(* Title: HOL/Fun.thy ID: $Id: Fun.thy,v 1.72 2008/04/09 06:10:11 haftmann Exp $ 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 <-> (∀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 ==> (!!x. P x ==> g (f x) = x) ==> P x ==> x = g u" by auto subsection {* The Identity Function @{text id} *} definition id :: "'a => 'a" where "id = (λx. 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 o g"} *} definition comp :: "('b => 'c) => ('a => 'b) => 'a => 'c" (infixl "o" 55) where "f o g = (λx. f (g x))" notation (xsymbols) comp (infixl "o" 55) notation (HTML output) comp (infixl "o" 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 => 'b) => ('b => 'c) => 'a => 'c" (infixl "o>" 60) where "f o> g = (λx. 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) 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" "bij_betw f A B <-> 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 "!!x y. f x = f y ==> x = y" shows "inj f" using assms unfolding inj_on_def by auto 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) 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 ==> 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 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 ==> inj_on f A" by (simp add: bij_betw_def) 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 ∧ f a = b" let ?g = "%b. The (?P b)" { fix a b assume P: "?P b a" hence ex1: "∃a. ?P b a" using s unfolding image_def by blast hence uex1: "∃!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 ∈ 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 "∃b∈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_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: "[| 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} *} definition override_on :: "('a => 'b) => ('a => 'b) => 'a set => 'a => 'b" where "override_on f g A = (λa. if a ∈ 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 => 'a => ('a => 'b) => ('a => '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 ∈ 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) 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 [] [] (Term.equals T $ 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 ("arbitrary", aT --> bT))) end; *} code_const "op o" (SML infixl 5 "o") (Haskell infixr 9 ".") code_const "id" (Haskell "id") end
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 image_ident:
(λx. x) ` Y = Y
lemma image_id:
id ` Y = Y
lemma vimage_ident:
(λx. x) -` Y = Y
lemma vimage_id:
id -` A = A
lemma o_def:
f o g = (λx. f (g x))
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 UN_o:
UNION A (g o f) = UNION (f ` A) g
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:
id o> g = g
lemma fcomp_id:
f o> id = f
lemma injI:
(!!x y. f x = f y ==> x = y) ==> inj f
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)
lemma inj_on_id:
inj_on id A
lemma inj_on_id2:
inj_on (λx. x) A
lemma surj_id:
surj id
lemma bij_id:
bij 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
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 (λs. {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)
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)
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 ==> inj_on f A
lemma bij_betw_inv:
bij_betw f A B ==> ∃g. bij_betw g B A
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 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_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
lemma fun_upd_idem_iff:
(f(x := y) = f) = (f x = y)
lemma fun_upd_idem:
f x = y ==> f(x := y) = f
lemma 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)
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
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