src/HOL/Fun.thy
 author haftmann Mon, 21 Dec 2009 08:32:03 +0100 changeset 34150 22acb8b38639 parent 34101 d689f0b33047 child 34153 5da0f7abbe29 permissions -rw-r--r--
moved lemmas o_eq_dest, o_eq_elim here
```
(*  Title:      HOL/Fun.thy
Author:     Tobias Nipkow, Cambridge University Computer Laboratory
*)

theory Fun
imports Complete_Lattice
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"

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

lemma image_id [simp]: "id ` Y = Y"

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

lemma vimage_id [simp]: "id -` A = A"

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)"

lemma o_assoc: "f o (g o h) = f o g o h"

lemma id_o [simp]: "id o g = g"

lemma o_id [simp]: "f o id = f"

lemma o_eq_dest:
"a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
by (simp only: o_def) (fact fun_cong)

lemma o_eq_elim:
"a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
by (erule meta_mp) (fact o_eq_dest)

lemma image_compose: "(f o g) ` r = f`(g`r)"

lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
by auto

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)"

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

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

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

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)"

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"

lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"

lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"

lemma inj_on_id[simp]: "inj_on id A"

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

lemma surj_id[simp]: "surj id"

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"

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"

lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
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})"

lemma inj_on_empty[iff]: "inj_on f {}"

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 (blast intro: sym)
done

lemma surj_range: "surj f ==> range f = UNIV"

lemma surjD: "surj f ==> EX 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)"
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"

lemma bij_is_inj: "bij f ==> inj f"

lemma bij_is_surj: "bij f ==> surj f"

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

lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)

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

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"

lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"

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"
done

lemma inj_on_image_set_diff:
"[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
done

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

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)"
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)"
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: inj_on_def surj_def, blast)
done

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)"
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 (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)"

(* 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)"

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"

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

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

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 [simp]: "swap a a f = f"

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 comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
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"

hide (open) const swap

subsection {* Inversion of injective functions *}

definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
"the_inv_into A f == %x. THE y. y : A & f y = x"

lemma the_inv_into_f_f:
"[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
apply (blast intro: the_equality)
done

lemma f_the_inv_into_f:
"inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
apply (rule the1I2)
apply(blast dest: inj_onD)
apply blast
done

lemma the_inv_into_into:
"[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
apply (rule the1I2)
apply(blast dest: inj_onD)
apply blast
done

lemma the_inv_into_onto[simp]:
"inj_on f A ==> the_inv_into A f ` (f ` A) = A"
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])

lemma the_inv_into_f_eq:
"[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
apply (erule subst)
apply (erule the_inv_into_f_f, assumption)
done

lemma the_inv_into_comp:
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
apply (rule the_inv_into_f_eq)
apply (fast intro: comp_inj_on)
done

lemma inj_on_the_inv_into:
"inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
by (auto intro: inj_onI simp: image_def the_inv_into_f_f)

lemma bij_betw_the_inv_into:
"bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)

abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
"the_inv f \<equiv> the_inv_into UNIV f"

lemma the_inv_f_f:
assumes "inj f"
shows "the_inv f (f x) = x" using assms UNIV_I
by (rule the_inv_into_f_f)

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 = Unsynchronized.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")