# Theory Fun

Up to index of Isabelle/HOL

theory Fun
imports Typedef
begin

    (*  Title:      HOL/Fun.thy
        ID:         $Id: Fun.thy,v 1.44 2005/09/22 21:56:15 nipkow Exp $
        Author:     Tobias Nipkow, Cambridge University Computer Laboratory
        Copyright   1994  University of Cambridge

    Notions about functions.
    *)

    theory Fun
    imports Typedef
    begin

    instance set :: (type) order
      by (intro_classes,
          (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)

    constdefs
      fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
       "fun_upd f a b == % x. if x=a then b else f x"

    nonterminals
      updbinds updbind
    syntax
      "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
      ""         :: "updbind => updbinds"             ("_")
      "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
      "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)

    translations
      "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
      "f(x:=y)"                     == "fun_upd f x y"

    (* Hint: to define the sum of two functions (or maps), use sum_case.
             A nice infix syntax could be defined (in Datatype.thy or below) by
    consts
      fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
    translations
     "fun_sum" == sum_case
    *)

    constdefs
     override_on :: "('a => 'b) => ('a => 'b) => 'a set => ('a => 'b)"
    "override_on f g A == %a. if a : A then g a else f a"

     id :: "'a => 'a"
    "id == %x. x"

     comp :: "['b => 'c, 'a => 'b, 'a] => 'c"   (infixl "o" 55)
    "f o g == %x. f(g(x))"

    text{*compatibility*}
    lemmas o_def = comp_def

    syntax (xsymbols)
      comp :: "['b => 'c, 'a => 'b, 'a] => 'c"        (infixl "o" 55)
    syntax (HTML output)
      comp :: "['b => 'c, 'a => 'b, 'a] => 'c"        (infixl "o" 55)

    constdefs
      inj_on :: "['a => 'b, 'a set] => bool"         (*injective*)
        "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"

    text{*A common special case: functions injective over the entire domain type.*}
    syntax inj   :: "('a => 'b) => bool"
    translations
      "inj f" == "inj_on f UNIV"

    constdefs
      surj :: "('a => 'b) => bool"                   (*surjective*)
        "surj f == ! y. ? x. y=f(x)"

      bij :: "('a => 'b) => bool"                    (*bijective*)
        "bij f == inj f & surj f"

    text{*As a simplification rule, it replaces all function equalities by
      first-order equalities.*}
    lemma expand_fun_eq: "(f = g) = (! x. f(x)=g(x))"
    apply (rule iffI)
    apply (simp (no_asm_simp))
    apply (rule ext, simp (no_asm_simp))
    done

    lemma apply_inverse:
        "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)"
    by auto

    text{*The Identity Function: @{term id}*}
    lemma id_apply [simp]: "id x = x"
    by (simp add: id_def)

    lemma inj_on_id[simp]: "inj_on id A"
    by (simp add: inj_on_def)

    lemma inj_on_id2[simp]: "inj_on (%x. x) A"
    by (simp add: inj_on_def)

    lemma surj_id[simp]: "surj id"
    by (simp add: surj_def)

    lemma bij_id[simp]: "bij id"
    by (simp add: bij_def inj_on_id surj_id)

    subsection{*The Composition Operator: @{term "f o g"}*}

    lemma o_apply [simp]: "(f o g) x = f (g x)"
    by (simp add: comp_def)

    lemma o_assoc: "f o (g o h) = f o g o h"
    by (simp add: comp_def)

    lemma id_o [simp]: "id o g = g"
    by (simp add: comp_def)

    lemma o_id [simp]: "f o id = f"
    by (simp add: comp_def)

    lemma image_compose: "(f o g) ` r = f`(g`r)"
    by (simp add: comp_def, blast)

    lemma image_eq_UN: "f`A = (UN x:A. {f x})"
    by blast

    lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
    by (unfold comp_def, blast)

    subsection{*The Injectivity Predicate, @{term inj}*}

    text{*NB: @{term inj} now just translates to @{term inj_on}*}

    text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*}
    lemma datatype_injI:
        "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
    by (simp add: inj_on_def)

    theorem range_ex1_eq: "inj f ==> b : range f = (EX! x. b = f x)"
      by (unfold inj_on_def, blast)

    lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
    by (simp add: inj_on_def)

    (*Useful with the simplifier*)
    lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
    by (force simp add: inj_on_def)

    subsection{*The Predicate @{term inj_on}: Injectivity On A Restricted Domain*}

    lemma inj_onI:
        "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
    by (simp add: inj_on_def)

    lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
    by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)

    lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
    by (unfold inj_on_def, blast)

    lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
    by (blast dest!: inj_onD)

    lemma comp_inj_on:
         "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
    by (simp add: comp_def inj_on_def)

    lemma inj_on_imageI: "inj_on (g o f) A ==> inj_on g (f ` A)"
    apply(simp add:inj_on_def image_def)
    apply blast
    done

    lemma inj_on_image_iff: "[| ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
      inj_on f A |] ==> inj_on g (f ` A) = inj_on g A"
    apply(unfold inj_on_def)
    apply blast
    done

    lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
    by (unfold inj_on_def, blast)

    lemma inj_singleton: "inj (%s. {s})"
    by (simp add: inj_on_def)

    lemma inj_on_empty[iff]: "inj_on f {}"
    by(simp add: inj_on_def)

    lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
    by (unfold inj_on_def, blast)

    lemma inj_on_Un:
     "inj_on f (A Un B) =
      (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
    apply(unfold inj_on_def)
    apply (blast intro:sym)
    done

    lemma inj_on_insert[iff]:
      "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
    apply(unfold inj_on_def)
    apply (blast intro:sym)
    done

    lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
    apply(unfold inj_on_def)
    apply (blast)
    done

    subsection{*The Predicate @{term surj}: Surjectivity*}

    lemma surjI: "(!! x. g(f x) = x) ==> surj g"
    apply (simp add: surj_def)
    apply (blast intro: sym)
    done

    lemma surj_range: "surj f ==> range f = UNIV"
    by (auto simp add: surj_def)

    lemma surjD: "surj f ==> EX x. y = f x"
    by (simp add: surj_def)

    lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
    by (simp add: surj_def, blast)

    lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
    apply (simp add: comp_def surj_def, clarify)
    apply (drule_tac x = y in spec, clarify)
    apply (drule_tac x = x in spec, blast)
    done

    subsection{*The Predicate @{term bij}: Bijectivity*}

    lemma bijI: "[| inj f; surj f |] ==> bij f"
    by (simp add: bij_def)

    lemma bij_is_inj: "bij f ==> inj f"
    by (simp add: bij_def)

    lemma bij_is_surj: "bij f ==> surj f"
    by (simp add: bij_def)

    subsection{*Facts About the Identity Function*}

    text{*We seem to need both the @{term id} forms and the @{term "λx. x"}
    forms. The latter can arise by rewriting, while @{term id} may be used
    explicitly.*}

    lemma image_ident [simp]: "(%x. x) ` Y = Y"
    by blast

    lemma image_id [simp]: "id ` Y = Y"
    by (simp add: id_def)

    lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
    by blast

    lemma vimage_id [simp]: "id -` A = A"
    by (simp add: id_def)

    lemma vimage_image_eq: "f -` (f ` A) = {y. EX x:A. f x = f y}"
    by (blast intro: sym)

    lemma image_vimage_subset: "f ` (f -` A) <= A"
    by blast

    lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
    by blast

    lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
    by (simp add: surj_range)

    lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
    by (simp add: inj_on_def, blast)

    lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
    apply (unfold surj_def)
    apply (blast intro: sym)
    done

    lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
    by (unfold inj_on_def, blast)

    lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
    apply (unfold bij_def)
    apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
    done

    lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
    by blast

    lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
    by blast

    lemma inj_on_image_Int:
       "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
    apply (simp add: inj_on_def, blast)
    done

    lemma inj_on_image_set_diff:
       "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
    apply (simp add: inj_on_def, blast)
    done

    lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
    by (simp add: inj_on_def, blast)

    lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
    by (simp add: inj_on_def, blast)

    lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
    by (blast dest: injD)

    lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
    by (simp add: inj_on_def, blast)

    lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
    by (blast dest: injD)

    lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
    by blast

    (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
    lemma image_INT:
       "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
        ==> f ` (INTER A B) = (INT x:A. f ` B x)"
    apply (simp add: inj_on_def, blast)
    done

    (*Compare with image_INT: no use of inj_on, and if f is surjective then
      it doesn't matter whether A is empty*)
    lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
    apply (simp add: bij_def)
    apply (simp add: inj_on_def surj_def, blast)
    done

    lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
    by (auto simp add: surj_def)

    lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
    by (auto simp add: inj_on_def)

    lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
    apply (simp add: bij_def)
    apply (rule equalityI)
    apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
    done

    subsection{*Function Updating*}

    lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
    apply (simp add: fun_upd_def, safe)
    apply (erule subst)
    apply (rule_tac [2] ext, auto)
    done

    (* f x = y ==> f(x:=y) = f *)
    lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]

    (* f(x := f x) = f *)
    lemmas fun_upd_triv = refl [THEN fun_upd_idem]
    declare fun_upd_triv [iff]

    lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
    by (simp add: fun_upd_def)

    (* fun_upd_apply supersedes these two,   but they are useful
       if fun_upd_apply is intentionally removed from the simpset *)
    lemma fun_upd_same: "(f(x:=y)) x = y"
    by simp

    lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
    by simp

    lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
    by (simp add: expand_fun_eq)

    lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
    by (rule ext, auto)

    lemma inj_on_fun_updI: "[| inj_on f A; y ∉ f`A |] ==> inj_on (f(x:=y)) A"
    by(fastsimp simp:inj_on_def image_def)

    lemma fun_upd_image:
         "f(x:=y) ` A = (if x ∈ A then insert y (f ` (A-{x})) else f ` A)"
    by auto

    subsection{* @{text override_on} *}

    lemma override_on_emptyset[simp]: "override_on f g {} = f"
    by(simp add:override_on_def)

    lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
    by(simp add:override_on_def)

    lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
    by(simp add:override_on_def)

    subsection{* swap *}

    constdefs
      swap :: "['a, 'a, 'a => 'b] => ('a => 'b)"
       "swap a b f == f(a := f b, b:= f a)"

    lemma swap_self: "swap a a f = f"
    by (simp add: swap_def)

    lemma swap_commute: "swap a b f = swap b a f"
    by (rule ext, simp add: fun_upd_def swap_def)

    lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
    by (rule ext, simp add: fun_upd_def swap_def)

    lemma inj_on_imp_inj_on_swap:
         "[|inj_on f A; a ∈ A; b ∈ A|] ==> inj_on (swap a b f) A"
    by (simp add: inj_on_def swap_def, blast)

    lemma inj_on_swap_iff [simp]:
      assumes A: "a ∈ A" "b ∈ A" shows "inj_on (swap a b f) A = inj_on f A"
    proof
      assume "inj_on (swap a b f) A"
      with A have "inj_on (swap a b (swap a b f)) A"
        by (iprover intro: inj_on_imp_inj_on_swap)
      thus "inj_on f A" by simp
    next
      assume "inj_on f A"
      with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
    qed

    lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
    apply (simp add: surj_def swap_def, clarify)
    apply (rule_tac P = "y = f b" in case_split_thm, blast)
    apply (rule_tac P = "y = f a" in case_split_thm, auto)
      --{*We don't yet have @{text case_tac}*}
    done

    lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
    proof
      assume "surj (swap a b f)"
      hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap)
      thus "surj f" by simp
    next
      assume "surj f"
      thus "surj (swap a b f)" by (rule surj_imp_surj_swap)
    qed

    lemma bij_swap_iff: "bij (swap a b f) = bij f"
    by (simp add: bij_def)

    text{*The ML section includes some compatibility bindings and a simproc
    for function updates, in addition to the usual ML-bindings of theorems.*}
    ML
    {*
    val id_def = thm "id_def";
    val inj_on_def = thm "inj_on_def";
    val surj_def = thm "surj_def";
    val bij_def = thm "bij_def";
    val fun_upd_def = thm "fun_upd_def";

    val o_def = thm "comp_def";
    val injI = thm "inj_onI";
    val inj_inverseI = thm "inj_on_inverseI";
    val set_cs = claset() delrules [equalityI];

    val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];

    (* simplifies terms of the form f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *)
    local
      fun gen_fun_upd NONE T _ _ = NONE
        | gen_fun_upd (SOME f) T x y = SOME (Const ("Fun.fun_upd",T) $ f $ x $ y)
      fun dest_fun_T1 (Type (_, T :: Ts)) = T
      fun find_double (t as Const ("Fun.fun_upd",T) $ f $ x $ y) =
        let
          fun find (Const ("Fun.fun_upd",T) $ g $ v $ w) =
                if v aconv x then SOME g else gen_fun_upd (find g) T v w
            | find t = NONE
        in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end

      val current_ss = simpset ()
      fun fun_upd_prover ss =
        rtac eq_reflection 1 THEN rtac ext 1 THEN
        simp_tac (Simplifier.inherit_bounds ss current_ss) 1
    in
      val fun_upd2_simproc =
        Simplifier.simproc (Theory.sign_of (the_context ()))
          "fun_upd2" ["f(v := w, x := y)"]
          (fn sg => fn ss => fn t =>
            case find_double t of (T, NONE) => NONE
            | (T, SOME rhs) =>
                SOME (Tactic.prove sg [] [] (Term.equals T $ t $ rhs) (K (fun_upd_prover ss))))
    end;
    Addsimprocs[fun_upd2_simproc];

    val expand_fun_eq = thm "expand_fun_eq";
    val apply_inverse = thm "apply_inverse";
    val id_apply = thm "id_apply";
    val o_apply = thm "o_apply";
    val o_assoc = thm "o_assoc";
    val id_o = thm "id_o";
    val o_id = thm "o_id";
    val image_compose = thm "image_compose";
    val image_eq_UN = thm "image_eq_UN";
    val UN_o = thm "UN_o";
    val datatype_injI = thm "datatype_injI";
    val injD = thm "injD";
    val inj_eq = thm "inj_eq";
    val inj_onI = thm "inj_onI";
    val inj_on_inverseI = thm "inj_on_inverseI";
    val inj_onD = thm "inj_onD";
    val inj_on_iff = thm "inj_on_iff";
    val comp_inj_on = thm "comp_inj_on";
    val inj_on_contraD = thm "inj_on_contraD";
    val inj_singleton = thm "inj_singleton";
    val subset_inj_on = thm "subset_inj_on";
    val surjI = thm "surjI";
    val surj_range = thm "surj_range";
    val surjD = thm "surjD";
    val surjE = thm "surjE";
    val comp_surj = thm "comp_surj";
    val bijI = thm "bijI";
    val bij_is_inj = thm "bij_is_inj";
    val bij_is_surj = thm "bij_is_surj";
    val image_ident = thm "image_ident";
    val image_id = thm "image_id";
    val vimage_ident = thm "vimage_ident";
    val vimage_id = thm "vimage_id";
    val vimage_image_eq = thm "vimage_image_eq";
    val image_vimage_subset = thm "image_vimage_subset";
    val image_vimage_eq = thm "image_vimage_eq";
    val surj_image_vimage_eq = thm "surj_image_vimage_eq";
    val inj_vimage_image_eq = thm "inj_vimage_image_eq";
    val vimage_subsetD = thm "vimage_subsetD";
    val vimage_subsetI = thm "vimage_subsetI";
    val vimage_subset_eq = thm "vimage_subset_eq";
    val image_Int_subset = thm "image_Int_subset";
    val image_diff_subset = thm "image_diff_subset";
    val inj_on_image_Int = thm "inj_on_image_Int";
    val inj_on_image_set_diff = thm "inj_on_image_set_diff";
    val image_Int = thm "image_Int";
    val image_set_diff = thm "image_set_diff";
    val inj_image_mem_iff = thm "inj_image_mem_iff";
    val inj_image_subset_iff = thm "inj_image_subset_iff";
    val inj_image_eq_iff = thm "inj_image_eq_iff";
    val image_UN = thm "image_UN";
    val image_INT = thm "image_INT";
    val bij_image_INT = thm "bij_image_INT";
    val surj_Compl_image_subset = thm "surj_Compl_image_subset";
    val inj_image_Compl_subset = thm "inj_image_Compl_subset";
    val bij_image_Compl_eq = thm "bij_image_Compl_eq";
    val fun_upd_idem_iff = thm "fun_upd_idem_iff";
    val fun_upd_idem = thm "fun_upd_idem";
    val fun_upd_apply = thm "fun_upd_apply";
    val fun_upd_same = thm "fun_upd_same";
    val fun_upd_other = thm "fun_upd_other";
    val fun_upd_upd = thm "fun_upd_upd";
    val fun_upd_twist = thm "fun_upd_twist";
    val range_ex1_eq = thm "range_ex1_eq";
    *}

    end

lemmas o_def:

      f o g == %x. f (g x)

lemmas o_def:

      f o g == %x. f (g x)

lemma expand_fun_eq:

      (f = g) = (∀x. f x = g x)

lemma apply_inverse:

      [| f x = u; !!x. P x ==> g (f x) = x; P x |] ==> x = g u

lemma id_apply:

      id x = x

lemma inj_on_id:

      inj_on id A

lemma inj_on_id2:

      inj_on (%x. x) A

lemma surj_id:

      surj id

lemma bij_id:

      bij id

### The Composition Operator: @{term "f o g"}

lemma o_apply:

      (f o g) x = f (g x)

lemma o_assoc:

      f o (g o h) = f o g o h

lemma id_o:

      id o g = g

lemma o_id:

      f o id = f

lemma image_compose:

      (f o g) ` r = f ` g ` r

lemma image_eq_UN:

      f ` A = (UN x:A. {f x})

lemma UN_o:

      UNION A (g o f) = UNION (f ` A) g

### The Injectivity Predicate, @{term inj}

lemma datatype_injI:

      (!!x. ∀y. f x = f y --> x = y) ==> inj f

theorem range_ex1_eq:

      inj f ==> (b ∈ range f) = (∃!x. b = f x)

lemma injD:

      [| inj f; f x = f y |] ==> x = y

lemma inj_eq:

      inj f ==> (f x = f y) = (x = y)

### The Predicate @{term inj_on}: Injectivity On A Restricted Domain

lemma inj_onI:

      (!!x y. [| x ∈ A; y ∈ A; f x = f y |] ==> x = y) ==> inj_on f A

lemma inj_on_inverseI:

      (!!x. x ∈ A ==> g (f x) = x) ==> inj_on f A

lemma inj_onD:

      [| inj_on f A; f x = f y; x ∈ A; y ∈ A |] ==> x = y

lemma inj_on_iff:

      [| inj_on f A; x ∈ A; y ∈ A |] ==> (f x = f y) = (x = y)

lemma comp_inj_on:

      [| inj_on f A; inj_on g (f ` A) |] ==> inj_on (g o f) A

lemma inj_on_imageI:

      inj_on (g o f) A ==> inj_on g (f ` A)

lemma inj_on_image_iff:

      [| ∀x∈A. ∀y∈A. (g (f x) = g (f y)) = (g x = g y); inj_on f A |]
      ==> inj_on g (f ` A) = inj_on g A

lemma inj_on_contraD:

      [| inj_on f A; x ≠ y; x ∈ A; y ∈ A |] ==> f x ≠ f y

lemma inj_singleton:

      inj (%s. {s})

lemma inj_on_empty:

      inj_on f {}

lemma subset_inj_on:

      [| inj_on f B; A ⊆ B |] ==> inj_on f A

lemma inj_on_Un:

      inj_on f (A ∪ B) = (inj_on f A ∧ inj_on f B ∧ f ` (A - B) ∩ f ` (B - A) = {})

lemma inj_on_insert:

      inj_on f (insert a A) = (inj_on f A ∧ f a ∉ f ` (A - {a}))

lemma inj_on_diff:

      inj_on f A ==> inj_on f (A - B)

### The Predicate @{term surj}: Surjectivity

lemma surjI:

      (!!x. g (f x) = x) ==> surj g

lemma surj_range:

      surj f ==> range f = UNIV

lemma surjD:

      surj f ==> ∃x. y = f x

lemma surjE:

      [| surj f; !!x. y = f x ==> C |] ==> C

lemma comp_surj:

      [| surj f; surj g |] ==> surj (g o f)

### The Predicate @{term bij}: Bijectivity

lemma bijI:

      [| inj f; surj f |] ==> bij f

lemma bij_is_inj:

      bij f ==> inj f

lemma bij_is_surj:

      bij f ==> surj f

### Facts About the Identity Function

lemma image_ident:

      (%x. x) ` Y = Y

lemma image_id:

      id ` Y = Y

lemma vimage_ident:

      (%x. x) -` Y = Y

lemma vimage_id:

      id -` A = A

lemma vimage_image_eq:

      f -` f ` A = {y. ∃x∈A. f x = f y}

lemma image_vimage_subset:

      f ` f -` A ⊆ A

lemma image_vimage_eq:

      f ` f -` A = A ∩ range f

lemma surj_image_vimage_eq:

      surj f ==> f ` f -` A = A

lemma inj_vimage_image_eq:

      inj f ==> f -` f ` A = A

lemma vimage_subsetD:

      [| surj f; f -` B ⊆ A |] ==> B ⊆ f ` A

lemma vimage_subsetI:

      [| inj f; B ⊆ f ` A |] ==> f -` B ⊆ A

lemma vimage_subset_eq:

      bij f ==> (f -` B ⊆ A) = (B ⊆ f ` A)

lemma image_Int_subset:

      f ` (A ∩ B) ⊆ f ` A ∩ f ` B

lemma image_diff_subset:

      f ` A - f ` B ⊆ f ` (A - B)

lemma inj_on_image_Int:

      [| inj_on f C; A ⊆ C; B ⊆ C |] ==> f ` (A ∩ B) = f ` A ∩ f ` B

lemma inj_on_image_set_diff:

      [| inj_on f C; A ⊆ C; B ⊆ C |] ==> f ` (A - B) = f ` A - f ` B

lemma image_Int:

      inj f ==> f ` (A ∩ B) = f ` A ∩ f ` B

lemma image_set_diff:

      inj f ==> f ` (A - B) = f ` A - f ` B

lemma inj_image_mem_iff:

      inj f ==> (f a ∈ f ` A) = (a ∈ A)

lemma inj_image_subset_iff:

      inj f ==> (f ` A ⊆ f ` B) = (A ⊆ B)

lemma inj_image_eq_iff:

      inj f ==> (f ` A = f ` B) = (A = B)

lemma image_UN:

      f ` UNION A B = (UN x:A. f ` B x)

lemma image_INT:

      [| inj_on f C; ∀x∈A. B x ⊆ C; j ∈ A |] ==> f ` INTER A B = (INT x:A. f ` B x)

lemma bij_image_INT:

      bij f ==> f ` INTER A B = (INT x:A. f ` B x)

lemma surj_Compl_image_subset:

      surj f ==> - f ` A ⊆ f ` (- A)

lemma inj_image_Compl_subset:

      inj f ==> f ` (- A) ⊆ - f ` A

lemma bij_image_Compl_eq:

      bij f ==> f ` (- A) = - f ` A

### Function Updating

lemma fun_upd_idem_iff:

      (f(x := y) = f) = (f x = y)

lemmas fun_upd_idem:

      f x = y ==> f(x := y) = f

lemmas fun_upd_idem:

      f x = y ==> f(x := y) = f

lemmas fun_upd_triv:

      f(x := f x) = f

lemmas fun_upd_triv:

      f(x := f x) = f

lemma fun_upd_apply:

      (f(x := y)) z = (if z = x then y else f z)

lemma fun_upd_same:

      (f(x := y)) x = y

lemma fun_upd_other:

      z ≠ x ==> (f(x := y)) z = f z

lemma fun_upd_upd:

      f(x := y, x := z) = f(x := z)

lemma fun_upd_twist:

      a ≠ c ==> m(a := b, c := d) = m(c := d, a := b)

lemma inj_on_fun_updI:

      [| inj_on f A; y ∉ f ` A |] ==> inj_on (f(x := y)) A

lemma fun_upd_image:

      f(x := y) ` A = (if x ∈ A then insert y (f ` (A - {x})) else f ` A)

### @{text override_on}

lemma override_on_emptyset:

      override_on f g {} = f

lemma override_on_apply_notin:

      a ∉ A ==> override_on f g A a = f a

lemma override_on_apply_in:

      a ∈ A ==> override_on f g A a = g a

### swap

lemma swap_self:

      swap a a f = f

lemma swap_commute:

      swap a b f = swap b a f

lemma swap_nilpotent:

      swap a b (swap a b f) = f

lemma inj_on_imp_inj_on_swap:

      [| inj_on f A; a ∈ A; b ∈ A |] ==> inj_on (swap a b f) A

lemma inj_on_swap_iff:

      [| a ∈ A; b ∈ A |] ==> inj_on (swap a b f) A = inj_on f A

lemma surj_imp_surj_swap:

      surj f ==> surj (swap a b f)

lemma surj_swap_iff:

      surj (swap a b f) = surj f

lemma bij_swap_iff:

      bij (swap a b f) = bij f