
clasohm@1475: (*  Title:      HOL/Fun.thy
clasohm@1475:     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
clasohm@923:     Copyright   1994  University of Cambridge
huffman@18154: *)
clasohm@923: 
huffman@18154: header {* Notions about functions *}
clasohm@923: 
paulson@15510: theory Fun
haftmann@32139: imports Complete_Lattice
haftmann@32554: uses ("Tools/transfer.ML")
nipkow@15131: begin
nipkow@2912: 
haftmann@26147: text{*As a simplification rule, it replaces all function equalities by
haftmann@26147:   first-order equalities.*}
haftmann@26147: lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
haftmann@26147: apply (rule iffI)
haftmann@26147: apply (simp (no_asm_simp))
haftmann@26147: apply (rule ext)
haftmann@26147: apply (simp (no_asm_simp))
haftmann@26147: done
paulson@6171: 
haftmann@26147: lemma apply_inverse:
haftmann@26357:   "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
haftmann@26147:   by auto
oheimb@5305: 
nipkow@2912: 
haftmann@26147: subsection {* The Identity Function @{text id} *}
nipkow@2912: 
haftmann@22744: definition
haftmann@22744:   id :: "'a \<Rightarrow> 'a"
haftmann@22744: where
haftmann@22744:   "id = (\<lambda>x. x)"
nipkow@13910: 
haftmann@26147: lemma id_apply [simp]: "id x = x"
haftmann@26147:   by (simp add: id_def)
haftmann@26147: 
haftmann@26147: lemma image_ident [simp]: "(%x. x) ` Y = Y"
haftmann@26147: by blast
haftmann@26147: 
haftmann@26147: lemma image_id [simp]: "id ` Y = Y"
haftmann@26147: by (simp add: id_def)
haftmann@26147: 
haftmann@26147: lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
haftmann@26147: by blast
haftmann@26147: 
haftmann@26147: lemma vimage_id [simp]: "id -` A = A"
haftmann@26147: by (simp add: id_def)
haftmann@26147: 
haftmann@26147: 
haftmann@26147: subsection {* The Composition Operator @{text "f \<circ> g"} *}
haftmann@26147: 
haftmann@22744: definition
haftmann@22744:   comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
haftmann@22744: where
haftmann@22744:   "f o g = (\<lambda>x. f (g x))"
oheimb@11123: 
wenzelm@21210: notation (xsymbols)
wenzelm@19656:   comp  (infixl "\<circ>" 55)
wenzelm@19656: 
wenzelm@21210: notation (HTML output)
wenzelm@19656:   comp  (infixl "\<circ>" 55)
wenzelm@19656: 
paulson@13585: text{*compatibility*}
paulson@13585: lemmas o_def = comp_def
paulson@13585: 
paulson@13585: lemma o_apply [simp]: "(f o g) x = f (g x)"
paulson@13585: by (simp add: comp_def)
paulson@13585: 
paulson@13585: lemma o_assoc: "f o (g o h) = f o g o h"
paulson@13585: by (simp add: comp_def)
paulson@13585: 
paulson@13585: lemma id_o [simp]: "id o g = g"
paulson@13585: by (simp add: comp_def)
paulson@13585: 
paulson@13585: lemma o_id [simp]: "f o id = f"
paulson@13585: by (simp add: comp_def)
paulson@13585: 
paulson@13585: lemma image_compose: "(f o g) ` r = f`(g`r)"
paulson@13585: by (simp add: comp_def, blast)
paulson@13585: 
paulson@33044: lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
paulson@33044:   by auto
paulson@33044: 
paulson@13585: lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
paulson@13585: by (unfold comp_def, blast)
paulson@13585: 
paulson@13585: 
haftmann@26588: subsection {* The Forward Composition Operator @{text fcomp} *}
haftmann@26357: 
haftmann@26357: definition
haftmann@26357:   fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60)
haftmann@26357: where
haftmann@26357:   "f o> g = (\<lambda>x. g (f x))"
haftmann@26357: 
haftmann@26357: lemma fcomp_apply:  "(f o> g) x = g (f x)"
haftmann@26357:   by (simp add: fcomp_def)
haftmann@26357: 
haftmann@26357: lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"
haftmann@26357:   by (simp add: fcomp_def)
haftmann@26357: 
haftmann@26357: lemma id_fcomp [simp]: "id o> g = g"
haftmann@26357:   by (simp add: fcomp_def)
haftmann@26357: 
haftmann@26357: lemma fcomp_id [simp]: "f o> id = f"
haftmann@26357:   by (simp add: fcomp_def)
haftmann@26357: 
haftmann@31202: code_const fcomp
haftmann@31202:   (Eval infixl 1 "#>")
haftmann@31202: 
haftmann@26588: no_notation fcomp (infixl "o>" 60)
haftmann@26588: 
haftmann@26357: 
haftmann@26147: subsection {* Injectivity and Surjectivity *}
paulson@13585: 
haftmann@26147: constdefs
haftmann@26147:   inj_on :: "['a => 'b, 'a set] => bool"  -- "injective"
haftmann@26147:   "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
paulson@13585: 
haftmann@26147: text{*A common special case: functions injective over the entire domain type.*}
haftmann@26147: 
haftmann@26147: abbreviation
haftmann@26147:   "inj f == inj_on f UNIV"
haftmann@26147: 
haftmann@26147: definition
haftmann@26147:   bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
haftmann@28562:   [code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
haftmann@26147: 
haftmann@26147: constdefs
haftmann@26147:   surj :: "('a => 'b) => bool"                   (*surjective*)
haftmann@26147:   "surj f == ! y. ? x. y=f(x)"
haftmann@26147: 
haftmann@26147:   bij :: "('a => 'b) => bool"                    (*bijective*)
haftmann@26147:   "bij f == inj f & surj f"
haftmann@26147: 
haftmann@26147: lemma injI:
haftmann@26147:   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
haftmann@26147:   shows "inj f"
haftmann@26147:   using assms unfolding inj_on_def by auto
paulson@13585: 
haftmann@31771: text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
paulson@13585: lemma datatype_injI:
paulson@13585:     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
paulson@13585: by (simp add: inj_on_def)
paulson@13585: 
berghofe@13637: theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
berghofe@13637:   by (unfold inj_on_def, blast)
berghofe@13637: 
paulson@13585: lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
paulson@13585: by (simp add: inj_on_def)
paulson@13585: 
nipkow@32988: lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
paulson@13585: by (force simp add: inj_on_def)
paulson@13585: 
nipkow@32988: lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
nipkow@32988: by (simp add: inj_on_eq_iff)
nipkow@32988: 
haftmann@26147: lemma inj_on_id[simp]: "inj_on id A"
haftmann@26147:   by (simp add: inj_on_def) 
paulson@13585: 
haftmann@26147: lemma inj_on_id2[simp]: "inj_on (%x. x) A"
haftmann@26147: by (simp add: inj_on_def) 
haftmann@26147: 
haftmann@26147: lemma surj_id[simp]: "surj id"
haftmann@26147: by (simp add: surj_def) 
haftmann@26147: 
haftmann@26147: lemma bij_id[simp]: "bij id"
haftmann@26147: by (simp add: bij_def inj_on_id surj_id) 
paulson@13585: 
paulson@13585: lemma inj_onI:
paulson@13585:     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
paulson@13585: by (simp add: inj_on_def)
paulson@13585: 
paulson@13585: lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
paulson@13585: by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
paulson@13585: 
paulson@13585: lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
paulson@13585: by (unfold inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
paulson@13585: by (blast dest!: inj_onD)
paulson@13585: 
paulson@13585: lemma comp_inj_on:
paulson@13585:      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
paulson@13585: by (simp add: comp_def inj_on_def)
paulson@13585: 
nipkow@15303: lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
nipkow@15303: apply(simp add:inj_on_def image_def)
nipkow@15303: apply blast
nipkow@15303: done
nipkow@15303: 
nipkow@15439: lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
nipkow@15439:   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
nipkow@15439: apply(unfold inj_on_def)
nipkow@15439: apply blast
nipkow@15439: done
nipkow@15439: 
paulson@13585: lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
paulson@13585: by (unfold inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma inj_singleton: "inj (%s. {s})"
paulson@13585: by (simp add: inj_on_def)
paulson@13585: 
nipkow@15111: lemma inj_on_empty[iff]: "inj_on f {}"
nipkow@15111: by(simp add: inj_on_def)
nipkow@15111: 
nipkow@15303: lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
paulson@13585: by (unfold inj_on_def, blast)
paulson@13585: 
nipkow@15111: lemma inj_on_Un:
nipkow@15111:  "inj_on f (A Un B) =
nipkow@15111:   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
nipkow@15111: apply(unfold inj_on_def)
nipkow@15111: apply (blast intro:sym)
nipkow@15111: done
nipkow@15111: 
nipkow@15111: lemma inj_on_insert[iff]:
nipkow@15111:   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
nipkow@15111: apply(unfold inj_on_def)
nipkow@15111: apply (blast intro:sym)
nipkow@15111: done
nipkow@15111: 
nipkow@15111: lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
nipkow@15111: apply(unfold inj_on_def)
nipkow@15111: apply (blast)
nipkow@15111: done
nipkow@15111: 
paulson@13585: lemma surjI: "(!! x. g(f x) = x) ==> surj g"
paulson@13585: apply (simp add: surj_def)
paulson@13585: apply (blast intro: sym)
paulson@13585: done
paulson@13585: 
paulson@13585: lemma surj_range: "surj f ==> range f = UNIV"
paulson@13585: by (auto simp add: surj_def)
paulson@13585: 
paulson@13585: lemma surjD: "surj f ==> EX x. y = f x"
paulson@13585: by (simp add: surj_def)
paulson@13585: 
paulson@13585: lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
paulson@13585: by (simp add: surj_def, blast)
paulson@13585: 
paulson@13585: lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
paulson@13585: apply (simp add: comp_def surj_def, clarify)
paulson@13585: apply (drule_tac x = y in spec, clarify)
paulson@13585: apply (drule_tac x = x in spec, blast)
paulson@13585: done
paulson@13585: 
paulson@13585: lemma bijI: "[| inj f; surj f |] ==> bij f"
paulson@13585: by (simp add: bij_def)
paulson@13585: 
paulson@13585: lemma bij_is_inj: "bij f ==> inj f"
paulson@13585: by (simp add: bij_def)
paulson@13585: 
paulson@13585: lemma bij_is_surj: "bij f ==> surj f"
paulson@13585: by (simp add: bij_def)
paulson@13585: 
nipkow@26105: lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
nipkow@26105: by (simp add: bij_betw_def)
nipkow@26105: 
nipkow@32337: lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
nipkow@32337: by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
nipkow@32337: 
nipkow@31424: lemma bij_betw_trans:
nipkow@31424:   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
nipkow@31424: by(auto simp add:bij_betw_def comp_inj_on)
nipkow@31424: 
nipkow@26105: lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
nipkow@26105: proof -
nipkow@26105:   have i: "inj_on f A" and s: "f ` A = B"
nipkow@26105:     using assms by(auto simp:bij_betw_def)
nipkow@26105:   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
nipkow@26105:   { fix a b assume P: "?P b a"
nipkow@26105:     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
nipkow@26105:     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
nipkow@26105:     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
nipkow@26105:   } note g = this
nipkow@26105:   have "inj_on ?g B"
nipkow@26105:   proof(rule inj_onI)
nipkow@26105:     fix x y assume "x:B" "y:B" "?g x = ?g y"
nipkow@26105:     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
nipkow@26105:     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
nipkow@26105:     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
nipkow@26105:   qed
nipkow@26105:   moreover have "?g ` B = A"
nipkow@26105:   proof(auto simp:image_def)
nipkow@26105:     fix b assume "b:B"
nipkow@26105:     with s obtain a where P: "?P b a" unfolding image_def by blast
nipkow@26105:     thus "?g b \<in> A" using g[OF P] by auto
nipkow@26105:   next
nipkow@26105:     fix a assume "a:A"
nipkow@26105:     then obtain b where P: "?P b a" using s unfolding image_def by blast
nipkow@26105:     then have "b:B" using s unfolding image_def by blast
nipkow@26105:     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
nipkow@26105:   qed
nipkow@26105:   ultimately show ?thesis by(auto simp:bij_betw_def)
nipkow@26105: qed
nipkow@26105: 
paulson@13585: lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
paulson@13585: by (simp add: surj_range)
paulson@13585: 
paulson@13585: lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
paulson@13585: by (simp add: inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
paulson@13585: apply (unfold surj_def)
paulson@13585: apply (blast intro: sym)
paulson@13585: done
paulson@13585: 
paulson@13585: lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
paulson@13585: by (unfold inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
paulson@13585: apply (unfold bij_def)
paulson@13585: apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585: done
paulson@13585: 
nipkow@31424: lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
nipkow@31424: by(blast dest: inj_onD)
nipkow@31424: 
paulson@13585: lemma inj_on_image_Int:
paulson@13585:    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
paulson@13585: apply (simp add: inj_on_def, blast)
paulson@13585: done
paulson@13585: 
paulson@13585: lemma inj_on_image_set_diff:
paulson@13585:    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
paulson@13585: apply (simp add: inj_on_def, blast)
paulson@13585: done
paulson@13585: 
paulson@13585: lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
paulson@13585: by (simp add: inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
paulson@13585: by (simp add: inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
paulson@13585: by (blast dest: injD)
paulson@13585: 
paulson@13585: lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
paulson@13585: by (simp add: inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
paulson@13585: by (blast dest: injD)
paulson@13585: 
paulson@13585: (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
paulson@13585: lemma image_INT:
paulson@13585:    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
paulson@13585:     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
paulson@13585: apply (simp add: inj_on_def, blast)
paulson@13585: done
paulson@13585: 
paulson@13585: (*Compare with image_INT: no use of inj_on, and if f is surjective then
paulson@13585:   it doesn't matter whether A is empty*)
paulson@13585: lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
paulson@13585: apply (simp add: bij_def)
paulson@13585: apply (simp add: inj_on_def surj_def, blast)
paulson@13585: done
paulson@13585: 
paulson@13585: lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
paulson@13585: by (auto simp add: surj_def)
paulson@13585: 
paulson@13585: lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
paulson@13585: by (auto simp add: inj_on_def)
paulson@13585: 
paulson@13585: lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
paulson@13585: apply (simp add: bij_def)
paulson@13585: apply (rule equalityI)
paulson@13585: apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
paulson@13585: done
paulson@13585: 
paulson@13585: 
paulson@13585: subsection{*Function Updating*}
paulson@13585: 
haftmann@26147: constdefs
haftmann@26147:   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
haftmann@26147:   "fun_upd f a b == % x. if x=a then b else f x"
haftmann@26147: 
haftmann@26147: nonterminals
haftmann@26147:   updbinds updbind
haftmann@26147: syntax
haftmann@26147:   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
haftmann@26147:   ""         :: "updbind => updbinds"             ("_")
haftmann@26147:   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
haftmann@26147:   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
haftmann@26147: 
haftmann@26147: translations
haftmann@26147:   "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
haftmann@26147:   "f(x:=y)"                     == "fun_upd f x y"
haftmann@26147: 
haftmann@26147: (* Hint: to define the sum of two functions (or maps), use sum_case.
haftmann@26147:          A nice infix syntax could be defined (in Datatype.thy or below) by
haftmann@26147: consts
haftmann@26147:   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
haftmann@26147: translations
haftmann@26147:  "fun_sum" == sum_case
haftmann@26147: *)
haftmann@26147: 
paulson@13585: lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
paulson@13585: apply (simp add: fun_upd_def, safe)
paulson@13585: apply (erule subst)
paulson@13585: apply (rule_tac [2] ext, auto)
paulson@13585: done
paulson@13585: 
paulson@13585: (* f x = y ==> f(x:=y) = f *)
paulson@13585: lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
paulson@13585: 
paulson@13585: (* f(x := f x) = f *)
paulson@17084: lemmas fun_upd_triv = refl [THEN fun_upd_idem]
paulson@17084: declare fun_upd_triv [iff]
paulson@13585: 
paulson@13585: lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
paulson@17084: by (simp add: fun_upd_def)
paulson@13585: 
paulson@13585: (* fun_upd_apply supersedes these two,   but they are useful
paulson@13585:    if fun_upd_apply is intentionally removed from the simpset *)
paulson@13585: lemma fun_upd_same: "(f(x:=y)) x = y"
paulson@13585: by simp
paulson@13585: 
paulson@13585: lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
paulson@13585: by simp
paulson@13585: 
paulson@13585: lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
paulson@13585: by (simp add: expand_fun_eq)
paulson@13585: 
paulson@13585: lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
paulson@13585: by (rule ext, auto)
paulson@13585: 
nipkow@15303: lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
nipkow@15303: by(fastsimp simp:inj_on_def image_def)
nipkow@15303: 
paulson@15510: lemma fun_upd_image:
paulson@15510:      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
paulson@15510: by auto
paulson@15510: 
nipkow@31080: lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
nipkow@31080: by(auto intro: ext)
nipkow@31080: 
haftmann@26147: 
haftmann@26147: subsection {* @{text override_on} *}
haftmann@26147: 
haftmann@26147: definition
haftmann@26147:   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
haftmann@26147: where
haftmann@26147:   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910: 
nipkow@15691: lemma override_on_emptyset[simp]: "override_on f g {} = f"
nipkow@15691: by(simp add:override_on_def)
nipkow@13910: 
nipkow@15691: lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
nipkow@15691: by(simp add:override_on_def)
nipkow@13910: 
nipkow@15691: lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
nipkow@15691: by(simp add:override_on_def)
nipkow@13910: 
haftmann@26147: 
haftmann@26147: subsection {* @{text swap} *}
paulson@15510: 
haftmann@22744: definition
haftmann@22744:   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@22744: where
haftmann@22744:   "swap a b f = f (a := f b, b:= f a)"
paulson@15510: 
paulson@15510: lemma swap_self: "swap a a f = f"
nipkow@15691: by (simp add: swap_def)
paulson@15510: 
paulson@15510: lemma swap_commute: "swap a b f = swap b a f"
paulson@15510: by (rule ext, simp add: fun_upd_def swap_def)
paulson@15510: 
paulson@15510: lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
paulson@15510: by (rule ext, simp add: fun_upd_def swap_def)
paulson@15510: 
paulson@15510: lemma inj_on_imp_inj_on_swap:
haftmann@22744:   "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
paulson@15510: by (simp add: inj_on_def swap_def, blast)
paulson@15510: 
paulson@15510: lemma inj_on_swap_iff [simp]:
paulson@15510:   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
paulson@15510: proof 
paulson@15510:   assume "inj_on (swap a b f) A"
paulson@15510:   with A have "inj_on (swap a b (swap a b f)) A" 
nipkow@17589:     by (iprover intro: inj_on_imp_inj_on_swap) 
paulson@15510:   thus "inj_on f A" by simp 
paulson@15510: next
paulson@15510:   assume "inj_on f A"
nipkow@27165:   with A show "inj_on (swap a b f) A" by(iprover intro: inj_on_imp_inj_on_swap)
paulson@15510: qed
paulson@15510: 
paulson@15510: lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
paulson@15510: apply (simp add: surj_def swap_def, clarify)
wenzelm@27125: apply (case_tac "y = f b", blast)
wenzelm@27125: apply (case_tac "y = f a", auto)
paulson@15510: done
paulson@15510: 
paulson@15510: lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
paulson@15510: proof 
paulson@15510:   assume "surj (swap a b f)"
paulson@15510:   hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
paulson@15510:   thus "surj f" by simp 
paulson@15510: next
paulson@15510:   assume "surj f"
paulson@15510:   thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
paulson@15510: qed
paulson@15510: 
paulson@15510: lemma bij_swap_iff: "bij (swap a b f) = bij f"
paulson@15510: by (simp add: bij_def)
haftmann@21547: 
nipkow@27188: hide (open) const swap
haftmann@21547: 
haftmann@31949: 
haftmann@31949: subsection {* Inversion of injective functions *}
haftmann@31949: 
nipkow@33057: definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
nipkow@33057: "the_inv_into A f == %x. THE y. y : A & f y = x"
nipkow@32957: 
nipkow@33057: lemma the_inv_into_f_f:
nipkow@33057:   "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
nipkow@33057: apply (simp add: the_inv_into_def inj_on_def)
nipkow@32957: apply (blast intro: the_equality)
nipkow@32957: done
nipkow@32957: 
nipkow@33057: lemma f_the_inv_into_f:
nipkow@33057:   "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
nipkow@33057: apply (simp add: the_inv_into_def)
nipkow@32957: apply (rule the1I2)
nipkow@32957:  apply(blast dest: inj_onD)
nipkow@32957: apply blast
nipkow@32957: done
nipkow@32957: 
nipkow@33057: lemma the_inv_into_into:
nipkow@33057:   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
nipkow@33057: apply (simp add: the_inv_into_def)
nipkow@32957: apply (rule the1I2)
nipkow@32957:  apply(blast dest: inj_onD)
nipkow@32957: apply blast
nipkow@32957: done
nipkow@32957: 
nipkow@33057: lemma the_inv_into_onto[simp]:
nipkow@33057:   "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
nipkow@33057: by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
nipkow@32957: 
nipkow@33057: lemma the_inv_into_f_eq:
nipkow@33057:   "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
nipkow@32957:   apply (erule subst)
nipkow@33057:   apply (erule the_inv_into_f_f, assumption)
nipkow@32957:   done
nipkow@32957: 
nipkow@33057: lemma the_inv_into_comp:
nipkow@32957:   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057:   the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
nipkow@33057: apply (rule the_inv_into_f_eq)
nipkow@32957:   apply (fast intro: comp_inj_on)
nipkow@33057:  apply (simp add: f_the_inv_into_f the_inv_into_into)
nipkow@33057: apply (simp add: the_inv_into_into)
nipkow@32957: done
nipkow@32957: 
nipkow@33057: lemma inj_on_the_inv_into:
nipkow@33057:   "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
nipkow@33057: by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
nipkow@32957: 
nipkow@33057: lemma bij_betw_the_inv_into:
nipkow@33057:   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
nipkow@33057: by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32957: 
berghofe@32998: abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
nipkow@33057:   "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998: 
berghofe@32998: lemma the_inv_f_f:
berghofe@32998:   assumes "inj f"
berghofe@32998:   shows "the_inv f (f x) = x" using assms UNIV_I
nipkow@33057:   by (rule the_inv_into_f_f)
berghofe@32998: 
haftmann@31949: 
haftmann@22845: subsection {* Proof tool setup *} 
haftmann@22845: 
haftmann@22845: text {* simplifies terms of the form
haftmann@22845:   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
haftmann@22845: 
wenzelm@24017: simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
haftmann@22845: let
haftmann@22845:   fun gen_fun_upd NONE T _ _ = NONE
wenzelm@24017:     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
haftmann@22845:   fun dest_fun_T1 (Type (_, T :: Ts)) = T
haftmann@22845:   fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
haftmann@22845:     let
haftmann@22845:       fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
haftmann@22845:             if v aconv x then SOME g else gen_fun_upd (find g) T v w
haftmann@22845:         | find t = NONE
haftmann@22845:     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017: 
wenzelm@24017:   fun proc ss ct =
wenzelm@24017:     let
wenzelm@24017:       val ctxt = Simplifier.the_context ss
wenzelm@24017:       val t = Thm.term_of ct
wenzelm@24017:     in
wenzelm@24017:       case find_double t of
wenzelm@24017:         (T, NONE) => NONE
wenzelm@24017:       | (T, SOME rhs) =>
wenzelm@27330:           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@24017:             (fn _ =>
wenzelm@24017:               rtac eq_reflection 1 THEN
wenzelm@24017:               rtac ext 1 THEN
wenzelm@24017:               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
wenzelm@24017:     end
wenzelm@24017: in proc end
haftmann@22845: *}
haftmann@22845: 
haftmann@22845: 
haftmann@32554: subsection {* Generic transfer procedure *}
haftmann@32554: 
haftmann@32554: definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool"
haftmann@32554:   where "TransferMorphism a B \<longleftrightarrow> True"
haftmann@32554: 
haftmann@32554: use "Tools/transfer.ML"
haftmann@32554: 
haftmann@32554: setup Transfer.setup
haftmann@32554: 
haftmann@32554: 
haftmann@21870: subsection {* Code generator setup *}
haftmann@21870: 
berghofe@25886: types_code
berghofe@25886:   "fun"  ("(_ ->/ _)")
berghofe@25886: attach (term_of) {*
berghofe@25886: fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
berghofe@25886: *}
berghofe@25886: attach (test) {*
berghofe@25886: fun gen_fun_type aF aT bG bT i =
berghofe@25886:   let
wenzelm@32740:     val tab = Unsynchronized.ref [];
berghofe@25886:     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
berghofe@25886:       (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()
berghofe@25886:   in
berghofe@25886:     (fn x =>
berghofe@25886:        case AList.lookup op = (!tab) x of
berghofe@25886:          NONE =>
berghofe@25886:            let val p as (y, _) = bG i
berghofe@25886:            in (tab := (x, p) :: !tab; y) end
berghofe@25886:        | SOME (y, _) => y,
berghofe@28711:      fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
berghofe@25886:   end;
berghofe@25886: *}
berghofe@25886: 
haftmann@21870: code_const "op \<circ>"
haftmann@21870:   (SML infixl 5 "o")
haftmann@21870:   (Haskell infixr 9 ".")
haftmann@21870: 
haftmann@21906: code_const "id"
haftmann@21906:   (Haskell "id")
haftmann@21906: 
nipkow@2912: end




