# Theory Product_Type

Up to index of Isabelle/HOL

theory Product_Type
imports Fun
uses (Tools/split_rule.ML)
begin

    (*  Title:      HOL/Product_Type.thy
        ID:         $Id: Product_Type.thy,v 1.39 2005/08/16 16:53:12 paulson Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1992  University of Cambridge
    *)

    header {* Cartesian products *}

    theory Product_Type
    imports Fun
    uses ("Tools/split_rule.ML")
    begin

    subsection {* Unit *}

    typedef unit = "{True}"
    proof
      show "True : ?unit" by blast
    qed

    constdefs
      Unity :: unit    ("'(')")
      "() == Abs_unit True"

    lemma unit_eq: "u = ()"
      by (induct u) (simp add: unit_def Unity_def)

    text {*
      Simplification procedure for @{thm [source] unit_eq}.  Cannot use
      this rule directly --- it loops!
    *}

    ML_setup {*
      val unit_eq_proc =
        let val unit_meta_eq = mk_meta_eq (thm "unit_eq") in
          Simplifier.simproc (Theory.sign_of (the_context ())) "unit_eq" ["x::unit"]
          (fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq)
        end;

      Addsimprocs [unit_eq_proc];
    *}

    lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
      by simp

    lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
      by (rule triv_forall_equality)

    lemma unit_induct [induct type: unit]: "P () ==> P x"
      by simp

    text {*
      This rewrite counters the effect of @{text unit_eq_proc} on @{term
      [source] "%u::unit. f u"}, replacing it by @{term [source]
      f} rather than by @{term [source] "%u. f ()"}.
    *}

    lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f"
      by (rule ext) simp

    subsection {* Pairs *}

    subsubsection {* Type definition *}

    constdefs
      Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
      "Pair_Rep == (%a b. %x y. x=a & y=b)"

    global

    typedef (Prod)
      ('a, 'b) "*"    (infixr 20)
        = "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}"
    proof
      fix a b show "Pair_Rep a b : ?Prod"
        by blast
    qed

    syntax (xsymbols)
      "*"      :: "[type, type] => type"         ("(_ ×/ _)" [21, 20] 20)
    syntax (HTML output)
      "*"      :: "[type, type] => type"         ("(_ ×/ _)" [21, 20] 20)

    local

    subsubsection {* Abstract constants and syntax *}

    global

    consts
      fst      :: "'a * 'b => 'a"
      snd      :: "'a * 'b => 'b"
      split    :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
      curry    :: "['a * 'b => 'c, 'a, 'b] => 'c"
      prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
      Pair     :: "['a, 'b] => 'a * 'b"
      Sigma    :: "['a set, 'a => 'b set] => ('a * 'b) set"

    local

    text {*
      Patterns -- extends pre-defined type @{typ pttrn} used in
      abstractions.
    *}

    nonterminals
      tuple_args patterns

    syntax
      "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
      "_tuple_arg"  :: "'a => tuple_args"                   ("_")
      "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
      "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
      ""            :: "pttrn => patterns"                  ("_")
      "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
      "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10)
      "@Times" ::"['a set,  'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80)

    translations
      "(x, y)"       == "Pair x y"
      "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
      "%(x,y,zs).b"  == "split(%x (y,zs).b)"
      "%(x,y).b"     == "split(%x y. b)"
      "_abs (Pair x y) t" => "%(x,y).t"
      (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
         The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)

      "SIGMA x:A. B" => "Sigma A (%x. B)"
      "A <*> B"      => "Sigma A (_K B)"

    (* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*)
    (* works best with enclosing "let", if "let" does not avoid eta-contraction   *)
    print_translation {*
    let fun split_tr' [Abs (x,T,t as (Abs abs))] =
          (* split (%x y. t) => %(x,y) t *)
          let val (y,t') = atomic_abs_tr' abs;
              val (x',t'') = atomic_abs_tr' (x,T,t');

          in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end
        | split_tr' [Abs (x,T,(s as Const ("split",_)$t))] =
           (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
           let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t];
               val (x',t'') = atomic_abs_tr' (x,T,t');
           in Syntax.const "_abs"$
               (Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end
        | split_tr' [Const ("split",_)$t] =
           (* split (split (%x y z. t)) => %((x,y),z). t *)
           split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
        | split_tr' [Const ("_abs",_)$x_y$(Abs abs)] =
           (* split (%pttrn z. t) => %(pttrn,z). t *)
           let val (z,t) = atomic_abs_tr' abs;
           in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end
        | split_tr' _ =  raise Match;
    in [("split", split_tr')]
    end
    *}

    (* print "split f" as "λ(x,y). f x y" and "split (λx. f x)" as "λ(x,y). f x y" *)
    typed_print_translation {*
    let
      fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match
        | split_guess_names_tr' _ T  [Abs (x,xT,t)] =
            (case (head_of t) of
               Const ("split",_) => raise Match
             | _ => let
                      val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
                      val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0);
                      val (x',t'') = atomic_abs_tr' (x,xT,t');
                    in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
        | split_guess_names_tr' _ T [t] =
           (case (head_of t) of
               Const ("split",_) => raise Match
             | _ => let
                      val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match;
                      val (y,t') =
                            atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0);
                      val (x',t'') = atomic_abs_tr' ("x",xT,t');
                    in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end)
        | split_guess_names_tr' _ _ _ = raise Match;
    in [("split", split_guess_names_tr')]
    end
    *}

    text{*Deleted x-symbol and html support using @{text"Σ"} (Sigma) because of the danger of confusion with Sum.*}

    syntax (xsymbols)
      "@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ × _" [81, 80] 80)

    syntax (HTML output)
      "@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ × _" [81, 80] 80)

    print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *}

    subsubsection {* Definitions *}

    defs
      Pair_def:     "Pair a b == Abs_Prod(Pair_Rep a b)"
      fst_def:      "fst p == THE a. EX b. p = (a, b)"
      snd_def:      "snd p == THE b. EX a. p = (a, b)"
      split_def:    "split == (%c p. c (fst p) (snd p))"
      curry_def:    "curry == (%c x y. c (x,y))"
      prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))"
      Sigma_def:    "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"

    subsubsection {* Lemmas and proof tool setup *}

    lemma ProdI: "Pair_Rep a b : Prod"
      by (unfold Prod_def) blast

    lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'"
      apply (unfold Pair_Rep_def)
      apply (drule fun_cong [THEN fun_cong], blast)
      done

    lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
      apply (rule inj_on_inverseI)
      apply (erule Abs_Prod_inverse)
      done

    lemma Pair_inject:
      "(a, b) = (a', b') ==> (a = a' ==> b = b' ==> R) ==> R"
    proof -
      case rule_context [unfolded Pair_def]
      show ?thesis
        apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
        apply (rule rule_context ProdI)+
        .
    qed

    lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')"
      by (blast elim!: Pair_inject)

    lemma fst_conv [simp]: "fst (a, b) = a"
      by (unfold fst_def) blast

    lemma snd_conv [simp]: "snd (a, b) = b"
      by (unfold snd_def) blast

    lemma fst_eqD: "fst (x, y) = a ==> x = a"
      by simp

    lemma snd_eqD: "snd (x, y) = a ==> y = a"
      by simp

    lemma PairE_lemma: "EX x y. p = (x, y)"
      apply (unfold Pair_def)
      apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
      apply (erule exE, erule exE, rule exI, rule exI)
      apply (rule Rep_Prod_inverse [symmetric, THEN trans])
      apply (erule arg_cong)
      done

    lemma PairE [cases type: *]: "(!!x y. p = (x, y) ==> Q) ==> Q"
      by (insert PairE_lemma [of p]) blast

    ML {*
      local val PairE = thm "PairE" in
        fun pair_tac s =
          EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac];
      end;
    *}

    lemma surjective_pairing: "p = (fst p, snd p)"
      -- {* Do not add as rewrite rule: invalidates some proofs in IMP *}
      by (cases p) simp

    lemmas pair_collapse = surjective_pairing [symmetric]
    declare pair_collapse [simp]

    lemma surj_pair [simp]: "EX x y. z = (x, y)"
      apply (rule exI)
      apply (rule exI)
      apply (rule surjective_pairing)
      done

    lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
    proof
      fix a b
      assume "!!x. PROP P x"
      thus "PROP P (a, b)" .
    next
      fix x
      assume "!!a b. PROP P (a, b)"
      hence "PROP P (fst x, snd x)" .
      thus "PROP P x" by simp
    qed

    lemmas split_tupled_all = split_paired_all unit_all_eq2

    text {*
      The rule @{thm [source] split_paired_all} does not work with the
      Simplifier because it also affects premises in congrence rules,
      where this can lead to premises of the form @{text "!!a b. ... =
      ?P(a, b)"} which cannot be solved by reflexivity.
    *}

    ML_setup {*
      (* replace parameters of product type by individual component parameters *)
      val safe_full_simp_tac = generic_simp_tac true (true, false, false);
      local (* filtering with exists_paired_all is an essential optimization *)
        fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
              can HOLogic.dest_prodT T orelse exists_paired_all t
          | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
          | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
          | exists_paired_all _ = false;
        val ss = HOL_basic_ss
          addsimps [thm "split_paired_all", thm "unit_all_eq2", thm "unit_abs_eta_conv"]
          addsimprocs [unit_eq_proc];
      in
        val split_all_tac = SUBGOAL (fn (t, i) =>
          if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
        val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
          if exists_paired_all t then full_simp_tac ss i else no_tac);
        fun split_all th =
       if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th;
      end;

    claset_ref() := claset() addSbefore ("split_all_tac", split_all_tac);
    *}

    lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
      -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
      by fast

    lemma curry_split [simp]: "curry (split f) = f"
      by (simp add: curry_def split_def)

    lemma split_curry [simp]: "split (curry f) = f"
      by (simp add: curry_def split_def)

    lemma curryI [intro!]: "f (a,b) ==> curry f a b"
      by (simp add: curry_def)

    lemma curryD [dest!]: "curry f a b ==> f (a,b)"
      by (simp add: curry_def)

    lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"
      by (simp add: curry_def)

    lemma curry_conv [simp]: "curry f a b = f (a,b)"
      by (simp add: curry_def)

    lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
      by fast

    lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
      by fast

    lemma split_conv [simp]: "split c (a, b) = c a b"
      by (simp add: split_def)

    lemmas split = split_conv  -- {* for backwards compatibility *}

    lemmas splitI = split_conv [THEN iffD2, standard]
    lemmas splitD = split_conv [THEN iffD1, standard]

    lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
      -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
      apply (rule ext)
      apply (tactic {* pair_tac "x" 1 *}, simp)
      done

    lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
      -- {* Can't be added to simpset: loops! *}
      by (simp add: split_Pair_apply)

    lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
      by (simp add: split_def)

    lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
    by (simp only: split_tupled_all, simp)

    lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"
      by (simp add: Pair_fst_snd_eq)

    lemma split_weak_cong: "p = q ==> split c p = split c q"
      -- {* Prevents simplification of @{term c}: much faster *}
      by (erule arg_cong)

    lemma split_eta: "(%(x, y). f (x, y)) = f"
      apply (rule ext)
      apply (simp only: split_tupled_all)
      apply (rule split_conv)
      done

    lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
      by (simp add: split_eta)

    text {*
      Simplification procedure for @{thm [source] cond_split_eta}.  Using
      @{thm [source] split_eta} as a rewrite rule is not general enough,
      and using @{thm [source] cond_split_eta} directly would render some
      existing proofs very inefficient; similarly for @{text
      split_beta}. *}

    ML_setup {*

    local
      val cond_split_eta = thm "cond_split_eta";
      fun  Pair_pat k 0 (Bound m) = (m = k)
      |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
                            m = k+i andalso Pair_pat k (i-1) t
      |    Pair_pat _ _ _ = false;
      fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
      |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
      |   no_args k i (Bound m) = m < k orelse m > k+i
      |   no_args _ _ _ = true;
      fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE
      |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
      |   split_pat tp i _ = NONE;
      fun metaeq thy ss lhs rhs = mk_meta_eq (Tactic.prove thy [] []
            (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
            (K (simp_tac (Simplifier.inherit_bounds ss HOL_basic_ss addsimps [cond_split_eta]) 1)));

      fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
      |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
                            (beta_term_pat k i t andalso beta_term_pat k i u)
      |   beta_term_pat k i t = no_args k i t;
      fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
      |    eta_term_pat _ _ _ = false;
      fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
      |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
                                  else (subst arg k i t $ subst arg k i u)
      |   subst arg k i t = t;
      fun beta_proc thy ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
            (case split_pat beta_term_pat 1 t of
            SOME (i,f) => SOME (metaeq thy ss s (subst arg 0 i f))
            | NONE => NONE)
      |   beta_proc _ _ _ = NONE;
      fun eta_proc thy ss (s as Const ("split", _) $ Abs (_, _, t)) =
            (case split_pat eta_term_pat 1 t of
              SOME (_,ft) => SOME (metaeq thy ss s (let val (f $ arg) = ft in f end))
            | NONE => NONE)
      |   eta_proc _ _ _ = NONE;
    in
      val split_beta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
        "split_beta" ["split f z"] beta_proc;
      val split_eta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
        "split_eta" ["split f"] eta_proc;
    end;

    Addsimprocs [split_beta_proc, split_eta_proc];
    *}

    lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
      by (subst surjective_pairing, rule split_conv)

    lemma split_split: "R (split c p) = (ALL x y. p = (x, y) --> R (c x y))"
      -- {* For use with @{text split} and the Simplifier. *}
      by (insert surj_pair [of p], clarify, simp)

    text {*
      @{thm [source] split_split} could be declared as @{text "[split]"}
      done after the Splitter has been speeded up significantly;
      precompute the constants involved and don't do anything unless the
      current goal contains one of those constants.
    *}

    lemma split_split_asm: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
    by (subst split_split, simp)

    text {*
      \medskip @{term split} used as a logical connective or set former.

      \medskip These rules are for use with @{text blast}; could instead
      call @{text simp} using @{thm [source] split} as rewrite. *}

    lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
      apply (simp only: split_tupled_all)
      apply (simp (no_asm_simp))
      done

    lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
      apply (simp only: split_tupled_all)
      apply (simp (no_asm_simp))
      done

    lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
      by (induct p) (auto simp add: split_def)

    lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
      by (induct p) (auto simp add: split_def)

    lemma splitE2:
      "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
    proof -
      assume q: "Q (split P z)"
      assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
      show R
        apply (rule r surjective_pairing)+
        apply (rule split_beta [THEN subst], rule q)
        done
    qed

    lemma splitD': "split R (a,b) c ==> R a b c"
      by simp

    lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
      by simp

    lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
    by (simp only: split_tupled_all, simp)

    lemma mem_splitE: "[| z: split c p; !!x y. [| p = (x,y); z: c x y |] ==> Q |] ==> Q"
    proof -
      case rule_context [unfolded split_def]
      show ?thesis by (rule rule_context surjective_pairing)+
    qed

    declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
    declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]

    ML_setup {*
    local (* filtering with exists_p_split is an essential optimization *)
      fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true
        | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
        | exists_p_split (Abs (_, _, t)) = exists_p_split t
        | exists_p_split _ = false;
      val ss = HOL_basic_ss addsimps [thm "split_conv"];
    in
    val split_conv_tac = SUBGOAL (fn (t, i) =>
        if exists_p_split t then safe_full_simp_tac ss i else no_tac);
    end;
    (* This prevents applications of splitE for already splitted arguments leading
       to quite time-consuming computations (in particular for nested tuples) *)
    claset_ref() := claset() addSbefore ("split_conv_tac", split_conv_tac);
    *}

    lemma split_eta_SetCompr [simp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
    by (rule ext, fast)

    lemma split_eta_SetCompr2 [simp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
    by (rule ext, fast)

    lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
      -- {* Allows simplifications of nested splits in case of independent predicates. *}
      apply (rule ext, blast)
      done

    (* Do NOT make this a simp rule as it
       a) only helps in special situations
       b) can lead to nontermination in the presence of split_def
    *)
    lemma split_comp_eq:
    "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
    by (rule ext, auto)

    lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
      by blast

    (*
    the following  would be slightly more general,
    but cannot be used as rewrite rule:
    ### Cannot add premise as rewrite rule because it contains (type) unknowns:
    ### ?y = .x
    Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
    by (rtac some_equality 1)
    by ( Simp_tac 1)
    by (split_all_tac 1)
    by (Asm_full_simp_tac 1)
    qed "The_split_eq";
    *)

    lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
      by auto

    text {*
      \bigskip @{term prod_fun} --- action of the product functor upon
      functions.
    *}

    lemma prod_fun [simp]: "prod_fun f g (a, b) = (f a, g b)"
      by (simp add: prod_fun_def)

    lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
      apply (rule ext)
      apply (tactic {* pair_tac "x" 1 *}, simp)
      done

    lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
      apply (rule ext)
      apply (tactic {* pair_tac "z" 1 *}, simp)
      done

    lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
      apply (rule image_eqI)
      apply (rule prod_fun [symmetric], assumption)
      done

    lemma prod_fun_imageE [elim!]:
      "[| c: (prod_fun f g)`r;  !!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P
        |] ==> P"
    proof -
      case rule_context
      assume major: "c: (prod_fun f g)`r"
      show ?thesis
        apply (rule major [THEN imageE])
        apply (rule_tac p = x in PairE)
        apply (rule rule_context)
         prefer 2
         apply blast
        apply (blast intro: prod_fun)
        done
    qed

    constdefs
      upd_fst :: "('a => 'c) => 'a * 'b => 'c * 'b"
     "upd_fst f == prod_fun f id"

      upd_snd :: "('b => 'c) => 'a * 'b => 'a * 'c"
     "upd_snd f == prod_fun id f"

    lemma upd_fst_conv [simp]: "upd_fst f (x,y) = (f x,y)"
    by (simp add: upd_fst_def)

    lemma upd_snd_conv [simp]: "upd_snd f (x,y) = (x,f y)"
    by (simp add: upd_snd_def)

    text {*
      \bigskip Disjoint union of a family of sets -- Sigma.
    *}

    lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
      by (unfold Sigma_def) blast

    lemma SigmaE [elim!]:
        "[| c: Sigma A B;
            !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
         |] ==> P"
      -- {* The general elimination rule. *}
      by (unfold Sigma_def) blast

    text {*
      Elimination of @{term "(a, b) : A × B"} -- introduces no
      eigenvariables.
    *}

    lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
    by blast

    lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
    by blast

    lemma SigmaE2:
        "[| (a, b) : Sigma A B;
            [| a:A;  b:B(a) |] ==> P
         |] ==> P"
      by blast

    lemma Sigma_cong:
         "[|A = B; !!x. x ∈ B ==> C x = D x|]
          ==> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
    by auto

    lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
      by blast

    lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
      by blast

    lemma Sigma_empty2 [simp]: "A <*> {} = {}"
      by blast

    lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
      by auto

    lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
      by auto

    lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
      by auto

    lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
      by blast

    lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
      by blast

    lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
      by (blast elim: equalityE)

    lemma SetCompr_Sigma_eq:
        "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
      by blast

    text {*
      \bigskip Complex rules for Sigma.
    *}

    lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
      by blast

    lemma UN_Times_distrib:
      "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
      -- {* Suggested by Pierre Chartier *}
      by blast

    lemma split_paired_Ball_Sigma [simp]:
        "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
      by blast

    lemma split_paired_Bex_Sigma [simp]:
        "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
      by blast

    lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
      by blast

    lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
      by blast

    lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
      by blast

    lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
      by blast

    lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
      by blast

    lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
      by blast

    lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
      by blast

    text {*
      Non-dependent versions are needed to avoid the need for higher-order
      matching, especially when the rules are re-oriented.
    *}

    lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
      by blast

    lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
      by blast

    lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
      by blast

    lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
      apply (rule_tac x = "(a, b)" in image_eqI)
       apply auto
      done

    text {*
      Setup of internal @{text split_rule}.
    *}

    constdefs
      internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
      "internal_split == split"

    lemma internal_split_conv: "internal_split c (a, b) = c a b"
      by (simp only: internal_split_def split_conv)

    hide const internal_split

    use "Tools/split_rule.ML"
    setup SplitRule.setup

    subsection {* Code generator setup *}

    types_code
      "*"     ("(_ */ _)")
    attach (term_of) {*
    fun term_of_id_42 f T g U (x, y) = HOLogic.pair_const T U $ f x $ g y;
    *}
    attach (test) {*
    fun gen_id_42 aG bG i = (aG i, bG i);
    *}

    consts_code
      "Pair"    ("(_,/ _)")
      "fst"     ("fst")
      "snd"     ("snd")

    ML {*
    local

    fun strip_abs 0 t = ([], t)
      | strip_abs i (Abs (s, T, t)) =
        let
          val s' = Codegen.new_name t s;
          val v = Free (s', T)
        in apfst (cons v) (strip_abs (i-1) (subst_bound (v, t))) end
      | strip_abs i (u as Const ("split", _) $ t) = (case strip_abs (i+1) t of
            (v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u)
          | _ => ([], u))
      | strip_abs i t = ([], t);

    fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of
        (t1 as Const ("Let", _), t2 :: t3 :: ts) =>
        let
          fun dest_let (l as Const ("Let", _) $ t $ u) =
              (case strip_abs 1 u of
                 ([p], u') => apfst (cons (p, t)) (dest_let u')
               | _ => ([], l))
            | dest_let t = ([], t);
          fun mk_code (gr, (l, r)) =
            let
              val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l);
              val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r);
            in (gr2, (pl, pr)) end
        in case dest_let (t1 $ t2 $ t3) of
            ([], _) => NONE
          | (ps, u) =>
              let
                val (gr1, qs) = foldl_map mk_code (gr, ps);
                val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
                val (gr3, pargs) = foldl_map
                  (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
              in
                SOME (gr3, Codegen.mk_app brack
                  (Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat
                      (separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) =>
                        [Pretty.block [Pretty.str "val ", pl, Pretty.str " =",
                           Pretty.brk 1, pr]]) qs))),
                    Pretty.brk 1, Pretty.str "in ", pu,
                    Pretty.brk 1, Pretty.str "end"])) pargs)
              end
        end
      | _ => NONE);

    fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of
        (t1 as Const ("split", _), t2 :: ts) =>
          (case strip_abs 1 (t1 $ t2) of
             ([p], u) =>
               let
                 val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p);
                 val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u);
                 val (gr3, pargs) = foldl_map
                   (Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts)
               in
                 SOME (gr2, Codegen.mk_app brack
                   (Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>",
                     Pretty.brk 1, pu, Pretty.str ")"]) pargs)
               end
           | _ => NONE)
      | _ => NONE);

    in

    val prod_codegen_setup =
      [Codegen.add_codegen "let_codegen" let_codegen,
       Codegen.add_codegen "split_codegen" split_codegen];

    end;
    *}

    setup prod_codegen_setup

    ML
    {*
    val Collect_split = thm "Collect_split";
    val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1";
    val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2";
    val PairE = thm "PairE";
    val PairE_lemma = thm "PairE_lemma";
    val Pair_Rep_inject = thm "Pair_Rep_inject";
    val Pair_def = thm "Pair_def";
    val Pair_eq = thm "Pair_eq";
    val Pair_fst_snd_eq = thm "Pair_fst_snd_eq";
    val Pair_inject = thm "Pair_inject";
    val ProdI = thm "ProdI";
    val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq";
    val SigmaD1 = thm "SigmaD1";
    val SigmaD2 = thm "SigmaD2";
    val SigmaE = thm "SigmaE";
    val SigmaE2 = thm "SigmaE2";
    val SigmaI = thm "SigmaI";
    val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1";
    val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2";
    val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1";
    val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2";
    val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1";
    val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2";
    val Sigma_Union = thm "Sigma_Union";
    val Sigma_def = thm "Sigma_def";
    val Sigma_empty1 = thm "Sigma_empty1";
    val Sigma_empty2 = thm "Sigma_empty2";
    val Sigma_mono = thm "Sigma_mono";
    val The_split = thm "The_split";
    val The_split_eq = thm "The_split_eq";
    val The_split_eq = thm "The_split_eq";
    val Times_Diff_distrib1 = thm "Times_Diff_distrib1";
    val Times_Int_distrib1 = thm "Times_Int_distrib1";
    val Times_Un_distrib1 = thm "Times_Un_distrib1";
    val Times_eq_cancel2 = thm "Times_eq_cancel2";
    val Times_subset_cancel2 = thm "Times_subset_cancel2";
    val UNIV_Times_UNIV = thm "UNIV_Times_UNIV";
    val UN_Times_distrib = thm "UN_Times_distrib";
    val Unity_def = thm "Unity_def";
    val cond_split_eta = thm "cond_split_eta";
    val fst_conv = thm "fst_conv";
    val fst_def = thm "fst_def";
    val fst_eqD = thm "fst_eqD";
    val inj_on_Abs_Prod = thm "inj_on_Abs_Prod";
    val injective_fst_snd = thm "injective_fst_snd";
    val mem_Sigma_iff = thm "mem_Sigma_iff";
    val mem_splitE = thm "mem_splitE";
    val mem_splitI = thm "mem_splitI";
    val mem_splitI2 = thm "mem_splitI2";
    val prod_eqI = thm "prod_eqI";
    val prod_fun = thm "prod_fun";
    val prod_fun_compose = thm "prod_fun_compose";
    val prod_fun_def = thm "prod_fun_def";
    val prod_fun_ident = thm "prod_fun_ident";
    val prod_fun_imageE = thm "prod_fun_imageE";
    val prod_fun_imageI = thm "prod_fun_imageI";
    val prod_induct = thm "prod_induct";
    val snd_conv = thm "snd_conv";
    val snd_def = thm "snd_def";
    val snd_eqD = thm "snd_eqD";
    val split = thm "split";
    val splitD = thm "splitD";
    val splitD' = thm "splitD'";
    val splitE = thm "splitE";
    val splitE' = thm "splitE'";
    val splitE2 = thm "splitE2";
    val splitI = thm "splitI";
    val splitI2 = thm "splitI2";
    val splitI2' = thm "splitI2'";
    val split_Pair_apply = thm "split_Pair_apply";
    val split_beta = thm "split_beta";
    val split_conv = thm "split_conv";
    val split_def = thm "split_def";
    val split_eta = thm "split_eta";
    val split_eta_SetCompr = thm "split_eta_SetCompr";
    val split_eta_SetCompr2 = thm "split_eta_SetCompr2";
    val split_paired_All = thm "split_paired_All";
    val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma";
    val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma";
    val split_paired_Ex = thm "split_paired_Ex";
    val split_paired_The = thm "split_paired_The";
    val split_paired_all = thm "split_paired_all";
    val split_part = thm "split_part";
    val split_split = thm "split_split";
    val split_split_asm = thm "split_split_asm";
    val split_tupled_all = thms "split_tupled_all";
    val split_weak_cong = thm "split_weak_cong";
    val surj_pair = thm "surj_pair";
    val surjective_pairing = thm "surjective_pairing";
    val unit_abs_eta_conv = thm "unit_abs_eta_conv";
    val unit_all_eq1 = thm "unit_all_eq1";
    val unit_all_eq2 = thm "unit_all_eq2";
    val unit_eq = thm "unit_eq";
    val unit_induct = thm "unit_induct";
    *}

    end

### Unit

lemma unit_eq:

      u = ()

lemma unit_all_eq1:

      (!!x. PROP P x) == PROP P ()

lemma unit_all_eq2:

      (!!x. PROP P) == PROP P

lemma unit_induct:

      P () ==> P x

lemma unit_abs_eta_conv:

      (%u. f ()) = f

### Pairs

#### Type definition

#### Abstract constants and syntax

#### Definitions

#### Lemmas and proof tool setup

lemma ProdI:

      Pair_Rep a b ∈ Prod

lemma Pair_Rep_inject:

      Pair_Rep a b = Pair_Rep a' b' ==> a = a' ∧ b = b'

lemma inj_on_Abs_Prod:

      inj_on Abs_Prod Prod

lemma Pair_inject:

      [| (a, b) = (a', b'); [| a = a'; b = b' |] ==> R |] ==> R

lemma Pair_eq:

      ((a, b) = (a', b')) = (a = a' ∧ b = b')

lemma fst_conv:

      fst (a, b) = a

lemma snd_conv:

      snd (a, b) = b

lemma fst_eqD:

      fst (x, y) = a ==> x = a

lemma snd_eqD:

      snd (x, y) = a ==> y = a

lemma PairE_lemma:

      ∃x y. p = (x, y)

lemma PairE:

      (!!x y. p = (x, y) ==> Q) ==> Q

lemma surjective_pairing:

      p = (fst p, snd p)

lemmas pair_collapse:

      (fst s, snd s) = s

lemmas pair_collapse:

      (fst s, snd s) = s

lemma surj_pair:

      ∃x y. z = (x, y)

lemma split_paired_all:

      (!!x. PROP P x) == (!!a b. PROP P (a, b))

lemmas split_tupled_all:

      (!!x. PROP P x) == (!!a b. PROP P (a, b))

      (!!x. PROP P) == PROP P

lemmas split_tupled_all:

      (!!x. PROP P x) == (!!a b. PROP P (a, b))

      (!!x. PROP P) == PROP P

lemma split_paired_All:

      (∀x. P x) = (∀a b. P (a, b))

lemma curry_split:

      curry (%(x, y). f x y) = f

lemma split_curry:

      (%(x, y). curry f x y) = f

lemma curryI:

      f (a, b) ==> curry f a b

lemma curryD:

      curry f a b ==> f (a, b)

lemma curryE:

      [| curry f a b; f (a, b) ==> Q |] ==> Q

lemma curry_conv:

      curry f a b = f (a, b)

lemma prod_induct:

      (!!a b. P (a, b)) ==> P x

lemma split_paired_Ex:

      (∃x. P x) = (∃a b. P (a, b))

lemma split_conv:

      split c (a, b) = c a b

lemmas split:

      split c (a, b) = c a b

lemmas split:

      split c (a, b) = c a b

lemmas splitI:

      c a b ==> split c (a, b)

lemmas splitI:

      c a b ==> split c (a, b)

lemmas splitD:

      split c (a, b) ==> c a b

lemmas splitD:

      split c (a, b) ==> c a b

lemma split_Pair_apply:

      (%(x, y). f (x, y)) = f

lemma split_paired_The:

      (THE x. P x) = (THE (a, b). P (a, b))

lemma The_split:

      (THE (x, y). P x y) = (THE xy. P (fst xy) (snd xy))

lemma Pair_fst_snd_eq:

      (s = t) = (fst s = fst t ∧ snd s = snd t)

lemma prod_eqI:

      [| fst p = fst q; snd p = snd q |] ==> p = q

lemma split_weak_cong:

      p = q ==> split c p = split c q

lemma split_eta:

      (%(x, y). f (x, y)) = f

lemma cond_split_eta:

      (!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g

lemma split_beta:

      split P z = P (fst z) (snd z)

lemma split_split:

      R (split c p) = (∀x y. p = (x, y) --> R (c x y))

lemma split_split_asm:

      R (split c p) = (¬ (∃x y. p = (x, y) ∧ ¬ R (c x y)))

lemma splitI2:

      (!!a b. p = (a, b) ==> c a b) ==> split c p

lemma splitI2':

      (!!a b. (a, b) = p ==> c a b x) ==> split c p x

lemma splitE:

      [| split c p; !!x y. [| p = (x, y); c x y |] ==> Q |] ==> Q

lemma splitE':

      [| split c p z; !!x y. [| p = (x, y); c x y z |] ==> Q |] ==> Q

lemma splitE2:

      [| Q (split P z); !!x y. [| z = (x, y); Q (P x y) |] ==> R |] ==> R

lemma splitD':

      split R (a, b) c ==> R a b c

lemma mem_splitI:

      z ∈ c a b ==> z ∈ split c (a, b)

lemma mem_splitI2:

      (!!a b. p = (a, b) ==> z ∈ c a b) ==> z ∈ split c p

lemma mem_splitE:

      [| z ∈ split c p; !!x y. [| p = (x, y); z ∈ c x y |] ==> Q |] ==> Q

lemma split_eta_SetCompr:

      (%u. ∃x y. u = (x, y) ∧ P (x, y)) = P

lemma split_eta_SetCompr2:

      (%u. ∃x y. u = (x, y) ∧ P x y) = (%(x, y). P x y)

lemma split_part:

      (%(a, b). P ∧ Q a b) = (%ab. P ∧ split Q ab)

lemma split_comp_eq:

      (%u. f (g (fst u)) (snd u)) = (%(x, y). f (g x) y)

lemma The_split_eq:

      (THE (x', y'). x = x' ∧ y = y') = (x, y)

lemma injective_fst_snd:

      [| fst x = fst y; snd x = snd y |] ==> x = y

lemma prod_fun:

      prod_fun f g (a, b) = (f a, g b)

lemma prod_fun_compose:

      prod_fun (f1.0 o f2.0) (g1.0 o g2.0) = prod_fun f1.0 g1.0 o prod_fun f2.0 g2.0

lemma prod_fun_ident:

      prod_fun (%x. x) (%y. y) = (%z. z)

lemma prod_fun_imageI:

      (a, b) ∈ r ==> (f a, g b) ∈ prod_fun f g ` r

lemma prod_fun_imageE:

      [| c ∈ prod_fun f g ` r; !!x y. [| c = (f x, g y); (x, y) ∈ r |] ==> P |] ==> P

lemma upd_fst_conv:

      upd_fst f (x, y) = (f x, y)

lemma upd_snd_conv:

      upd_snd f (x, y) = (x, f y)

lemma SigmaI:

      [| a ∈ A; b ∈ B a |] ==> (a, b) ∈ Sigma A B

lemma SigmaE:

      [| c ∈ Sigma A B; !!x y. [| x ∈ A; y ∈ B x; c = (x, y) |] ==> P |] ==> P

lemma SigmaD1:

      (a, b) ∈ Sigma A B ==> a ∈ A

lemma SigmaD2:

      (a, b) ∈ Sigma A B ==> b ∈ B a

lemma SigmaE2:

      [| (a, b) ∈ Sigma A B; [| a ∈ A; b ∈ B a |] ==> P |] ==> P

lemma Sigma_cong:

      [| A = B; !!x. x ∈ B ==> C x = D x |] ==> (SIGMA x:A. C x) = (SIGMA x:B. D x)

lemma Sigma_mono:

      [| A ⊆ C; !!x. x ∈ A ==> B x ⊆ D x |] ==> Sigma A B ⊆ Sigma C D

lemma Sigma_empty1:

      Sigma {} B = {}

lemma Sigma_empty2:

      A × {} = {}

lemma UNIV_Times_UNIV:

      UNIV × UNIV = UNIV

lemma Compl_Times_UNIV1:

      - (UNIV × A) = UNIV × - A

lemma Compl_Times_UNIV2:

      - (A × UNIV) = (- A) × UNIV

lemma mem_Sigma_iff:

      ((a, b) ∈ Sigma A B) = (a ∈ A ∧ b ∈ B a)

lemma Times_subset_cancel2:

      x ∈ C ==> (A × C ⊆ B × C) = (A ⊆ B)

lemma Times_eq_cancel2:

      x ∈ C ==> (A × C = B × C) = (A = B)

lemma SetCompr_Sigma_eq:

      {(x, y). P x ∧ Q x y} = (SIGMA x:Collect P. Collect (Q x))

lemma Collect_split:

      {(a, b). P a ∧ Q b} = Collect P × Collect Q

lemma UN_Times_distrib:

      (UN (a, b):A × B. E a × F b) = UNION A E × UNION B F

lemma split_paired_Ball_Sigma:

      (∀z∈Sigma A B. P z) = (∀x∈A. ∀y∈B x. P (x, y))

lemma split_paired_Bex_Sigma:

      (∃z∈Sigma A B. P z) = (∃x∈A. ∃y∈B x. P (x, y))

lemma Sigma_Un_distrib1:

      (SIGMA i:I ∪ J. C i) = (SIGMA i:I. C i) ∪ (SIGMA j:J. C j)

lemma Sigma_Un_distrib2:

      (SIGMA i:I. A i ∪ B i) = (SIGMA i:I. A i) ∪ (SIGMA i:I. B i)

lemma Sigma_Int_distrib1:

      (SIGMA i:I ∩ J. C i) = (SIGMA i:I. C i) ∩ (SIGMA j:J. C j)

lemma Sigma_Int_distrib2:

      (SIGMA i:I. A i ∩ B i) = (SIGMA i:I. A i) ∩ (SIGMA i:I. B i)

lemma Sigma_Diff_distrib1:

      (SIGMA i:I - J. C i) = (SIGMA i:I. C i) - (SIGMA j:J. C j)

lemma Sigma_Diff_distrib2:

      (SIGMA i:I. A i - B i) = (SIGMA i:I. A i) - (SIGMA i:I. B i)

lemma Sigma_Union:

      Sigma (Union X) B = (UN A:X. Sigma A B)

lemma Times_Un_distrib1:

      (A ∪ B) × C = A × C ∪ B × C

lemma Times_Int_distrib1:

      (A ∩ B) × C = A × C ∩ B × C

lemma Times_Diff_distrib1:

      (A - B) × C = A × C - B × C

lemma pair_imageI:

      (a, b) ∈ A ==> f a b ∈ (%(a, b). f a b) ` A

lemma internal_split_conv:

      internal_split c (a, b) = c a b

### Code generator setup