(* Title: HOL/Fun.thy
ID: $Id$
Author: Tobias Nipkow, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
*)
header {* 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 "\<circ>" 55)
syntax (HTML output)
comp :: "['b => 'c, 'a => 'b, 'a] => 'c" (infixl "\<circ>" 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 \<circ> 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 \<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)
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 \<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
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 "\<lambda>x. 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: "\<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
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 \<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 (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_context 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 (Goal.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