Theory FinFun

theory FinFun
imports Cardinality
(* Author: Andreas Lochbihler, Uni Karlsruhe *)

header {* Almost everywhere constant functions *}

theory FinFun
imports Cardinality
begin

text {*
  This theory defines functions which are constant except for finitely
  many points (FinFun) and introduces a type finfin along with a
  number of operators for them. The code generator is set up such that
  such functions can be represented as data in the generated code and
  all operators are executable.

  For details, see Formalising FinFuns - Generating Code for Functions as Data by A. Lochbihler in TPHOLs 2009.
*}


subsection {* The @{text "map_default"} operation *}

definition map_default :: "'b => ('a \<rightharpoonup> 'b) => 'a => 'b"
where "map_default b f a ≡ case f a of None => b | Some b' => b'"

lemma map_default_delete [simp]:
  "map_default b (f(a := None)) = (map_default b f)(a := b)"
by(simp add: map_default_def fun_eq_iff)

lemma map_default_insert:
  "map_default b (f(a \<mapsto> b')) = (map_default b f)(a := b')"
by(simp add: map_default_def fun_eq_iff)

lemma map_default_empty [simp]: "map_default b empty = (λa. b)"
by(simp add: fun_eq_iff map_default_def)

lemma map_default_inject:
  fixes g g' :: "'a \<rightharpoonup> 'b"
  assumes infin_eq: "¬ finite (UNIV :: 'a set) ∨ b = b'"
  and fin: "finite (dom g)" and b: "b ∉ ran g"
  and fin': "finite (dom g')" and b': "b' ∉ ran g'"
  and eq': "map_default b g = map_default b' g'"
  shows "b = b'" "g = g'"
proof -
  from infin_eq show bb': "b = b'"
  proof
    assume infin: "¬ finite (UNIV :: 'a set)"
    from fin fin' have "finite (dom g ∪ dom g')" by auto
    with infin have "UNIV - (dom g ∪ dom g') ≠ {}" by(auto dest: finite_subset)
    then obtain a where a: "a ∉ dom g ∪ dom g'" by auto
    hence "map_default b g a = b" "map_default b' g' a = b'" by(auto simp add: map_default_def)
    with eq' show "b = b'" by simp
  qed

  show "g = g'"
  proof
    fix x
    show "g x = g' x"
    proof(cases "g x")
      case None
      hence "map_default b g x = b" by(simp add: map_default_def)
      with bb' eq' have "map_default b' g' x = b'" by simp
      with b' have "g' x = None" by(simp add: map_default_def ran_def split: option.split_asm)
      with None show ?thesis by simp
    next
      case (Some c)
      with b have cb: "c ≠ b" by(auto simp add: ran_def)
      moreover from Some have "map_default b g x = c" by(simp add: map_default_def)
      with eq' have "map_default b' g' x = c" by simp
      ultimately have "g' x = Some c" using b' bb' by(auto simp add: map_default_def split: option.splits)
      with Some show ?thesis by simp
    qed
  qed
qed

subsection {* The finfun type *}

definition "finfun = {f::'a=>'b. ∃b. finite {a. f a ≠ b}}"

typedef ('a,'b) finfun  ("(_ =>f /_)" [22, 21] 21) = "finfun :: ('a => 'b) set"
  morphisms finfun_apply Abs_finfun
proof -
  have "∃f. finite {x. f x ≠ undefined}"
  proof
    show "finite {x. (λy. undefined) x ≠ undefined}" by auto
  qed
  then show ?thesis unfolding finfun_def by auto
qed

type_notation finfun ("(_ =>f /_)" [22, 21] 21)

setup_lifting type_definition_finfun

lemma fun_upd_finfun: "y(a := b) ∈ finfun <-> y ∈ finfun"
proof -
  { fix b'
    have "finite {a'. (y(a := b)) a' ≠ b'} = finite {a'. y a' ≠ b'}"
    proof(cases "b = b'")
      case True
      hence "{a'. (y(a := b)) a' ≠ b'} = {a'. y a' ≠ b'} - {a}" by auto
      thus ?thesis by simp
    next
      case False
      hence "{a'. (y(a := b)) a' ≠ b'} = insert a {a'. y a' ≠ b'}" by auto
      thus ?thesis by simp
    qed }
  thus ?thesis unfolding finfun_def by blast
qed

lemma const_finfun: "(λx. a) ∈ finfun"
by(auto simp add: finfun_def)

lemma finfun_left_compose:
  assumes "y ∈ finfun"
  shows "g o y ∈ finfun"
proof -
  from assms obtain b where "finite {a. y a ≠ b}"
    unfolding finfun_def by blast
  hence "finite {c. g (y c) ≠ g b}"
  proof(induct "{a. y a ≠ b}" arbitrary: y)
    case empty
    hence "y = (λa. b)" by(auto)
    thus ?case by(simp)
  next
    case (insert x F)
    note IH = `!!y. F = {a. y a ≠ b} ==> finite {c. g (y c) ≠ g b}`
    from `insert x F = {a. y a ≠ b}` `x ∉ F`
    have F: "F = {a. (y(x := b)) a ≠ b}" by(auto)
    show ?case
    proof(cases "g (y x) = g b")
      case True
      hence "{c. g ((y(x := b)) c) ≠ g b} = {c. g (y c) ≠ g b}" by auto
      with IH[OF F] show ?thesis by simp
    next
      case False
      hence "{c. g (y c) ≠ g b} = insert x {c. g ((y(x := b)) c) ≠ g b}" by auto
      with IH[OF F] show ?thesis by(simp)
    qed
  qed
  thus ?thesis unfolding finfun_def by auto
qed

lemma assumes "y ∈ finfun"
  shows fst_finfun: "fst o y ∈ finfun"
  and snd_finfun: "snd o y ∈ finfun"
proof -
  from assms obtain b c where bc: "finite {a. y a ≠ (b, c)}"
    unfolding finfun_def by auto
  have "{a. fst (y a) ≠ b} ⊆ {a. y a ≠ (b, c)}"
    and "{a. snd (y a) ≠ c} ⊆ {a. y a ≠ (b, c)}" by auto
  hence "finite {a. fst (y a) ≠ b}" 
    and "finite {a. snd (y a) ≠ c}" using bc by(auto intro: finite_subset)
  thus "fst o y ∈ finfun" "snd o y ∈ finfun"
    unfolding finfun_def by auto
qed

lemma map_of_finfun: "map_of xs ∈ finfun"
unfolding finfun_def
by(induct xs)(auto simp add: Collect_neg_eq Collect_conj_eq Collect_imp_eq intro: finite_subset)

lemma Diag_finfun: "(λx. (f x, g x)) ∈ finfun <-> f ∈ finfun ∧ g ∈ finfun"
by(auto intro: finite_subset simp add: Collect_neg_eq Collect_imp_eq Collect_conj_eq finfun_def)

lemma finfun_right_compose:
  assumes g: "g ∈ finfun" and inj: "inj f"
  shows "g o f ∈ finfun"
proof -
  from g obtain b where b: "finite {a. g a ≠ b}" unfolding finfun_def by blast
  moreover have "f ` {a. g (f a) ≠ b} ⊆ {a. g a ≠ b}" by auto
  moreover from inj have "inj_on f {a.  g (f a) ≠ b}" by(rule subset_inj_on) blast
  ultimately have "finite {a. g (f a) ≠ b}"
    by(blast intro: finite_imageD[where f=f] finite_subset)
  thus ?thesis unfolding finfun_def by auto
qed

lemma finfun_curry:
  assumes fin: "f ∈ finfun"
  shows "curry f ∈ finfun" "curry f a ∈ finfun"
proof -
  from fin obtain c where c: "finite {ab. f ab ≠ c}" unfolding finfun_def by blast
  moreover have "{a. ∃b. f (a, b) ≠ c} = fst ` {ab. f ab ≠ c}" by(force)
  hence "{a. curry f a ≠ (λb. c)} = fst ` {ab. f ab ≠ c}"
    by(auto simp add: curry_def fun_eq_iff)
  ultimately have "finite {a. curry f a ≠ (λb. c)}" by simp
  thus "curry f ∈ finfun" unfolding finfun_def by blast
  
  have "snd ` {ab. f ab ≠ c} = {b. ∃a. f (a, b) ≠ c}" by(force)
  hence "{b. f (a, b) ≠ c} ⊆ snd ` {ab. f ab ≠ c}" by auto
  hence "finite {b. f (a, b) ≠ c}" by(rule finite_subset)(rule finite_imageI[OF c])
  thus "curry f a ∈ finfun" unfolding finfun_def by auto
qed

bundle finfun =
  fst_finfun[simp] snd_finfun[simp] Abs_finfun_inverse[simp] 
  finfun_apply_inverse[simp] Abs_finfun_inject[simp] finfun_apply_inject[simp]
  Diag_finfun[simp] finfun_curry[simp]
  const_finfun[iff] fun_upd_finfun[iff] finfun_apply[iff] map_of_finfun[iff]
  finfun_left_compose[intro] fst_finfun[intro] snd_finfun[intro]

lemma Abs_finfun_inject_finite:
  fixes x y :: "'a => 'b"
  assumes fin: "finite (UNIV :: 'a set)"
  shows "Abs_finfun x = Abs_finfun y <-> x = y"
proof
  assume "Abs_finfun x = Abs_finfun y"
  moreover have "x ∈ finfun" "y ∈ finfun" unfolding finfun_def
    by(auto intro: finite_subset[OF _ fin])
  ultimately show "x = y" by(simp add: Abs_finfun_inject)
qed simp

lemma Abs_finfun_inject_finite_class:
  fixes x y :: "('a :: finite) => 'b"
  shows "Abs_finfun x = Abs_finfun y <-> x = y"
using finite_UNIV
by(simp add: Abs_finfun_inject_finite)

lemma Abs_finfun_inj_finite:
  assumes fin: "finite (UNIV :: 'a set)"
  shows "inj (Abs_finfun :: ('a => 'b) => 'a =>f 'b)"
proof(rule inj_onI)
  fix x y :: "'a => 'b"
  assume "Abs_finfun x = Abs_finfun y"
  moreover have "x ∈ finfun" "y ∈ finfun" unfolding finfun_def
    by(auto intro: finite_subset[OF _ fin])
  ultimately show "x = y" by(simp add: Abs_finfun_inject)
qed

lemma Abs_finfun_inverse_finite:
  fixes x :: "'a => 'b"
  assumes fin: "finite (UNIV :: 'a set)"
  shows "finfun_apply (Abs_finfun x) = x"
  including finfun
proof -
  from fin have "x ∈ finfun"
    by(auto simp add: finfun_def intro: finite_subset)
  thus ?thesis by simp
qed

lemma Abs_finfun_inverse_finite_class:
  fixes x :: "('a :: finite) => 'b"
  shows "finfun_apply (Abs_finfun x) = x"
using finite_UNIV by(simp add: Abs_finfun_inverse_finite)

lemma finfun_eq_finite_UNIV: "finite (UNIV :: 'a set) ==> (finfun :: ('a => 'b) set) = UNIV"
unfolding finfun_def by(auto intro: finite_subset)

lemma finfun_finite_UNIV_class: "finfun = (UNIV :: ('a :: finite => 'b) set)"
by(simp add: finfun_eq_finite_UNIV)

lemma map_default_in_finfun:
  assumes fin: "finite (dom f)"
  shows "map_default b f ∈ finfun"
unfolding finfun_def
proof(intro CollectI exI)
  from fin show "finite {a. map_default b f a ≠ b}"
    by(auto simp add: map_default_def dom_def Collect_conj_eq split: option.splits)
qed

lemma finfun_cases_map_default:
  obtains b g where "f = Abs_finfun (map_default b g)" "finite (dom g)" "b ∉ ran g"
proof -
  obtain y where f: "f = Abs_finfun y" and y: "y ∈ finfun" by(cases f)
  from y obtain b where b: "finite {a. y a ≠ b}" unfolding finfun_def by auto
  let ?g = "(λa. if y a = b then None else Some (y a))"
  have "map_default b ?g = y" by(simp add: fun_eq_iff map_default_def)
  with f have "f = Abs_finfun (map_default b ?g)" by simp
  moreover from b have "finite (dom ?g)" by(auto simp add: dom_def)
  moreover have "b ∉ ran ?g" by(auto simp add: ran_def)
  ultimately show ?thesis by(rule that)
qed


subsection {* Kernel functions for type @{typ "'a =>f 'b"} *}

lift_definition finfun_const :: "'b => 'a =>f 'b" ("K$/ _" [0] 1)
is "λ b x. b" by (rule const_finfun)

lift_definition finfun_update :: "'a =>f 'b => 'a => 'b => 'a =>f 'b" ("_'(_ $:= _')" [1000,0,0] 1000) is "fun_upd"
by (simp add: fun_upd_finfun)

lemma finfun_update_twist: "a ≠ a' ==> f(a $:= b)(a' $:= b') = f(a' $:= b')(a $:= b)"
by transfer (simp add: fun_upd_twist)

lemma finfun_update_twice [simp]:
  "f(a $:= b)(a $:= b') = f(a $:= b')"
by transfer simp

lemma finfun_update_const_same: "(K$ b)(a $:= b) = (K$ b)"
by transfer (simp add: fun_eq_iff)

subsection {* Code generator setup *}

definition finfun_update_code :: "'a =>f 'b => 'a => 'b => 'a =>f 'b"
where [simp, code del]: "finfun_update_code = finfun_update"

code_datatype finfun_const finfun_update_code

lemma finfun_update_const_code [code]:
  "(K$ b)(a $:= b') = (if b = b' then (K$ b) else finfun_update_code (K$ b) a b')"
by(simp add: finfun_update_const_same)

lemma finfun_update_update_code [code]:
  "(finfun_update_code f a b)(a' $:= b') = (if a = a' then f(a $:= b') else finfun_update_code (f(a' $:= b')) a b)"
by(simp add: finfun_update_twist)


subsection {* Setup for quickcheck *}

quickcheck_generator finfun constructors: finfun_update_code, "finfun_const :: 'b => 'a =>f 'b"

subsection {* @{text "finfun_update"} as instance of @{text "comp_fun_commute"} *}

interpretation finfun_update: comp_fun_commute "λa f. f(a :: 'a $:= b')"
  including finfun
proof
  fix a a' :: 'a
  show "(λf. f(a $:= b')) o (λf. f(a' $:= b')) = (λf. f(a' $:= b')) o (λf. f(a $:= b'))"
  proof
    fix b
    have "(finfun_apply b)(a := b', a' := b') = (finfun_apply b)(a' := b', a := b')"
      by(cases "a = a'")(auto simp add: fun_upd_twist)
    then have "b(a $:= b')(a' $:= b') = b(a' $:= b')(a $:= b')"
      by(auto simp add: finfun_update_def fun_upd_twist)
    then show "((λf. f(a $:= b')) o (λf. f(a' $:= b'))) b = ((λf. f(a' $:= b')) o (λf. f(a $:= b'))) b"
      by (simp add: fun_eq_iff)
  qed
qed

lemma fold_finfun_update_finite_univ:
  assumes fin: "finite (UNIV :: 'a set)"
  shows "Finite_Set.fold (λa f. f(a $:= b')) (K$ b) (UNIV :: 'a set) = (K$ b')"
proof -
  { fix A :: "'a set"
    from fin have "finite A" by(auto intro: finite_subset)
    hence "Finite_Set.fold (λa f. f(a $:= b')) (K$ b) A = Abs_finfun (λa. if a ∈ A then b' else b)"
    proof(induct)
      case (insert x F)
      have "(λa. if a = x then b' else (if a ∈ F then b' else b)) = (λa. if a = x ∨ a ∈ F then b' else b)"
        by(auto)
      with insert show ?case
        by(simp add: finfun_const_def fun_upd_def)(simp add: finfun_update_def Abs_finfun_inverse_finite[OF fin] fun_upd_def)
    qed(simp add: finfun_const_def) }
  thus ?thesis by(simp add: finfun_const_def)
qed


subsection {* Default value for FinFuns *}

definition finfun_default_aux :: "('a => 'b) => 'b"
where [code del]: "finfun_default_aux f = (if finite (UNIV :: 'a set) then undefined else THE b. finite {a. f a ≠ b})"

lemma finfun_default_aux_infinite:
  fixes f :: "'a => 'b"
  assumes infin: "¬ finite (UNIV :: 'a set)"
  and fin: "finite {a. f a ≠ b}"
  shows "finfun_default_aux f = b"
proof -
  let ?B = "{a. f a ≠ b}"
  from fin have "(THE b. finite {a. f a ≠ b}) = b"
  proof(rule the_equality)
    fix b'
    assume "finite {a. f a ≠ b'}" (is "finite ?B'")
    with infin fin have "UNIV - (?B' ∪ ?B) ≠ {}" by(auto dest: finite_subset)
    then obtain a where a: "a ∉ ?B' ∪ ?B" by auto
    thus "b' = b" by auto
  qed
  thus ?thesis using infin by(simp add: finfun_default_aux_def)
qed


lemma finite_finfun_default_aux:
  fixes f :: "'a => 'b"
  assumes fin: "f ∈ finfun"
  shows "finite {a. f a ≠ finfun_default_aux f}"
proof(cases "finite (UNIV :: 'a set)")
  case True thus ?thesis using fin
    by(auto simp add: finfun_def finfun_default_aux_def intro: finite_subset)
next
  case False
  from fin obtain b where b: "finite {a. f a ≠ b}" (is "finite ?B")
    unfolding finfun_def by blast
  with False show ?thesis by(simp add: finfun_default_aux_infinite)
qed

lemma finfun_default_aux_update_const:
  fixes f :: "'a => 'b"
  assumes fin: "f ∈ finfun"
  shows "finfun_default_aux (f(a := b)) = finfun_default_aux f"
proof(cases "finite (UNIV :: 'a set)")
  case False
  from fin obtain b' where b': "finite {a. f a ≠ b'}" unfolding finfun_def by blast
  hence "finite {a'. (f(a := b)) a' ≠ b'}"
  proof(cases "b = b' ∧ f a ≠ b'") 
    case True
    hence "{a. f a ≠ b'} = insert a {a'. (f(a := b)) a' ≠ b'}" by auto
    thus ?thesis using b' by simp
  next
    case False
    moreover
    { assume "b ≠ b'"
      hence "{a'. (f(a := b)) a' ≠ b'} = insert a {a. f a ≠ b'}" by auto
      hence ?thesis using b' by simp }
    moreover
    { assume "b = b'" "f a = b'"
      hence "{a'. (f(a := b)) a' ≠ b'} = {a. f a ≠ b'}" by auto
      hence ?thesis using b' by simp }
    ultimately show ?thesis by blast
  qed
  with False b' show ?thesis by(auto simp del: fun_upd_apply simp add: finfun_default_aux_infinite)
next
  case True thus ?thesis by(simp add: finfun_default_aux_def)
qed

lift_definition finfun_default :: "'a =>f 'b => 'b"
is "finfun_default_aux" .

lemma finite_finfun_default: "finite {a. finfun_apply f a ≠ finfun_default f}"
by transfer (erule finite_finfun_default_aux)

lemma finfun_default_const: "finfun_default ((K$ b) :: 'a =>f 'b) = (if finite (UNIV :: 'a set) then undefined else b)"
by(transfer)(auto simp add: finfun_default_aux_infinite finfun_default_aux_def)

lemma finfun_default_update_const:
  "finfun_default (f(a $:= b)) = finfun_default f"
by transfer (simp add: finfun_default_aux_update_const)

lemma finfun_default_const_code [code]:
  "finfun_default ((K$ c) :: 'a :: card_UNIV =>f 'b) = (if CARD('a) = 0 then c else undefined)"
by(simp add: finfun_default_const)

lemma finfun_default_update_code [code]:
  "finfun_default (finfun_update_code f a b) = finfun_default f"
by(simp add: finfun_default_update_const)

subsection {* Recursion combinator and well-formedness conditions *}

definition finfun_rec :: "('b => 'c) => ('a => 'b => 'c => 'c) => ('a =>f 'b) => 'c"
where [code del]:
  "finfun_rec cnst upd f ≡
   let b = finfun_default f;
       g = THE g. f = Abs_finfun (map_default b g) ∧ finite (dom g) ∧ b ∉ ran g
   in Finite_Set.fold (λa. upd a (map_default b g a)) (cnst b) (dom g)"

locale finfun_rec_wf_aux =
  fixes cnst :: "'b => 'c"
  and upd :: "'a => 'b => 'c => 'c"
  assumes upd_const_same: "upd a b (cnst b) = cnst b"
  and upd_commute: "a ≠ a' ==> upd a b (upd a' b' c) = upd a' b' (upd a b c)"
  and upd_idemp: "b ≠ b' ==> upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)"
begin


lemma upd_left_comm: "comp_fun_commute (λa. upd a (f a))"
by(unfold_locales)(auto intro: upd_commute simp add: fun_eq_iff)

lemma upd_upd_twice: "upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)"
by(cases "b ≠ b'")(auto simp add: fun_upd_def upd_const_same upd_idemp)

lemma map_default_update_const:
  assumes fin: "finite (dom f)"
  and anf: "a ∉ dom f"
  and fg: "f ⊆m g"
  shows "upd a d  (Finite_Set.fold (λa. upd a (map_default d g a)) (cnst d) (dom f)) =
         Finite_Set.fold (λa. upd a (map_default d g a)) (cnst d) (dom f)"
proof -
  let ?upd = "λa. upd a (map_default d g a)"
  let ?fr = "λA. Finite_Set.fold ?upd (cnst d) A"
  interpret gwf: comp_fun_commute "?upd" by(rule upd_left_comm)
  
  from fin anf fg show ?thesis
  proof(induct "dom f" arbitrary: f)
    case empty
    from `{} = dom f` have "f = empty" by(auto simp add: dom_def)
    thus ?case by(simp add: finfun_const_def upd_const_same)
  next
    case (insert a' A)
    note IH = `!!f.  [| A = dom f; a ∉ dom f; f ⊆m g |] ==> upd a d (?fr (dom f)) = ?fr (dom f)`
    note fin = `finite A` note anf = `a ∉ dom f` note a'nA = `a' ∉ A`
    note domf = `insert a' A = dom f` note fg = `f ⊆m g`
    
    from domf obtain b where b: "f a' = Some b" by auto
    let ?f' = "f(a' := None)"
    have "upd a d (?fr (insert a' A)) = upd a d (upd a' (map_default d g a') (?fr A))"
      by(subst gwf.fold_insert[OF fin a'nA]) rule
    also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec)
    hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def)
    also from anf domf have "a ≠ a'" by auto note upd_commute[OF this]
    also from domf a'nA anf fg have "a ∉ dom ?f'" "?f' ⊆m g" and A: "A = dom ?f'" by(auto simp add: ran_def map_le_def)
    note A also note IH[OF A `a ∉ dom ?f'` `?f' ⊆m g`]
    also have "upd a' (map_default d f a') (?fr (dom (f(a' := None)))) = ?fr (dom f)"
      unfolding domf[symmetric] gwf.fold_insert[OF fin a'nA] ga' unfolding A ..
    also have "insert a' (dom ?f') = dom f" using domf by auto
    finally show ?case .
  qed
qed

lemma map_default_update_twice:
  assumes fin: "finite (dom f)"
  and anf: "a ∉ dom f"
  and fg: "f ⊆m g"
  shows "upd a d'' (upd a d' (Finite_Set.fold (λa. upd a (map_default d g a)) (cnst d) (dom f))) =
         upd a d'' (Finite_Set.fold (λa. upd a (map_default d g a)) (cnst d) (dom f))"
proof -
  let ?upd = "λa. upd a (map_default d g a)"
  let ?fr = "λA. Finite_Set.fold ?upd (cnst d) A"
  interpret gwf: comp_fun_commute "?upd" by(rule upd_left_comm)
  
  from fin anf fg show ?thesis
  proof(induct "dom f" arbitrary: f)
    case empty
    from `{} = dom f` have "f = empty" by(auto simp add: dom_def)
    thus ?case by(auto simp add: finfun_const_def finfun_update_def upd_upd_twice)
  next
    case (insert a' A)
    note IH = `!!f. [|A = dom f; a ∉ dom f; f ⊆m g|] ==> upd a d'' (upd a d' (?fr (dom f))) = upd a d'' (?fr (dom f))`
    note fin = `finite A` note anf = `a ∉ dom f` note a'nA = `a' ∉ A`
    note domf = `insert a' A = dom f` note fg = `f ⊆m g`
    
    from domf obtain b where b: "f a' = Some b" by auto
    let ?f' = "f(a' := None)"
    let ?b' = "case f a' of None => d | Some b => b"
    from domf have "upd a d'' (upd a d' (?fr (dom f))) = upd a d'' (upd a d' (?fr (insert a' A)))" by simp
    also note gwf.fold_insert[OF fin a'nA]
    also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec)
    hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def)
    also from anf domf have ana': "a ≠ a'" by auto note upd_commute[OF this]
    also note upd_commute[OF ana']
    also from domf a'nA anf fg have "a ∉ dom ?f'" "?f' ⊆m g" and A: "A = dom ?f'" by(auto simp add: ran_def map_le_def)
    note A also note IH[OF A `a ∉ dom ?f'` `?f' ⊆m g`]
    also note upd_commute[OF ana'[symmetric]] also note ga'[symmetric] also note A[symmetric]
    also note gwf.fold_insert[symmetric, OF fin a'nA] also note domf
    finally show ?case .
  qed
qed

lemma map_default_eq_id [simp]: "map_default d ((λa. Some (f a)) |` {a. f a ≠ d}) = f"
by(auto simp add: map_default_def restrict_map_def)

lemma finite_rec_cong1:
  assumes f: "comp_fun_commute f" and g: "comp_fun_commute g"
  and fin: "finite A"
  and eq: "!!a. a ∈ A ==> f a = g a"
  shows "Finite_Set.fold f z A = Finite_Set.fold g z A"
proof -
  interpret f: comp_fun_commute f by(rule f)
  interpret g: comp_fun_commute g by(rule g)
  { fix B
    assume BsubA: "B ⊆ A"
    with fin have "finite B" by(blast intro: finite_subset)
    hence "B ⊆ A ==> Finite_Set.fold f z B = Finite_Set.fold g z B"
    proof(induct)
      case empty thus ?case by simp
    next
      case (insert a B)
      note finB = `finite B` note anB = `a ∉ B` note sub = `insert a B ⊆ A`
      note IH = `B ⊆ A ==> Finite_Set.fold f z B = Finite_Set.fold g z B`
      from sub anB have BpsubA: "B ⊂ A" and BsubA: "B ⊆ A" and aA: "a ∈ A" by auto
      from IH[OF BsubA] eq[OF aA] finB anB
      show ?case by(auto)
    qed
    with BsubA have "Finite_Set.fold f z B = Finite_Set.fold g z B" by blast }
  thus ?thesis by blast
qed

lemma finfun_rec_upd [simp]:
  "finfun_rec cnst upd (f(a' $:= b')) = upd a' b' (finfun_rec cnst upd f)"
  including finfun
proof -
  obtain b where b: "b = finfun_default f" by auto
  let ?the = "λf g. f = Abs_finfun (map_default b g) ∧ finite (dom g) ∧ b ∉ ran g"
  obtain g where g: "g = The (?the f)" by blast
  obtain y where f: "f = Abs_finfun y" and y: "y ∈ finfun" by (cases f)
  from f y b have bfin: "finite {a. y a ≠ b}" by(simp add: finfun_default_def finite_finfun_default_aux)

  let ?g = "(λa. Some (y a)) |` {a. y a ≠ b}"
  from bfin have fing: "finite (dom ?g)" by auto
  have bran: "b ∉ ran ?g" by(auto simp add: ran_def restrict_map_def)
  have yg: "y = map_default b ?g" by simp
  have gg: "g = ?g" unfolding g
  proof(rule the_equality)
    from f y bfin show "?the f ?g"
      by(auto)(simp add: restrict_map_def ran_def split: split_if_asm)
  next
    fix g'
    assume "?the f g'"
    hence fin': "finite (dom g')" and ran': "b ∉ ran g'"
      and eq: "Abs_finfun (map_default b ?g) = Abs_finfun (map_default b g')" using f yg by auto
    from fin' fing have "map_default b ?g ∈ finfun" "map_default b g' ∈ finfun" by(blast intro: map_default_in_finfun)+
    with eq have "map_default b ?g = map_default b g'" by simp
    with fing bran fin' ran' show "g' = ?g" by(rule map_default_inject[OF disjI2[OF refl], THEN sym])
  qed

  show ?thesis
  proof(cases "b' = b")
    case True
    note b'b = True

    let ?g' = "(λa. Some ((y(a' := b)) a)) |` {a. (y(a' := b)) a ≠ b}"
    from bfin b'b have fing': "finite (dom ?g')"
      by(auto simp add: Collect_conj_eq Collect_imp_eq intro: finite_subset)
    have brang': "b ∉ ran ?g'" by(auto simp add: ran_def restrict_map_def)

    let ?b' = "λa. case ?g' a of None => b | Some b => b"
    let ?b = "map_default b ?g"
    from upd_left_comm upd_left_comm fing'
    have "Finite_Set.fold (λa. upd a (?b' a)) (cnst b) (dom ?g') = Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g')"
      by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b b map_default_def)
    also interpret gwf: comp_fun_commute "λa. upd a (?b a)" by(rule upd_left_comm)
    have "Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g') = upd a' b' (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g))"
    proof(cases "y a' = b")
      case True
      with b'b have g': "?g' = ?g" by(auto simp add: restrict_map_def)
      from True have a'ndomg: "a' ∉ dom ?g" by auto
      from f b'b b show ?thesis unfolding g'
        by(subst map_default_update_const[OF fing a'ndomg map_le_refl, symmetric]) simp
    next
      case False
      hence domg: "dom ?g = insert a' (dom ?g')" by auto
      from False b'b have a'ndomg': "a' ∉ dom ?g'" by auto
      have "Finite_Set.fold (λa. upd a (?b a)) (cnst b) (insert a' (dom ?g')) = 
            upd a' (?b a') (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g'))"
        using fing' a'ndomg' unfolding b'b by(rule gwf.fold_insert)
      hence "upd a' b (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (insert a' (dom ?g'))) =
             upd a' b (upd a' (?b a') (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g')))" by simp
      also from b'b have g'leg: "?g' ⊆m ?g" by(auto simp add: restrict_map_def map_le_def)
      note map_default_update_twice[OF fing' a'ndomg' this, of b "?b a'" b]
      also note map_default_update_const[OF fing' a'ndomg' g'leg, of b]
      finally show ?thesis unfolding b'b domg[unfolded b'b] by(rule sym)
    qed
    also have "The (?the (f(a' $:= b'))) = ?g'"
    proof(rule the_equality)
      from f y b b'b brang' fing' show "?the (f(a' $:= b')) ?g'"
        by(auto simp del: fun_upd_apply simp add: finfun_update_def)
    next
      fix g'
      assume "?the (f(a' $:= b')) g'"
      hence fin': "finite (dom g')" and ran': "b ∉ ran g'"
        and eq: "f(a' $:= b') = Abs_finfun (map_default b g')" 
        by(auto simp del: fun_upd_apply)
      from fin' fing' have "map_default b g' ∈ finfun" "map_default b ?g' ∈ finfun"
        by(blast intro: map_default_in_finfun)+
      with eq f b'b b have "map_default b ?g' = map_default b g'"
        by(simp del: fun_upd_apply add: finfun_update_def)
      with fing' brang' fin' ran' show "g' = ?g'"
        by(rule map_default_inject[OF disjI2[OF refl], THEN sym])
    qed
    ultimately show ?thesis unfolding finfun_rec_def Let_def b gg[unfolded g b] using bfin b'b b
      by(simp only: finfun_default_update_const map_default_def)
  next
    case False
    note b'b = this
    let ?g' = "?g(a' \<mapsto> b')"
    let ?b' = "map_default b ?g'"
    let ?b = "map_default b ?g"
    from fing have fing': "finite (dom ?g')" by auto
    from bran b'b have bnrang': "b ∉ ran ?g'" by(auto simp add: ran_def)
    have ffmg': "map_default b ?g' = y(a' := b')" by(auto simp add: map_default_def restrict_map_def)
    with f y have f_Abs: "f(a' $:= b') = Abs_finfun (map_default b ?g')" by(auto simp add: finfun_update_def)
    have g': "The (?the (f(a' $:= b'))) = ?g'"
    proof (rule the_equality)
      from fing' bnrang' f_Abs show "?the (f(a' $:= b')) ?g'"
        by(auto simp add: finfun_update_def restrict_map_def)
    next
      fix g' assume "?the (f(a' $:= b')) g'"
      hence f': "f(a' $:= b') = Abs_finfun (map_default b g')"
        and fin': "finite (dom g')" and brang': "b ∉ ran g'" by auto
      from fing' fin' have "map_default b ?g' ∈ finfun" "map_default b g' ∈ finfun"
        by(auto intro: map_default_in_finfun)
      with f' f_Abs have "map_default b g' = map_default b ?g'" by simp
      with fin' brang' fing' bnrang' show "g' = ?g'"
        by(rule map_default_inject[OF disjI2[OF refl]])
    qed
    have dom: "dom (((λa. Some (y a)) |` {a. y a ≠ b})(a' \<mapsto> b')) = insert a' (dom ((λa. Some (y a)) |` {a. y a ≠ b}))"
      by auto
    show ?thesis
    proof(cases "y a' = b")
      case True
      hence a'ndomg: "a' ∉ dom ?g" by auto
      from f y b'b True have yff: "y = map_default b (?g' |` dom ?g)"
        by(auto simp add: restrict_map_def map_default_def intro!: ext)
      hence f': "f = Abs_finfun (map_default b (?g' |` dom ?g))" using f by simp
      interpret g'wf: comp_fun_commute "λa. upd a (?b' a)" by(rule upd_left_comm)
      from upd_left_comm upd_left_comm fing
      have "Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g) = Finite_Set.fold (λa. upd a (?b' a)) (cnst b) (dom ?g)"
        by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b True map_default_def)
      thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric]
        unfolding g' g[symmetric] gg g'wf.fold_insert[OF fing a'ndomg, of "cnst b", folded dom]
        by -(rule arg_cong2[where f="upd a'"], simp_all add: map_default_def)
    next
      case False
      hence "insert a' (dom ?g) = dom ?g" by auto
      moreover {
        let ?g'' = "?g(a' := None)"
        let ?b'' = "map_default b ?g''"
        from False have domg: "dom ?g = insert a' (dom ?g'')" by auto
        from False have a'ndomg'': "a' ∉ dom ?g''" by auto
        have fing'': "finite (dom ?g'')" by(rule finite_subset[OF _ fing]) auto
        have bnrang'': "b ∉ ran ?g''" by(auto simp add: ran_def restrict_map_def)
        interpret gwf: comp_fun_commute "λa. upd a (?b a)" by(rule upd_left_comm)
        interpret g'wf: comp_fun_commute "λa. upd a (?b' a)" by(rule upd_left_comm)
        have "upd a' b' (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (insert a' (dom ?g''))) =
              upd a' b' (upd a' (?b a') (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g'')))"
          unfolding gwf.fold_insert[OF fing'' a'ndomg''] f ..
        also have g''leg: "?g |` dom ?g'' ⊆m ?g" by(auto simp add: map_le_def)
        have "dom (?g |` dom ?g'') = dom ?g''" by auto
        note map_default_update_twice[where d=b and f = "?g |` dom ?g''" and a=a' and d'="?b a'" and d''=b' and g="?g",
                                     unfolded this, OF fing'' a'ndomg'' g''leg]
        also have b': "b' = ?b' a'" by(auto simp add: map_default_def)
        from upd_left_comm upd_left_comm fing''
        have "Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g'') =
          Finite_Set.fold (λa. upd a (?b' a)) (cnst b) (dom ?g'')"
          by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b map_default_def)
        with b' have "upd a' b' (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g'')) =
                     upd a' (?b' a') (Finite_Set.fold (λa. upd a (?b' a)) (cnst b) (dom ?g''))" by simp
        also note g'wf.fold_insert[OF fing'' a'ndomg'', symmetric]
        finally have "upd a' b' (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g)) =
                   Finite_Set.fold (λa. upd a (?b' a)) (cnst b) (dom ?g)"
          unfolding domg . }
      ultimately have "Finite_Set.fold (λa. upd a (?b' a)) (cnst b) (insert a' (dom ?g)) =
                    upd a' b' (Finite_Set.fold (λa. upd a (?b a)) (cnst b) (dom ?g))" by simp
      thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric] g[symmetric] g' dom[symmetric]
        using b'b gg by(simp add: map_default_insert)
    qed
  qed
qed

end

locale finfun_rec_wf = finfun_rec_wf_aux + 
  assumes const_update_all:
  "finite (UNIV :: 'a set) ==> Finite_Set.fold (λa. upd a b') (cnst b) (UNIV :: 'a set) = cnst b'"
begin

lemma finfun_rec_const [simp]: includes finfun shows
  "finfun_rec cnst upd (K$ c) = cnst c"
proof(cases "finite (UNIV :: 'a set)")
  case False
  hence "finfun_default ((K$ c) :: 'a =>f 'b) = c" by(simp add: finfun_default_const)
  moreover have "(THE g :: 'a \<rightharpoonup> 'b. (K$ c) = Abs_finfun (map_default c g) ∧ finite (dom g) ∧ c ∉ ran g) = empty"
  proof (rule the_equality)
    show "(K$ c) = Abs_finfun (map_default c empty) ∧ finite (dom empty) ∧ c ∉ ran empty"
      by(auto simp add: finfun_const_def)
  next
    fix g :: "'a \<rightharpoonup> 'b"
    assume "(K$ c) = Abs_finfun (map_default c g) ∧ finite (dom g) ∧ c ∉ ran g"
    hence g: "(K$ c) = Abs_finfun (map_default c g)" and fin: "finite (dom g)" and ran: "c ∉ ran g" by blast+
    from g map_default_in_finfun[OF fin, of c] have "map_default c g = (λa. c)"
      by(simp add: finfun_const_def)
    moreover have "map_default c empty = (λa. c)" by simp
    ultimately show "g = empty" by-(rule map_default_inject[OF disjI2[OF refl] fin ran], auto)
  qed
  ultimately show ?thesis by(simp add: finfun_rec_def)
next
  case True
  hence default: "finfun_default ((K$ c) :: 'a =>f 'b) = undefined" by(simp add: finfun_default_const)
  let ?the = "λg :: 'a \<rightharpoonup> 'b. (K$ c) = Abs_finfun (map_default undefined g) ∧ finite (dom g) ∧ undefined ∉ ran g"
  show ?thesis
  proof(cases "c = undefined")
    case True
    have the: "The ?the = empty"
    proof (rule the_equality)
      from True show "?the empty" by(auto simp add: finfun_const_def)
    next
      fix g'
      assume "?the g'"
      hence fg: "(K$ c) = Abs_finfun (map_default undefined g')"
        and fin: "finite (dom g')" and g: "undefined ∉ ran g'" by simp_all
      from fin have "map_default undefined g' ∈ finfun" by(rule map_default_in_finfun)
      with fg have "map_default undefined g' = (λa. c)"
        by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1, symmetric])
      with True show "g' = empty"
        by -(rule map_default_inject(2)[OF _ fin g], auto)
    qed
    show ?thesis unfolding finfun_rec_def using `finite UNIV` True
      unfolding Let_def the default by(simp)
  next
    case False
    have the: "The ?the = (λa :: 'a. Some c)"
    proof (rule the_equality)
      from False True show "?the (λa :: 'a. Some c)"
        by(auto simp add: map_default_def [abs_def] finfun_const_def dom_def ran_def)
    next
      fix g' :: "'a \<rightharpoonup> 'b"
      assume "?the g'"
      hence fg: "(K$ c) = Abs_finfun (map_default undefined g')"
        and fin: "finite (dom g')" and g: "undefined ∉ ran g'" by simp_all
      from fin have "map_default undefined g' ∈ finfun" by(rule map_default_in_finfun)
      with fg have "map_default undefined g' = (λa. c)"
        by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1])
      with True False show "g' = (λa::'a. Some c)"
        by - (rule map_default_inject(2)[OF _ fin g],
          auto simp add: dom_def ran_def map_default_def [abs_def])
    qed
    show ?thesis unfolding finfun_rec_def using True False
      unfolding Let_def the default by(simp add: dom_def map_default_def const_update_all)
  qed
qed

end

subsection {* Weak induction rule and case analysis for FinFuns *}

lemma finfun_weak_induct [consumes 0, case_names const update]:
  assumes const: "!!b. P (K$ b)"
  and update: "!!f a b. P f ==> P (f(a $:= b))"
  shows "P x"
  including finfun
proof(induct x rule: Abs_finfun_induct)
  case (Abs_finfun y)
  then obtain b where "finite {a. y a ≠ b}" unfolding finfun_def by blast
  thus ?case using `y ∈ finfun`
  proof(induct "{a. y a ≠ b}" arbitrary: y rule: finite_induct)
    case empty
    hence "!!a. y a = b" by blast
    hence "y = (λa. b)" by(auto)
    hence "Abs_finfun y = finfun_const b" unfolding finfun_const_def by simp
    thus ?case by(simp add: const)
  next
    case (insert a A)
    note IH = `!!y. [| A = {a. y a ≠ b}; y ∈ finfun  |] ==> P (Abs_finfun y)`
    note y = `y ∈ finfun`
    with `insert a A = {a. y a ≠ b}` `a ∉ A`
    have "A = {a'. (y(a := b)) a' ≠ b}" "y(a := b) ∈ finfun" by auto
    from IH[OF this] have "P (finfun_update (Abs_finfun (y(a := b))) a (y a))" by(rule update)
    thus ?case using y unfolding finfun_update_def by simp
  qed
qed

lemma finfun_exhaust_disj: "(∃b. x = finfun_const b) ∨ (∃f a b. x = finfun_update f a b)"
by(induct x rule: finfun_weak_induct) blast+

lemma finfun_exhaust:
  obtains b where "x = (K$ b)"
        | f a b where "x = f(a $:= b)"
by(atomize_elim)(rule finfun_exhaust_disj)

lemma finfun_rec_unique:
  fixes f :: "'a =>f 'b => 'c"
  assumes c: "!!c. f (K$ c) = cnst c"
  and u: "!!g a b. f (g(a $:= b)) = upd g a b (f g)"
  and c': "!!c. f' (K$ c) = cnst c"
  and u': "!!g a b. f' (g(a $:= b)) = upd g a b (f' g)"
  shows "f = f'"
proof
  fix g :: "'a =>f 'b"
  show "f g = f' g"
    by(induct g rule: finfun_weak_induct)(auto simp add: c u c' u')
qed


subsection {* Function application *}

notation finfun_apply (infixl "$" 999)

interpretation finfun_apply_aux: finfun_rec_wf_aux "λb. b" "λa' b c. if (a = a') then b else c"
by(unfold_locales) auto

interpretation finfun_apply: finfun_rec_wf "λb. b" "λa' b c. if (a = a') then b else c"
proof(unfold_locales)
  fix b' b :: 'a
  assume fin: "finite (UNIV :: 'b set)"
  { fix A :: "'b set"
    interpret comp_fun_commute "λa'. If (a = a') b'" by(rule finfun_apply_aux.upd_left_comm)
    from fin have "finite A" by(auto intro: finite_subset)
    hence "Finite_Set.fold (λa'. If (a = a') b') b A = (if a ∈ A then b' else b)"
      by induct auto }
  from this[of UNIV] show "Finite_Set.fold (λa'. If (a = a') b') b UNIV = b'" by simp
qed

lemma finfun_apply_def: "op $ = (λf a. finfun_rec (λb. b) (λa' b c. if (a = a') then b else c) f)"
proof(rule finfun_rec_unique)
  fix c show "op $ (K$ c) = (λa. c)" by(simp add: finfun_const.rep_eq)
next
  fix g a b show "op $ g(a $:= b) = (λc. if c = a then b else g $ c)"
    by(auto simp add: finfun_update_def fun_upd_finfun Abs_finfun_inverse finfun_apply)
qed auto

lemma finfun_upd_apply: "f(a $:= b) $ a' = (if a = a' then b else f $ a')"
  and finfun_upd_apply_code [code]: "(finfun_update_code f a b) $ a' = (if a = a' then b else f $ a')"
by(simp_all add: finfun_apply_def)

lemma finfun_const_apply [simp, code]: "(K$ b) $ a = b"
by(simp add: finfun_apply_def)

lemma finfun_upd_apply_same [simp]:
  "f(a $:= b) $ a = b"
by(simp add: finfun_upd_apply)

lemma finfun_upd_apply_other [simp]:
  "a ≠ a' ==> f(a $:= b) $ a' = f $ a'"
by(simp add: finfun_upd_apply)

lemma finfun_ext: "(!!a. f $ a = g $ a) ==> f = g"
by(auto simp add: finfun_apply_inject[symmetric] simp del: finfun_apply_inject)

lemma expand_finfun_eq: "(f = g) = (op $ f = op $ g)"
by(auto intro: finfun_ext)

lemma finfun_upd_triv [simp]: "f(x $:= f $ x) = f"
by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply)

lemma finfun_const_inject [simp]: "(K$ b) = (K$ b') ≡ b = b'"
by(simp add: expand_finfun_eq fun_eq_iff)

lemma finfun_const_eq_update:
  "((K$ b) = f(a $:= b')) = (b = b' ∧ (∀a'. a ≠ a' --> f $ a' = b))"
by(auto simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply)

subsection {* Function composition *}

definition finfun_comp :: "('a => 'b) => 'c =>f 'a => 'c =>f 'b" (infixr "o$" 55)
where [code del]: "g o$ f  = finfun_rec (λb. (K$ g b)) (λa b c. c(a $:= g b)) f"

notation (xsymbols) finfun_comp (infixr "o$" 55)

interpretation finfun_comp_aux: finfun_rec_wf_aux "(λb. (K$ g b))" "(λa b c. c(a $:= g b))"
by(unfold_locales)(auto simp add: finfun_upd_apply intro: finfun_ext)

interpretation finfun_comp: finfun_rec_wf "(λb. (K$ g b))" "(λa b c. c(a $:= g b))"
proof
  fix b' b :: 'a
  assume fin: "finite (UNIV :: 'c set)"
  { fix A :: "'c set"
    from fin have "finite A" by(auto intro: finite_subset)
    hence "Finite_Set.fold (λ(a :: 'c) c. c(a $:= g b')) (K$ g b) A =
      Abs_finfun (λa. if a ∈ A then g b' else g b)"
      by induct (simp_all add: finfun_const_def, auto simp add: finfun_update_def Abs_finfun_inverse_finite fun_upd_def Abs_finfun_inject_finite fun_eq_iff fin) }
  from this[of UNIV] show "Finite_Set.fold (λ(a :: 'c) c. c(a $:= g b')) (K$ g b) UNIV = (K$ g b')"
    by(simp add: finfun_const_def)
qed

lemma finfun_comp_const [simp, code]:
  "g o$ (K$ c) = (K$ g c)"
by(simp add: finfun_comp_def)

lemma finfun_comp_update [simp]: "g o$ (f(a $:= b)) = (g o$ f)(a $:= g b)"
  and finfun_comp_update_code [code]: 
  "g o$ (finfun_update_code f a b) = finfun_update_code (g o$ f) a (g b)"
by(simp_all add: finfun_comp_def)

lemma finfun_comp_apply [simp]:
  "op $ (g o$ f) = g o op $ f"
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_upd_apply)

lemma finfun_comp_comp_collapse [simp]: "f o$ g o$ h = (f o g) o$ h"
by(induct h rule: finfun_weak_induct) simp_all

lemma finfun_comp_const1 [simp]: "(λx. c) o$ f = (K$ c)"
by(induct f rule: finfun_weak_induct)(auto intro: finfun_ext simp add: finfun_upd_apply)

lemma finfun_comp_id1 [simp]: "(λx. x) o$ f = f" "id o$ f = f"
by(induct f rule: finfun_weak_induct) auto

lemma finfun_comp_conv_comp: "g o$ f = Abs_finfun (g o op $ f)"
  including finfun
proof -
  have "(λf. g o$ f) = (λf. Abs_finfun (g o op $ f))"
  proof(rule finfun_rec_unique)
    { fix c show "Abs_finfun (g o op $ (K$ c)) = (K$ g c)"
        by(simp add: finfun_comp_def o_def)(simp add: finfun_const_def) }
    { fix g' a b show "Abs_finfun (g o op $ g'(a $:= b)) = (Abs_finfun (g o op $ g'))(a $:= g b)"
      proof -
        obtain y where y: "y ∈ finfun" and g': "g' = Abs_finfun y" by(cases g')
        moreover from g' have "(g o op $ g') ∈ finfun" by(simp add: finfun_left_compose)
        moreover have "g o y(a := b) = (g o y)(a := g b)" by(auto)
        ultimately show ?thesis by(simp add: finfun_comp_def finfun_update_def)
      qed }
  qed auto
  thus ?thesis by(auto simp add: fun_eq_iff)
qed

definition finfun_comp2 :: "'b =>f 'c => ('a => 'b) => 'a =>f 'c" (infixr "$o" 55)
where [code del]: "g $o f = Abs_finfun (op $ g o f)"

notation (xsymbol) finfun_comp2 (infixr "$o" 55)

lemma finfun_comp2_const [code, simp]: "finfun_comp2 (K$ c) f = (K$ c)"
  including finfun
by(simp add: finfun_comp2_def finfun_const_def comp_def)

lemma finfun_comp2_update:
  includes finfun
  assumes inj: "inj f"
  shows "finfun_comp2 (g(b $:= c)) f = (if b ∈ range f then (finfun_comp2 g f)(inv f b $:= c) else finfun_comp2 g f)"
proof(cases "b ∈ range f")
  case True
  from inj have "!!x. (op $ g)(f x := c) o f = (op $ g o f)(x := c)" by(auto intro!: ext dest: injD)
  with inj True show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def finfun_right_compose)
next
  case False
  hence "(op $ g)(b := c) o f = op $ g o f" by(auto simp add: fun_eq_iff)
  with False show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def)
qed

subsection {* Universal quantification *}

definition finfun_All_except :: "'a list => 'a =>f bool => bool"
where [code del]: "finfun_All_except A P ≡ ∀a. a ∈ set A ∨ P $ a"

lemma finfun_All_except_const: "finfun_All_except A (K$ b) <-> b ∨ set A = UNIV"
by(auto simp add: finfun_All_except_def)

lemma finfun_All_except_const_finfun_UNIV_code [code]:
  "finfun_All_except A (K$ b) = (b ∨ is_list_UNIV A)"
by(simp add: finfun_All_except_const is_list_UNIV_iff)

lemma finfun_All_except_update:
  "finfun_All_except A f(a $:= b) = ((a ∈ set A ∨ b) ∧ finfun_All_except (a # A) f)"
by(fastforce simp add: finfun_All_except_def finfun_upd_apply)

lemma finfun_All_except_update_code [code]:
  fixes a :: "'a :: card_UNIV"
  shows "finfun_All_except A (finfun_update_code f a b) = ((a ∈ set A ∨ b) ∧ finfun_All_except (a # A) f)"
by(simp add: finfun_All_except_update)

definition finfun_All :: "'a =>f bool => bool"
where "finfun_All = finfun_All_except []"

lemma finfun_All_const [simp]: "finfun_All (K$ b) = b"
by(simp add: finfun_All_def finfun_All_except_def)

lemma finfun_All_update: "finfun_All f(a $:= b) = (b ∧ finfun_All_except [a] f)"
by(simp add: finfun_All_def finfun_All_except_update)

lemma finfun_All_All: "finfun_All P = All (op $ P)"
by(simp add: finfun_All_def finfun_All_except_def)


definition finfun_Ex :: "'a =>f bool => bool"
where "finfun_Ex P = Not (finfun_All (Not o$ P))"

lemma finfun_Ex_Ex: "finfun_Ex P = Ex (op $ P)"
unfolding finfun_Ex_def finfun_All_All by simp

lemma finfun_Ex_const [simp]: "finfun_Ex (K$ b) = b"
by(simp add: finfun_Ex_def)


subsection {* A diagonal operator for FinFuns *}

definition finfun_Diag :: "'a =>f 'b => 'a =>f 'c => 'a =>f ('b × 'c)" ("(1'($_,/ _$'))" [0, 0] 1000)
where [code del]: "($f, g$) = finfun_rec (λb. Pair b o$ g) (λa b c. c(a $:= (b, g $ a))) f"

interpretation finfun_Diag_aux: finfun_rec_wf_aux "λb. Pair b o$ g" "λa b c. c(a $:= (b, g $ a))"
by(unfold_locales)(simp_all add: expand_finfun_eq fun_eq_iff finfun_upd_apply)

interpretation finfun_Diag: finfun_rec_wf "λb. Pair b o$ g" "λa b c. c(a $:= (b, g $ a))"
proof
  fix b' b :: 'a
  assume fin: "finite (UNIV :: 'c set)"
  { fix A :: "'c set"
    interpret comp_fun_commute "λa c. c(a $:= (b', g $ a))" by(rule finfun_Diag_aux.upd_left_comm)
    from fin have "finite A" by(auto intro: finite_subset)
    hence "Finite_Set.fold (λa c. c(a $:= (b', g $ a))) (Pair b o$ g) A =
      Abs_finfun (λa. (if a ∈ A then b' else b, g $ a))"
      by(induct)(simp_all add: finfun_const_def finfun_comp_conv_comp o_def,
                 auto simp add: finfun_update_def Abs_finfun_inverse_finite fun_upd_def Abs_finfun_inject_finite fun_eq_iff fin) }
  from this[of UNIV] show "Finite_Set.fold (λa c. c(a $:= (b', g $ a))) (Pair b o$ g) UNIV = Pair b' o$ g"
    by(simp add: finfun_const_def finfun_comp_conv_comp o_def)
qed

lemma finfun_Diag_const1: "($K$ b, g$) = Pair b o$ g"
by(simp add: finfun_Diag_def)

text {*
  Do not use @{thm finfun_Diag_const1} for the code generator because @{term "Pair b"} is injective, i.e. if @{term g} is free of redundant updates, there is no need to check for redundant updates as is done for @{term "op o$"}.
*}

lemma finfun_Diag_const_code [code]:
  "($K$ b, K$ c$) = (K$ (b, c))"
  "($K$ b, finfun_update_code g a c$) = finfun_update_code ($K$ b, g$) a (b, c)"
by(simp_all add: finfun_Diag_const1)

lemma finfun_Diag_update1: "($f(a $:= b), g$) = ($f, g$)(a $:= (b, g $ a))"
  and finfun_Diag_update1_code [code]: "($finfun_update_code f a b, g$) = ($f, g$)(a $:= (b, g $ a))"
by(simp_all add: finfun_Diag_def)

lemma finfun_Diag_const2: "($f, K$ c$) = (λb. (b, c)) o$ f"
by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1)

lemma finfun_Diag_update2: "($f, g(a $:= c)$) = ($f, g$)(a $:= (f $ a, c))"
by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1)

lemma finfun_Diag_const_const [simp]: "($K$ b, K$ c$) = (K$ (b, c))"
by(simp add: finfun_Diag_const1)

lemma finfun_Diag_const_update:
  "($K$ b, g(a $:= c)$) = ($K$ b, g$)(a $:= (b, c))"
by(simp add: finfun_Diag_const1)

lemma finfun_Diag_update_const:
  "($f(a $:= b), K$ c$) = ($f, K$ c$)(a $:= (b, c))"
by(simp add: finfun_Diag_def)

lemma finfun_Diag_update_update:
  "($f(a $:= b), g(a' $:= c)$) = (if a = a' then ($f, g$)(a $:= (b, c)) else ($f, g$)(a $:= (b, g $ a))(a' $:= (f $ a', c)))"
by(auto simp add: finfun_Diag_update1 finfun_Diag_update2)

lemma finfun_Diag_apply [simp]: "op $ ($f, g$) = (λx. (f $ x, g $ x))"
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_Diag_const1 finfun_Diag_update1 finfun_upd_apply)

lemma finfun_Diag_conv_Abs_finfun:
  "($f, g$) = Abs_finfun ((λx. (f $ x, g $ x)))"
  including finfun
proof -
  have "(λf :: 'a =>f 'b. ($f, g$)) = (λf. Abs_finfun ((λx. (f $ x, g $ x))))"
  proof(rule finfun_rec_unique)
    { fix c show "Abs_finfun (λx. ((K$ c) $ x, g $ x)) = Pair c o$ g"
        by(simp add: finfun_comp_conv_comp o_def finfun_const_def) }
    { fix g' a b
      show "Abs_finfun (λx. (g'(a $:= b) $ x, g $ x)) =
            (Abs_finfun (λx. (g' $ x, g $ x)))(a $:= (b, g $ a))"
        by(auto simp add: finfun_update_def fun_eq_iff simp del: fun_upd_apply) simp }
  qed(simp_all add: finfun_Diag_const1 finfun_Diag_update1)
  thus ?thesis by(auto simp add: fun_eq_iff)
qed

lemma finfun_Diag_eq: "($f, g$) = ($f', g'$) <-> f = f' ∧ g = g'"
by(auto simp add: expand_finfun_eq fun_eq_iff)

definition finfun_fst :: "'a =>f ('b × 'c) => 'a =>f 'b"
where [code]: "finfun_fst f = fst o$ f"

lemma finfun_fst_const: "finfun_fst (K$ bc) = (K$ fst bc)"
by(simp add: finfun_fst_def)

lemma finfun_fst_update: "finfun_fst (f(a $:= bc)) = (finfun_fst f)(a $:= fst bc)"
  and finfun_fst_update_code: "finfun_fst (finfun_update_code f a bc) = (finfun_fst f)(a $:= fst bc)"
by(simp_all add: finfun_fst_def)

lemma finfun_fst_comp_conv: "finfun_fst (f o$ g) = (fst o f) o$ g"
by(simp add: finfun_fst_def)

lemma finfun_fst_conv [simp]: "finfun_fst ($f, g$) = f"
by(induct f rule: finfun_weak_induct)(simp_all add: finfun_Diag_const1 finfun_fst_comp_conv o_def finfun_Diag_update1 finfun_fst_update)

lemma finfun_fst_conv_Abs_finfun: "finfun_fst = (λf. Abs_finfun (fst o op $ f))"
by(simp add: finfun_fst_def [abs_def] finfun_comp_conv_comp)


definition finfun_snd :: "'a =>f ('b × 'c) => 'a =>f 'c"
where [code]: "finfun_snd f = snd o$ f"

lemma finfun_snd_const: "finfun_snd (K$ bc) = (K$ snd bc)"
by(simp add: finfun_snd_def)

lemma finfun_snd_update: "finfun_snd (f(a $:= bc)) = (finfun_snd f)(a $:= snd bc)"
  and finfun_snd_update_code [code]: "finfun_snd (finfun_update_code f a bc) = (finfun_snd f)(a $:= snd bc)"
by(simp_all add: finfun_snd_def)

lemma finfun_snd_comp_conv: "finfun_snd (f o$ g) = (snd o f) o$ g"
by(simp add: finfun_snd_def)

lemma finfun_snd_conv [simp]: "finfun_snd ($f, g$) = g"
apply(induct f rule: finfun_weak_induct)
apply(auto simp add: finfun_Diag_const1 finfun_snd_comp_conv o_def finfun_Diag_update1 finfun_snd_update finfun_upd_apply intro: finfun_ext)
done

lemma finfun_snd_conv_Abs_finfun: "finfun_snd = (λf. Abs_finfun (snd o op $ f))"
by(simp add: finfun_snd_def [abs_def] finfun_comp_conv_comp)

lemma finfun_Diag_collapse [simp]: "($finfun_fst f, finfun_snd f$) = f"
by(induct f rule: finfun_weak_induct)(simp_all add: finfun_fst_const finfun_snd_const finfun_fst_update finfun_snd_update finfun_Diag_update_update)

subsection {* Currying for FinFuns *}

definition finfun_curry :: "('a × 'b) =>f 'c => 'a =>f 'b =>f 'c"
where [code del]: "finfun_curry = finfun_rec (finfun_const o finfun_const) (λ(a, b) c f. f(a $:= (f $ a)(b $:= c)))"

interpretation finfun_curry_aux: finfun_rec_wf_aux "finfun_const o finfun_const" "λ(a, b) c f. f(a $:= (f $ a)(b $:= c))"
apply(unfold_locales)
apply(auto simp add: split_def finfun_update_twist finfun_upd_apply split_paired_all finfun_update_const_same)
done

interpretation finfun_curry: finfun_rec_wf "finfun_const o finfun_const" "λ(a, b) c f. f(a $:= (f $ a)(b $:= c))"
proof(unfold_locales)
  fix b' b :: 'b
  assume fin: "finite (UNIV :: ('c × 'a) set)"
  hence fin1: "finite (UNIV :: 'c set)" and fin2: "finite (UNIV :: 'a set)"
    unfolding UNIV_Times_UNIV[symmetric]
    by(fastforce dest: finite_cartesian_productD1 finite_cartesian_productD2)+
  note [simp] = Abs_finfun_inverse_finite[OF fin] Abs_finfun_inverse_finite[OF fin1] Abs_finfun_inverse_finite[OF fin2]
  { fix A :: "('c × 'a) set"
    interpret comp_fun_commute "λa :: 'c × 'a. (λ(a, b) c f. f(a $:= (f $ a)(b $:= c))) a b'"
      by(rule finfun_curry_aux.upd_left_comm)
    from fin have "finite A" by(auto intro: finite_subset)
    hence "Finite_Set.fold (λa :: 'c × 'a. (λ(a, b) c f. f(a $:= (f $ a)(b $:= c))) a b') ((finfun_const o finfun_const) b) A = Abs_finfun (λa. Abs_finfun (λb''. if (a, b'') ∈ A then b' else b))"
      by induct (simp_all, auto simp add: finfun_update_def finfun_const_def split_def intro!: arg_cong[where f="Abs_finfun"] ext) }
  from this[of UNIV]
  show "Finite_Set.fold (λa :: 'c × 'a. (λ(a, b) c f. f(a $:= (f $ a)(b $:= c))) a b') ((finfun_const o finfun_const) b) UNIV = (finfun_const o finfun_const) b'"
    by(simp add: finfun_const_def)
qed

lemma finfun_curry_const [simp, code]: "finfun_curry (K$ c) = (K$ K$ c)"
by(simp add: finfun_curry_def)

lemma finfun_curry_update [simp]:
  "finfun_curry (f((a, b) $:= c)) = (finfun_curry f)(a $:= (finfun_curry f $ a)(b $:= c))"
  and finfun_curry_update_code [code]:
  "finfun_curry (finfun_update_code f (a, b) c) = (finfun_curry f)(a $:= (finfun_curry f $ a)(b $:= c))"
by(simp_all add: finfun_curry_def)

lemma finfun_Abs_finfun_curry: assumes fin: "f ∈ finfun"
  shows "(λa. Abs_finfun (curry f a)) ∈ finfun"
  including finfun
proof -
  from fin obtain c where c: "finite {ab. f ab ≠ c}" unfolding finfun_def by blast
  have "{a. ∃b. f (a, b) ≠ c} = fst ` {ab. f ab ≠ c}" by(force)
  hence "{a. curry f a ≠ (λx. c)} = fst ` {ab. f ab ≠ c}"
    by(auto simp add: curry_def fun_eq_iff)
  with fin c have "finite {a.  Abs_finfun (curry f a) ≠ (K$ c)}"
    by(simp add: finfun_const_def finfun_curry)
  thus ?thesis unfolding finfun_def by auto
qed

lemma finfun_curry_conv_curry:
  fixes f :: "('a × 'b) =>f 'c"
  shows "finfun_curry f = Abs_finfun (λa. Abs_finfun (curry (finfun_apply f) a))"
  including finfun
proof -
  have "finfun_curry = (λf :: ('a × 'b) =>f 'c. Abs_finfun (λa. Abs_finfun (curry (finfun_apply f) a)))"
  proof(rule finfun_rec_unique)
    fix c show "finfun_curry (K$ c) = (K$ K$ c)" by simp
    fix f a
    show "finfun_curry (f(a $:= c)) = (finfun_curry f)(fst a $:= (finfun_curry f $ (fst a))(snd a $:= c))"
      by(cases a) simp
    show "Abs_finfun (λa. Abs_finfun (curry (finfun_apply (K$ c)) a)) = (K$ K$ c)"
      by(simp add: finfun_curry_def finfun_const_def curry_def)
    fix g b
    show "Abs_finfun (λaa. Abs_finfun (curry (op $ g(a $:= b)) aa)) =
      (Abs_finfun (λa. Abs_finfun (curry (op $ g) a)))(
      fst a $:= ((Abs_finfun (λa. Abs_finfun (curry (op $ g) a))) $ (fst a))(snd a $:= b))"
      by(cases a)(auto intro!: ext arg_cong[where f=Abs_finfun] simp add: finfun_curry_def finfun_update_def finfun_Abs_finfun_curry)
  qed
  thus ?thesis by(auto simp add: fun_eq_iff)
qed

subsection {* Executable equality for FinFuns *}

lemma eq_finfun_All_ext: "(f = g) <-> finfun_All ((λ(x, y). x = y) o$ ($f, g$))"
by(simp add: expand_finfun_eq fun_eq_iff finfun_All_All o_def)

instantiation finfun :: ("{card_UNIV,equal}",equal) equal begin
definition eq_finfun_def [code]: "HOL.equal f g <-> finfun_All ((λ(x, y). x = y) o$ ($f, g$))"
instance by(intro_classes)(simp add: eq_finfun_All_ext eq_finfun_def)
end

lemma [code nbe]:
  "HOL.equal (f :: _ =>f _) f <-> True"
  by (fact equal_refl)

subsection {* An operator that explicitly removes all redundant updates in the generated representations *}

definition finfun_clearjunk :: "'a =>f 'b => 'a =>f 'b"
where [simp, code del]: "finfun_clearjunk = id"

lemma finfun_clearjunk_const [code]: "finfun_clearjunk (K$ b) = (K$ b)"
by simp

lemma finfun_clearjunk_update [code]: 
  "finfun_clearjunk (finfun_update_code f a b) = f(a $:= b)"
by simp

subsection {* The domain of a FinFun as a FinFun *}

definition finfun_dom :: "('a =>f 'b) => ('a =>f bool)"
where [code del]: "finfun_dom f = Abs_finfun (λa. f $ a ≠ finfun_default f)"

lemma finfun_dom_const:
  "finfun_dom ((K$ c) :: 'a =>f 'b) = (K$ finite (UNIV :: 'a set) ∧ c ≠ undefined)"
unfolding finfun_dom_def finfun_default_const
by(auto)(simp_all add: finfun_const_def)

text {*
  @{term "finfun_dom" } raises an exception when called on a FinFun whose domain is a finite type. 
  For such FinFuns, the default value (and as such the domain) is undefined.
*}

lemma finfun_dom_const_code [code]:
  "finfun_dom ((K$ c) :: ('a :: card_UNIV) =>f 'b) = 
   (if CARD('a) = 0 then (K$ False) else Code.abort (STR ''finfun_dom called on finite type'') (λ_. finfun_dom (K$ c)))"
by(simp add: finfun_dom_const card_UNIV card_eq_0_iff)

lemma finfun_dom_finfunI: "(λa. f $ a ≠ finfun_default f) ∈ finfun"
using finite_finfun_default[of f]
by(simp add: finfun_def exI[where x=False])

lemma finfun_dom_update [simp]:
  "finfun_dom (f(a $:= b)) = (finfun_dom f)(a $:= (b ≠ finfun_default f))"
including finfun unfolding finfun_dom_def finfun_update_def
apply(simp add: finfun_default_update_const fun_upd_apply finfun_dom_finfunI)
apply(fold finfun_update.rep_eq)
apply(simp add: finfun_upd_apply fun_eq_iff fun_upd_def finfun_default_update_const)
done

lemma finfun_dom_update_code [code]:
  "finfun_dom (finfun_update_code f a b) = finfun_update_code (finfun_dom f) a (b ≠ finfun_default f)"
by(simp)

lemma finite_finfun_dom: "finite {x. finfun_dom f $ x}"
proof(induct f rule: finfun_weak_induct)
  case (const b)
  thus ?case
    by (cases "finite (UNIV :: 'a set) ∧ b ≠ undefined")
      (auto simp add: finfun_dom_const UNIV_def [symmetric] Set.empty_def [symmetric])
next
  case (update f a b)
  have "{x. finfun_dom f(a $:= b) $ x} =
    (if b = finfun_default f then {x. finfun_dom f $ x} - {a} else insert a {x. finfun_dom f $ x})"
    by (auto simp add: finfun_upd_apply split: split_if_asm)
  thus ?case using update by simp
qed


subsection {* The domain of a FinFun as a sorted list *}

definition finfun_to_list :: "('a :: linorder) =>f 'b => 'a list"
where
  "finfun_to_list f = (THE xs. set xs = {x. finfun_dom f $ x} ∧ sorted xs ∧ distinct xs)"

lemma set_finfun_to_list [simp]: "set (finfun_to_list f) = {x. finfun_dom f $ x}" (is ?thesis1)
  and sorted_finfun_to_list: "sorted (finfun_to_list f)" (is ?thesis2)
  and distinct_finfun_to_list: "distinct (finfun_to_list f)" (is ?thesis3)
proof -
  have "?thesis1 ∧ ?thesis2 ∧ ?thesis3"
    unfolding finfun_to_list_def
    by(rule theI')(rule finite_sorted_distinct_unique finite_finfun_dom)+
  thus ?thesis1 ?thesis2 ?thesis3 by simp_all
qed

lemma finfun_const_False_conv_bot: "op $ (K$ False) = bot"
by auto

lemma finfun_const_True_conv_top: "op $ (K$ True) = top"
by auto

lemma finfun_to_list_const:
  "finfun_to_list ((K$ c) :: ('a :: {linorder} =>f 'b)) = 
  (if ¬ finite (UNIV :: 'a set) ∨ c = undefined then [] else THE xs. set xs = UNIV ∧ sorted xs ∧ distinct xs)"
by(auto simp add: finfun_to_list_def finfun_const_False_conv_bot finfun_const_True_conv_top finfun_dom_const)

lemma finfun_to_list_const_code [code]:
  "finfun_to_list ((K$ c) :: ('a :: {linorder, card_UNIV} =>f 'b)) =
   (if CARD('a) = 0 then [] else Code.abort (STR ''finfun_to_list called on finite type'') (λ_. finfun_to_list ((K$ c) :: ('a =>f 'b))))"
by(auto simp add: finfun_to_list_const card_UNIV card_eq_0_iff)

lemma remove1_insort_insert_same:
  "x ∉ set xs ==> remove1 x (insort_insert x xs) = xs"
by (metis insort_insert_insort remove1_insort)

lemma finfun_dom_conv:
  "finfun_dom f $ x <-> f $ x ≠ finfun_default f"
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_dom_const finfun_default_const finfun_default_update_const finfun_upd_apply)

lemma finfun_to_list_update:
  "finfun_to_list (f(a $:= b)) = 
  (if b = finfun_default f then List.remove1 a (finfun_to_list f) else List.insort_insert a (finfun_to_list f))"
proof(subst finfun_to_list_def, rule the_equality)
  fix xs
  assume "set xs = {x. finfun_dom f(a $:= b) $ x} ∧ sorted xs ∧ distinct xs"
  hence eq: "set xs = {x. finfun_dom f(a $:= b) $ x}"
    and [simp]: "sorted xs" "distinct xs" by simp_all
  show "xs = (if b = finfun_default f then remove1 a (finfun_to_list f) else insort_insert a (finfun_to_list f))"
  proof(cases "b = finfun_default f")
    case True [simp]
    show ?thesis
    proof(cases "finfun_dom f $ a")
      case True
      have "finfun_to_list f = insort_insert a xs"
        unfolding finfun_to_list_def
      proof(rule the_equality)
        have "set (insort_insert a xs) = insert a (set xs)" by(simp add: set_insort_insert)
        also note eq also
        have "insert a {x. finfun_dom f(a $:= b) $ x} = {x. finfun_dom f $ x}" using True
          by(auto simp add: finfun_upd_apply split: split_if_asm)
        finally show 1: "set (insort_insert a xs) = {x. finfun_dom f $ x} ∧ sorted (insort_insert a xs) ∧ distinct (insort_insert a xs)"
          by(simp add: sorted_insort_insert distinct_insort_insert)

        fix xs'
        assume "set xs' = {x. finfun_dom f $ x} ∧ sorted xs' ∧ distinct xs'"
        thus "xs' = insort_insert a xs" using 1 by(auto dest: sorted_distinct_set_unique)
      qed
      with eq True show ?thesis by(simp add: remove1_insort_insert_same)
    next
      case False
      hence "f $ a = b" by(auto simp add: finfun_dom_conv)
      hence f: "f(a $:= b) = f" by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply)
      from eq have "finfun_to_list f = xs" unfolding f finfun_to_list_def
        by(auto elim: sorted_distinct_set_unique intro!: the_equality)
      with eq False show ?thesis unfolding f by(simp add: remove1_idem)
    qed
  next
    case False
    show ?thesis
    proof(cases "finfun_dom f $ a")
      case True
      have "finfun_to_list f = xs"
        unfolding finfun_to_list_def
      proof(rule the_equality)
        have "finfun_dom f = finfun_dom f(a $:= b)" using False True
          by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply)
        with eq show 1: "set xs = {x. finfun_dom f $ x} ∧ sorted xs ∧ distinct xs"
          by(simp del: finfun_dom_update)
        
        fix xs'
        assume "set xs' = {x. finfun_dom f $ x} ∧ sorted xs' ∧ distinct xs'"
        thus "xs' = xs" using 1 by(auto elim: sorted_distinct_set_unique)
      qed
      thus ?thesis using False True eq by(simp add: insort_insert_triv)
    next
      case False
      have "finfun_to_list f = remove1 a xs"
        unfolding finfun_to_list_def
      proof(rule the_equality)
        have "set (remove1 a xs) = set xs - {a}" by simp
        also note eq also
        have "{x. finfun_dom f(a $:= b) $ x} - {a} = {x. finfun_dom f $ x}" using False
          by(auto simp add: finfun_upd_apply split: split_if_asm)
        finally show 1: "set (remove1 a xs) = {x. finfun_dom f $ x} ∧ sorted (remove1 a xs) ∧ distinct (remove1 a xs)"
          by(simp add: sorted_remove1)
        
        fix xs'
        assume "set xs' = {x. finfun_dom f $ x} ∧ sorted xs' ∧ distinct xs'"
        thus "xs' = remove1 a xs" using 1 by(blast intro: sorted_distinct_set_unique)
      qed
      thus ?thesis using False eq `b ≠ finfun_default f` 
        by (simp add: insort_insert_insort insort_remove1)
    qed
  qed
qed (auto simp add: distinct_finfun_to_list sorted_finfun_to_list sorted_remove1 set_insort_insert sorted_insort_insert distinct_insort_insert finfun_upd_apply split: split_if_asm)

lemma finfun_to_list_update_code [code]:
  "finfun_to_list (finfun_update_code f a b) = 
  (if b = finfun_default f then List.remove1 a (finfun_to_list f) else List.insort_insert a (finfun_to_list f))"
by(simp add: finfun_to_list_update)

text {* More type class instantiations *}

lemma card_eq_1_iff: "card A = 1 <-> A ≠ {} ∧ (∀x∈A. ∀y∈A. x = y)"
  (is "?lhs <-> ?rhs")
proof
  assume ?lhs
  moreover {
    fix x y
    assume A: "x ∈ A" "y ∈ A" and neq: "x ≠ y"
    have "finite A" using `?lhs` by(simp add: card_ge_0_finite)
    from neq have "2 = card {x, y}" by simp
    also have "… ≤ card A" using A `finite A`
      by(auto intro: card_mono)
    finally have False using `?lhs` by simp }
  ultimately show ?rhs by auto
next
  assume ?rhs
  hence "A = {THE x. x ∈ A}"
    by safe (auto intro: theI the_equality[symmetric])
  also have "card … = 1" by simp
  finally show ?lhs .
qed

lemma card_UNIV_finfun: 
  defines "F == finfun :: ('a => 'b) set"
  shows "CARD('a =>f 'b) = (if CARD('a) ≠ 0 ∧ CARD('b) ≠ 0 ∨ CARD('b) = 1 then CARD('b) ^ CARD('a) else 0)"
proof(cases "0 < CARD('a) ∧ 0 < CARD('b) ∨ CARD('b) = 1")
  case True
  from True have "F = UNIV"
  proof
    assume b: "CARD('b) = 1"
    hence "∀x :: 'b. x = undefined"
      by(auto simp add: card_eq_1_iff simp del: One_nat_def)
    thus ?thesis by(auto simp add: finfun_def F_def intro: exI[where x=undefined])
  qed(auto simp add: finfun_def card_gt_0_iff F_def intro: finite_subset[where B=UNIV])
  moreover have "CARD('a =>f 'b) = card F"
    unfolding type_definition.Abs_image[OF type_definition_finfun, symmetric]
    by(auto intro!: card_image inj_onI simp add: Abs_finfun_inject F_def)
  ultimately show ?thesis by(simp add: card_fun)
next
  case False
  hence infinite: "¬ (finite (UNIV :: 'a set) ∧ finite (UNIV :: 'b set))"
    and b: "CARD('b) ≠ 1" by(simp_all add: card_eq_0_iff)
  from b obtain b1 b2 :: 'b where "b1 ≠ b2"
    by(auto simp add: card_eq_1_iff simp del: One_nat_def)
  let ?f = "λa a' :: 'a. if a = a' then b1 else b2"
  from infinite have "¬ finite (UNIV :: ('a =>f 'b) set)"
  proof(rule contrapos_nn[OF _ conjI])
    assume finite: "finite (UNIV :: ('a =>f 'b) set)"
    hence "finite F"
      unfolding type_definition.Abs_image[OF type_definition_finfun, symmetric] F_def
      by(rule finite_imageD)(auto intro: inj_onI simp add: Abs_finfun_inject)
    hence "finite (range ?f)" 
      by(rule finite_subset[rotated 1])(auto simp add: F_def finfun_def `b1 ≠ b2` intro!: exI[where x=b2])
    thus "finite (UNIV :: 'a set)"
      by(rule finite_imageD)(auto intro: inj_onI simp add: fun_eq_iff `b1 ≠ b2` split: split_if_asm)
    
    from finite have "finite (range (λb. ((K$ b) :: 'a =>f 'b)))"
      by(rule finite_subset[rotated 1]) simp
    thus "finite (UNIV :: 'b set)"
      by(rule finite_imageD)(auto intro!: inj_onI)
  qed
  with False show ?thesis by simp
qed

lemma finite_UNIV_finfun:
  "finite (UNIV :: ('a =>f 'b) set) <->
  (finite (UNIV :: 'a set) ∧ finite (UNIV :: 'b set) ∨ CARD('b) = 1)"
  (is "?lhs <-> ?rhs")
proof -
  have "?lhs <-> CARD('a =>f 'b) > 0" by(simp add: card_gt_0_iff)
  also have "… <-> CARD('a) > 0 ∧ CARD('b) > 0 ∨ CARD('b) = 1"
    by(simp add: card_UNIV_finfun)
  also have "… = ?rhs" by(simp add: card_gt_0_iff)
  finally show ?thesis .
qed

instantiation finfun :: (finite_UNIV, card_UNIV) finite_UNIV begin
definition "finite_UNIV = Phantom('a =>f 'b)
  (let cb = of_phantom (card_UNIV :: 'b card_UNIV)
   in cb = 1 ∨ of_phantom (finite_UNIV :: 'a finite_UNIV) ∧ cb ≠ 0)"
instance
  by intro_classes (auto simp add: finite_UNIV_finfun_def Let_def card_UNIV finite_UNIV finite_UNIV_finfun card_gt_0_iff)
end

instantiation finfun :: (card_UNIV, card_UNIV) card_UNIV begin
definition "card_UNIV = Phantom('a =>f 'b)
  (let ca = of_phantom (card_UNIV :: 'a card_UNIV);
       cb = of_phantom (card_UNIV :: 'b card_UNIV)
   in if ca ≠ 0 ∧ cb ≠ 0 ∨ cb = 1 then cb ^ ca else 0)"
instance by intro_classes (simp add: card_UNIV_finfun_def card_UNIV Let_def card_UNIV_finfun)
end

text {* Deactivate syntax again. Import theory @{text FinFun_Syntax} to reactivate it again *}

no_type_notation
  finfun ("(_ =>f /_)" [22, 21] 21)

no_type_notation (xsymbols)
  finfun ("(_ =>f /_)" [22, 21] 21)

no_notation
  finfun_const ("K$/ _" [0] 1) and
  finfun_update ("_'(_ $:= _')" [1000,0,0] 1000) and
  finfun_apply (infixl "$" 999) and
  finfun_comp (infixr "o$" 55) and
  finfun_comp2 (infixr "$o" 55) and
  finfun_Diag ("(1'($_,/ _$'))" [0, 0] 1000)

no_notation (xsymbols) 
  finfun_comp (infixr "o$" 55) and
  finfun_comp2 (infixr "$o" 55)

end