(* Title: HOL/Map.thy ID: $Id: Map.thy,v 1.42 2005/09/29 15:02:57 paulson Exp $ Author: Tobias Nipkow, based on a theory by David von Oheimb Copyright 1997-2003 TU Muenchen The datatype of `maps' (written ~=>); strongly resembles maps in VDM. *) header {* Maps *} theory Map imports List begin types ('a,'b) "~=>" = "'a => 'b option" (infixr 0) translations (type) "a ~=> b " <= (type) "a => b option" consts chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)" map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "|`" 110) dom :: "('a ~=> 'b) => 'a set" ran :: "('a ~=> 'b) => 'b set" map_of :: "('a * 'b)list => 'a ~=> 'b" map_upds:: "('a ~=> 'b) => 'a list => 'b list => ('a ~=> 'b)" map_upd_s::"('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)" ("_/'(_{|->}_/')" [900,0,0]900) map_subst::"('a ~=> 'b) => 'b => 'b => ('a ~=> 'b)" ("_/'(_~>_/')" [900,0,0]900) map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "⊆m" 50) constdefs map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55) "f o_m g == (λk. case g k of None => None | Some v => f v)" nonterminals maplets maplet syntax empty :: "'a ~=> 'b" "_maplet" :: "['a, 'a] => maplet" ("_ /|->/ _") "_maplets" :: "['a, 'a] => maplet" ("_ /[|->]/ _") "" :: "maplet => maplets" ("_") "_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _") "_MapUpd" :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900) "_Map" :: "maplets => 'a ~=> 'b" ("(1[_])") syntax (xsymbols) "~=>" :: "[type, type] => type" (infixr "\<rightharpoonup>" 0) map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "om" 55) "_maplet" :: "['a, 'a] => maplet" ("_ /\<mapsto>/ _") "_maplets" :: "['a, 'a] => maplet" ("_ /[\<mapsto>]/ _") map_upd_s :: "('a ~=> 'b) => 'a set => 'b => ('a ~=> 'b)" ("_/'(_/{\<mapsto>}/_')" [900,0,0]900) map_subst :: "('a ~=> 'b) => 'b => 'b => ('a ~=> 'b)" ("_/'(_\<leadsto>_/')" [900,0,0]900) "@chg_map" :: "('a ~=> 'b) => 'a => ('b => 'b) => ('a ~=> 'b)" ("_/'(_/\<mapsto>λ_. _')" [900,0,0,0] 900) syntax (latex output) restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" ("_\<restriction>_" [111,110] 110) --"requires amssymb!" translations "empty" => "_K None" "empty" <= "%x. None" "m(x\<mapsto>λy. f)" == "chg_map (λy. f) x m" "_MapUpd m (_Maplets xy ms)" == "_MapUpd (_MapUpd m xy) ms" "_MapUpd m (_maplet x y)" == "m(x:=Some y)" "_MapUpd m (_maplets x y)" == "map_upds m x y" "_Map ms" == "_MapUpd empty ms" "_Map (_Maplets ms1 ms2)" <= "_MapUpd (_Map ms1) ms2" "_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3" defs chg_map_def: "chg_map f a m == case m a of None => m | Some b => m(a|->f b)" map_add_def: "m1++m2 == %x. case m2 x of None => m1 x | Some y => Some y" restrict_map_def: "m|`A == %x. if x : A then m x else None" map_upds_def: "m(xs [|->] ys) == m ++ map_of (rev(zip xs ys))" map_upd_s_def: "m(as{|->}b) == %x. if x : as then Some b else m x" map_subst_def: "m(a~>b) == %x. if m x = Some a then Some b else m x" dom_def: "dom(m) == {a. m a ~= None}" ran_def: "ran(m) == {b. EX a. m a = Some b}" map_le_def: "m1 ⊆m m2 == ALL a : dom m1. m1 a = m2 a" primrec "map_of [] = empty" "map_of (p#ps) = (map_of ps)(fst p |-> snd p)" subsection {* @{term [source] empty} *} lemma empty_upd_none[simp]: "empty(x := None) = empty" apply (rule ext) apply (simp (no_asm)) done (* FIXME: what is this sum_case nonsense?? *) lemma sum_case_empty_empty[simp]: "sum_case empty empty = empty" apply (rule ext) apply (simp (no_asm) split add: sum.split) done subsection {* @{term [source] map_upd} *} lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t" apply (rule ext) apply (simp (no_asm_simp)) done lemma map_upd_nonempty[simp]: "t(k|->x) ~= empty" apply safe apply (drule_tac x = k in fun_cong) apply (simp (no_asm_use)) done lemma map_upd_eqD1: "m(a\<mapsto>x) = n(a\<mapsto>y) ==> x = y" by (drule fun_cong [of _ _ a], auto) lemma map_upd_Some_unfold: "((m(a|->b)) x = Some y) = (x = a ∧ b = y ∨ x ≠ a ∧ m x = Some y)" by auto lemma image_map_upd[simp]: "x ∉ A ==> m(x \<mapsto> y) ` A = m ` A" by fastsimp lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))" apply (unfold image_def) apply (simp (no_asm_use) add: full_SetCompr_eq) apply (rule finite_subset) prefer 2 apply assumption apply auto done (* FIXME: what is this sum_case nonsense?? *) subsection {* @{term [source] sum_case} and @{term [source] empty}/@{term [source] map_upd} *} lemma sum_case_map_upd_empty[simp]: "sum_case (m(k|->y)) empty = (sum_case m empty)(Inl k|->y)" apply (rule ext) apply (simp (no_asm) split add: sum.split) done lemma sum_case_empty_map_upd[simp]: "sum_case empty (m(k|->y)) = (sum_case empty m)(Inr k|->y)" apply (rule ext) apply (simp (no_asm) split add: sum.split) done lemma sum_case_map_upd_map_upd[simp]: "sum_case (m1(k1|->y1)) (m2(k2|->y2)) = (sum_case (m1(k1|->y1)) m2)(Inr k2|->y2)" apply (rule ext) apply (simp (no_asm) split add: sum.split) done subsection {* @{term [source] chg_map} *} lemma chg_map_new[simp]: "m a = None ==> chg_map f a m = m" by (unfold chg_map_def, auto) lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)" by (unfold chg_map_def, auto) lemma chg_map_other [simp]: "a ≠ b ==> chg_map f a m b = m b" by (auto simp: chg_map_def split add: option.split) subsection {* @{term [source] map_of} *} lemma map_of_eq_None_iff: "(map_of xys x = None) = (x ∉ fst ` (set xys))" by (induct xys) simp_all lemma map_of_is_SomeD: "map_of xys x = Some y ==> (x,y) ∈ set xys" apply(induct xys) apply simp apply(clarsimp split:if_splits) done lemma map_of_eq_Some_iff[simp]: "distinct(map fst xys) ==> (map_of xys x = Some y) = ((x,y) ∈ set xys)" apply(induct xys) apply(simp) apply(auto simp:map_of_eq_None_iff[symmetric]) done lemma Some_eq_map_of_iff[simp]: "distinct(map fst xys) ==> (Some y = map_of xys x) = ((x,y) ∈ set xys)" by(auto simp del:map_of_eq_Some_iff simp add:map_of_eq_Some_iff[symmetric]) lemma map_of_is_SomeI [simp]: "[| distinct(map fst xys); (x,y) ∈ set xys |] ==> map_of xys x = Some y" apply (induct xys) apply simp apply force done lemma map_of_zip_is_None[simp]: "length xs = length ys ==> (map_of (zip xs ys) x = None) = (x ∉ set xs)" by (induct rule:list_induct2, simp_all) lemma finite_range_map_of: "finite (range (map_of xys))" apply (induct xys) apply (simp_all (no_asm) add: image_constant) apply (rule finite_subset) prefer 2 apply assumption apply auto done lemma map_of_SomeD [rule_format]: "map_of xs k = Some y --> (k,y):set xs" by (induct "xs", auto) lemma map_of_mapk_SomeI [rule_format]: "inj f ==> map_of t k = Some x --> map_of (map (split (%k. Pair (f k))) t) (f k) = Some x" apply (induct "t") apply (auto simp add: inj_eq) done lemma weak_map_of_SomeI [rule_format]: "(k, x) : set l --> (∃x. map_of l k = Some x)" by (induct "l", auto) lemma map_of_filter_in: "[| map_of xs k = Some z; P k z |] ==> map_of (filter (split P) xs) k = Some z" apply (rule mp) prefer 2 apply assumption apply (erule thin_rl) apply (induct "xs", auto) done lemma map_of_map: "map_of (map (%(a,b). (a,f b)) xs) x = option_map f (map_of xs x)" by (induct "xs", auto) subsection {* @{term [source] option_map} related *} lemma option_map_o_empty[simp]: "option_map f o empty = empty" apply (rule ext) apply (simp (no_asm)) done lemma option_map_o_map_upd[simp]: "option_map f o m(a|->b) = (option_map f o m)(a|->f b)" apply (rule ext) apply (simp (no_asm)) done subsection {* @{term [source] map_comp} related *} lemma map_comp_empty [simp]: "m om empty = empty" "empty om m = empty" by (auto simp add: map_comp_def intro: ext split: option.splits) lemma map_comp_simps [simp]: "m2 k = None ==> (m1 om m2) k = None" "m2 k = Some k' ==> (m1 om m2) k = m1 k'" by (auto simp add: map_comp_def) lemma map_comp_Some_iff: "((m1 om m2) k = Some v) = (∃k'. m2 k = Some k' ∧ m1 k' = Some v)" by (auto simp add: map_comp_def split: option.splits) lemma map_comp_None_iff: "((m1 om m2) k = None) = (m2 k = None ∨ (∃k'. m2 k = Some k' ∧ m1 k' = None)) " by (auto simp add: map_comp_def split: option.splits) subsection {* @{text "++"} *} lemma map_add_empty[simp]: "m ++ empty = m" apply (unfold map_add_def) apply (simp (no_asm)) done lemma empty_map_add[simp]: "empty ++ m = m" apply (unfold map_add_def) apply (rule ext) apply (simp split add: option.split) done lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" apply(rule ext) apply(simp add: map_add_def split:option.split) done lemma map_add_Some_iff: "((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)" apply (unfold map_add_def) apply (simp (no_asm) split add: option.split) done lemmas map_add_SomeD = map_add_Some_iff [THEN iffD1, standard] declare map_add_SomeD [dest!] lemma map_add_find_right[simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx" by (subst map_add_Some_iff, fast) lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)" apply (unfold map_add_def) apply (simp (no_asm) split add: option.split) done lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)" apply (unfold map_add_def) apply (rule ext, auto) done lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)" by(simp add:map_upds_def) lemma map_of_append[simp]: "map_of (xs@ys) = map_of ys ++ map_of xs" apply (unfold map_add_def) apply (induct "xs") apply (simp (no_asm)) apply (rule ext) apply (simp (no_asm_simp) split add: option.split) done declare fun_upd_apply [simp del] lemma finite_range_map_of_map_add: "finite (range f) ==> finite (range (f ++ map_of l))" apply (induct "l", auto) apply (erule finite_range_updI) done declare fun_upd_apply [simp] lemma inj_on_map_add_dom[iff]: "inj_on (m ++ m') (dom m') = inj_on m' (dom m')" by(fastsimp simp add:map_add_def dom_def inj_on_def split:option.splits) subsection {* @{term [source] restrict_map} *} lemma restrict_map_to_empty[simp]: "m|`{} = empty" by(simp add: restrict_map_def) lemma restrict_map_empty[simp]: "empty|`D = empty" by(simp add: restrict_map_def) lemma restrict_in [simp]: "x ∈ A ==> (m|`A) x = m x" by (auto simp: restrict_map_def) lemma restrict_out [simp]: "x ∉ A ==> (m|`A) x = None" by (auto simp: restrict_map_def) lemma ran_restrictD: "y ∈ ran (m|`A) ==> ∃x∈A. m x = Some y" by (auto simp: restrict_map_def ran_def split: split_if_asm) lemma dom_restrict [simp]: "dom (m|`A) = dom m ∩ A" by (auto simp: restrict_map_def dom_def split: split_if_asm) lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})" by (rule ext, auto simp: restrict_map_def) lemma restrict_restrict [simp]: "m|`A|`B = m|`(A∩B)" by (rule ext, auto simp: restrict_map_def) lemma restrict_fun_upd[simp]: "m(x := y)|`D = (if x ∈ D then (m|`(D-{x}))(x := y) else m|`D)" by(simp add: restrict_map_def expand_fun_eq) lemma fun_upd_None_restrict[simp]: "(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)" by(simp add: restrict_map_def expand_fun_eq) lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)" by(simp add: restrict_map_def expand_fun_eq) lemma fun_upd_restrict_conv[simp]: "x ∈ D ==> (m|`D)(x := y) = (m|`(D-{x}))(x := y)" by(simp add: restrict_map_def expand_fun_eq) subsection {* @{term [source] map_upds} *} lemma map_upds_Nil1[simp]: "m([] [|->] bs) = m" by(simp add:map_upds_def) lemma map_upds_Nil2[simp]: "m(as [|->] []) = m" by(simp add:map_upds_def) lemma map_upds_Cons[simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)" by(simp add:map_upds_def) lemma map_upds_append1[simp]: "!!ys m. size xs < size ys ==> m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" apply(induct xs) apply(clarsimp simp add:neq_Nil_conv) apply (case_tac ys, simp, simp) done lemma map_upds_list_update2_drop[simp]: "!!m ys i. [|size xs ≤ i; i < size ys|] ==> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" apply (induct xs, simp) apply (case_tac ys, simp) apply(simp split:nat.split) done lemma map_upd_upds_conv_if: "!!x y ys f. (f(x|->y))(xs [|->] ys) = (if x : set(take (length ys) xs) then f(xs [|->] ys) else (f(xs [|->] ys))(x|->y))" apply (induct xs, simp) apply(case_tac ys) apply(auto split:split_if simp:fun_upd_twist) done lemma map_upds_twist [simp]: "a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)" apply(insert set_take_subset) apply (fastsimp simp add: map_upd_upds_conv_if) done lemma map_upds_apply_nontin[simp]: "!!ys. x ~: set xs ==> (f(xs[|->]ys)) x = f x" apply (induct xs, simp) apply(case_tac ys) apply(auto simp: map_upd_upds_conv_if) done lemma fun_upds_append_drop[simp]: "!!m ys. size xs = size ys ==> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" apply(induct xs) apply (simp) apply(case_tac ys) apply simp_all done lemma fun_upds_append2_drop[simp]: "!!m ys. size xs = size ys ==> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" apply(induct xs) apply (simp) apply(case_tac ys) apply simp_all done lemma restrict_map_upds[simp]: "!!m ys. [| length xs = length ys; set xs ⊆ D |] ==> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)" apply (induct xs, simp) apply (case_tac ys, simp) apply(simp add:Diff_insert[symmetric] insert_absorb) apply(simp add: map_upd_upds_conv_if) done subsection {* @{term [source] map_upd_s} *} lemma map_upd_s_apply [simp]: "(m(as{|->}b)) x = (if x : as then Some b else m x)" by (simp add: map_upd_s_def) lemma map_subst_apply [simp]: "(m(a~>b)) x = (if m x = Some a then Some b else m x)" by (simp add: map_subst_def) subsection {* @{term [source] dom} *} lemma domI: "m a = Some b ==> a : dom m" by (unfold dom_def, auto) (* declare domI [intro]? *) lemma domD: "a : dom m ==> ∃b. m a = Some b" by (unfold dom_def, auto) lemma domIff[iff]: "(a : dom m) = (m a ~= None)" by (unfold dom_def, auto) declare domIff [simp del] lemma dom_empty[simp]: "dom empty = {}" apply (unfold dom_def) apply (simp (no_asm)) done lemma dom_fun_upd[simp]: "dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))" by (simp add:dom_def) blast lemma dom_map_of: "dom(map_of xys) = {x. ∃y. (x,y) : set xys}" apply(induct xys) apply(auto simp del:fun_upd_apply) done lemma dom_map_of_conv_image_fst: "dom(map_of xys) = fst ` (set xys)" by(force simp: dom_map_of) lemma dom_map_of_zip[simp]: "[| length xs = length ys; distinct xs |] ==> dom(map_of(zip xs ys)) = set xs" by(induct rule: list_induct2, simp_all) lemma finite_dom_map_of: "finite (dom (map_of l))" apply (unfold dom_def) apply (induct "l") apply (auto simp add: insert_Collect [symmetric]) done lemma dom_map_upds[simp]: "!!m ys. dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m" apply (induct xs, simp) apply (case_tac ys, auto) done lemma dom_map_add[simp]: "dom(m++n) = dom n Un dom m" by (unfold dom_def, auto) lemma dom_override_on[simp]: "dom(override_on f g A) = (dom f - {a. a : A - dom g}) Un {a. a : A Int dom g}" by(auto simp add: dom_def override_on_def) lemma map_add_comm: "dom m1 ∩ dom m2 = {} ==> m1++m2 = m2++m1" apply(rule ext) apply(fastsimp simp:map_add_def split:option.split) done subsection {* @{term [source] ran} *} lemma ranI: "m a = Some b ==> b : ran m" by (auto simp add: ran_def) (* declare ranI [intro]? *) lemma ran_empty[simp]: "ran empty = {}" apply (unfold ran_def) apply (simp (no_asm)) done lemma ran_map_upd[simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)" apply (unfold ran_def, auto) apply (subgoal_tac "~ (aa = a) ") apply auto done subsection {* @{text "map_le"} *} lemma map_le_empty [simp]: "empty ⊆m g" by(simp add:map_le_def) lemma upd_None_map_le [simp]: "f(x := None) ⊆m f" by(force simp add:map_le_def) lemma map_le_upd[simp]: "f ⊆m g ==> f(a := b) ⊆m g(a := b)" by(fastsimp simp add:map_le_def) lemma map_le_imp_upd_le [simp]: "m1 ⊆m m2 ==> m1(x := None) ⊆m m2(x \<mapsto> y)" by(force simp add:map_le_def) lemma map_le_upds[simp]: "!!f g bs. f ⊆m g ==> f(as [|->] bs) ⊆m g(as [|->] bs)" apply (induct as, simp) apply (case_tac bs, auto) done lemma map_le_implies_dom_le: "(f ⊆m g) ==> (dom f ⊆ dom g)" by (fastsimp simp add: map_le_def dom_def) lemma map_le_refl [simp]: "f ⊆m f" by (simp add: map_le_def) lemma map_le_trans[trans]: "[| m1 ⊆m m2; m2 ⊆m m3|] ==> m1 ⊆m m3" by(force simp add:map_le_def) lemma map_le_antisym: "[| f ⊆m g; g ⊆m f |] ==> f = g" apply (unfold map_le_def) apply (rule ext) apply (case_tac "x ∈ dom f", simp) apply (case_tac "x ∈ dom g", simp, fastsimp) done lemma map_le_map_add [simp]: "f ⊆m (g ++ f)" by (fastsimp simp add: map_le_def) lemma map_le_iff_map_add_commute: "(f ⊆m f ++ g) = (f++g = g++f)" by(fastsimp simp add:map_add_def map_le_def expand_fun_eq split:option.splits) lemma map_add_le_mapE: "f++g ⊆m h ==> g ⊆m h" by (fastsimp simp add: map_le_def map_add_def dom_def) lemma map_add_le_mapI: "[| f ⊆m h; g ⊆m h; f ⊆m f++g |] ==> f++g ⊆m h" by (clarsimp simp add: map_le_def map_add_def dom_def split:option.splits) end
lemma empty_upd_none:
empty(x := None) = empty
lemma sum_case_empty_empty:
sum_case empty empty = empty
lemma map_upd_triv:
t k = Some x ==> t(k |-> x) = t
lemma map_upd_nonempty:
t(k |-> x) ≠ empty
lemma map_upd_eqD1:
m(a |-> x) = n(a |-> y) ==> x = y
lemma map_upd_Some_unfold:
((m(a |-> b)) x = Some y) = (x = a ∧ b = y ∨ x ≠ a ∧ m x = Some y)
lemma image_map_upd:
x ∉ A ==> m(x |-> y) ` A = m ` A
lemma finite_range_updI:
finite (range f) ==> finite (range (f(a |-> b)))
lemma sum_case_map_upd_empty:
sum_case (m(k |-> y)) empty = sum_case m empty(Inl k |-> y)
lemma sum_case_empty_map_upd:
sum_case empty (m(k |-> y)) = sum_case empty m(Inr k |-> y)
lemma sum_case_map_upd_map_upd:
sum_case (m1.0(k1.0 |-> y1.0)) (m2.0(k2.0 |-> y2.0)) = sum_case (m1.0(k1.0 |-> y1.0)) m2.0(Inr k2.0 |-> y2.0)
lemma chg_map_new:
m a = None ==> chg_map f a m = m
lemma chg_map_upd:
m a = Some b ==> chg_map f a m = m(a |-> f b)
lemma chg_map_other:
a ≠ b ==> chg_map f a m b = m b
lemma map_of_eq_None_iff:
(map_of xys x = None) = (x ∉ fst ` set xys)
lemma map_of_is_SomeD:
map_of xys x = Some y ==> (x, y) ∈ set xys
lemma map_of_eq_Some_iff:
distinct (map fst xys) ==> (map_of xys x = Some y) = ((x, y) ∈ set xys)
lemma Some_eq_map_of_iff:
distinct (map fst xys) ==> (Some y = map_of xys x) = ((x, y) ∈ set xys)
lemma map_of_is_SomeI:
[| distinct (map fst xys); (x, y) ∈ set xys |] ==> map_of xys x = Some y
lemma map_of_zip_is_None:
length xs = length ys ==> (map_of (zip xs ys) x = None) = (x ∉ set xs)
lemma finite_range_map_of:
finite (range (map_of xys))
lemma map_of_SomeD:
map_of xs k = Some y ==> (k, y) ∈ set xs
lemma map_of_mapk_SomeI:
[| inj f; map_of t k = Some x |] ==> map_of (map (%(k, y). (f k, y)) t) (f k) = Some x
lemma weak_map_of_SomeI:
(k, x) ∈ set l ==> ∃x. map_of l k = Some x
lemma map_of_filter_in:
[| map_of xs k = Some z; P k z |] ==> map_of [(x, y)∈xs . P x y] k = Some z
lemma map_of_map:
map_of (map (%(a, b). (a, f b)) xs) x = option_map f (map_of xs x)
lemma option_map_o_empty:
option_map f o empty = empty
lemma option_map_o_map_upd:
option_map f o m(a |-> b) = (option_map f o m)(a |-> f b)
lemma map_comp_empty:
m o_m empty = empty
empty o_m m = empty
lemma map_comp_simps:
m2.0 k = None ==> (m1.0 o_m m2.0) k = None
m2.0 k = Some k' ==> (m1.0 o_m m2.0) k = m1.0 k'
lemma map_comp_Some_iff:
((m1.0 o_m m2.0) k = Some v) = (∃k'. m2.0 k = Some k' ∧ m1.0 k' = Some v)
lemma map_comp_None_iff:
((m1.0 o_m m2.0) k = None) = (m2.0 k = None ∨ (∃k'. m2.0 k = Some k' ∧ m1.0 k' = None))
lemma map_add_empty:
m ++ empty = m
lemma empty_map_add:
empty ++ m = m
lemma map_add_assoc:
m1.0 ++ (m2.0 ++ m3.0) = m1.0 ++ m2.0 ++ m3.0
lemma map_add_Some_iff:
((m ++ n) k = Some x) = (n k = Some x ∨ n k = None ∧ m k = Some x)
lemmas map_add_SomeD:
(m ++ n) k = Some x ==> n k = Some x ∨ n k = None ∧ m k = Some x
lemmas map_add_SomeD:
(m ++ n) k = Some x ==> n k = Some x ∨ n k = None ∧ m k = Some x
lemma map_add_find_right:
n k = Some xx ==> (m ++ n) k = Some xx
lemma map_add_None:
((m ++ n) k = None) = (n k = None ∧ m k = None)
lemma map_add_upd:
f ++ g(x |-> y) = (f ++ g)(x |-> y)
lemma map_add_upds:
m1.0 ++ m2.0(xs [|->] ys) = (m1.0 ++ m2.0)(xs [|->] ys)
lemma map_of_append:
map_of (xs @ ys) = map_of ys ++ map_of xs
lemma finite_range_map_of_map_add:
finite (range f) ==> finite (range (f ++ map_of l))
lemma inj_on_map_add_dom:
inj_on (m ++ m') (dom m') = inj_on m' (dom m')
lemma restrict_map_to_empty:
m |` {} = empty
lemma restrict_map_empty:
empty |` D = empty
lemma restrict_in:
x ∈ A ==> (m |` A) x = m x
lemma restrict_out:
x ∉ A ==> (m |` A) x = None
lemma ran_restrictD:
y ∈ ran (m |` A) ==> ∃x∈A. m x = Some y
lemma dom_restrict:
dom (m |` A) = dom m ∩ A
lemma restrict_upd_same:
m(x |-> y) |` (- {x}) = m |` (- {x})
lemma restrict_restrict:
m |` A |` B = m |` (A ∩ B)
lemma restrict_fun_upd:
m(x := y) |` D = (if x ∈ D then (m |` (D - {x}))(x := y) else m |` D)
lemma fun_upd_None_restrict:
(m |` D)(x := None) = (if x ∈ D then m |` (D - {x}) else m |` D)
lemma fun_upd_restrict:
(m |` D)(x := y) = (m |` (D - {x}))(x := y)
lemma fun_upd_restrict_conv:
x ∈ D ==> (m |` D)(x := y) = (m |` (D - {x}))(x := y)
lemma map_upds_Nil1:
m([] [|->] bs) = m
lemma map_upds_Nil2:
m(as [|->] []) = m
lemma map_upds_Cons:
m(a # as [|->] b # bs) = m(a |-> b, as [|->] bs)
lemma map_upds_append1:
length xs < length ys ==> m(xs @ [x] [|->] ys) = m(xs [|->] ys, x |-> ys ! length xs)
lemma map_upds_list_update2_drop:
[| length xs ≤ i; i < length ys |] ==> m(xs [|->] ys[i := y]) = m(xs [|->] ys)
lemma map_upd_upds_conv_if:
f(x |-> y, xs [|->] ys) = (if x ∈ set (take (length ys) xs) then f(xs [|->] ys) else f(xs [|->] ys, x |-> y))
lemma map_upds_twist:
a ∉ set as ==> m(a |-> b, as [|->] bs) = m(as [|->] bs, a |-> b)
lemma map_upds_apply_nontin:
x ∉ set xs ==> (f(xs [|->] ys)) x = f x
lemma fun_upds_append_drop:
length xs = length ys ==> m(xs @ zs [|->] ys) = m(xs [|->] ys)
lemma fun_upds_append2_drop:
length xs = length ys ==> m(xs [|->] ys @ zs) = m(xs [|->] ys)
lemma restrict_map_upds:
[| length xs = length ys; set xs ⊆ D |] ==> m(xs [|->] ys) |` D = (m |` (D - set xs))(xs [|->] ys)
lemma map_upd_s_apply:
(m(as{|->}b)) x = (if x ∈ as then Some b else m x)
lemma map_subst_apply:
(m(a~>b)) x = (if m x = Some a then Some b else m x)
lemma domI:
m a = Some b ==> a ∈ dom m
lemma domD:
a ∈ dom m ==> ∃b. m a = Some b
lemma domIff:
(a ∈ dom m) = (m a ≠ None)
lemma dom_empty:
dom empty = {}
lemma dom_fun_upd:
dom (f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))
lemma dom_map_of:
dom (map_of xys) = {x. ∃y. (x, y) ∈ set xys}
lemma dom_map_of_conv_image_fst:
dom (map_of xys) = fst ` set xys
lemma dom_map_of_zip:
[| length xs = length ys; distinct xs |] ==> dom (map_of (zip xs ys)) = set xs
lemma finite_dom_map_of:
finite (dom (map_of l))
lemma dom_map_upds:
dom (m(xs [|->] ys)) = set (take (length ys) xs) ∪ dom m
lemma dom_map_add:
dom (m ++ n) = dom n ∪ dom m
lemma dom_override_on:
dom (override_on f g A) = dom f - {a. a ∈ A - dom g} ∪ {a. a ∈ A ∩ dom g}
lemma map_add_comm:
dom m1.0 ∩ dom m2.0 = {} ==> m1.0 ++ m2.0 = m2.0 ++ m1.0
lemma ranI:
m a = Some b ==> b ∈ ran m
lemma ran_empty:
ran empty = {}
lemma ran_map_upd:
m a = None ==> ran (m(a |-> b)) = insert b (ran m)
lemma map_le_empty:
empty ⊆m g
lemma upd_None_map_le:
f(x := None) ⊆m f
lemma map_le_upd:
f ⊆m g ==> f(a := b) ⊆m g(a := b)
lemma map_le_imp_upd_le:
m1.0 ⊆m m2.0 ==> m1.0(x := None) ⊆m m2.0(x |-> y)
lemma map_le_upds:
f ⊆m g ==> f(as [|->] bs) ⊆m g(as [|->] bs)
lemma map_le_implies_dom_le:
f ⊆m g ==> dom f ⊆ dom g
lemma map_le_refl:
f ⊆m f
lemma map_le_trans:
[| m1.0 ⊆m m2.0; m2.0 ⊆m m3.0 |] ==> m1.0 ⊆m m3.0
lemma map_le_antisym:
[| f ⊆m g; g ⊆m f |] ==> f = g
lemma map_le_map_add:
f ⊆m g ++ f
lemma map_le_iff_map_add_commute:
(f ⊆m f ++ g) = (f ++ g = g ++ f)
lemma map_add_le_mapE:
f ++ g ⊆m h ==> g ⊆m h
lemma map_add_le_mapI:
[| f ⊆m h; g ⊆m h; f ⊆m f ++ g |] ==> f ++ g ⊆m h