924
diff changeset
   747
  (Eval infixl 3 "#->")
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 30924
diff changeset
   748
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   749
no_notation fcomp (infixl "o>" 60)
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26480
diff changeset
   750
no_notation scomp (infixl "o\<rightarrow>" 60)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   751
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   752
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   753
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   754
  @{term prod_fun} --- action of the product functor upon
36664
6302f9ad7047 repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents: 36622
diff changeset
   755
  functions.
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   756
*}
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   757
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   758
definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28537
diff changeset
   759
  [code del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   760
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28537
diff changeset
   761
lemma prod_fun [simp, code]: "prod_fun f g (a, b) = (f a, g b)"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   762
  by (simp add: prod_fun_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   763
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   764
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   765
  by (rule ext) auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   766
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   767
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   768
  by (rule ext) auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   769
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   770
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   771
  apply (rule image_eqI)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   772
  apply (rule prod_fun [symmetric], assumption)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   773
  done
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   774
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   775
lemma prod_fun_imageE [elim!]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   776
  assumes major: "c: (prod_fun f g)`r"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   777
    and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   778
  shows P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   779
  apply (rule major [THEN imageE])
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   780
  apply (rule_tac p = x in PairE)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   781
  apply (rule cases)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   782
   apply (blast intro: prod_fun)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   783
  apply blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   784
  done
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   785
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   786
definition
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   787
  apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   788
where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28537
diff changeset
   789
  [code del]: "apfst f = prod_fun f id"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   790
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   791
definition
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   792
  apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   793
where
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28537
diff changeset
   794
  [code del]: "apsnd f = prod_fun id f"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   795
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   796
lemma apfst_conv [simp, code]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   797
  "apfst f (x, y) = (f x, y)" 
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   798
  by (simp add: apfst_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   799
33638
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   800
lemma apsnd_conv [simp, code]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   801
  "apsnd f (x, y) = (x, f y)" 
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   802
  by (simp add: apsnd_def)
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   803
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   804
lemma fst_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   805
  "fst (apfst f x) = f (fst x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   806
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   807
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   808
lemma fst_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   809
  "fst (apsnd f x) = fst x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   810
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   811
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   812
lemma snd_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   813
  "snd (apfst f x) = snd x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   814
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   815
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   816
lemma snd_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   817
  "snd (apsnd f x) = f (snd x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   818
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   819
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   820
lemma apfst_compose:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   821
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   822
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   823
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   824
lemma apsnd_compose:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   825
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   826
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   827
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   828
lemma apfst_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   829
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   830
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   831
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   832
lemma apsnd_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   833
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   834
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   835
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   836
lemma apfst_id [simp] :
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   837
  "apfst id = id"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   838
  by (simp add: expand_fun_eq)
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   839
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   840
lemma apsnd_id [simp] :
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   841
  "apsnd id = id"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   842
  by (simp add: expand_fun_eq)
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   843
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   844
lemma apfst_eq_conv [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   845
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   846
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   847
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   848
lemma apsnd_eq_conv [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   849
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   850
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   851
33638
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   852
lemma apsnd_apfst_commute:
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   853
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   854
  by simp
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   855
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   856
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   857
  Disjoint union of a family of sets -- Sigma.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   858
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   859
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   860
definition  Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   861
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   862
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   863
abbreviation
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   864
  Times :: "['a set, 'b set] => ('a * 'b) set"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   865
    (infixr "<*>" 80) where
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   866
  "A <*> B == Sigma A (%_. B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   867
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   868
notation (xsymbols)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   869
  Times  (infixr "\<times>" 80)
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
   870
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   871
notation (HTML output)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   872
  Times  (infixr "\<times>" 80)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   873
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   874
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34900
diff changeset
   875
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   876
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34900
diff changeset
   877
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   878
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   879
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   880
  by (unfold Sigma_def) blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   881
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   882
lemma SigmaE [elim!]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   883
    "[| c: Sigma A B;
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   884
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   885
     |] ==> P"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   886
  -- {* The general elimination rule. *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   887
  by (unfold Sigma_def) blast
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   888
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   889
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   890
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   891
  eigenvariables.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   892
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   893
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   894
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   895
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   896
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   897
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   898
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   899
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   900
lemma SigmaE2:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   901
    "[| (a, b) : Sigma A B;
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   902
        [| a:A;  b:B(a) |] ==> P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   903
     |] ==> P"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   904
  by blast
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   905
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   906
lemma Sigma_cong:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   907
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   908
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   909
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   910
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   911
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   912
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   913
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   914
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   915
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   916
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   917
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   918
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   919
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   920
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   921
  by auto
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
   922
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   923
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   924
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   925
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   926
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   927
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   928
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   929
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   930
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   931
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   932
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   933
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   934
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   935
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   936
  by (blast elim: equalityE)
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   937
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   938
lemma SetCompr_Sigma_eq:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   939
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   940
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   941
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   942
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   943
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   944
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   945
lemma UN_Times_distrib:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   946
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   947
  -- {* Suggested by Pierre Chartier *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   948
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   949
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   950
lemma split_paired_Ball_Sigma [simp,no_atp]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   951
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   952
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   953
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   954
lemma split_paired_Bex_Sigma [simp,no_atp]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   955
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   956
  by blast
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
   957
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   958
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   959
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   960
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   961
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   962
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   963
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   964
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   965
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   966
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   967
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   968
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   969
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   970
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   971
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   972
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   973
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   974
  by blast
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
   975
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   976
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   977
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   978
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   979
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   980
  Non-dependent versions are needed to avoid the need for higher-order
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   981
  matching, especially when the rules are re-oriented.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   982
*}
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
   983
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   984
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
   985
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   986
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   987
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
   988
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   989
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   990
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
   991
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   992
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
   993
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
   994
  by auto
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
   995
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
   996
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
   997
  by (auto intro!: image_eqI)
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
   998
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
   999
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1000
  by (auto intro!: image_eqI)
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1001
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1002
lemma insert_times_insert[simp]:
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1003
  "insert a A \<times> insert b B =
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1004
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1005
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1006
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1007
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1008
  by (auto, rule_tac p = "f x" in PairE, auto)
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1009
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1010
lemma swap_inj_on:
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1011
  "inj_on (\<lambda>(i, j). (j, i)) A"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1012
  by (auto intro!: inj_onI)
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1013
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1014
lemma swap_product:
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1015
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1016
  by (simp add: split_def image_def) blast
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1017
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1018
lemma image_split_eq_Sigma:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1019
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1020
proof (safe intro!: imageI vimageI)
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1021
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1022
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1023
    using * eq[symmetric] by auto
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1024
qed simp_all
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1025
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1026
subsubsection {* Code generator setup *}
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1027
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28537
diff changeset
  1028
lemma [code]:
28346
b8390cd56b8f discontinued special treatment of op = vs. eq_class.eq
haftmann
parents: 28262
diff changeset
  1029
  "eq_class.eq (x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by (simp add: eq)
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1030
24844
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1031
lemma split_case_cert:
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1032
  assumes "CASE \<equiv> split f"
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1033
  shows "CASE (a, b) \<equiv> f a b"
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1034
  using assms by simp
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1035
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1036
setup {*
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1037
  Code.add_case @{thm split_case_cert}
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1038
*}
98c006a30218 certificates for code generator case expressions
haftmann
parents: 24699
diff changeset
  1039
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1040
code_type *
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1041
  (SML infix 2 "*")
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1042
  (OCaml infix 2 "*")
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1043
  (Haskell "!((_),/ (_))")
34900
9b12b0824bfe tuned for products vs. tupled functions
haftmann
parents: 34886
diff changeset
  1044
  (Scala "((_),/ (_))")
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1045
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1046
code_instance * :: eq
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1047
  (Haskell -)
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1048
28346
b8390cd56b8f discontinued special treatment of op = vs. eq_class.eq
haftmann
parents: 28262
diff changeset
  1049
code_const "eq_class.eq \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1050
  (Haskell infixl 4 "==")
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1051
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1052
code_const Pair
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1053
  (SML "!((_),/ (_))")
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1054
  (OCaml "!((_),/ (_))")
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1055
  (Haskell "!((_),/ (_))")
34886
873c31d9f10d some syntax setup for Scala
haftmann
parents: 33959
diff changeset
  1056
  (Scala "!((_),/ (_))")
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1057
22389
bdf16741d039 using "fst" "snd" for Haskell code
haftmann
parents: 22349
diff changeset
  1058
code_const fst and snd
bdf16741d039 using "fst" "snd" for Haskell code
haftmann
parents: 22349
diff changeset
  1059
  (Haskell "fst" and "snd")
bdf16741d039 using "fst" "snd" for Haskell code
haftmann
parents: 22349
diff changeset
  1060
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1061
types_code
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1062
  "*"     ("(_ */ _)")
16770
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16634
diff changeset
  1063
attach (term_of) {*
25885
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25534
diff changeset
  1064
fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
16770
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16634
diff changeset
  1065
*}
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16634
diff changeset
  1066
attach (test) {*
25885
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25534
diff changeset
  1067
fun gen_id_42 aG aT bG bT i =
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25534
diff changeset
  1068
  let
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25534
diff changeset
  1069
    val (x, t) = aG i;
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25534
diff changeset
  1070
    val (y, u) = bG i
6fbc3f54f819 New interface for test data generators.
berghofe
parents: 25534
diff changeset
  1071
  in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
16770
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16634
diff changeset
  1072
*}
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1073
18706
1e7562c7afe6 Re-inserted consts_code declaration accidentally deleted
berghofe
parents: 18702
diff changeset
  1074
consts_code
1e7562c7afe6 Re-inserted consts_code declaration accidentally deleted
berghofe
parents: 18702
diff changeset
  1075
  "Pair"    ("(_,/ _)")
1e7562c7afe6 Re-inserted consts_code declaration accidentally deleted
berghofe
parents: 18702
diff changeset
  1076
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1077
setup {*
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1078
let
18013
3f5d0acdfdba added extraction interface for code generator
haftmann
parents: 17956
diff changeset
  1079
19039
8eae46249628 added theory of executable rational numbers
haftmann
parents: 19008
diff changeset
  1080
fun strip_abs_split 0 t = ([], t)
8eae46249628 added theory of executable rational numbers
haftmann
parents: 19008
diff changeset
  1081
  | strip_abs_split i (Abs (s, T, t)) =
18013
3f5d0acdfdba added extraction interface for code generator
haftmann
parents: 17956
diff changeset
  1082
      let
3f5d0acdfdba added extraction interface for code generator
haftmann
parents: 17956
diff changeset
  1083
        val s' = Codegen.new_name t s;
3f5d0acdfdba added extraction interface for code generator
haftmann
parents: 17956
diff changeset
  1084
        val v = Free (s', T)
19039
8eae46249628 added theory of executable rational numbers
haftmann
parents: 19008
diff changeset
  1085
      in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
  1086
  | strip_abs_split i (u as Const (@{const_name split}, _) $ t) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
  1087
      (case strip_abs_split (i+1) t of
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1088
        (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1089
      | _ => ([], u))
30604
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1090
  | strip_abs_split i t =
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1091
      strip_abs_split i (Abs ("x", hd (binder_types (fastype_of t)), t $ Bound 0));
18013
3f5d0acdfdba added extraction interface for code generator
haftmann
parents: 17956
diff changeset
  1092
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
  1093
fun let_codegen thy defs dep thyname brack t gr =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
  1094
  (case strip_comb t of
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
  1095
    (t1 as Const (@{const_name Let}, _), t2 :: t3 :: ts) =>
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1096
    let
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
  1097
      fun dest_let (l as Const (@{const_name Let}, _) $ t $ u) =
19039
8eae46249628 added theory of executable rational numbers
haftmann
parents: 19008
diff changeset
  1098
          (case strip_abs_split 1 u of
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1099
             ([p], u') => apfst (cons (p, t)) (dest_let u')
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1100
           | _ => ([], l))
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1101
        | dest_let t = ([], t);
28537
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1102
      fun mk_code (l, r) gr =
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1103
        let
28537
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1104
          val (pl, gr1) = Codegen.invoke_codegen thy defs dep thyname false l gr;
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1105
          val (pr, gr2) = Codegen.invoke_codegen thy defs dep thyname false r gr1;
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1106
        in ((pl, pr), gr2) end
16634
f19d58cfb47a Adapted to new interface of code generator.
berghofe
parents: 16417
diff changeset
  1107
    in case dest_let (t1 $ t2 $ t3) of
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
  1108
        ([], _) => NONE
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1109
      | (ps, u) =>
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1110
          let
28537
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1111
            val (qs, gr1) = fold_map mk_code ps gr;
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1112
            val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1113
            val (pargs, gr3) = fold_map
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1114
              (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1115
          in
28537
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1116
            SOME (Codegen.mk_app brack
32952
aeb1e44fbc19 replaced String.concat by implode;
wenzelm
parents: 32010
diff changeset
  1117
              (Pretty.blk (0, [Codegen.str "let ", Pretty.blk (0, flat
26975
103dca19ef2e Replaced Pretty.str and Pretty.string_of by specific functions (from Codegen) that
berghofe
parents: 26798
diff changeset
  1118
                  (separate [Codegen.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
103dca19ef2e Replaced Pretty.str and Pretty.string_of by specific functions (from Codegen) that
berghofe
parents: 26798
diff changeset
  1119
                    [Pretty.block [Codegen.str "val ", pl, Codegen.str " =",
16634
f19d58cfb47a Adapted to new interface of code generator.
berghofe
parents: 16417
diff changeset
  1120
                       Pretty.brk 1, pr]]) qs))),
26975
103dca19ef2e Replaced Pretty.str and Pretty.string_of by specific functions (from Codegen) that
berghofe
parents: 26798
diff changeset
  1121
                Pretty.brk 1, Codegen.str "in ", pu,
28537
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1122
                Pretty.brk 1, Codegen.str "end"])) pargs, gr3)
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1123
          end
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1124
    end
16634
f19d58cfb47a Adapted to new interface of code generator.
berghofe
parents: 16417
diff changeset
  1125
  | _ => NONE);
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1126
28537
1e84256d1a8a established canonical argument order in SML code generators
haftmann
parents: 28346
diff changeset
  1127
fun split_codegen thy defs dep thyname brack t gr = (case strip_comb t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
  1128
    (t1 as Const (@{const_name split}, _), t2 :: ts) =>
30604
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1129
      let
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1130
        val ([p], u) = strip_abs_split 1 (t1 $ t2);
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1131
        val (q, gr1) = Codegen.invoke_codegen thy defs dep thyname false p gr;
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1132
        val (pu, gr2) = Codegen.invoke_codegen thy defs dep thyname false u gr1;
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1133
        val (pargs, gr3) = fold_map
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1134
          (Codegen.invoke_codegen thy defs dep thyname true) ts gr2
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1135
      in
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1136
        SOME (Codegen.mk_app brack
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1137
          (Pretty.block [Codegen.str "(fn ", q, Codegen.str " =>",
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1138
            Pretty.brk 1, pu, Codegen.str ")"]) pargs, gr2)
2a9911f4b0a3 split_codegen now eta-expands terms on-the-fly.
berghofe
parents: 28719
diff changeset
  1139
      end
16634
f19d58cfb47a Adapted to new interface of code generator.
berghofe
parents: 16417
diff changeset
  1140
  | _ => NONE);
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1141
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1142
in
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1143
20105
454f4be984b7 adaptions in codegen
haftmann
parents: 20044
diff changeset
  1144
  Codegen.add_codegen "let_codegen" let_codegen
454f4be984b7 adaptions in codegen
haftmann
parents: 20044
diff changeset
  1145
  #> Codegen.add_codegen "split_codegen" split_codegen
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1146
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1147
end
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1148
*}
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1149
31723
f5cafe803b55 discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents: 31667
diff changeset
  1150
use "Tools/inductive_set.ML"
f5cafe803b55 discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents: 31667
diff changeset
  1151
setup Inductive_Set.setup
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
  1152
10213
01c2744a3786 *** empty log message ***
nipkow
parents:
diff changeset
  1153
end