(*  Title:      HOL/Fun.thy

    Author:     Tobias Nipkow, Cambridge University Computer Laboratory

    Copyright   1994  University of Cambridge

*)

header {* Notions about functions *}

theory Fun

imports Set

begin

text{*As a simplification rule, it replaces all function equalities by

  first-order equalities.*}

lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"

apply (rule iffI)

apply (simp (no_asm_simp))

apply (rule ext)

apply (simp (no_asm_simp))

done

lemma apply_inverse:

  "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"

  by auto

subsection {* The Identity Function @{text id} *}

definition

  id :: "'a \<Rightarrow> 'a"

where

  "id = (\<lambda>x. x)"

lemma id_apply [simp]: "id x = x"

  by (simp add: id_def)

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)

subsection {* The Composition Operator @{text "f \<circ> g"} *}

definition

  comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)

where

  "f o g = (\<lambda>x. f (g x))"

notation (xsymbols)

  comp  (infixl "\<circ>" 55)

notation (HTML output)

  comp  (infixl "\<circ>" 55)

text{*compatibility*}

lemmas o_def = comp_def

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 UN_o: "UNION A (g o f) = UNION (f`A) g"

by (unfold comp_def, blast)

subsection {* The Forward Composition Operator @{text fcomp} *}

definition

  fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60)

where

  "f o> g = (\<lambda>x. g (f x))"

lemma fcomp_apply:  "(f o> g) x = g (f x)"

  by (simp add: fcomp_def)

lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"

  by (simp add: fcomp_def)

lemma id_fcomp [simp]: "id o> g = g"

  by (simp add: fcomp_def)

lemma fcomp_id [simp]: "f o> id = f"

  by (simp add: fcomp_def)

code_const fcomp

  (Eval infixl 1 "#>")

no_notation fcomp (infixl "o>" 60)

subsection {* Injectivity and Surjectivity *}

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.*}

abbreviation

  "inj f == inj_on f UNIV"

definition

  bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"

  [code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"

constdefs

  surj :: "('a => 'b) => bool"                   (*surjective*)

  "surj f == ! y. ? x. y=f(x)"

  bij :: "('a => 'b) => bool"                    (*bijective*)

  "bij f == inj f & surj f"

lemma injI:

  assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"

  shows "inj f"

  using assms unfolding inj_on_def by auto

text{*For Proofs in @{text "Tools/Datatype/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 \<Longrightarrow> 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)

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) 

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 \<Longrightarrow> inj_on g (f ` A)"

apply(simp add:inj_on_def image_def)

apply blast

done

lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);

  inj_on f A \<rbrakk> \<Longrightarrow> 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

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

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)

lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"

by (simp add: bij_betw_def)

lemma bij_betw_trans:

  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"

by(auto simp add:bij_betw_def comp_inj_on)

lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"

proof -

  have i: "inj_on f A" and s: "f ` A = B"

    using assms by(auto simp:bij_betw_def)

  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"

  { fix a b assume P: "?P b a"

    hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast

    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])

    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp

  } note g = this

  have "inj_on ?g B"

  proof(rule inj_onI)

    fix x y assume "x:B" "y:B" "?g x = ?g y"

    from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast

    from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast

    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp

  qed

  moreover have "?g ` B = A"

  proof(auto simp:image_def)

    fix b assume "b:B"

    with s obtain a where P: "?P b a" unfolding image_def by blast

    thus "?g b \<in> A" using g[OF P] by auto

  next

    fix a assume "a:A"

    then obtain b where P: "?P b a" using s unfolding image_def by blast

    then have "b:B" using s unfolding image_def by blast

    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast

  qed

  ultimately show ?thesis by(auto simp:bij_betw_def)

qed

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 inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"

by(blast dest: inj_onD)

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)

(*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*}

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

*)

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: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"

by(fastsimp simp:inj_on_def image_def)

lemma fun_upd_image:

     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"

by auto

lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"

by(auto intro: ext)

subsection {* @{text override_on} *}

definition

  override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"

where

  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"

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 {* @{text swap} *}

definition

  swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"

where

  "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 \<in> A; b \<in> 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 \<in> A" "b \<in> 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 (case_tac "y = f b", blast)

apply (case_tac "y = f a", auto)

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)

hide (open) const swap

subsection {* Proof tool setup *} 

text {* simplifies terms of the form

  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}

simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>

let

  fun gen_fun_upd NONE T _ _ = NONE

    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)

  fun dest_fun_T1 (Type (_, T :: Ts)) = T

  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =

    let

      fun find (Const (@{const_name 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

  fun proc ss ct =

    let

      val ctxt = Simplifier.the_context ss

      val t = Thm.term_of ct

    in

      case find_double t of

        (T, NONE) => NONE

      | (T, SOME rhs) =>

          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))

            (fn _ =>

              rtac eq_reflection 1 THEN

              rtac ext 1 THEN

              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))

    end

in proc end

*}

subsection {* Code generator setup *}

types_code

  "fun"  ("(_ ->/ _)")

attach (term_of) {*

fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);

*}

attach (test) {*

fun gen_fun_type aF aT bG bT i =

  let

    val tab = ref [];

    fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",

      (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()

  in

    (fn x =>

       case AList.lookup op = (!tab) x of

         NONE =>

           let val p as (y, _) = bG i

           in (tab := (x, p) :: !tab; y) end

       | SOME (y, _) => y,

     fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))

  end;

*}

code_const "op \<circ>"

  (SML infixl 5 "o")

  (Haskell infixr 9 ".")

code_const "id"

  (Haskell "id")

end