Theory HOL-Library.Complete_Partial_Order2

(*  Title:      HOL/Library/Complete_Partial_Order2.thy
    Author:     Andreas Lochbihler, ETH Zurich
*)

section ‹Formalisation of chain-complete partial orders, continuity and admissibility›

theory Complete_Partial_Order2 imports 
  Main
begin

unbundle lattice_syntax

lemma chain_transfer [transfer_rule]:
  includes lifting_syntax
  shows "((A ===> A ===> (=)) ===> rel_set A ===> (=)) Complete_Partial_Order.chain Complete_Partial_Order.chain"
unfolding chain_def[abs_def] by transfer_prover
                             
lemma linorder_chain [simp, intro!]:
  fixes Y :: "_ :: linorder set"
  shows "Complete_Partial_Order.chain (≤) Y"
by(auto intro: chainI)

lemma fun_lub_apply: "Sup. fun_lub Sup Y x = Sup ((λf. f x) ` Y)"
by(simp add: fun_lub_def image_def)

lemma fun_lub_empty [simp]: "fun_lub lub {} = (λ_. lub {})"
by(rule ext)(simp add: fun_lub_apply)

lemma chain_fun_ordD: 
  assumes "Complete_Partial_Order.chain (fun_ord le) Y"
  shows "Complete_Partial_Order.chain le ((λf. f x) ` Y)"
by(rule chainI)(auto dest: chainD[OF assms] simp add: fun_ord_def)

lemma chain_Diff:
  "Complete_Partial_Order.chain ord A
   Complete_Partial_Order.chain ord (A - B)"
by(erule chain_subset) blast

lemma chain_rel_prodD1:
  "Complete_Partial_Order.chain (rel_prod orda ordb) Y
   Complete_Partial_Order.chain orda (fst ` Y)"
by(auto 4 3 simp add: chain_def)

lemma chain_rel_prodD2:
  "Complete_Partial_Order.chain (rel_prod orda ordb) Y
   Complete_Partial_Order.chain ordb (snd ` Y)"
by(auto 4 3 simp add: chain_def)


context ccpo begin

lemma ccpo_fun: "class.ccpo (fun_lub Sup) (fun_ord (≤)) (mk_less (fun_ord (≤)))"
  by standard (auto 4 3 simp add: mk_less_def fun_ord_def fun_lub_apply
    intro: order.trans order.antisym chain_imageI ccpo_Sup_upper ccpo_Sup_least)

lemma ccpo_Sup_below_iff: "Complete_Partial_Order.chain (≤) Y  Sup Y  x  (yY. y  x)"
by(fast intro: order_trans[OF ccpo_Sup_upper] ccpo_Sup_least)

lemma Sup_minus_bot: 
  assumes chain: "Complete_Partial_Order.chain (≤) A"
  shows "(A - {{}}) = A"
    (is "?lhs = ?rhs")
proof (rule order.antisym)
  show "?lhs  ?rhs"
    by (blast intro: ccpo_Sup_least chain_Diff[OF chain] ccpo_Sup_upper[OF chain])
  show "?rhs  ?lhs"
  proof (rule ccpo_Sup_least [OF chain])
    show "x  A  x  ?lhs" for x
      by (cases "x = {}")
        (blast intro: ccpo_Sup_least chain_empty ccpo_Sup_upper[OF chain_Diff[OF chain]])+
  qed
qed

lemma mono_lub:
  fixes le_b (infix "" 60)
  assumes chain: "Complete_Partial_Order.chain (fun_ord (≤)) Y"
  and mono: "f. f  Y  monotone le_b (≤) f"
  shows "monotone (⊑) (≤) (fun_lub Sup Y)"
proof(rule monotoneI)
  fix x y
  assume "x  y"

  have chain'': "x. Complete_Partial_Order.chain (≤) ((λf. f x) ` Y)"
    using chain by(rule chain_imageI)(simp add: fun_ord_def)
  then show "fun_lub Sup Y x  fun_lub Sup Y y" unfolding fun_lub_apply
  proof(rule ccpo_Sup_least)
    fix x'
    assume "x'  (λf. f x) ` Y"
    then obtain f where "f  Y" "x' = f x" by blast
    note x' = f x also
    from f  Y x  y have "f x  f y" by(blast dest: mono monotoneD)
    also have "  ((λf. f y) ` Y)" using chain''
      by(rule ccpo_Sup_upper)(simp add: f  Y)
    finally show "x'  ((λf. f y) ` Y)" .
  qed
qed

context
  fixes le_b (infix "" 60) and Y f
  assumes chain: "Complete_Partial_Order.chain le_b Y" 
  and mono1: "y. y  Y  monotone le_b (≤) (λx. f x y)"
  and mono2: "x a b.  x  Y; a  b; a  Y; b  Y   f x a  f x b"
begin

lemma Sup_mono: 
  assumes le: "x  y" and x: "x  Y" and y: "y  Y"
  shows "(f x ` Y)  (f y ` Y)" (is "_  ?rhs")
proof(rule ccpo_Sup_least)
  from chain show chain': "Complete_Partial_Order.chain (≤) (f x ` Y)" when "x  Y" for x
    by(rule chain_imageI) (insert that, auto dest: mono2)

  fix x'
  assume "x'  f x ` Y"
  then obtain y' where "y'  Y" "x' = f x y'" by blast note this(2)
  also from mono1[OF y'  Y] le have "  f y y'" by(rule monotoneD)
  also have "  ?rhs" using chain'[OF y]
    by (auto intro!: ccpo_Sup_upper simp add: y'  Y)
  finally show "x'  ?rhs" .
qed(rule x)

lemma diag_Sup: "((λx. (f x ` Y)) ` Y) = ((λx. f x x) ` Y)" (is "?lhs = ?rhs")
proof(rule order.antisym)
  have chain1: "Complete_Partial_Order.chain (≤) ((λx. (f x ` Y)) ` Y)"
    using chain by(rule chain_imageI)(rule Sup_mono)
  have chain2: "y'. y'  Y  Complete_Partial_Order.chain (≤) (f y' ` Y)" using chain
    by(rule chain_imageI)(auto dest: mono2)
  have chain3: "Complete_Partial_Order.chain (≤) ((λx. f x x) ` Y)"
    using chain by(rule chain_imageI)(auto intro: monotoneD[OF mono1] mono2 order.trans)

  show "?lhs  ?rhs" using chain1
  proof(rule ccpo_Sup_least)
    fix x'
    assume "x'  (λx. (f x ` Y)) ` Y"
    then obtain y' where "y'  Y" "x' = (f y' ` Y)" by blast note this(2)
    also have "  ?rhs" using chain2[OF y'  Y]
    proof(rule ccpo_Sup_least)
      fix x
      assume "x  f y' ` Y"
      then obtain y where "y  Y" and x: "x = f y' y" by blast
      define y'' where "y'' = (if y  y' then y' else y)"
      from chain y  Y y'  Y have "y  y'  y'  y" by(rule chainD)
      hence "f y' y  f y'' y''" using y  Y y'  Y
        by(auto simp add: y''_def intro: mono2 monotoneD[OF mono1])
      also from y  Y y'  Y have "y''  Y" by(simp add: y''_def)
      from chain3 have "f y'' y''  ?rhs" by(rule ccpo_Sup_upper)(simp add: y''  Y)
      finally show "x  ?rhs" by(simp add: x)
    qed
    finally show "x'  ?rhs" .
  qed

  show "?rhs  ?lhs" using chain3
  proof(rule ccpo_Sup_least)
    fix y
    assume "y  (λx. f x x) ` Y"
    then obtain x where "x  Y" and "y = f x x" by blast note this(2)
    also from chain2[OF x  Y] have "  (f x ` Y)"
      by(rule ccpo_Sup_upper)(simp add: x  Y)
    also have "  ?lhs" by(rule ccpo_Sup_upper[OF chain1])(simp add: x  Y)
    finally show "y  ?lhs" .
  qed
qed

end

lemma Sup_image_mono_le:
  fixes le_b (infix "" 60) and Sup_b ("")
  assumes ccpo: "class.ccpo Sup_b (⊑) lt_b"
  assumes chain: "Complete_Partial_Order.chain (⊑) Y"
  and mono: "x y.  x  y; x  Y   f x  f y"
  shows "Sup (f ` Y)  f (Y)"
proof(rule ccpo_Sup_least)
  show "Complete_Partial_Order.chain (≤) (f ` Y)"
    using chain by(rule chain_imageI)(rule mono)

  fix x
  assume "x  f ` Y"
  then obtain y where "y  Y" and "x = f y" by blast note this(2)
  also have "y  Y" using ccpo chain y  Y by(rule ccpo.ccpo_Sup_upper)
  hence "f y  f (Y)" using y  Y by(rule mono)
  finally show "x  " .
qed

lemma swap_Sup:
  fixes le_b (infix "" 60)
  assumes Y: "Complete_Partial_Order.chain (⊑) Y"
  and Z: "Complete_Partial_Order.chain (fun_ord (≤)) Z"
  and mono: "f. f  Z  monotone (⊑) (≤) f"
  shows "((λx. (x ` Y)) ` Z) = ((λx. ((λf. f x) ` Z)) ` Y)"
  (is "?lhs = ?rhs")
proof(cases "Y = {}")
  case True
  then show ?thesis
    by (simp add: image_constant_conv cong del: SUP_cong_simp)
next
  case False
  have chain1: "f. f  Z  Complete_Partial_Order.chain (≤) (f ` Y)"
    by(rule chain_imageI[OF Y])(rule monotoneD[OF mono])
  have chain2: "Complete_Partial_Order.chain (≤) ((λx. (x ` Y)) ` Z)" using Z
  proof(rule chain_imageI)
    fix f g
    assume "f  Z" "g  Z"
      and "fun_ord (≤) f g"
    from chain1[OF f  Z] show "(f ` Y)  (g ` Y)"
    proof(rule ccpo_Sup_least)
      fix x
      assume "x  f ` Y"
      then obtain y where "y  Y" "x = f y" by blast note this(2)
      also have "  g y" using fun_ord (≤) f g by(simp add: fun_ord_def)
      also have "  (g ` Y)" using chain1[OF g  Z]
        by(rule ccpo_Sup_upper)(simp add: y  Y)
      finally show "x  (g ` Y)" .
    qed
  qed
  have chain3: "x. Complete_Partial_Order.chain (≤) ((λf. f x) ` Z)"
    using Z by(rule chain_imageI)(simp add: fun_ord_def)
  have chain4: "Complete_Partial_Order.chain (≤) ((λx. ((λf. f x) ` Z)) ` Y)"
    using Y
  proof(rule chain_imageI)
    fix f x y
    assume "x  y"
    show "((λf. f x) ` Z)  ((λf. f y) ` Z)" (is "_  ?rhs") using chain3
    proof(rule ccpo_Sup_least)
      fix x'
      assume "x'  (λf. f x) ` Z"
      then obtain f where "f  Z" "x' = f x" by blast note this(2)
      also have "f x  f y" using f  Z x  y by(rule monotoneD[OF mono])
      also have "f y  ?rhs" using chain3
        by(rule ccpo_Sup_upper)(simp add: f  Z)
      finally show "x'  ?rhs" .
    qed
  qed

  from chain2 have "?lhs  ?rhs"
  proof(rule ccpo_Sup_least)
    fix x
    assume "x  (λx. (x ` Y)) ` Z"
    then obtain f where "f  Z" "x = (f ` Y)" by blast note this(2)
    also have "  ?rhs" using chain1[OF f  Z]
    proof(rule ccpo_Sup_least)
      fix x'
      assume "x'  f ` Y"
      then obtain y where "y  Y" "x' = f y" by blast note this(2)
      also have "f y  ((λf. f y) ` Z)" using chain3
        by(rule ccpo_Sup_upper)(simp add: f  Z)
      also have "  ?rhs" using chain4 by(rule ccpo_Sup_upper)(simp add: y  Y)
      finally show "x'  ?rhs" .
    qed
    finally show "x  ?rhs" .
  qed
  moreover
  have "?rhs  ?lhs" using chain4
  proof(rule ccpo_Sup_least)
    fix x
    assume "x  (λx. ((λf. f x) ` Z)) ` Y"
    then obtain y where "y  Y" "x = ((λf. f y) ` Z)" by blast note this(2)
    also have "  ?lhs" using chain3
    proof(rule ccpo_Sup_least)
      fix x'
      assume "x'  (λf. f y) ` Z"
      then obtain f where "f  Z" "x' = f y" by blast note this(2)
      also have "f y  (f ` Y)" using chain1[OF f  Z]
        by(rule ccpo_Sup_upper)(simp add: y  Y)
      also have "  ?lhs" using chain2
        by(rule ccpo_Sup_upper)(simp add: f  Z)
      finally show "x'  ?lhs" .
    qed
    finally show "x  ?lhs" .
  qed
  ultimately show "?lhs = ?rhs"
    by (rule order.antisym)
qed

lemma fixp_mono:
  assumes fg: "fun_ord (≤) f g"
  and f: "monotone (≤) (≤) f"
  and g: "monotone (≤) (≤) g"
  shows "ccpo_class.fixp f  ccpo_class.fixp g"
unfolding fixp_def
proof(rule ccpo_Sup_least)
  fix x
  assume "x  ccpo_class.iterates f"
  thus "x  ccpo_class.iterates g"
  proof induction
    case (step x)
    from f step.IH have "f x  f (ccpo_class.iterates g)" by(rule monotoneD)
    also have "  g (ccpo_class.iterates g)" using fg by(simp add: fun_ord_def)
    also have " = ccpo_class.iterates g" by(fold fixp_def fixp_unfold[OF g]) simp
    finally show ?case .
  qed(blast intro: ccpo_Sup_least)
qed(rule chain_iterates[OF f])

context fixes ordb :: "'b  'b  bool" (infix "" 60) begin

lemma iterates_mono:
  assumes f: "f  ccpo.iterates (fun_lub Sup) (fun_ord (≤)) F"
  and mono: "f. monotone (⊑) (≤) f  monotone (⊑) (≤) (F f)"
  shows "monotone (⊑) (≤) f"
using f
by(induction rule: ccpo.iterates.induct[OF ccpo_fun, consumes 1, case_names step Sup])(blast intro: mono mono_lub)+

lemma fixp_preserves_mono:
  assumes mono: "x. monotone (fun_ord (≤)) (≤) (λf. F f x)"
  and mono2: "f. monotone (⊑) (≤) f  monotone (⊑) (≤) (F f)"
  shows "monotone (⊑) (≤) (ccpo.fixp (fun_lub Sup) (fun_ord (≤)) F)"
  (is "monotone _ _ ?fixp")
proof(rule monotoneI)
  have mono: "monotone (fun_ord (≤)) (fun_ord (≤)) F"
    by(rule monotoneI)(auto simp add: fun_ord_def intro: monotoneD[OF mono])
  let ?iter = "ccpo.iterates (fun_lub Sup) (fun_ord (≤)) F"
  have chain: "x. Complete_Partial_Order.chain (≤) ((λf. f x) ` ?iter)"
    by(rule chain_imageI[OF ccpo.chain_iterates[OF ccpo_fun mono]])(simp add: fun_ord_def)

  fix x y
  assume "x  y"
  show "?fixp x  ?fixp y"
    apply (simp only: ccpo.fixp_def[OF ccpo_fun] fun_lub_apply)
    using chain
  proof(rule ccpo_Sup_least)
    fix x'
    assume "x'  (λf. f x) ` ?iter"
    then obtain f where "f  ?iter" "x' = f x" by blast note this(2)
    also have "f x  f y"
      by(rule monotoneD[OF iterates_mono[OF f  ?iter mono2]])(blast intro: x  y)+
    also have "f y  ((λf. f y) ` ?iter)" using chain
      by(rule ccpo_Sup_upper)(simp add: f  ?iter)
    finally show "x'  " .
  qed
qed

end

end

lemma monotone2monotone:
  assumes 2: "x. monotone ordb ordc (λy. f x y)"
  and t: "monotone orda ordb (λx. t x)"
  and 1: "y. monotone orda ordc (λx. f x y)"
  and trans: "transp ordc"
  shows "monotone orda ordc (λx. f x (t x))"
by(blast intro: monotoneI transpD[OF trans] monotoneD[OF t] monotoneD[OF 2] monotoneD[OF 1])

subsection ‹Continuity›

definition cont :: "('a set  'a)  ('a  'a  bool)  ('b set  'b)  ('b  'b  bool)  ('a  'b)  bool"
where
  "cont luba orda lubb ordb f  
  (Y. Complete_Partial_Order.chain orda Y  Y  {}  f (luba Y) = lubb (f ` Y))"

definition mcont :: "('a set  'a)  ('a  'a  bool)  ('b set  'b)  ('b  'b  bool)  ('a  'b)  bool"
where
  "mcont luba orda lubb ordb f 
   monotone orda ordb f  cont luba orda lubb ordb f"

subsubsection ‹Theorem collection cont_intro›

named_theorems cont_intro "continuity and admissibility intro rules"
ML (* apply cont_intro rules as intro and try to solve 
   the remaining of the emerging subgoals with simp *)
fun cont_intro_tac ctxt =
  REPEAT_ALL_NEW (resolve_tac ctxt (rev (Named_Theorems.get ctxt named_theorems‹cont_intro›)))
  THEN_ALL_NEW (SOLVED' (simp_tac ctxt))

fun cont_intro_simproc ctxt ct =
  let
    fun mk_stmt t = t
      |> HOLogic.mk_Trueprop
      |> Thm.cterm_of ctxt
      |> Goal.init
    fun mk_thm t =
      if exists_subterm Term.is_Var t then
        NONE
      else
        case SINGLE (cont_intro_tac ctxt 1) (mk_stmt t) of
          SOME thm => SOME (Goal.finish ctxt thm RS @{thm Eq_TrueI})
        | NONE => NONE
  in
    case Thm.term_of ct of
      t as Const_ccpo.admissible _ for _ _ _ => mk_thm t
    | t as Const_mcont _ _ for _ _ _ _ _ => mk_thm t
    | t as Const_monotone_on _ _ for _ _ _ _ => mk_thm t
    | _ => NONE
  end
  handle THM _ => NONE 
  | TYPE _ => NONE

simproc_setup "cont_intro"
  ( "ccpo.admissible lub ord P"
  | "mcont lub ord lub' ord' f"
  | "monotone ord ord' f"
  ) = K cont_intro_simproc

lemmas [cont_intro] =
  call_mono
  let_mono
  if_mono
  option.const_mono
  tailrec.const_mono
  bind_mono

experiment begin

text ‹The following proof by simplification diverges if variables are not handled properly.›

lemma "(f. monotone R S f  thesis)  monotone R S g  thesis"
  by simp

end

declare if_mono[simp]

lemma monotone_id' [cont_intro]: "monotone ord ord (λx. x)"
by(simp add: monotone_def)

lemma monotone_applyI:
  "monotone orda ordb F  monotone (fun_ord orda) ordb (λf. F (f x))"
by(rule monotoneI)(auto simp add: fun_ord_def dest: monotoneD)

lemma monotone_if_fun [partial_function_mono]:
  " monotone (fun_ord orda) (fun_ord ordb) F; monotone (fun_ord orda) (fun_ord ordb) G 
   monotone (fun_ord orda) (fun_ord ordb) (λf n. if c n then F f n else G f n)"
by(simp add: monotone_def fun_ord_def)

lemma monotone_fun_apply_fun [partial_function_mono]: 
  "monotone (fun_ord (fun_ord ord)) (fun_ord ord) (λf n. f t (g n))"
by(rule monotoneI)(simp add: fun_ord_def)

lemma monotone_fun_ord_apply: 
  "monotone orda (fun_ord ordb) f  (x. monotone orda ordb (λy. f y x))"
by(auto simp add: monotone_def fun_ord_def)

context preorder begin

declare transp_on_le[cont_intro]

lemma monotone_const [simp, cont_intro]: "monotone ord (≤) (λ_. c)"
by(rule monotoneI) simp

end

lemma transp_le [cont_intro, simp]:
  "class.preorder ord (mk_less ord)  transp ord"
by(rule preorder.transp_on_le)

context partial_function_definitions begin

declare const_mono [cont_intro, simp]

lemma transp_le [cont_intro, simp]: "transp leq"
by(rule transpI)(rule leq_trans)

lemma preorder [cont_intro, simp]: "class.preorder leq (mk_less leq)"
by(unfold_locales)(auto simp add: mk_less_def intro: leq_refl leq_trans)

declare ccpo[cont_intro, simp]

end

lemma contI [intro?]:
  "(Y.  Complete_Partial_Order.chain orda Y; Y  {}   f (luba Y) = lubb (f ` Y)) 
   cont luba orda lubb ordb f"
unfolding cont_def by blast

lemma contD:
  " cont luba orda lubb ordb f; Complete_Partial_Order.chain orda Y; Y  {}  
   f (luba Y) = lubb (f ` Y)"
unfolding cont_def by blast

lemma cont_id [simp, cont_intro]: "Sup. cont Sup ord Sup ord id"
by(rule contI) simp

lemma cont_id' [simp, cont_intro]: "Sup. cont Sup ord Sup ord (λx. x)"
using cont_id[unfolded id_def] .

lemma cont_applyI [cont_intro]:
  assumes cont: "cont luba orda lubb ordb g"
  shows "cont (fun_lub luba) (fun_ord orda) lubb ordb (λf. g (f x))"
by(rule contI)(drule chain_fun_ordD[where x=x], simp add: fun_lub_apply image_image contD[OF cont])

lemma call_cont: "cont (fun_lub lub) (fun_ord ord) lub ord (λf. f t)"
by(simp add: cont_def fun_lub_apply)

lemma cont_if [cont_intro]:
  " cont luba orda lubb ordb f; cont luba orda lubb ordb g 
   cont luba orda lubb ordb (λx. if c then f x else g x)"
by(cases c) simp_all

lemma mcontI [intro?]:
   " monotone orda ordb f; cont luba orda lubb ordb f   mcont luba orda lubb ordb f"
by(simp add: mcont_def)

lemma mcont_mono: "mcont luba orda lubb ordb f  monotone orda ordb f"
by(simp add: mcont_def)

lemma mcont_cont [simp]: "mcont luba orda lubb ordb f  cont luba orda lubb ordb f"
by(simp add: mcont_def)

lemma mcont_monoD:
  " mcont luba orda lubb ordb f; orda x y   ordb (f x) (f y)"
by(auto simp add: mcont_def dest: monotoneD)

lemma mcont_contD:
  " mcont luba orda lubb ordb f; Complete_Partial_Order.chain orda Y; Y  {} 
   f (luba Y) = lubb (f ` Y)"
by(auto simp add: mcont_def dest: contD)

lemma mcont_call [cont_intro, simp]:
  "mcont (fun_lub lub) (fun_ord ord) lub ord (λf. f t)"
by(simp add: mcont_def call_mono call_cont)

lemma mcont_id' [cont_intro, simp]: "mcont lub ord lub ord (λx. x)"
by(simp add: mcont_def monotone_id')

lemma mcont_applyI:
  "mcont luba orda lubb ordb (λx. F x)  mcont (fun_lub luba) (fun_ord orda) lubb ordb (λf. F (f x))"
by(simp add: mcont_def monotone_applyI cont_applyI)

lemma mcont_if [cont_intro, simp]:
  " mcont luba orda lubb ordb (λx. f x); mcont luba orda lubb ordb (λx. g x) 
   mcont luba orda lubb ordb (λx. if c then f x else g x)"
by(simp add: mcont_def cont_if)

lemma cont_fun_lub_apply: 
  "cont luba orda (fun_lub lubb) (fun_ord ordb) f  (x. cont luba orda lubb ordb (λy. f y x))"
by(simp add: cont_def fun_lub_def fun_eq_iff)(auto simp add: image_def)

lemma mcont_fun_lub_apply: 
  "mcont luba orda (fun_lub lubb) (fun_ord ordb) f  (x. mcont luba orda lubb ordb (λy. f y x))"
by(auto simp add: monotone_fun_ord_apply cont_fun_lub_apply mcont_def)

context ccpo begin

lemma cont_const [simp, cont_intro]: "cont luba orda Sup (≤) (λx. c)"
by (rule contI) (simp add: image_constant_conv cong del: SUP_cong_simp)

lemma mcont_const [cont_intro, simp]:
  "mcont luba orda Sup (≤) (λx. c)"
by(simp add: mcont_def)

lemma cont_apply:
  assumes 2: "x. cont lubb ordb Sup (≤) (λy. f x y)"
  and t: "cont luba orda lubb ordb (λx. t x)"
  and 1: "y. cont luba orda Sup (≤) (λx. f x y)"
  and mono: "monotone orda ordb (λx. t x)"
  and mono2: "x. monotone ordb (≤) (λy. f x y)"
  and mono1: "y. monotone orda (≤) (λx. f x y)"
  shows "cont luba orda Sup (≤) (λx. f x (t x))"
proof
  fix Y
  assume chain: "Complete_Partial_Order.chain orda Y" and "Y  {}"
  moreover from chain have chain': "Complete_Partial_Order.chain ordb (t ` Y)"
    by(rule chain_imageI)(rule monotoneD[OF mono])
  ultimately show "f (luba Y) (t (luba Y)) = ((λx. f x (t x)) ` Y)"
    by(simp add: contD[OF 1] contD[OF t] contD[OF 2] image_image)
      (rule diag_Sup[OF chain], auto intro: monotone2monotone[OF mono2 mono monotone_const transpI] monotoneD[OF mono1])
qed

lemma mcont2mcont':
  " x. mcont lub' ord' Sup (≤) (λy. f x y);
     y. mcont lub ord Sup (≤) (λx. f x y);
     mcont lub ord lub' ord' (λy. t y) 
   mcont lub ord Sup (≤) (λx. f x (t x))"
unfolding mcont_def by(blast intro: transp_on_le monotone2monotone cont_apply)

lemma mcont2mcont:
  "mcont lub' ord' Sup (≤) (λx. f x); mcont lub ord lub' ord' (λx. t x) 
   mcont lub ord Sup (≤) (λx. f (t x))"
by(rule mcont2mcont'[OF _ mcont_const]) 

context
  fixes ord :: "'b  'b  bool" (infix "" 60) 
  and lub :: "'b set  'b" ("")
begin

lemma cont_fun_lub_Sup:
  assumes chainM: "Complete_Partial_Order.chain (fun_ord (≤)) M"
  and mcont [rule_format]: "fM. mcont lub (⊑) Sup (≤) f"
  shows "cont lub (⊑) Sup (≤) (fun_lub Sup M)"
proof(rule contI)
  fix Y
  assume chain: "Complete_Partial_Order.chain (⊑) Y"
    and Y: "Y  {}"
  from swap_Sup[OF chain chainM mcont[THEN mcont_mono]]
  show "fun_lub Sup M (Y) = (fun_lub Sup M ` Y)"
    by(simp add: mcont_contD[OF mcont chain Y] fun_lub_apply cong: image_cong)
qed

lemma mcont_fun_lub_Sup:
  " Complete_Partial_Order.chain (fun_ord (≤)) M;
    fM. mcont lub ord Sup (≤) f 
   mcont lub (⊑) Sup (≤) (fun_lub Sup M)"
by(simp add: mcont_def cont_fun_lub_Sup mono_lub)

lemma iterates_mcont:
  assumes f: "f  ccpo.iterates (fun_lub Sup) (fun_ord (≤)) F"
  and mono: "f. mcont lub (⊑) Sup (≤) f  mcont lub (⊑) Sup (≤) (F f)"
  shows "mcont lub (⊑) Sup (≤) f"
using f
by(induction rule: ccpo.iterates.induct[OF ccpo_fun, consumes 1, case_names step Sup])(blast intro: mono mcont_fun_lub_Sup)+

lemma fixp_preserves_mcont:
  assumes mono: "x. monotone (fun_ord (≤)) (≤) (λf. F f x)"
  and mcont: "f. mcont lub (⊑) Sup (≤) f  mcont lub (⊑) Sup (≤) (F f)"
  shows "mcont lub (⊑) Sup (≤) (ccpo.fixp (fun_lub Sup) (fun_ord (≤)) F)"
  (is "mcont _ _ _ _ ?fixp")
unfolding mcont_def
proof(intro conjI monotoneI contI)
  have mono: "monotone (fun_ord (≤)) (fun_ord (≤)) F"
    by(rule monotoneI)(auto simp add: fun_ord_def intro: monotoneD[OF mono])
  let ?iter = "ccpo.iterates (fun_lub Sup) (fun_ord (≤)) F"
  have chain: "x. Complete_Partial_Order.chain (≤) ((λf. f x) ` ?iter)"
    by(rule chain_imageI[OF ccpo.chain_iterates[OF ccpo_fun mono]])(simp add: fun_ord_def)

  {
    fix x y
    assume "x  y"
    show "?fixp x  ?fixp y"
      apply (simp only: ccpo.fixp_def[OF ccpo_fun] fun_lub_apply)
      using chain
    proof(rule ccpo_Sup_least)
      fix x'
      assume "x'  (λf. f x) ` ?iter"
      then obtain f where "f  ?iter" "x' = f x" by blast note this(2)
      also from _ x  y have "f x  f y"
        by(rule mcont_monoD[OF iterates_mcont[OF f  ?iter mcont]])
      also have "f y  ((λf. f y) ` ?iter)" using chain
        by(rule ccpo_Sup_upper)(simp add: f  ?iter)
      finally show "x'  " .
    qed
  next
    fix Y
    assume chain: "Complete_Partial_Order.chain (⊑) Y"
      and Y: "Y  {}"
    { fix f
      assume "f  ?iter"
      hence "f (Y) = (f ` Y)"
        using mcont chain Y by(rule mcont_contD[OF iterates_mcont]) }
    moreover have "((λf. (f ` Y)) ` ?iter) = ((λx. ((λf. f x) ` ?iter)) ` Y)"
      using chain ccpo.chain_iterates[OF ccpo_fun mono]
      by(rule swap_Sup)(rule mcont_mono[OF iterates_mcont[OF _ mcont]])
    ultimately show "?fixp (Y) = (?fixp ` Y)" unfolding ccpo.fixp_def[OF ccpo_fun]
      by(simp add: fun_lub_apply cong: image_cong)
  }
qed

end

context
  fixes F :: "'c  'c" and U :: "'c  'b  'a" and C :: "('b  'a)  'c" and f
  assumes mono: "x. monotone (fun_ord (≤)) (≤) (λf. U (F (C f)) x)"
  and eq: "f  C (ccpo.fixp (fun_lub Sup) (fun_ord (≤)) (λf. U (F (C f))))"
  and inverse: "f. U (C f) = f"
begin

lemma fixp_preserves_mono_uc:
  assumes mono2: "f. monotone ord (≤) (U f)  monotone ord (≤) (U (F f))"
  shows "monotone ord (≤) (U f)"
using fixp_preserves_mono[OF mono mono2] by(subst eq)(simp add: inverse)

lemma fixp_preserves_mcont_uc:
  assumes mcont: "f. mcont lubb ordb Sup (≤) (U f)  mcont lubb ordb Sup (≤) (U (F f))"
  shows "mcont lubb ordb Sup (≤) (U f)"
using fixp_preserves_mcont[OF mono mcont] by(subst eq)(simp add: inverse)

end

lemmas fixp_preserves_mono1 = fixp_preserves_mono_uc[of "λx. x" _ "λx. x", OF _ _ refl]
lemmas fixp_preserves_mono2 =
  fixp_preserves_mono_uc[of "case_prod" _ "curry", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
lemmas fixp_preserves_mono3 =
  fixp_preserves_mono_uc[of "λf. case_prod (case_prod f)" _ "λf. curry (curry f)", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
lemmas fixp_preserves_mono4 =
  fixp_preserves_mono_uc[of "λf. case_prod (case_prod (case_prod f))" _ "λf. curry (curry (curry f))", unfolded case_prod_curry curry_case_prod, OF _ _ refl]

lemmas fixp_preserves_mcont1 = fixp_preserves_mcont_uc[of "λx. x" _ "λx. x", OF _ _ refl]
lemmas fixp_preserves_mcont2 =
  fixp_preserves_mcont_uc[of "case_prod" _ "curry", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
lemmas fixp_preserves_mcont3 =
  fixp_preserves_mcont_uc[of "λf. case_prod (case_prod f)" _ "λf. curry (curry f)", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
lemmas fixp_preserves_mcont4 =
  fixp_preserves_mcont_uc[of "λf. case_prod (case_prod (case_prod f))" _ "λf. curry (curry (curry f))", unfolded case_prod_curry curry_case_prod, OF _ _ refl]

end

lemma (in preorder) monotone_if_bot:
  fixes bot
  assumes mono: "x y.  x  y; ¬ (x  bound)   ord (f x) (f y)"
  and bot: "x. ¬ x  bound  ord bot (f x)" "ord bot bot"
  shows "monotone (≤) ord (λx. if x  bound then bot else f x)"
by(rule monotoneI)(auto intro: bot intro: mono order_trans)

lemma (in ccpo) mcont_if_bot:
  fixes bot and lub ("") and ord (infix "" 60)
  assumes ccpo: "class.ccpo lub (⊑) lt"
  and mono: "x y.  x  y; ¬ x  bound   f x  f y"
  and cont: "Y.  Complete_Partial_Order.chain (≤) Y; Y  {}; x. x  Y  ¬ x  bound   f (Y) = (f ` Y)"
  and bot: "x. ¬ x  bound  bot  f x"
  shows "mcont Sup (≤) lub (⊑) (λx. if x  bound then bot else f x)" (is "mcont _ _ _ _ ?g")
proof(intro mcontI contI)
  interpret c: ccpo lub "(⊑)" lt by(fact ccpo)
  show "monotone (≤) (⊑) ?g" by(rule monotone_if_bot)(simp_all add: mono bot)

  fix Y
  assume chain: "Complete_Partial_Order.chain (≤) Y" and Y: "Y  {}"
  show "?g (Y) = (?g ` Y)"
  proof(cases "Y  {x. x  bound}")
    case True
    hence "Y  bound" using chain by(auto intro: ccpo_Sup_least)
    moreover have "Y  {x. ¬ x  bound} = {}" using True by auto
    ultimately show ?thesis using True Y
      by (auto simp add: image_constant_conv cong del: c.SUP_cong_simp)
  next
    case False
    let ?Y = "Y  {x. ¬ x  bound}"
    have chain': "Complete_Partial_Order.chain (≤) ?Y"
      using chain by(rule chain_subset) simp

    from False obtain y where ybound: "¬ y  bound" and y: "y  Y" by blast
    hence "¬ Y  bound" by (metis ccpo_Sup_upper chain order.trans)
    hence "?g (Y) = f (Y)" by simp
    also have "Y  ?Y" using chain
    proof(rule ccpo_Sup_least)
      fix x
      assume x: "x  Y"
      show "x  ?Y"
      proof(cases "x  bound")
        case True
        with chainD[OF chain x y] have "x  y" using ybound by(auto intro: order_trans)
        thus ?thesis by(rule order_trans)(auto intro: ccpo_Sup_upper[OF chain'] simp add: y ybound)
      qed(auto intro: ccpo_Sup_upper[OF chain'] simp add: x)
    qed
    hence "Y = ?Y" by(rule order.antisym)(blast intro: ccpo_Sup_least[OF chain'] ccpo_Sup_upper[OF chain])
    hence "f (Y) = f (?Y)" by simp
    also have "f (?Y) = (f ` ?Y)" using chain' by(rule cont)(insert y ybound, auto)
    also have "(f ` ?Y) = (?g ` Y)"
    proof(cases "Y  {x. x  bound} = {}")
      case True
      hence "f ` ?Y = ?g ` Y" by auto
      thus ?thesis by(rule arg_cong)
    next
      case False
      have chain'': "Complete_Partial_Order.chain (⊑) (insert bot (f ` ?Y))"
        using chain by(auto intro!: chainI bot dest: chainD intro: mono)
      hence chain''': "Complete_Partial_Order.chain (⊑) (f ` ?Y)" by(rule chain_subset) blast
      have "bot  (f ` ?Y)" using y ybound by(blast intro: c.order_trans[OF bot] c.ccpo_Sup_upper[OF chain'''])
      hence "(insert bot (f ` ?Y))  (f ` ?Y)" using chain''
        by(auto intro: c.ccpo_Sup_least c.ccpo_Sup_upper[OF chain''']) 
      with _ have " = (insert bot (f ` ?Y))"
        by(rule c.order.antisym)(blast intro: c.ccpo_Sup_least[OF chain'''] c.ccpo_Sup_upper[OF chain''])
      also have "insert bot (f ` ?Y) = ?g ` Y" using False by auto
      finally show ?thesis .
    qed
    finally show ?thesis .
  qed
qed

context partial_function_definitions begin

lemma mcont_const [cont_intro, simp]:
  "mcont luba orda lub leq (λx. c)"
by(rule ccpo.mcont_const)(rule Partial_Function.ccpo[OF partial_function_definitions_axioms])

lemmas [cont_intro, simp] =
  ccpo.cont_const[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]

lemma mono2mono:
  assumes "monotone ordb leq (λy. f y)" "monotone orda ordb (λx. t x)"
  shows "monotone orda leq (λx. f (t x))"
using assms by(rule monotone2monotone) simp_all

lemmas mcont2mcont' = ccpo.mcont2mcont'[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
lemmas mcont2mcont = ccpo.mcont2mcont[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]

lemmas fixp_preserves_mono1 = ccpo.fixp_preserves_mono1[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
lemmas fixp_preserves_mono2 = ccpo.fixp_preserves_mono2[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
lemmas fixp_preserves_mono3 = ccpo.fixp_preserves_mono3[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
lemmas fixp_preserves_mono4 = ccpo.fixp_preserves_mono4[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
lemmas fixp_preserves_mcont1 = ccpo.fixp_preserves_mcont1[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
lemmas fixp_preserves_mcont2 = ccpo.fixp_preserves_mcont2[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
lemmas fixp_preserves_mcont3 = ccpo.fixp_preserves_mcont3[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
lemmas fixp_preserves_mcont4 = ccpo.fixp_preserves_mcont4[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]

lemma monotone_if_bot:
  fixes bot
  assumes g: "x. g x = (if leq x bound then bot else f x)"
  and mono: "x y.  leq x y; ¬ leq x bound   ord (f x) (f y)"
  and bot: "x. ¬ leq x bound  ord bot (f x)" "ord bot bot"
  shows "monotone leq ord g"
unfolding g[abs_def] using preorder mono bot by(rule preorder.monotone_if_bot)

lemma mcont_if_bot:
  fixes bot
  assumes ccpo: "class.ccpo lub' ord (mk_less ord)"
  and bot: "x. ¬ leq x bound  ord bot (f x)"
  and g: "x. g x = (if leq x bound then bot else f x)"
  and mono: "x y.  leq x y; ¬ leq x bound   ord (f x) (f y)"
  and cont: "Y.  Complete_Partial_Order.chain leq Y; Y  {}; x. x  Y  ¬ leq x bound   f (lub Y) = lub' (f ` Y)"
  shows "mcont lub leq lub' ord g"
unfolding g[abs_def] using ccpo mono cont bot by(rule ccpo.mcont_if_bot[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]])

end

subsection ‹Admissibility›

lemma admissible_subst:
  assumes adm: "ccpo.admissible luba orda (λx. P x)"
  and mcont: "mcont lubb ordb luba orda f"
  shows "ccpo.admissible lubb ordb (λx. P (f x))"
apply(rule ccpo.admissibleI)
apply(frule (1) mcont_contD[OF mcont])
apply(auto intro: ccpo.admissibleD[OF adm] chain_imageI dest: mcont_monoD[OF mcont])
done

lemmas [simp, cont_intro] = 
  admissible_all
  admissible_ball
  admissible_const
  admissible_conj

lemma admissible_disj' [simp, cont_intro]:
  " class.ccpo lub ord (mk_less ord); ccpo.admissible lub ord P; ccpo.admissible lub ord Q 
   ccpo.admissible lub ord (λx. P x  Q x)"
by(rule ccpo.admissible_disj)

lemma admissible_imp' [cont_intro]:
  " class.ccpo lub ord (mk_less ord);
     ccpo.admissible lub ord (λx. ¬ P x);
     ccpo.admissible lub ord (λx. Q x) 
   ccpo.admissible lub ord (λx. P x  Q x)"
unfolding imp_conv_disj by(rule ccpo.admissible_disj)

lemma admissible_imp [cont_intro]:
  "(Q  ccpo.admissible lub ord (λx. P x))
   ccpo.admissible lub ord (λx. Q  P x)"
by(rule ccpo.admissibleI)(auto dest: ccpo.admissibleD)

lemma admissible_not_mem' [THEN admissible_subst, cont_intro, simp]:
  shows admissible_not_mem: "ccpo.admissible Union (⊆) (λA. x  A)"
by(rule ccpo.admissibleI) auto

lemma admissible_eqI:
  assumes f: "cont luba orda lub ord (λx. f x)"
  and g: "cont luba orda lub ord (λx. g x)"
  shows "ccpo.admissible luba orda (λx. f x = g x)"
apply(rule ccpo.admissibleI)
apply(simp_all add: contD[OF f] contD[OF g] cong: image_cong)
done

corollary admissible_eq_mcontI [cont_intro]:
  " mcont luba orda lub ord (λx. f x); 
    mcont luba orda lub ord (λx. g x) 
   ccpo.admissible luba orda (λx. f x = g x)"
by(rule admissible_eqI)(auto simp add: mcont_def)

lemma admissible_iff [cont_intro, simp]:
  " ccpo.admissible lub ord (λx. P x  Q x); ccpo.admissible lub ord (λx. Q x  P x) 
   ccpo.admissible lub ord (λx. P x  Q x)"
by(subst iff_conv_conj_imp)(rule admissible_conj)

context ccpo begin

lemma admissible_leI:
  assumes f: "mcont luba orda Sup (≤) (λx. f x)"
  and g: "mcont luba orda Sup (≤) (λx. g x)"
  shows "ccpo.admissible luba orda (λx. f x  g x)"
proof(rule ccpo.admissibleI)
  fix A
  assume chain: "Complete_Partial_Order.chain orda A"
    and le: "xA. f x  g x"
    and False: "A  {}"
  have "f (luba A) = (f ` A)" by(simp add: mcont_contD[OF f] chain False)
  also have "  (g ` A)"
  proof(rule ccpo_Sup_least)
    from chain show "Complete_Partial_Order.chain (≤) (f ` A)"
      by(rule chain_imageI)(rule mcont_monoD[OF f])
    
    fix x
    assume "x  f ` A"
    then obtain y where "y  A" "x = f y" by blast note this(2)
    also have "f y  g y" using le y  A by simp
    also have "Complete_Partial_Order.chain (≤) (g ` A)"
      using chain by(rule chain_imageI)(rule mcont_monoD[OF g])
    hence "g y  (g ` A)" by(rule ccpo_Sup_upper)(simp add: y  A)
    finally show "x  " .
  qed
  also have " = g (luba A)" by(simp add: mcont_contD[OF g] chain False)
  finally show "f (luba A)  g (luba A)" .
qed

end

lemma admissible_leI:
  fixes ord (infix "" 60) and lub ("")
  assumes "class.ccpo lub (⊑) (mk_less (⊑))"
  and "mcont luba orda lub (⊑) (λx. f x)"
  and "mcont luba orda lub (⊑) (λx. g x)"
  shows "ccpo.admissible luba orda (λx. f x  g x)"
using assms by(rule ccpo.admissible_leI)

declare ccpo_class.admissible_leI[cont_intro]

context ccpo begin

lemma admissible_not_below: "ccpo.admissible Sup (≤) (λx. ¬ (≤) x y)"
by(rule ccpo.admissibleI)(simp add: ccpo_Sup_below_iff)

end

lemma (in preorder) preorder [cont_intro, simp]: "class.preorder (≤) (mk_less (≤))"
by(unfold_locales)(auto simp add: mk_less_def intro: order_trans)

context partial_function_definitions begin

lemmas [cont_intro, simp] =
  admissible_leI[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
  ccpo.admissible_not_below[THEN admissible_subst, OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]

end

setup Sign.map_naming (Name_Space.mandatory_path "ccpo")

inductive compact :: "('a set  'a)  ('a  'a  bool)  'a  bool"
  for lub ord x 
where compact:
  " ccpo.admissible lub ord (λy. ¬ ord x y);
     ccpo.admissible lub ord (λy. x  y) 
   compact lub ord x"

setup Sign.map_naming Name_Space.parent_path

context ccpo begin

lemma compactI:
  assumes "ccpo.admissible Sup (≤) (λy. ¬ x  y)"
  shows "ccpo.compact Sup (≤) x"
using assms
proof(rule ccpo.compact.intros)
  have neq: "(λy. x  y) = (λy. ¬ x  y  ¬ y  x)" by(auto)
  show "ccpo.admissible Sup (≤) (λy. x  y)"
    by(subst neq)(rule admissible_disj admissible_not_below assms)+
qed

lemma compact_bot:
  assumes "x = Sup {}"
  shows "ccpo.compact Sup (≤) x"
proof(rule compactI)
  show "ccpo.admissible Sup (≤) (λy. ¬ x  y)" using assms
    by(auto intro!: ccpo.admissibleI intro: ccpo_Sup_least chain_empty)
qed

end

lemma admissible_compact_neq' [THEN admissible_subst, cont_intro, simp]:
  shows admissible_compact_neq: "ccpo.compact lub ord k  ccpo.admissible lub ord (λx. k  x)"
by(simp add: ccpo.compact.simps)

lemma admissible_neq_compact' [THEN admissible_subst, cont_intro, simp]:
  shows admissible_neq_compact: "ccpo.compact lub ord k  ccpo.admissible lub ord (λx. x  k)"
by(subst eq_commute)(rule admissible_compact_neq)

context partial_function_definitions begin

lemmas [cont_intro, simp] = ccpo.compact_bot[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]

end

context ccpo begin

lemma fixp_strong_induct:
  assumes [cont_intro]: "ccpo.admissible Sup (≤) P"
  and mono: "monotone (≤) (≤) f"
  and bot: "P ({})"
  and step: "x.  x  ccpo_class.fixp f; P x   P (f x)"
  shows "P (ccpo_class.fixp f)"
proof(rule fixp_induct[where P="λx. x  ccpo_class.fixp f  P x", THEN conjunct2])
  note [cont_intro] = admissible_leI
  show "ccpo.admissible Sup (≤) (λx. x  ccpo_class.fixp f  P x)" by simp
next
  show "{}  ccpo_class.fixp f  P ({})"
    by(auto simp add: bot intro: ccpo_Sup_least chain_empty)
next
  fix x
  assume "x  ccpo_class.fixp f  P x"
  thus "f x  ccpo_class.fixp f  P (f x)"
    by(subst fixp_unfold[OF mono])(auto dest: monotoneD[OF mono] intro: step)
qed(rule mono)

end

context partial_function_definitions begin

lemma fixp_strong_induct_uc:
  fixes F :: "'c  'c"
    and U :: "'c  'b  'a"
    and C :: "('b  'a)  'c"
    and P :: "('b  'a)  bool"
  assumes mono: "x. mono_body (λf. U (F (C f)) x)"
    and eq: "f  C (fixp_fun (λf. U (F (C f))))"
    and inverse: "f. U (C f) = f"
    and adm: "ccpo.admissible lub_fun le_fun P"
    and bot: "P (λ_. lub {})"
    and step: "f'.  P (U f'); le_fun (U f') (U f)   P (U (F f'))"
  shows "P (U f)"
unfolding eq inverse
apply (rule ccpo.fixp_strong_induct[OF ccpo adm])
apply (insert mono, auto simp: monotone_def fun_ord_def bot fun_lub_def)[2]
apply (rule_tac f'5="C x" in step)
apply (simp_all add: inverse eq)
done

end

subsection term(=) as order›

definition lub_singleton :: "('a set  'a)  bool"
where "lub_singleton lub  (a. lub {a} = a)"

definition the_Sup :: "'a set  'a"
where "the_Sup A = (THE a. a  A)"

lemma lub_singleton_the_Sup [cont_intro, simp]: "lub_singleton the_Sup"
by(simp add: lub_singleton_def the_Sup_def)

lemma (in ccpo) lub_singleton: "lub_singleton Sup"
by(simp add: lub_singleton_def)

lemma (in partial_function_definitions) lub_singleton [cont_intro, simp]: "lub_singleton lub"
by(rule ccpo.lub_singleton)(rule Partial_Function.ccpo[OF partial_function_definitions_axioms])

lemma preorder_eq [cont_intro, simp]:
  "class.preorder (=) (mk_less (=))"
by(unfold_locales)(simp_all add: mk_less_def)

lemma monotone_eqI [cont_intro]:
  assumes "class.preorder ord (mk_less ord)"
  shows "monotone (=) ord f"
proof -
  interpret preorder ord "mk_less ord" by fact
  show ?thesis by(simp add: monotone_def)
qed

lemma cont_eqI [cont_intro]: 
  fixes f :: "'a  'b"
  assumes "lub_singleton lub"
  shows "cont the_Sup (=) lub ord f"
proof(rule contI)
  fix Y :: "'a set"
  assume "Complete_Partial_Order.chain (=) Y" "Y  {}"
  then obtain a where "Y = {a}" by(auto simp add: chain_def)
  thus "f (the_Sup Y) = lub (f ` Y)" using assms
    by(simp add: the_Sup_def lub_singleton_def)
qed

lemma mcont_eqI [cont_intro, simp]:
  " class.preorder ord (mk_less ord); lub_singleton lub 
   mcont the_Sup (=) lub ord f"
by(simp add: mcont_def cont_eqI monotone_eqI)

subsection ‹ccpo for products›

definition prod_lub :: "('a set  'a)  ('b set  'b)  ('a × 'b) set  'a × 'b"
where "prod_lub Sup_a Sup_b Y = (Sup_a (fst ` Y), Sup_b (snd ` Y))"

lemma lub_singleton_prod_lub [cont_intro, simp]:
  " lub_singleton luba; lub_singleton lubb   lub_singleton (prod_lub luba lubb)"
by(simp add: lub_singleton_def prod_lub_def)

lemma prod_lub_empty [simp]: "prod_lub luba lubb {} = (luba {}, lubb {})"
by(simp add: prod_lub_def)

lemma preorder_rel_prodI [cont_intro, simp]:
  assumes "class.preorder orda (mk_less orda)"
  and "class.preorder ordb (mk_less ordb)"
  shows "class.preorder (rel_prod orda ordb) (mk_less (rel_prod orda ordb))"
proof -
  interpret a: preorder orda "mk_less orda" by fact
  interpret b: preorder ordb "mk_less ordb" by fact
  show ?thesis by(unfold_locales)(auto simp add: mk_less_def intro: a.order_trans b.order_trans)
qed

lemma order_rel_prodI:
  assumes a: "class.order orda (mk_less orda)"
  and b: "class.order ordb (mk_less ordb)"
  shows "class.order (rel_prod orda ordb) (mk_less (rel_prod orda ordb))"
  (is "class.order ?ord ?ord'")
proof(intro class.order.intro class.order_axioms.intro)
  interpret a: order orda "mk_less orda" by(fact a)
  interpret b: order ordb "mk_less ordb" by(fact b)
  show "class.preorder ?ord ?ord'" by(rule preorder_rel_prodI) unfold_locales

  fix x y
  assume "?ord x y" "?ord y x"
  thus "x = y" by(cases x y rule: prod.exhaust[case_product prod.exhaust]) auto
qed

lemma monotone_rel_prodI:
  assumes mono2: "a. monotone ordb ordc (λb. f (a, b))"
  and mono1: "b. monotone orda ordc (λa. f (a, b))"
  and a: "class.preorder orda (mk_less orda)"
  and b: "class.preorder ordb (mk_less ordb)"
  and c: "class.preorder ordc (mk_less ordc)"
  shows "monotone (rel_prod orda ordb) ordc f"
proof -
  interpret a: preorder orda "mk_less orda" by(rule a)
  interpret b: preorder ordb "mk_less ordb" by(rule b)
  interpret c: preorder ordc "mk_less ordc" by(rule c)
  show ?thesis using mono2 mono1
    by(auto 7 2 simp add: monotone_def intro: c.order_trans)
qed

lemma monotone_rel_prodD1:
  assumes mono: "monotone (rel_prod orda ordb) ordc f"
  and preorder: "class.preorder ordb (mk_less ordb)"
  shows "monotone orda ordc (λa. f (a, b))"
proof -
  interpret preorder ordb "mk_less ordb" by(rule preorder)
  show ?thesis using mono by(simp add: monotone_def)
qed

lemma monotone_rel_prodD2:
  assumes mono: "monotone (rel_prod orda ordb) ordc f"
  and preorder: "class.preorder orda (mk_less orda)"
  shows "monotone ordb ordc (λb. f (a, b))"
proof -
  interpret preorder orda "mk_less orda" by(rule preorder)
  show ?thesis using mono by(simp add: monotone_def)
qed

lemma monotone_case_prodI:
  " a. monotone ordb ordc (f a); b. monotone orda ordc (λa. f a b);
    class.preorder orda (mk_less orda); class.preorder ordb (mk_less ordb);
    class.preorder ordc (mk_less ordc) 
   monotone (rel_prod orda ordb) ordc (case_prod f)"
by(rule monotone_rel_prodI) simp_all

lemma monotone_case_prodD1:
  assumes mono: "monotone (rel_prod orda ordb) ordc (case_prod f)"
  and preorder: "class.preorder ordb (mk_less ordb)"
  shows "monotone orda ordc (λa. f a b)"
using monotone_rel_prodD1[OF assms] by simp

lemma monotone_case_prodD2:
  assumes mono: "monotone (rel_prod orda ordb) ordc (case_prod f)"
  and preorder: "class.preorder orda (mk_less orda)"
  shows "monotone ordb ordc (f a)"
using monotone_rel_prodD2[OF assms] by simp

context 
  fixes orda ordb ordc
  assumes a: "class.preorder orda (mk_less orda)"
  and b: "class.preorder ordb (mk_less ordb)"
  and c: "class.preorder ordc (mk_less ordc)"
begin

lemma monotone_rel_prod_iff:
  "monotone (rel_prod orda ordb) ordc f 
   (a. monotone ordb ordc (λb. f (a, b)))  
   (b. monotone orda ordc (λa. f (a, b)))"
using a b c by(blast intro: monotone_rel_prodI dest: monotone_rel_prodD1 monotone_rel_prodD2)

lemma monotone_case_prod_iff [simp]:
  "monotone (rel_prod orda ordb) ordc (case_prod f) 
   (a. monotone ordb ordc (f a))  (b. monotone orda ordc (λa. f a b))"
by(simp add: monotone_rel_prod_iff)

end

lemma monotone_case_prod_apply_iff:
  "monotone orda ordb (λx. (case_prod f x) y)  monotone orda ordb (case_prod (λa b. f a b y))"
by(simp add: monotone_def)

lemma monotone_case_prod_applyD:
  "monotone orda ordb (λx. (case_prod f x) y)
   monotone orda ordb (case_prod (λa b. f a b y))"
by(simp add: monotone_case_prod_apply_iff)

lemma monotone_case_prod_applyI:
  "monotone orda ordb (case_prod (λa b. f a b y))
   monotone orda ordb (λx. (case_prod f x) y)"
by(simp add: monotone_case_prod_apply_iff)


lemma cont_case_prod_apply_iff:
  "cont luba orda lubb ordb (λx. (case_prod f x) y)  cont luba orda lubb ordb (case_prod (λa b. f a b y))"
by(simp add: cont_def split_def)

lemma cont_case_prod_applyI:
  "cont luba orda lubb ordb (case_prod (λa b. f a b y))
   cont luba orda lubb ordb (λx. (case_prod f x) y)"
by(simp add: cont_case_prod_apply_iff)

lemma cont_case_prod_applyD:
  "cont luba orda lubb ordb (λx. (case_prod f x) y)
   cont luba orda lubb ordb (case_prod (λa b. f a b y))"
by(simp add: cont_case_prod_apply_iff)

lemma mcont_case_prod_apply_iff [simp]:
  "mcont luba orda lubb ordb (λx. (case_prod f x) y)  
   mcont luba orda lubb ordb (case_prod (λa b. f a b y))"
by(simp add: mcont_def monotone_case_prod_apply_iff cont_case_prod_apply_iff)

lemma cont_prodD1: 
  assumes cont: "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc f"
  and "class.preorder orda (mk_less orda)"
  and luba: "lub_singleton luba"
  shows "cont lubb ordb lubc ordc (λy. f (x, y))"
proof(rule contI)
  interpret preorder orda "mk_less orda" by fact

  fix Y :: "'b set"
  let ?Y = "{x} × Y"
  assume "Complete_Partial_Order.chain ordb Y" "Y  {}"
  hence "Complete_Partial_Order.chain (rel_prod orda ordb) ?Y" "?Y  {}" 
    by(simp_all add: chain_def)
  with cont have "f (prod_lub luba lubb ?Y) = lubc (f ` ?Y)" by(rule contD)
  moreover have "f ` ?Y = (λy. f (x, y)) ` Y" by auto
  ultimately show "f (x, lubb Y) = lubc ((λy. f (x, y)) ` Y)" using luba
    by(simp add: prod_lub_def Y  {} lub_singleton_def)
qed

lemma cont_prodD2: 
  assumes cont: "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc f"
  and "class.preorder ordb (mk_less ordb)"
  and lubb: "lub_singleton lubb"
  shows "cont luba orda lubc ordc (λx. f (x, y))"
proof(rule contI)
  interpret preorder ordb "mk_less ordb" by fact

  fix Y
  assume Y: "Complete_Partial_Order.chain orda Y" "Y  {}"
  let ?Y = "Y × {y}"
  have "f (luba Y, y) = f (prod_lub luba lubb ?Y)"
    using lubb by(simp add: prod_lub_def Y lub_singleton_def)
  also from Y have "Complete_Partial_Order.chain (rel_prod orda ordb) ?Y" "?Y  {}"
    by(simp_all add: chain_def)
  with cont have "f (prod_lub luba lubb ?Y) = lubc (f ` ?Y)" by(rule contD)
  also have "f ` ?Y = (λx. f (x, y)) ` Y" by auto
  finally show "f (luba Y, y) = lubc " .
qed

lemma cont_case_prodD1:
  assumes "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc (case_prod f)"
  and "class.preorder orda (mk_less orda)"
  and "lub_singleton luba"
  shows "cont lubb ordb lubc ordc (f x)"
using cont_prodD1[OF assms] by simp

lemma cont_case_prodD2:
  assumes "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc (case_prod f)"
  and "class.preorder ordb (mk_less ordb)"
  and "lub_singleton lubb"
  shows "cont luba orda lubc ordc (λx. f x y)"
using cont_prodD2[OF assms] by simp

context ccpo begin

lemma cont_prodI: 
  assumes mono: "monotone (rel_prod orda ordb) (≤) f"
  and cont1: "x. cont lubb ordb Sup (≤) (λy. f (x, y))"
  and cont2: "y. cont luba orda Sup (≤) (λx. f (x, y))"
  and "class.preorder orda (mk_less orda)"
  and "class.preorder ordb (mk_less ordb)"
  shows "cont (prod_lub luba lubb) (rel_prod orda ordb) Sup (≤) f"
proof(rule contI)
  interpret a: preorder orda "mk_less orda" by fact 
  interpret b: preorder ordb "mk_less ordb" by fact
  
  fix Y
  assume chain: "Complete_Partial_Order.chain (rel_prod orda ordb) Y"
    and "Y  {}"
  have "f (prod_lub luba lubb Y) = f (luba (fst ` Y), lubb (snd ` Y))"
    by(simp add: prod_lub_def)
  also from cont2 have "f (luba (fst ` Y), lubb (snd ` Y)) = ((λx. f (x, lubb (snd ` Y))) ` fst ` Y)"
    by(rule contD)(simp_all add: chain_rel_prodD1[OF chain] Y  {})
  also from cont1 have "x. f (x, lubb (snd ` Y)) = ((λy. f (x, y)) ` snd ` Y)"
    by(rule contD)(simp_all add: chain_rel_prodD2[OF chain] Y  {})
  hence "((λx. f (x, lubb (snd ` Y))) ` fst ` Y) = ((λx.  x) ` fst ` Y)" by simp
  also have " = ((λx. f (fst x, snd x)) ` Y)"
    unfolding image_image split_def using chain
    apply(rule diag_Sup)
    using monotoneD[OF mono]
    by(auto intro: monotoneI)
  finally show "f (prod_lub luba lubb Y) = (f ` Y)" by simp
qed

lemma cont_case_prodI:
  assumes "monotone (rel_prod orda ordb) (≤) (case_prod f)"
  and "x. cont lubb ordb Sup (≤) (λy. f x y)"
  and "y. cont luba orda Sup (≤) (λx. f x y)"
  and "class.preorder orda (mk_less orda)"
  and "class.preorder ordb (mk_less ordb)"
  shows "cont (prod_lub luba lubb) (rel_prod orda ordb) Sup (≤) (case_prod f)"
by(rule cont_prodI)(simp_all add: assms)

lemma cont_case_prod_iff:
  " monotone (rel_prod orda ordb) (≤) (case_prod f);
     class.preorder orda (mk_less orda); lub_singleton luba;
     class.preorder ordb (mk_less ordb); lub_singleton lubb 
   cont (prod_lub luba lubb) (rel_prod orda ordb) Sup (≤) (case_prod f) 
   (x. cont lubb ordb Sup (≤) (λy. f x y))  (y. cont luba orda Sup (≤) (λx. f x y))"
by(blast dest: cont_case_prodD1 cont_case_prodD2 intro: cont_case_prodI)

end

context partial_function_definitions begin

lemma mono2mono2:
  assumes f: "monotone (rel_prod ordb ordc) leq (λ(x, y). f x y)"
  and t: "monotone orda ordb (λx. t x)"
  and t': "monotone orda ordc (λx. t' x)"
  shows "monotone orda leq (λx. f (t x) (t' x))"
proof(rule monotoneI)
  fix x y
  assume "orda x y"
  hence "rel_prod ordb ordc (t x, t' x) (t y, t' y)"
    using t t' by(auto dest: monotoneD)
  from monotoneD[OF f this] show "leq (f (t x) (t' x)) (f (t y) (t' y))" by simp
qed

lemma cont_case_prodI [cont_intro]:
  " monotone (rel_prod orda ordb) leq (case_prod f);
    x. cont lubb ordb lub leq (λy. f x y);
    y. cont luba orda lub leq (λx. f x y);
    class.preorder orda (mk_less orda);
    class.preorder ordb (mk_less ordb) 
   cont (prod_lub luba lubb) (rel_prod orda ordb) lub leq (case_prod f)"
by(rule ccpo.cont_case_prodI)(rule Partial_Function.ccpo[OF partial_function_definitions_axioms])

lemma cont_case_prod_iff:
  " monotone (rel_prod orda ordb) leq (case_prod f);
     class.preorder orda (mk_less orda); lub_singleton luba;
     class.preorder ordb (mk_less ordb); lub_singleton lubb 
   cont (prod_lub luba lubb) (rel_prod orda ordb) lub leq (case_prod f) 
   (x. cont lubb ordb lub leq (λy. f x y))  (y. cont luba orda lub leq (λx. f x y))"
by(blast dest: cont_case_prodD1 cont_case_prodD2 intro: cont_case_prodI)

lemma mcont_case_prod_iff [simp]:
  " class.preorder orda (mk_less orda); lub_singleton luba;
     class.preorder ordb (mk_less ordb); lub_singleton lubb 
   mcont (prod_lub luba lubb) (rel_prod orda ordb) lub leq (case_prod f) 
   (x. mcont lubb ordb lub leq (λy. f x y))  (y. mcont luba orda lub leq (λx. f x y))"
unfolding mcont_def by(auto simp add: cont_case_prod_iff)

end

lemma mono2mono_case_prod [cont_intro]:
  assumes "x y. monotone orda ordb (λf. pair f x y)"
  shows "monotone orda ordb (λf. case_prod (pair f) x)"
by(rule monotoneI)(auto split: prod.split dest: monotoneD[OF assms])

subsection ‹Complete lattices as ccpo›

context complete_lattice begin

lemma complete_lattice_ccpo: "class.ccpo Sup (≤) (<)"
by(unfold_locales)(fast intro: Sup_upper Sup_least)+

lemma complete_lattice_ccpo': "class.ccpo Sup (≤) (mk_less (≤))"
by(unfold_locales)(auto simp add: mk_less_def intro: Sup_upper Sup_least)

lemma complete_lattice_partial_function_definitions: 
  "partial_function_definitions (≤) Sup"
by(unfold_locales)(auto intro: Sup_least Sup_upper)

lemma complete_lattice_partial_function_definitions_dual:
  "partial_function_definitions (≥) Inf"
by(unfold_locales)(auto intro: Inf_lower Inf_greatest)

lemmas [cont_intro, simp] =
  Partial_Function.ccpo[OF complete_lattice_partial_function_definitions]
  Partial_Function.ccpo[OF complete_lattice_partial_function_definitions_dual]

lemma mono2mono_inf:
  assumes f: "monotone ord (≤) (λx. f x)" 
  and g: "monotone ord (≤) (λx. g x)"
  shows "monotone ord (≤) (λx. f x  g x)"
by(auto 4 3 dest: monotoneD[OF f] monotoneD[OF g] intro: le_infI1 le_infI2 intro!: monotoneI)

lemma mcont_const [simp]: "mcont lub ord Sup (≤) (λ_. c)"
by(rule ccpo.mcont_const[OF complete_lattice_ccpo])

lemma mono2mono_sup:
  assumes f: "monotone ord (≤) (λx. f x)"
  and g: "monotone ord (≤) (λx. g x)"
  shows "monotone ord (≤) (λx. f x  g x)"
by(auto 4 3 intro!: monotoneI intro: sup.coboundedI1 sup.coboundedI2 dest: monotoneD[OF f] monotoneD[OF g])

lemma Sup_image_sup: 
  assumes "Y  {}"
  shows "((⊔) x ` Y) = x  Y"
proof(rule Sup_eqI)
  fix y
  assume "y  (⊔) x ` Y"
  then obtain z where "y = x  z" and "z  Y" by blast
  from z  Y have "z  Y" by(rule Sup_upper)
  with _ show "y  x  Y" unfolding y = x  z by(rule sup_mono) simp
next
  fix y
  assume upper: "z. z  (⊔) x ` Y  z  y"
  show "x  Y  y" unfolding Sup_insert[symmetric]
  proof(rule Sup_least)
    fix z
    assume "z  insert x Y"
    from assms obtain z' where "z'  Y" by blast
    let ?z = "if z  Y then x  z else x  z'"
    have "z  x  ?z" using z'  Y z  insert x Y by auto
    also have "  y" by(rule upper)(auto split: if_split_asm intro: z'  Y)
    finally show "z  y" .
  qed
qed

lemma mcont_sup1: "mcont Sup (≤) Sup (≤) (λy. x  y)"
by(auto 4 3 simp add: mcont_def sup.coboundedI1 sup.coboundedI2 intro!: monotoneI contI intro: Sup_image_sup[symmetric])

lemma mcont_sup2: "mcont Sup (≤) Sup (≤) (λx. x  y)"
by(subst sup_commute)(rule mcont_sup1)

lemma mcont2mcont_sup [cont_intro, simp]:
  " mcont lub ord Sup (≤) (λx. f x);
     mcont lub ord Sup (≤) (λx. g x) 
   mcont lub ord Sup (≤) (λx. f x  g x)"
by(best intro: ccpo.mcont2mcont'[OF complete_lattice_ccpo] mcont_sup1 mcont_sup2 ccpo.mcont_const[OF complete_lattice_ccpo])

end

lemmas [cont_intro] = admissible_leI[OF complete_lattice_ccpo']

context complete_distrib_lattice begin

lemma mcont_inf1: "mcont Sup (≤) Sup (≤) (λy. x  y)"
by(auto intro: monotoneI contI simp add: le_infI2 inf_Sup mcont_def)

lemma mcont_inf2: "mcont Sup (≤) Sup (≤) (λx. x  y)"
by(auto intro: monotoneI contI simp add: le_infI1 Sup_inf mcont_def)

lemma mcont2mcont_inf [cont_intro, simp]:
  " mcont lub ord Sup (≤) (λx. f x);
    mcont lub ord Sup (≤) (λx. g x) 
   mcont lub ord Sup (≤) (λx. f x  g x)"
by(best intro: ccpo.mcont2mcont'[OF complete_lattice_ccpo] mcont_inf1 mcont_inf2 ccpo.mcont_const[OF complete_lattice_ccpo])

end

interpretation lfp: partial_function_definitions "(≤) :: _ :: complete_lattice  _" Sup
by(rule complete_lattice_partial_function_definitions)

declaration Partial_Function.init "lfp" termlfp.fixp_fun termlfp.mono_body
  @{thm lfp.fixp_rule_uc} @{thm lfp.fixp_induct_uc} NONE