
nipkow@10213: (*  Title:      HOL/Product_Type.thy
nipkow@10213:     ID:         $Id$
nipkow@10213:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
nipkow@10213:     Copyright   1992  University of Cambridge
wenzelm@11777: *)
nipkow@10213: 
wenzelm@11838: header {* Cartesian products *}
nipkow@10213: 
nipkow@15131: theory Product_Type
haftmann@24699: imports Inductive
haftmann@24699: uses
haftmann@24699:   ("Tools/split_rule.ML")
haftmann@24699:   ("Tools/inductive_set_package.ML")
haftmann@24699:   ("Tools/inductive_realizer.ML")
haftmann@24699:   ("Tools/datatype_realizer.ML")
nipkow@15131: begin
wenzelm@11838: 
haftmann@24699: subsection {* @{typ bool} is a datatype *}
haftmann@24699: 
haftmann@24699: rep_datatype bool
haftmann@24699:   distinct True_not_False False_not_True
haftmann@24699:   induction bool_induct
haftmann@24699: 
haftmann@24699: declare case_split [cases type: bool]
haftmann@24699:   -- "prefer plain propositional version"
haftmann@24699: 
haftmann@25534: lemma [code func]:
haftmann@25534:   shows "False = P \<longleftrightarrow> \<not> P"
haftmann@25534:     and "True = P \<longleftrightarrow> P" 
haftmann@25534:     and "P = False \<longleftrightarrow> \<not> P" 
haftmann@25534:     and "P = True \<longleftrightarrow> P" by simp_all
haftmann@25534: 
haftmann@25534: code_const "op = \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
haftmann@25534:   (Haskell infixl 4 "==")
haftmann@25534: 
haftmann@25534: code_instance bool :: eq
haftmann@25534:   (Haskell -)
haftmann@24699: 
haftmann@26358: 
wenzelm@11838: subsection {* Unit *}
wenzelm@11838: 
wenzelm@11838: typedef unit = "{True}"
wenzelm@11838: proof
haftmann@20588:   show "True : ?unit" ..
wenzelm@11838: qed
wenzelm@11838: 
haftmann@24699: definition
wenzelm@11838:   Unity :: unit    ("'(')")
haftmann@24699: where
haftmann@24699:   "() = Abs_unit True"
wenzelm@11838: 
paulson@24286: lemma unit_eq [noatp]: "u = ()"
wenzelm@11838:   by (induct u) (simp add: unit_def Unity_def)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   Simplification procedure for @{thm [source] unit_eq}.  Cannot use
wenzelm@11838:   this rule directly --- it loops!
wenzelm@11838: *}
wenzelm@11838: 
wenzelm@11838: ML_setup {*
wenzelm@13462:   val unit_eq_proc =
haftmann@24699:     let val unit_meta_eq = mk_meta_eq @{thm unit_eq} in
haftmann@24699:       Simplifier.simproc @{theory} "unit_eq" ["x::unit"]
skalberg@15531:       (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)
wenzelm@13462:     end;
wenzelm@11838: 
wenzelm@11838:   Addsimprocs [unit_eq_proc];
wenzelm@11838: *}
wenzelm@11838: 
haftmann@24699: lemma unit_induct [noatp,induct type: unit]: "P () ==> P x"
haftmann@24699:   by simp
haftmann@24699: 
haftmann@24699: rep_datatype unit
haftmann@24699:   induction unit_induct
haftmann@24699: 
wenzelm@11838: lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
wenzelm@11838:   by simp
wenzelm@11838: 
wenzelm@11838: lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
wenzelm@11838:   by (rule triv_forall_equality)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   This rewrite counters the effect of @{text unit_eq_proc} on @{term
wenzelm@11838:   [source] "%u::unit. f u"}, replacing it by @{term [source]
wenzelm@11838:   f} rather than by @{term [source] "%u. f ()"}.
wenzelm@11838: *}
wenzelm@11838: 
paulson@24286: lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f"
wenzelm@11838:   by (rule ext) simp
nipkow@10213: 
nipkow@10213: 
haftmann@26358: text {* code generator setup *}
haftmann@26358: 
haftmann@26358: instance unit :: eq ..
haftmann@26358: 
haftmann@26358: lemma [code func]:
haftmann@26358:   "(u\<Colon>unit) = v \<longleftrightarrow> True" unfolding unit_eq [of u] unit_eq [of v] by rule+
haftmann@26358: 
haftmann@26358: code_type unit
haftmann@26358:   (SML "unit")
haftmann@26358:   (OCaml "unit")
haftmann@26358:   (Haskell "()")
haftmann@26358: 
haftmann@26358: code_instance unit :: eq
haftmann@26358:   (Haskell -)
haftmann@26358: 
haftmann@26358: code_const "op = \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
haftmann@26358:   (Haskell infixl 4 "==")
haftmann@26358: 
haftmann@26358: code_const Unity
haftmann@26358:   (SML "()")
haftmann@26358:   (OCaml "()")
haftmann@26358:   (Haskell "()")
haftmann@26358: 
haftmann@26358: code_reserved SML
haftmann@26358:   unit
haftmann@26358: 
haftmann@26358: code_reserved OCaml
haftmann@26358:   unit
haftmann@26358: 
haftmann@26358: 
wenzelm@11838: subsection {* Pairs *}
nipkow@10213: 
haftmann@26358: subsubsection {* Product type, basic operations and concrete syntax *}
nipkow@10213: 
haftmann@26358: definition
haftmann@26358:   Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
haftmann@26358: where
haftmann@26358:   "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
nipkow@10213: 
nipkow@10213: global
nipkow@10213: 
nipkow@10213: typedef (Prod)
haftmann@22838:   ('a, 'b) "*"    (infixr "*" 20)
haftmann@26358:     = "{f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
oheimb@11025: proof
haftmann@26358:   fix a b show "Pair_Rep a b \<in> ?Prod"
haftmann@26358:     by rule+
oheimb@11025: qed
nipkow@10213: 
wenzelm@12114: syntax (xsymbols)
schirmer@15422:   "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
nipkow@10213: syntax (HTML output)
schirmer@15422:   "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
nipkow@10213: 
nipkow@10213: consts
haftmann@26358:   Pair     :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
haftmann@26358:   fst      :: "'a \<times> 'b \<Rightarrow> 'a"
haftmann@26358:   snd      :: "'a \<times> 'b \<Rightarrow> 'b"
haftmann@26358:   split    :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
haftmann@26358:   curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"
nipkow@10213: 
wenzelm@11777: local
nipkow@10213: 
wenzelm@19535: defs
wenzelm@19535:   Pair_def:     "Pair a b == Abs_Prod (Pair_Rep a b)"
wenzelm@19535:   fst_def:      "fst p == THE a. EX b. p = Pair a b"
wenzelm@19535:   snd_def:      "snd p == THE b. EX a. p = Pair a b"
haftmann@24162:   split_def:    "split == (%c p. c (fst p) (snd p))"
haftmann@24162:   curry_def:    "curry == (%c x y. c (Pair x y))"
wenzelm@19535: 
wenzelm@11777: text {*
wenzelm@11777:   Patterns -- extends pre-defined type @{typ pttrn} used in
wenzelm@11777:   abstractions.
wenzelm@11777: *}
nipkow@10213: 
nipkow@10213: nonterminals
nipkow@10213:   tuple_args patterns
nipkow@10213: 
nipkow@10213: syntax
nipkow@10213:   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
nipkow@10213:   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
nipkow@10213:   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
oheimb@11025:   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
oheimb@11025:   ""            :: "pttrn => patterns"                  ("_")
oheimb@11025:   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
nipkow@10213: 
nipkow@10213: translations
nipkow@10213:   "(x, y)"       == "Pair x y"
nipkow@10213:   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
nipkow@10213:   "%(x,y,zs).b"  == "split(%x (y,zs).b)"
nipkow@10213:   "%(x,y).b"     == "split(%x y. b)"
nipkow@10213:   "_abs (Pair x y) t" => "%(x,y).t"
nipkow@10213:   (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
nipkow@10213:      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
nipkow@10213: 
schirmer@14359: (* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*)
schirmer@14359: (* works best with enclosing "let", if "let" does not avoid eta-contraction   *)
schirmer@14359: print_translation {*
schirmer@14359: let fun split_tr' [Abs (x,T,t as (Abs abs))] =
schirmer@14359:       (* split (%x y. t) => %(x,y) t *)
schirmer@14359:       let val (y,t') = atomic_abs_tr' abs;
schirmer@14359:           val (x',t'') = atomic_abs_tr' (x,T,t');
schirmer@14359:     
schirmer@14359:       in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end
schirmer@14359:     | split_tr' [Abs (x,T,(s as Const ("split",_)$t))] =
schirmer@14359:        (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
schirmer@14359:        let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t];
schirmer@14359:            val (x',t'') = atomic_abs_tr' (x,T,t');
schirmer@14359:        in Syntax.const "_abs"$ 
schirmer@14359:            (Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end
schirmer@14359:     | split_tr' [Const ("split",_)$t] =
schirmer@14359:        (* split (split (%x y z. t)) => %((x,y),z). t *)   
schirmer@14359:        split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
schirmer@14359:     | split_tr' [Const ("_abs",_)$x_y$(Abs abs)] =
schirmer@14359:        (* split (%pttrn z. t) => %(pttrn,z). t *)
schirmer@14359:        let val (z,t) = atomic_abs_tr' abs;
schirmer@14359:        in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end
schirmer@14359:     | split_tr' _ =  raise Match;
schirmer@14359: in [("split", split_tr')]
schirmer@14359: end
schirmer@14359: *}
schirmer@14359: 
schirmer@15422: (* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
schirmer@15422: typed_print_translation {*
schirmer@15422: let
schirmer@15422:   fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match
schirmer@15422:     | split_guess_names_tr' _ T  [Abs (x,xT,t)] =
schirmer@15422:         (case (head_of t) of
schirmer@15422:            Const ("split",_) => raise Match
schirmer@15422:          | _ => let 
schirmer@15422:                   val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
schirmer@15422:                   val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); 
schirmer@15422:                   val (x',t'') = atomic_abs_tr' (x,xT,t');
schirmer@15422:                 in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
schirmer@15422:     | split_guess_names_tr' _ T [t] =
schirmer@15422:        (case (head_of t) of
schirmer@15422:            Const ("split",_) => raise Match 
schirmer@15422:          | _ => let 
schirmer@15422:                   val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
schirmer@15422:                   val (y,t') = 
schirmer@15422:                         atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); 
schirmer@15422:                   val (x',t'') = atomic_abs_tr' ("x",xT,t');
schirmer@15422:                 in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
schirmer@15422:     | split_guess_names_tr' _ _ _ = raise Match;
schirmer@15422: in [("split", split_guess_names_tr')]
schirmer@15422: end 
schirmer@15422: *}
schirmer@15422: 
nipkow@10213: 
haftmann@26358: text {* Towards a datatype declaration *}
wenzelm@11838: 
haftmann@26358: lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@26358:   apply (unfold Pair_def)
haftmann@26358:   apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
haftmann@26358:   apply (erule exE, erule exE, rule exI, rule exI)
haftmann@26358:   apply (rule Rep_Prod_inverse [symmetric, THEN trans])
haftmann@26358:   apply (erule arg_cong)
haftmann@26358:   done
wenzelm@11838: 
haftmann@26358: lemma PairE [cases type: *]:
haftmann@26358:   obtains x y where "p = (x, y)"
haftmann@26358:   using surj_pair [of p] by blast
haftmann@26358: 
haftmann@26358: 
haftmann@26358: lemma prod_induct [induct type: *]: "(\<And>a b. P (a, b)) \<Longrightarrow> P x"
haftmann@26358:   by (cases x) simp
haftmann@26358: 
haftmann@26358: lemma ProdI: "Pair_Rep a b \<in> Prod"
haftmann@26358:   unfolding Prod_def by rule+
haftmann@26358: 
haftmann@26358: lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' \<Longrightarrow> a = a' \<and> b = b'"
haftmann@26358:   unfolding Pair_Rep_def by (drule fun_cong, drule fun_cong) blast
nipkow@10213: 
wenzelm@11838: lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
wenzelm@11838:   apply (rule inj_on_inverseI)
wenzelm@11838:   apply (erule Abs_Prod_inverse)
wenzelm@11838:   done
wenzelm@11838: 
wenzelm@11838: lemma Pair_inject:
wenzelm@18372:   assumes "(a, b) = (a', b')"
wenzelm@18372:     and "a = a' ==> b = b' ==> R"
wenzelm@18372:   shows R
wenzelm@18372:   apply (insert prems [unfolded Pair_def])
wenzelm@18372:   apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
wenzelm@18372:   apply (assumption | rule ProdI)+
wenzelm@18372:   done
nipkow@10213: 
wenzelm@11838: lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')"
wenzelm@11838:   by (blast elim!: Pair_inject)
wenzelm@11838: 
haftmann@22886: lemma fst_conv [simp, code]: "fst (a, b) = a"
wenzelm@19535:   unfolding fst_def by blast
wenzelm@11838: 
haftmann@22886: lemma snd_conv [simp, code]: "snd (a, b) = b"
wenzelm@19535:   unfolding snd_def by blast
oheimb@11025: 
haftmann@26358: rep_datatype prod
haftmann@26358:   inject Pair_eq
haftmann@26358:   induction prod_induct
haftmann@26358: 
haftmann@26358: 
haftmann@26358: subsubsection {* Basic rules and proof tools *}
haftmann@26358: 
wenzelm@11838: lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838:   by simp
wenzelm@11838: 
wenzelm@11838: lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838:   by simp
wenzelm@11838: 
haftmann@26358: lemma pair_collapse [simp]: "(fst p, snd p) = p"
wenzelm@11838:   by (cases p) simp
wenzelm@11838: 
haftmann@26358: lemmas surjective_pairing = pair_collapse [symmetric]
wenzelm@11838: 
wenzelm@11838: lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820: proof
wenzelm@11820:   fix a b
wenzelm@11820:   assume "!!x. PROP P x"
wenzelm@19535:   then show "PROP P (a, b)" .
wenzelm@11820: next
wenzelm@11820:   fix x
wenzelm@11820:   assume "!!a b. PROP P (a, b)"
wenzelm@19535:   from `PROP P (fst x, snd x)` show "PROP P x" by simp
wenzelm@11820: qed
wenzelm@11820: 
wenzelm@11838: text {*
wenzelm@11838:   The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838:   Simplifier because it also affects premises in congrence rules,
wenzelm@11838:   where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838:   ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@11838: *}
wenzelm@11838: 
haftmann@26358: lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358: 
haftmann@26358: ML_setup {*
wenzelm@11838:   (* replace parameters of product type by individual component parameters *)
wenzelm@11838:   val safe_full_simp_tac = generic_simp_tac true (true, false, false);
wenzelm@11838:   local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@16121:     fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
wenzelm@11838:           can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838:       | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838:       | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838:       | exists_paired_all _ = false;
wenzelm@11838:     val ss = HOL_basic_ss
wenzelm@26340:       addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@11838:       addsimprocs [unit_eq_proc];
wenzelm@11838:   in
wenzelm@11838:     val split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838:       if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838:     val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838:       if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838:     fun split_all th =
wenzelm@26340:    if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
wenzelm@11838:   end;
wenzelm@26340: *}
wenzelm@11838: 
wenzelm@26340: declaration {* fn _ =>
wenzelm@26340:   Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
wenzelm@16121: *}
wenzelm@11838: 
wenzelm@11838: lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838:   -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838:   by fast
wenzelm@11838: 
haftmann@26358: lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358:   by fast
haftmann@26358: 
haftmann@26358: lemma Pair_fst_snd_eq: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@26358:   by (cases s, cases t) simp
haftmann@26358: 
haftmann@26358: lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
haftmann@26358:   by (simp add: Pair_fst_snd_eq)
haftmann@26358: 
haftmann@26358: 
haftmann@26358: subsubsection {* @{text split} and @{text curry} *}
haftmann@26358: 
haftmann@26358: lemma split_conv [simp, code func]: "split f (a, b) = f a b"
haftmann@26358:   by (simp add: split_def)
haftmann@26358: 
haftmann@26358: lemma curry_conv [simp, code func]: "curry f a b = f (a, b)"
haftmann@26358:   by (simp add: curry_def)
haftmann@26358: 
haftmann@26358: lemmas split = split_conv  -- {* for backwards compatibility *}
haftmann@26358: 
haftmann@26358: lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@26358:   by (rule split_conv [THEN iffD2])
haftmann@26358: 
haftmann@26358: lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@26358:   by (rule split_conv [THEN iffD1])
haftmann@26358: 
haftmann@26358: lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@26358:   by (simp add: curry_def)
haftmann@26358: 
haftmann@26358: lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@26358:   by (simp add: curry_def)
haftmann@26358: 
haftmann@26358: lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@26358:   by (simp add: curry_def)
haftmann@26358: 
skalberg@14189: lemma curry_split [simp]: "curry (split f) = f"
skalberg@14189:   by (simp add: curry_def split_def)
skalberg@14189: 
skalberg@14189: lemma split_curry [simp]: "split (curry f) = f"
skalberg@14189:   by (simp add: curry_def split_def)
skalberg@14189: 
haftmann@26358: lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
haftmann@26358:   by (simp add: split_def id_def)
wenzelm@11838: 
haftmann@26358: lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@26358:   -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
haftmann@26358:   by (rule ext) auto
wenzelm@11838: 
haftmann@26358: lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@26358:   by (cases x) simp
wenzelm@11838: 
haftmann@26358: lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@26358:   unfolding split_def ..
wenzelm@11838: 
wenzelm@11838: lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838:   -- {* Can't be added to simpset: loops! *}
haftmann@26358:   by (simp add: split_eta)
wenzelm@11838: 
wenzelm@11838: lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
wenzelm@11838:   by (simp add: split_def)
wenzelm@11838: 
haftmann@26358: lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
wenzelm@11838:   -- {* Prevents simplification of @{term c}: much faster *}
wenzelm@11838:   by (erule arg_cong)
wenzelm@11838: 
wenzelm@11838: lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
wenzelm@11838:   by (simp add: split_eta)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838:   @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838:   and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838:   existing proofs very inefficient; similarly for @{text
haftmann@26358:   split_beta}.
haftmann@26358: *}
wenzelm@11838: 
wenzelm@11838: ML_setup {*
wenzelm@11838: 
wenzelm@11838: local
wenzelm@18328:   val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"]
wenzelm@11838:   fun  Pair_pat k 0 (Bound m) = (m = k)
wenzelm@11838:   |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
wenzelm@11838:                         m = k+i andalso Pair_pat k (i-1) t
wenzelm@11838:   |    Pair_pat _ _ _ = false;
wenzelm@11838:   fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
wenzelm@11838:   |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@11838:   |   no_args k i (Bound m) = m < k orelse m > k+i
wenzelm@11838:   |   no_args _ _ _ = true;
skalberg@15531:   fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
wenzelm@11838:   |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
skalberg@15531:   |   split_pat tp i _ = NONE;
wenzelm@20044:   fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@13480:         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
wenzelm@18328:         (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
wenzelm@11838: 
wenzelm@11838:   fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
wenzelm@11838:   |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
wenzelm@11838:                         (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@11838:   |   beta_term_pat k i t = no_args k i t;
wenzelm@11838:   fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@11838:   |    eta_term_pat _ _ _ = false;
wenzelm@11838:   fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@11838:   |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@11838:                               else (subst arg k i t $ subst arg k i u)
wenzelm@11838:   |   subst arg k i t = t;
wenzelm@20044:   fun beta_proc ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
wenzelm@11838:         (case split_pat beta_term_pat 1 t of
wenzelm@20044:         SOME (i,f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531:         | NONE => NONE)
wenzelm@20044:   |   beta_proc _ _ = NONE;
wenzelm@20044:   fun eta_proc ss (s as Const ("split", _) $ Abs (_, _, t)) =
wenzelm@11838:         (case split_pat eta_term_pat 1 t of
wenzelm@20044:           SOME (_,ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531:         | NONE => NONE)
wenzelm@20044:   |   eta_proc _ _ = NONE;
wenzelm@11838: in
wenzelm@22577:   val split_beta_proc = Simplifier.simproc @{theory} "split_beta" ["split f z"] (K beta_proc);
wenzelm@22577:   val split_eta_proc = Simplifier.simproc @{theory} "split_eta" ["split f"] (K eta_proc);
wenzelm@11838: end;
wenzelm@11838: 
wenzelm@11838: Addsimprocs [split_beta_proc, split_eta_proc];
wenzelm@11838: *}
wenzelm@11838: 
wenzelm@11838: lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838:   by (subst surjective_pairing, rule split_conv)
wenzelm@11838: 
paulson@24286: lemma split_split [noatp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838:   -- {* For use with @{text split} and the Simplifier. *}
paulson@15481:   by (insert surj_pair [of p], clarify, simp)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838:   done after the Splitter has been speeded up significantly;
wenzelm@11838:   precompute the constants involved and don't do anything unless the
wenzelm@11838:   current goal contains one of those constants.
wenzelm@11838: *}
wenzelm@11838: 
paulson@24286: lemma split_split_asm [noatp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208: by (subst split_split, simp)
wenzelm@11838: 
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   \medskip @{term split} used as a logical connective or set former.
wenzelm@11838: 
wenzelm@11838:   \medskip These rules are for use with @{text blast}; could instead
wenzelm@11838:   call @{text simp} using @{thm [source] split} as rewrite. *}
wenzelm@11838: 
wenzelm@11838: lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838:   apply (simp only: split_tupled_all)
wenzelm@11838:   apply (simp (no_asm_simp))
wenzelm@11838:   done
wenzelm@11838: 
wenzelm@11838: lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838:   apply (simp only: split_tupled_all)
wenzelm@11838:   apply (simp (no_asm_simp))
wenzelm@11838:   done
wenzelm@11838: 
wenzelm@11838: lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
wenzelm@11838:   by (induct p) (auto simp add: split_def)
wenzelm@11838: 
wenzelm@11838: lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
wenzelm@11838:   by (induct p) (auto simp add: split_def)
wenzelm@11838: 
wenzelm@11838: lemma splitE2:
wenzelm@11838:   "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838: proof -
wenzelm@11838:   assume q: "Q (split P z)"
wenzelm@11838:   assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838:   show R
wenzelm@11838:     apply (rule r surjective_pairing)+
wenzelm@11838:     apply (rule split_beta [THEN subst], rule q)
wenzelm@11838:     done
wenzelm@11838: qed
wenzelm@11838: 
wenzelm@11838: lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838:   by simp
wenzelm@11838: 
wenzelm@11838: lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838:   by simp
wenzelm@11838: 
wenzelm@11838: lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208: by (simp only: split_tupled_all, simp)
wenzelm@11838: 
wenzelm@18372: lemma mem_splitE:
wenzelm@18372:   assumes major: "z: split c p"
wenzelm@18372:     and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q"
wenzelm@18372:   shows Q
wenzelm@18372:   by (rule major [unfolded split_def] cases surjective_pairing)+
wenzelm@11838: 
wenzelm@11838: declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838: declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838: 
wenzelm@26340: ML {*
wenzelm@11838: local (* filtering with exists_p_split is an essential optimization *)
wenzelm@16121:   fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
wenzelm@11838:     | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838:     | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838:     | exists_p_split _ = false;
wenzelm@16121:   val ss = HOL_basic_ss addsimps [thm "split_conv"];
wenzelm@11838: in
wenzelm@11838: val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838:     if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838: end;
wenzelm@26340: *}
wenzelm@26340: 
wenzelm@11838: (* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838:    to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@26340: declaration {* fn _ =>
wenzelm@26340:   Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
wenzelm@16121: *}
wenzelm@11838: 
paulson@24286: lemma split_eta_SetCompr [simp,noatp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372:   by (rule ext) fast
wenzelm@11838: 
paulson@24286: lemma split_eta_SetCompr2 [simp,noatp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372:   by (rule ext) fast
wenzelm@11838: 
wenzelm@11838: lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838:   -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372:   by (rule ext) blast
wenzelm@11838: 
nipkow@14337: (* Do NOT make this a simp rule as it
nipkow@14337:    a) only helps in special situations
nipkow@14337:    b) can lead to nontermination in the presence of split_def
nipkow@14337: *)
nipkow@14337: lemma split_comp_eq: 
paulson@20415:   fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415:   shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372:   by (rule ext) auto
oheimb@14101: 
haftmann@26358: lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358:   apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358:    apply auto
haftmann@26358:   done
haftmann@26358: 
wenzelm@11838: lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838:   by blast
wenzelm@11838: 
wenzelm@11838: (*
wenzelm@11838: the following  would be slightly more general,
wenzelm@11838: but cannot be used as rewrite rule:
wenzelm@11838: ### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838: ### ?y = .x
wenzelm@11838: Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208: by (rtac some_equality 1)
paulson@14208: by ( Simp_tac 1)
paulson@14208: by (split_all_tac 1)
paulson@14208: by (Asm_full_simp_tac 1)
wenzelm@11838: qed "The_split_eq";
wenzelm@11838: *)
wenzelm@11838: 
wenzelm@11838: text {*
wenzelm@11838:   Setup of internal @{text split_rule}.
wenzelm@11838: *}
wenzelm@11838: 
haftmann@25511: definition
haftmann@25511:   internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
haftmann@25511: where
wenzelm@11032:   "internal_split == split"
wenzelm@11032: 
wenzelm@11032: lemma internal_split_conv: "internal_split c (a, b) = c a b"
wenzelm@11032:   by (simp only: internal_split_def split_conv)
wenzelm@11032: 
wenzelm@11032: hide const internal_split
wenzelm@11032: 
oheimb@11025: use "Tools/split_rule.ML"
wenzelm@11032: setup SplitRule.setup
nipkow@10213: 
haftmann@24699: lemmas prod_caseI = prod.cases [THEN iffD2, standard]
haftmann@24699: 
haftmann@24699: lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@24699:   by auto
haftmann@24699: 
haftmann@24699: lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@24699:   by (auto simp: split_tupled_all)
haftmann@24699: 
haftmann@24699: lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@24699:   by (induct p) auto
haftmann@24699: 
haftmann@24699: lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@24699:   by (induct p) auto
haftmann@24699: 
haftmann@24699: lemma prod_case_unfold: "prod_case = (%c p. c (fst p) (snd p))"
haftmann@24699:   by (simp add: expand_fun_eq)
haftmann@24699: 
haftmann@24699: declare prod_caseI2' [intro!] prod_caseI2 [intro!] prod_caseI [intro!]
haftmann@24699: declare prod_caseE' [elim!] prod_caseE [elim!]
haftmann@24699: 
haftmann@24844: lemma prod_case_split:
haftmann@24699:   "prod_case = split"
haftmann@24699:   by (auto simp add: expand_fun_eq)
haftmann@24699: 
bulwahn@26143: lemma prod_case_beta:
bulwahn@26143:   "prod_case f p = f (fst p) (snd p)"
bulwahn@26143:   unfolding prod_case_split split_beta ..
bulwahn@26143: 
haftmann@24699: 
haftmann@24699: subsection {* Further cases/induct rules for tuples *}
haftmann@24699: 
haftmann@24699: lemma prod_cases3 [cases type]:
haftmann@24699:   obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699:   by (cases y, case_tac b) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
haftmann@24699: lemma prod_cases4 [cases type]:
haftmann@24699:   obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699:   by (cases y, case_tac c) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
haftmann@24699: lemma prod_cases5 [cases type]:
haftmann@24699:   obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699:   by (cases y, case_tac d) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
haftmann@24699: lemma prod_cases6 [cases type]:
haftmann@24699:   obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699:   by (cases y, case_tac e) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
haftmann@24699: lemma prod_cases7 [cases type]:
haftmann@24699:   obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699:   by (cases y, case_tac f) blast
haftmann@24699: 
haftmann@24699: lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699:     "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699:   by (cases x) blast
haftmann@24699: 
haftmann@24699: 
haftmann@26358: subsubsection {* Derived operations *}
haftmann@26358: 
haftmann@26358: text {*
haftmann@26358:   The composition-uncurry combinator.
haftmann@26358: *}
haftmann@26358: 
haftmann@26358: definition
haftmann@26358:   mbind :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "o->" 60)
haftmann@26358: where
haftmann@26358:   "f o-> g = (\<lambda>x. split g (f x))"
haftmann@26358: 
haftmann@26358: notation (xsymbols)
haftmann@26358:   mbind  (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358: 
haftmann@26358: notation (HTML output)
haftmann@26358:   mbind  (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358: 
haftmann@26358: lemma mbind_apply:  "(f \<circ>\<rightarrow> g) x = split g (f x)"
haftmann@26358:   by (simp add: mbind_def)
haftmann@26358: 
haftmann@26358: lemma Pair_mbind: "Pair x \<circ>\<rightarrow> f = f x"
haftmann@26358:   by (simp add: expand_fun_eq mbind_apply)
haftmann@26358: 
haftmann@26358: lemma mbind_Pair: "x \<circ>\<rightarrow> Pair = x"
haftmann@26358:   by (simp add: expand_fun_eq mbind_apply)
haftmann@26358: 
haftmann@26358: lemma mbind_mbind: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
haftmann@26358:   by (simp add: expand_fun_eq split_twice mbind_def)
haftmann@26358: 
haftmann@26358: lemma mbind_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
haftmann@26358:   by (simp add: expand_fun_eq mbind_apply fcomp_def split_def)
haftmann@26358: 
haftmann@26358: lemma fcomp_mbind: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
haftmann@26358:   by (simp add: expand_fun_eq mbind_apply fcomp_apply)
haftmann@26358: 
haftmann@26358: 
haftmann@26358: text {*
haftmann@26358:   @{term prod_fun} --- action of the product functor upon
haftmann@26358:   functions.
haftmann@26358: *}
haftmann@21195: 
haftmann@26358: definition prod_fun :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
haftmann@26358:   [code func del]: "prod_fun f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358: 
haftmann@26358: lemma prod_fun [simp, code func]: "prod_fun f g (a, b) = (f a, g b)"
haftmann@26358:   by (simp add: prod_fun_def)
haftmann@26358: 
haftmann@26358: lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
haftmann@26358:   by (rule ext) auto
haftmann@26358: 
haftmann@26358: lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
haftmann@26358:   by (rule ext) auto
haftmann@26358: 
haftmann@26358: lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
haftmann@26358:   apply (rule image_eqI)
haftmann@26358:   apply (rule prod_fun [symmetric], assumption)
haftmann@26358:   done
haftmann@21195: 
haftmann@26358: lemma prod_fun_imageE [elim!]:
haftmann@26358:   assumes major: "c: (prod_fun f g)`r"
haftmann@26358:     and cases: "!!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P"
haftmann@26358:   shows P
haftmann@26358:   apply (rule major [THEN imageE])
haftmann@26358:   apply (rule_tac p = x in PairE)
haftmann@26358:   apply (rule cases)
haftmann@26358:    apply (blast intro: prod_fun)
haftmann@26358:   apply blast
haftmann@26358:   done
haftmann@26358: 
haftmann@26358: definition
haftmann@26358:   apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
haftmann@26358: where
haftmann@26358:   [code func del]: "apfst f = prod_fun f id"
haftmann@26358: 
haftmann@26358: definition
haftmann@26358:   apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
haftmann@26358: where
haftmann@26358:   [code func del]: "apsnd f = prod_fun id f"
haftmann@26358: 
haftmann@26358: lemma apfst_conv [simp, code]:
haftmann@26358:   "apfst f (x, y) = (f x, y)" 
haftmann@26358:   by (simp add: apfst_def)
haftmann@26358: 
haftmann@26358: lemma upd_snd_conv [simp, code]:
haftmann@26358:   "apsnd f (x, y) = (x, f y)" 
haftmann@26358:   by (simp add: apsnd_def)
haftmann@21195: 
haftmann@21195: 
haftmann@26358: text {*
haftmann@26358:   Disjoint union of a family of sets -- Sigma.
haftmann@26358: *}
haftmann@26358: 
haftmann@26358: definition  Sigma :: "['a set, 'a => 'b set] => ('a \<times> 'b) set" where
haftmann@26358:   Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358: 
haftmann@26358: abbreviation
haftmann@26358:   Times :: "['a set, 'b set] => ('a * 'b) set"
haftmann@26358:     (infixr "<*>" 80) where
haftmann@26358:   "A <*> B == Sigma A (%_. B)"
haftmann@26358: 
haftmann@26358: notation (xsymbols)
haftmann@26358:   Times  (infixr "\<times>" 80)
berghofe@15394: 
haftmann@26358: notation (HTML output)
haftmann@26358:   Times  (infixr "\<times>" 80)
haftmann@26358: 
haftmann@26358: syntax
haftmann@26358:   "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358: 
haftmann@26358: translations
haftmann@26358:   "SIGMA x:A. B" == "Product_Type.Sigma A (%x. B)"
haftmann@26358: 
haftmann@26358: lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358:   by (unfold Sigma_def) blast
haftmann@26358: 
haftmann@26358: lemma SigmaE [elim!]:
haftmann@26358:     "[| c: Sigma A B;
haftmann@26358:         !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358:      |] ==> P"
haftmann@26358:   -- {* The general elimination rule. *}
haftmann@26358:   by (unfold Sigma_def) blast
haftmann@20588: 
haftmann@26358: text {*
haftmann@26358:   Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358:   eigenvariables.
haftmann@26358: *}
haftmann@26358: 
haftmann@26358: lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma SigmaE2:
haftmann@26358:     "[| (a, b) : Sigma A B;
haftmann@26358:         [| a:A;  b:B(a) |] ==> P
haftmann@26358:      |] ==> P"
haftmann@26358:   by blast
haftmann@20588: 
haftmann@26358: lemma Sigma_cong:
haftmann@26358:      "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358:       \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358:   by auto
haftmann@26358: 
haftmann@26358: lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358:   by auto
haftmann@21908: 
haftmann@26358: lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358:   by auto
haftmann@26358: 
haftmann@26358: lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358:   by auto
haftmann@26358: 
haftmann@26358: lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358:   by (blast elim: equalityE)
haftmann@20588: 
haftmann@26358: lemma SetCompr_Sigma_eq:
haftmann@26358:     "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma UN_Times_distrib:
haftmann@26358:   "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358:   -- {* Suggested by Pierre Chartier *}
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma split_paired_Ball_Sigma [simp,noatp]:
haftmann@26358:     "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma split_paired_Bex_Sigma [simp,noatp]:
haftmann@26358:     "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358:   by blast
haftmann@21908: 
haftmann@26358: lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358:   by blast
haftmann@21908: 
haftmann@26358: lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: text {*
haftmann@26358:   Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358:   matching, especially when the rules are re-oriented.
haftmann@26358: *}
haftmann@21908: 
haftmann@26358: lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
haftmann@26358:   by blast
haftmann@26358: 
haftmann@26358: 
haftmann@26358: subsubsection {* Code generator setup *}
haftmann@21908: 
haftmann@20588: instance * :: (eq, eq) eq ..
haftmann@20588: 
haftmann@20588: lemma [code func]:
haftmann@21454:   "(x1\<Colon>'a\<Colon>eq, y1\<Colon>'b\<Colon>eq) = (x2, y2) \<longleftrightarrow> x1 = x2 \<and> y1 = y2" by auto
haftmann@20588: 
haftmann@24844: lemma split_case_cert:
haftmann@24844:   assumes "CASE \<equiv> split f"
haftmann@24844:   shows "CASE (a, b) \<equiv> f a b"
haftmann@24844:   using assms by simp
haftmann@24844: 
haftmann@24844: setup {*
haftmann@24844:   Code.add_case @{thm split_case_cert}
haftmann@24844: *}
haftmann@24844: 
haftmann@21908: code_type *
haftmann@21908:   (SML infix 2 "*")
haftmann@21908:   (OCaml infix 2 "*")
haftmann@21908:   (Haskell "!((_),/ (_))")
haftmann@21908: 
haftmann@20588: code_instance * :: eq
haftmann@20588:   (Haskell -)
haftmann@20588: 
haftmann@21908: code_const "op = \<Colon> 'a\<Colon>eq \<times> 'b\<Colon>eq \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
haftmann@20588:   (Haskell infixl 4 "==")
haftmann@20588: 
haftmann@21908: code_const Pair
haftmann@21908:   (SML "!((_),/ (_))")
haftmann@21908:   (OCaml "!((_),/ (_))")
haftmann@21908:   (Haskell "!((_),/ (_))")
haftmann@20588: 
haftmann@22389: code_const fst and snd
haftmann@22389:   (Haskell "fst" and "snd")
haftmann@22389: 
berghofe@15394: types_code
berghofe@15394:   "*"     ("(_ */ _)")
berghofe@16770: attach (term_of) {*
berghofe@25885: fun term_of_id_42 aF aT bF bT (x, y) = HOLogic.pair_const aT bT $ aF x $ bF y;
berghofe@16770: *}
berghofe@16770: attach (test) {*
berghofe@25885: fun gen_id_42 aG aT bG bT i =
berghofe@25885:   let
berghofe@25885:     val (x, t) = aG i;
berghofe@25885:     val (y, u) = bG i
berghofe@25885:   in ((x, y), fn () => HOLogic.pair_const aT bT $ t () $ u ()) end;
berghofe@16770: *}
berghofe@15394: 
berghofe@18706: consts_code
berghofe@18706:   "Pair"    ("(_,/ _)")
berghofe@18706: 
haftmann@21908: setup {*
haftmann@21908: 
haftmann@21908: let
haftmann@18013: 
haftmann@19039: fun strip_abs_split 0 t = ([], t)
haftmann@19039:   | strip_abs_split i (Abs (s, T, t)) =
haftmann@18013:       let
haftmann@18013:         val s' = Codegen.new_name t s;
haftmann@18013:         val v = Free (s', T)
haftmann@19039:       in apfst (cons v) (strip_abs_split (i-1) (subst_bound (v, t))) end
haftmann@19039:   | strip_abs_split i (u as Const ("split", _) $ t) = (case strip_abs_split (i+1) t of
berghofe@15394:         (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
berghofe@15394:       | _ => ([], u))
haftmann@19039:   | strip_abs_split i t = ([], t);
haftmann@18013: 
berghofe@16634: fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of
berghofe@16634:     (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
berghofe@15394:     let
berghofe@15394:       fun dest_let (l as Const ("Let", _) $ t $ u) =
haftmann@19039:           (case strip_abs_split 1 u of
berghofe@15394:              ([p], u') => apfst (cons (p, t)) (dest_let u')
berghofe@15394:            | _ => ([], l))
berghofe@15394:         | dest_let t = ([], t);
berghofe@15394:       fun mk_code (gr, (l, r)) =
berghofe@15394:         let
berghofe@16634:           val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l);
berghofe@16634:           val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r);
berghofe@15394:         in (gr2, (pl, pr)) end
berghofe@16634:     in case dest_let (t1 $ t2 $ t3) of
skalberg@15531:         ([], _) => NONE
berghofe@15394:       | (ps, u) =>
berghofe@15394:           let
berghofe@15394:             val (gr1, qs) = foldl_map mk_code (gr, ps);
berghofe@16634:             val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
berghofe@16634:             val (gr3, pargs) = foldl_map
berghofe@17021:               (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
berghofe@15394:           in
berghofe@16634:             SOME (gr3, Codegen.mk_app brack
berghofe@16634:               (Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat
berghofe@16634:                   (separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
berghofe@16634:                     [Pretty.block [Pretty.str "val ", pl, Pretty.str " =",
berghofe@16634:                        Pretty.brk 1, pr]]) qs))),
berghofe@16634:                 Pretty.brk 1, Pretty.str "in ", pu,
berghofe@16634:                 Pretty.brk 1, Pretty.str "end"])) pargs)
berghofe@15394:           end
berghofe@15394:     end
berghofe@16634:   | _ => NONE);
berghofe@15394: 
berghofe@16634: fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of
berghofe@16634:     (t1 as Const ("split", _), t2 :: ts) =>
haftmann@19039:       (case strip_abs_split 1 (t1 $ t2) of
berghofe@16634:          ([p], u) =>
berghofe@16634:            let
berghofe@16634:              val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p);
berghofe@16634:              val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
berghofe@16634:              val (gr3, pargs) = foldl_map
berghofe@17021:                (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
berghofe@16634:            in
berghofe@16634:              SOME (gr2, Codegen.mk_app brack
berghofe@16634:                (Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>",
berghofe@16634:                  Pretty.brk 1, pu, Pretty.str ")"]) pargs)
berghofe@16634:            end
berghofe@16634:        | _ => NONE)
berghofe@16634:   | _ => NONE);
berghofe@15394: 
haftmann@21908: in
haftmann@21908: 
haftmann@20105:   Codegen.add_codegen "let_codegen" let_codegen
haftmann@20105:   #> Codegen.add_codegen "split_codegen" split_codegen
berghofe@15394: 
haftmann@21908: end
berghofe@15394: *}
berghofe@15394: 
haftmann@24699: 
haftmann@24699: subsection {* Legacy bindings *}
haftmann@24699: 
haftmann@21908: ML {*
paulson@15404: val Collect_split = thm "Collect_split";
paulson@15404: val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
paulson@15404: val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
paulson@15404: val PairE = thm "PairE";
paulson@15404: val Pair_Rep_inject = thm "Pair_Rep_inject";
paulson@15404: val Pair_def = thm "Pair_def";
paulson@15404: val Pair_eq = thm "Pair_eq";
paulson@15404: val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
paulson@15404: val ProdI = thm "ProdI";
paulson@15404: val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
paulson@15404: val SigmaD1 = thm "SigmaD1";
paulson@15404: val SigmaD2 = thm "SigmaD2";
paulson@15404: val SigmaE = thm "SigmaE";
paulson@15404: val SigmaE2 = thm "SigmaE2";
paulson@15404: val SigmaI = thm "SigmaI";
paulson@15404: val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
paulson@15404: val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
paulson@15404: val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
paulson@15404: val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
paulson@15404: val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
paulson@15404: val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
paulson@15404: val Sigma_Union = thm "Sigma_Union";
paulson@15404: val Sigma_def = thm "Sigma_def";
paulson@15404: val Sigma_empty1 = thm "Sigma_empty1";
paulson@15404: val Sigma_empty2 = thm "Sigma_empty2";
paulson@15404: val Sigma_mono = thm "Sigma_mono";
paulson@15404: val The_split = thm "The_split";
paulson@15404: val The_split_eq = thm "The_split_eq";
paulson@15404: val The_split_eq = thm "The_split_eq";
paulson@15404: val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
paulson@15404: val Times_Int_distrib1 = thm "Times_Int_distrib1";
paulson@15404: val Times_Un_distrib1 = thm "Times_Un_distrib1";
paulson@15404: val Times_eq_cancel2 = thm "Times_eq_cancel2";
paulson@15404: val Times_subset_cancel2 = thm "Times_subset_cancel2";
paulson@15404: val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
paulson@15404: val UN_Times_distrib = thm "UN_Times_distrib";
paulson@15404: val Unity_def = thm "Unity_def";
paulson@15404: val cond_split_eta = thm "cond_split_eta";
paulson@15404: val fst_conv = thm "fst_conv";
paulson@15404: val fst_def = thm "fst_def";
paulson@15404: val fst_eqD = thm "fst_eqD";
paulson@15404: val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
paulson@15404: val mem_Sigma_iff = thm "mem_Sigma_iff";
paulson@15404: val mem_splitE = thm "mem_splitE";
paulson@15404: val mem_splitI = thm "mem_splitI";
paulson@15404: val mem_splitI2 = thm "mem_splitI2";
paulson@15404: val prod_eqI = thm "prod_eqI";
paulson@15404: val prod_fun = thm "prod_fun";
paulson@15404: val prod_fun_compose = thm "prod_fun_compose";
paulson@15404: val prod_fun_def = thm "prod_fun_def";
paulson@15404: val prod_fun_ident = thm "prod_fun_ident";
paulson@15404: val prod_fun_imageE = thm "prod_fun_imageE";
paulson@15404: val prod_fun_imageI = thm "prod_fun_imageI";
paulson@15404: val prod_induct = thm "prod_induct";
paulson@15404: val snd_conv = thm "snd_conv";
paulson@15404: val snd_def = thm "snd_def";
paulson@15404: val snd_eqD = thm "snd_eqD";
paulson@15404: val split = thm "split";
paulson@15404: val splitD = thm "splitD";
paulson@15404: val splitD' = thm "splitD'";
paulson@15404: val splitE = thm "splitE";
paulson@15404: val splitE' = thm "splitE'";
paulson@15404: val splitE2 = thm "splitE2";
paulson@15404: val splitI = thm "splitI";
paulson@15404: val splitI2 = thm "splitI2";
paulson@15404: val splitI2' = thm "splitI2'";
paulson@15404: val split_beta = thm "split_beta";
paulson@15404: val split_conv = thm "split_conv";
paulson@15404: val split_def = thm "split_def";
paulson@15404: val split_eta = thm "split_eta";
paulson@15404: val split_eta_SetCompr = thm "split_eta_SetCompr";
paulson@15404: val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
paulson@15404: val split_paired_All = thm "split_paired_All";
paulson@15404: val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
paulson@15404: val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
paulson@15404: val split_paired_Ex = thm "split_paired_Ex";
paulson@15404: val split_paired_The = thm "split_paired_The";
paulson@15404: val split_paired_all = thm "split_paired_all";
paulson@15404: val split_part = thm "split_part";
paulson@15404: val split_split = thm "split_split";
paulson@15404: val split_split_asm = thm "split_split_asm";
paulson@15404: val split_tupled_all = thms "split_tupled_all";
paulson@15404: val split_weak_cong = thm "split_weak_cong";
paulson@15404: val surj_pair = thm "surj_pair";
paulson@15404: val surjective_pairing = thm "surjective_pairing";
paulson@15404: val unit_abs_eta_conv = thm "unit_abs_eta_conv";
paulson@15404: val unit_all_eq1 = thm "unit_all_eq1";
paulson@15404: val unit_all_eq2 = thm "unit_all_eq2";
paulson@15404: val unit_eq = thm "unit_eq";
paulson@15404: *}
paulson@15404: 
haftmann@24699: 
haftmann@24699: subsection {* Further inductive packages *}
haftmann@24699: 
haftmann@24699: use "Tools/inductive_realizer.ML"
haftmann@24699: setup InductiveRealizer.setup
haftmann@24699: 
haftmann@24699: use "Tools/inductive_set_package.ML"
haftmann@24699: setup InductiveSetPackage.setup
haftmann@24699: 
haftmann@24699: use "Tools/datatype_realizer.ML"
haftmann@24699: setup DatatypeRealizer.setup
haftmann@24699: 
nipkow@10213: end




