Theory Cfun

theory Cfun
imports Cpodef Fun_Cpo Product_Cpo
(*  Title:      HOL/HOLCF/Cfun.thy
    Author:     Franz Regensburger
    Author:     Brian Huffman
*)

section ‹The type of continuous functions›

theory Cfun
  imports Cpodef Fun_Cpo Product_Cpo
begin

default_sort cpo


subsection ‹Definition of continuous function type›

definition "cfun = {f::'a ⇒ 'b. cont f}"

cpodef ('a, 'b) cfun ("(_ →/ _)" [1, 0] 0) = "cfun :: ('a ⇒ 'b) set"
  by (auto simp: cfun_def intro: cont_const adm_cont)

type_notation (ASCII)
  cfun  (infixr "->" 0)

notation (ASCII)
  Rep_cfun  ("(_$/_)" [999,1000] 999)

notation
  Rep_cfun  ("(_⋅/_)" [999,1000] 999)


subsection ‹Syntax for continuous lambda abstraction›

syntax "_cabs" :: "[logic, logic] ⇒ logic"

parse_translation ‹
(* rewrite (_cabs x t) => (Abs_cfun (%x. t)) *)
  [Syntax_Trans.mk_binder_tr (@{syntax_const "_cabs"}, @{const_syntax Abs_cfun})];
›

print_translation ‹
  [(@{const_syntax Abs_cfun}, fn _ => fn [Abs abs] =>
      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
      in Syntax.const @{syntax_const "_cabs"} $ x $ t end)]
›  ― ‹To avoid eta-contraction of body›

text ‹Syntax for nested abstractions›

syntax (ASCII)
  "_Lambda" :: "[cargs, logic] ⇒ logic"  ("(3LAM _./ _)" [1000, 10] 10)

syntax
  "_Lambda" :: "[cargs, logic] ⇒ logic" ("(3Λ _./ _)" [1000, 10] 10)

parse_ast_translation ‹
(* rewrite (LAM x y z. t) => (_cabs x (_cabs y (_cabs z t))) *)
(* cf. Syntax.lambda_ast_tr from src/Pure/Syntax/syn_trans.ML *)
  let
    fun Lambda_ast_tr [pats, body] =
          Ast.fold_ast_p @{syntax_const "_cabs"}
            (Ast.unfold_ast @{syntax_const "_cargs"} (Ast.strip_positions pats), body)
      | Lambda_ast_tr asts = raise Ast.AST ("Lambda_ast_tr", asts);
  in [(@{syntax_const "_Lambda"}, K Lambda_ast_tr)] end;
›

print_ast_translation ‹
(* rewrite (_cabs x (_cabs y (_cabs z t))) => (LAM x y z. t) *)
(* cf. Syntax.abs_ast_tr' from src/Pure/Syntax/syn_trans.ML *)
  let
    fun cabs_ast_tr' asts =
      (case Ast.unfold_ast_p @{syntax_const "_cabs"}
          (Ast.Appl (Ast.Constant @{syntax_const "_cabs"} :: asts)) of
        ([], _) => raise Ast.AST ("cabs_ast_tr'", asts)
      | (xs, body) => Ast.Appl
          [Ast.Constant @{syntax_const "_Lambda"},
           Ast.fold_ast @{syntax_const "_cargs"} xs, body]);
  in [(@{syntax_const "_cabs"}, K cabs_ast_tr')] end
›

text ‹Dummy patterns for continuous abstraction›
translations
  "Λ _. t"  "CONST Abs_cfun (λ_. t)"


subsection ‹Continuous function space is pointed›

lemma bottom_cfun: "⊥ ∈ cfun"
  by (simp add: cfun_def inst_fun_pcpo)

instance cfun :: (cpo, discrete_cpo) discrete_cpo
  by intro_classes (simp add: below_cfun_def Rep_cfun_inject)

instance cfun :: (cpo, pcpo) pcpo
  by (rule typedef_pcpo [OF type_definition_cfun below_cfun_def bottom_cfun])

lemmas Rep_cfun_strict =
  typedef_Rep_strict [OF type_definition_cfun below_cfun_def bottom_cfun]

lemmas Abs_cfun_strict =
  typedef_Abs_strict [OF type_definition_cfun below_cfun_def bottom_cfun]

text ‹function application is strict in its first argument›

lemma Rep_cfun_strict1 [simp]: "⊥⋅x = ⊥"
  by (simp add: Rep_cfun_strict)

lemma LAM_strict [simp]: "(Λ x. ⊥) = ⊥"
  by (simp add: inst_fun_pcpo [symmetric] Abs_cfun_strict)

text ‹for compatibility with old HOLCF-Version›
lemma inst_cfun_pcpo: "⊥ = (Λ x. ⊥)"
  by simp


subsection ‹Basic properties of continuous functions›

text ‹Beta-equality for continuous functions›

lemma Abs_cfun_inverse2: "cont f ⟹ Rep_cfun (Abs_cfun f) = f"
  by (simp add: Abs_cfun_inverse cfun_def)

lemma beta_cfun: "cont f ⟹ (Λ x. f x)⋅u = f u"
  by (simp add: Abs_cfun_inverse2)


subsubsection ‹Beta-reduction simproc›

text ‹
  Given the term @{term "(Λ x. f x)⋅y"}, the procedure tries to
  construct the theorem @{term "(Λ x. f x)⋅y ≡ f y"}.  If this
  theorem cannot be completely solved by the cont2cont rules, then
  the procedure returns the ordinary conditional ‹beta_cfun›
  rule.

  The simproc does not solve any more goals that would be solved by
  using ‹beta_cfun› as a simp rule.  The advantage of the
  simproc is that it can avoid deeply-nested calls to the simplifier
  that would otherwise be caused by large continuity side conditions.

  Update: The simproc now uses rule ‹Abs_cfun_inverse2› instead
  of ‹beta_cfun›, to avoid problems with eta-contraction.
›

simproc_setup beta_cfun_proc ("Rep_cfun (Abs_cfun f)") = ‹
  fn phi => fn ctxt => fn ct =>
    let
      val f = #2 (Thm.dest_comb (#2 (Thm.dest_comb ct)));
      val [T, U] = Thm.dest_ctyp (Thm.ctyp_of_cterm f);
      val tr = Thm.instantiate' [SOME T, SOME U] [SOME f] (mk_meta_eq @{thm Abs_cfun_inverse2});
      val rules = Named_Theorems.get ctxt \<^named_theorems>‹cont2cont›;
      val tac = SOLVED' (REPEAT_ALL_NEW (match_tac ctxt (rev rules)));
    in SOME (perhaps (SINGLE (tac 1)) tr) end
›

text ‹Eta-equality for continuous functions›

lemma eta_cfun: "(Λ x. f⋅x) = f"
  by (rule Rep_cfun_inverse)

text ‹Extensionality for continuous functions›

lemma cfun_eq_iff: "f = g ⟷ (∀x. f⋅x = g⋅x)"
  by (simp add: Rep_cfun_inject [symmetric] fun_eq_iff)

lemma cfun_eqI: "(⋀x. f⋅x = g⋅x) ⟹ f = g"
  by (simp add: cfun_eq_iff)

text ‹Extensionality wrt. ordering for continuous functions›

lemma cfun_below_iff: "f ⊑ g ⟷ (∀x. f⋅x ⊑ g⋅x)"
  by (simp add: below_cfun_def fun_below_iff)

lemma cfun_belowI: "(⋀x. f⋅x ⊑ g⋅x) ⟹ f ⊑ g"
  by (simp add: cfun_below_iff)

text ‹Congruence for continuous function application›

lemma cfun_cong: "f = g ⟹ x = y ⟹ f⋅x = g⋅y"
  by simp

lemma cfun_fun_cong: "f = g ⟹ f⋅x = g⋅x"
  by simp

lemma cfun_arg_cong: "x = y ⟹ f⋅x = f⋅y"
  by simp


subsection ‹Continuity of application›

lemma cont_Rep_cfun1: "cont (λf. f⋅x)"
  by (rule cont_Rep_cfun [OF cont_id, THEN cont2cont_fun])

lemma cont_Rep_cfun2: "cont (λx. f⋅x)"
  using Rep_cfun [where x = f] by (simp add: cfun_def)

lemmas monofun_Rep_cfun = cont_Rep_cfun [THEN cont2mono]

lemmas monofun_Rep_cfun1 = cont_Rep_cfun1 [THEN cont2mono]
lemmas monofun_Rep_cfun2 = cont_Rep_cfun2 [THEN cont2mono]

text ‹contlub, cont properties of @{term Rep_cfun} in each argument›

lemma contlub_cfun_arg: "chain Y ⟹ f⋅(⨆i. Y i) = (⨆i. f⋅(Y i))"
  by (rule cont_Rep_cfun2 [THEN cont2contlubE])

lemma contlub_cfun_fun: "chain F ⟹ (⨆i. F i)⋅x = (⨆i. F i⋅x)"
  by (rule cont_Rep_cfun1 [THEN cont2contlubE])

text ‹monotonicity of application›

lemma monofun_cfun_fun: "f ⊑ g ⟹ f⋅x ⊑ g⋅x"
  by (simp add: cfun_below_iff)

lemma monofun_cfun_arg: "x ⊑ y ⟹ f⋅x ⊑ f⋅y"
  by (rule monofun_Rep_cfun2 [THEN monofunE])

lemma monofun_cfun: "f ⊑ g ⟹ x ⊑ y ⟹ f⋅x ⊑ g⋅y"
  by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg])

text ‹ch2ch - rules for the type @{typ "'a → 'b"}›

lemma chain_monofun: "chain Y ⟹ chain (λi. f⋅(Y i))"
  by (erule monofun_Rep_cfun2 [THEN ch2ch_monofun])

lemma ch2ch_Rep_cfunR: "chain Y ⟹ chain (λi. f⋅(Y i))"
  by (rule monofun_Rep_cfun2 [THEN ch2ch_monofun])

lemma ch2ch_Rep_cfunL: "chain F ⟹ chain (λi. (F i)⋅x)"
  by (rule monofun_Rep_cfun1 [THEN ch2ch_monofun])

lemma ch2ch_Rep_cfun [simp]: "chain F ⟹ chain Y ⟹ chain (λi. (F i)⋅(Y i))"
  by (simp add: chain_def monofun_cfun)

lemma ch2ch_LAM [simp]:
  "(⋀x. chain (λi. S i x)) ⟹ (⋀i. cont (λx. S i x)) ⟹ chain (λi. Λ x. S i x)"
  by (simp add: chain_def cfun_below_iff)

text ‹contlub, cont properties of @{term Rep_cfun} in both arguments›

lemma lub_APP: "chain F ⟹ chain Y ⟹ (⨆i. F i⋅(Y i)) = (⨆i. F i)⋅(⨆i. Y i)"
  by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub)

lemma lub_LAM:
  assumes "⋀x. chain (λi. F i x)"
    and "⋀i. cont (λx. F i x)"
  shows "(⨆i. Λ x. F i x) = (Λ x. ⨆i. F i x)"
  using assms by (simp add: lub_cfun lub_fun ch2ch_lambda)

lemmas lub_distribs = lub_APP lub_LAM

text ‹strictness›

lemma strictI: "f⋅x = ⊥ ⟹ f⋅⊥ = ⊥"
  apply (rule bottomI)
  apply (erule subst)
  apply (rule minimal [THEN monofun_cfun_arg])
  done

text ‹type @{typ "'a → 'b"} is chain complete›

lemma lub_cfun: "chain F ⟹ (⨆i. F i) = (Λ x. ⨆i. F i⋅x)"
  by (simp add: lub_cfun lub_fun ch2ch_lambda)


subsection ‹Continuity simplification procedure›

text ‹cont2cont lemma for @{term Rep_cfun}›

lemma cont2cont_APP [simp, cont2cont]:
  assumes f: "cont (λx. f x)"
  assumes t: "cont (λx. t x)"
  shows "cont (λx. (f x)⋅(t x))"
proof -
  from cont_Rep_cfun1 f have "cont (λx. (f x)⋅y)" for y
    by (rule cont_compose)
  with t cont_Rep_cfun2 show "cont (λx. (f x)⋅(t x))"
    by (rule cont_apply)
qed

text ‹
  Two specific lemmas for the combination of LCF and HOL terms.
  These lemmas are needed in theories that use types like @{typ "'a → 'b ⇒ 'c"}.
›

lemma cont_APP_app [simp]: "cont f ⟹ cont g ⟹ cont (λx. ((f x)⋅(g x)) s)"
  by (rule cont2cont_APP [THEN cont2cont_fun])

lemma cont_APP_app_app [simp]: "cont f ⟹ cont g ⟹ cont (λx. ((f x)⋅(g x)) s t)"
  by (rule cont_APP_app [THEN cont2cont_fun])


text ‹cont2mono Lemma for @{term "λx. LAM y. c1(x)(y)"}›

lemma cont2mono_LAM:
  "⟦⋀x. cont (λy. f x y); ⋀y. monofun (λx. f x y)⟧
    ⟹ monofun (λx. Λ y. f x y)"
  by (simp add: monofun_def cfun_below_iff)

text ‹cont2cont Lemma for @{term "λx. LAM y. f x y"}›

text ‹
  Not suitable as a cont2cont rule, because on nested lambdas
  it causes exponential blow-up in the number of subgoals.
›

lemma cont2cont_LAM:
  assumes f1: "⋀x. cont (λy. f x y)"
  assumes f2: "⋀y. cont (λx. f x y)"
  shows "cont (λx. Λ y. f x y)"
proof (rule cont_Abs_cfun)
  from f1 show "f x ∈ cfun" for x
    by (simp add: cfun_def)
  from f2 show "cont f"
    by (rule cont2cont_lambda)
qed

text ‹
  This version does work as a cont2cont rule, since it
  has only a single subgoal.
›

lemma cont2cont_LAM' [simp, cont2cont]:
  fixes f :: "'a::cpo ⇒ 'b::cpo ⇒ 'c::cpo"
  assumes f: "cont (λp. f (fst p) (snd p))"
  shows "cont (λx. Λ y. f x y)"
  using assms by (simp add: cont2cont_LAM prod_cont_iff)

lemma cont2cont_LAM_discrete [simp, cont2cont]:
  "(⋀y::'a::discrete_cpo. cont (λx. f x y)) ⟹ cont (λx. Λ y. f x y)"
  by (simp add: cont2cont_LAM)


subsection ‹Miscellaneous›

text ‹Monotonicity of @{term Abs_cfun}›

lemma monofun_LAM: "cont f ⟹ cont g ⟹ (⋀x. f x ⊑ g x) ⟹ (Λ x. f x) ⊑ (Λ x. g x)"
  by (simp add: cfun_below_iff)

text ‹some lemmata for functions with flat/chfin domain/range types›

lemma chfin_Rep_cfunR: "chain Y ⟹ ∀s. ∃n. (LUB i. Y i)⋅s = Y n⋅s"
  for Y :: "nat ⇒ 'a::cpo → 'b::chfin"
  apply (rule allI)
  apply (subst contlub_cfun_fun)
   apply assumption
  apply (fast intro!: lub_eqI chfin lub_finch2 chfin2finch ch2ch_Rep_cfunL)
  done

lemma adm_chfindom: "adm (λ(u::'a::cpo → 'b::chfin). P(u⋅s))"
  by (rule adm_subst, simp, rule adm_chfin)


subsection ‹Continuous injection-retraction pairs›

text ‹Continuous retractions are strict.›

lemma retraction_strict: "∀x. f⋅(g⋅x) = x ⟹ f⋅⊥ = ⊥"
  apply (rule bottomI)
  apply (drule_tac x="⊥" in spec)
  apply (erule subst)
  apply (rule monofun_cfun_arg)
  apply (rule minimal)
  done

lemma injection_eq: "∀x. f⋅(g⋅x) = x ⟹ (g⋅x = g⋅y) = (x = y)"
  apply (rule iffI)
   apply (drule_tac f=f in cfun_arg_cong)
   apply simp
  apply simp
  done

lemma injection_below: "∀x. f⋅(g⋅x) = x ⟹ (g⋅x ⊑ g⋅y) = (x ⊑ y)"
  apply (rule iffI)
   apply (drule_tac f=f in monofun_cfun_arg)
   apply simp
  apply (erule monofun_cfun_arg)
  done

lemma injection_defined_rev: "∀x. f⋅(g⋅x) = x ⟹ g⋅z = ⊥ ⟹ z = ⊥"
  apply (drule_tac f=f in cfun_arg_cong)
  apply (simp add: retraction_strict)
  done

lemma injection_defined: "∀x. f⋅(g⋅x) = x ⟹ z ≠ ⊥ ⟹ g⋅z ≠ ⊥"
  by (erule contrapos_nn, rule injection_defined_rev)


text ‹a result about functions with flat codomain›

lemma flat_eqI: "x ⊑ y ⟹ x ≠ ⊥ ⟹ x = y"
  for x y :: "'a::flat"
  by (drule ax_flat) simp

lemma flat_codom: "f⋅x = c ⟹ f⋅⊥ = ⊥ ∨ (∀z. f⋅z = c)"
  for c :: "'b::flat"
  apply (cases "f⋅x = ⊥")
   apply (rule disjI1)
   apply (rule bottomI)
   apply (erule_tac t="⊥" in subst)
   apply (rule minimal [THEN monofun_cfun_arg])
  apply clarify
  apply (rule_tac a = "f⋅⊥" in refl [THEN box_equals])
   apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
  apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
  done


subsection ‹Identity and composition›

definition ID :: "'a → 'a"
  where "ID = (Λ x. x)"

definition cfcomp  :: "('b → 'c) → ('a → 'b) → 'a → 'c"
  where oo_def: "cfcomp = (Λ f g x. f⋅(g⋅x))"

abbreviation cfcomp_syn :: "['b → 'c, 'a → 'b] ⇒ 'a → 'c"  (infixr "oo" 100)
  where "f oo g == cfcomp⋅f⋅g"

lemma ID1 [simp]: "ID⋅x = x"
  by (simp add: ID_def)

lemma cfcomp1: "(f oo g) = (Λ x. f⋅(g⋅x))"
  by (simp add: oo_def)

lemma cfcomp2 [simp]: "(f oo g)⋅x = f⋅(g⋅x)"
  by (simp add: cfcomp1)

lemma cfcomp_LAM: "cont g ⟹ f oo (Λ x. g x) = (Λ x. f⋅(g x))"
  by (simp add: cfcomp1)

lemma cfcomp_strict [simp]: "⊥ oo f = ⊥"
  by (simp add: cfun_eq_iff)

text ‹
  Show that interpretation of (pcpo, ‹_→_›) is a category.
  ▪ The class of objects is interpretation of syntactical class pcpo.
  ▪ The class of arrows  between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a → 'b"}.
  ▪ The identity arrow is interpretation of @{term ID}.
  ▪ The composition of f and g is interpretation of ‹oo›.
›

lemma ID2 [simp]: "f oo ID = f"
  by (rule cfun_eqI, simp)

lemma ID3 [simp]: "ID oo f = f"
  by (rule cfun_eqI) simp

lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h"
  by (rule cfun_eqI) simp


subsection ‹Strictified functions›

default_sort pcpo

definition seq :: "'a → 'b → 'b"
  where "seq = (Λ x. if x = ⊥ then ⊥ else ID)"

lemma cont2cont_if_bottom [cont2cont, simp]:
  assumes f: "cont (λx. f x)"
    and g: "cont (λx. g x)"
  shows "cont (λx. if f x = ⊥ then ⊥ else g x)"
proof (rule cont_apply [OF f])
  show "cont (λy. if y = ⊥ then ⊥ else g x)" for x
    unfolding cont_def is_lub_def is_ub_def ball_simps
    by (simp add: lub_eq_bottom_iff)
  show "cont (λx. if y = ⊥ then ⊥ else g x)" for y
    by (simp add: g)
qed

lemma seq_conv_if: "seq⋅x = (if x = ⊥ then ⊥ else ID)"
  by (simp add: seq_def)

lemma seq_simps [simp]:
  "seq⋅⊥ = ⊥"
  "seq⋅x⋅⊥ = ⊥"
  "x ≠ ⊥ ⟹ seq⋅x = ID"
  by (simp_all add: seq_conv_if)

definition strictify  :: "('a → 'b) → 'a → 'b"
  where "strictify = (Λ f x. seq⋅x⋅(f⋅x))"

lemma strictify_conv_if: "strictify⋅f⋅x = (if x = ⊥ then ⊥ else f⋅x)"
  by (simp add: strictify_def)

lemma strictify1 [simp]: "strictify⋅f⋅⊥ = ⊥"
  by (simp add: strictify_conv_if)

lemma strictify2 [simp]: "x ≠ ⊥ ⟹ strictify⋅f⋅x = f⋅x"
  by (simp add: strictify_conv_if)


subsection ‹Continuity of let-bindings›

lemma cont2cont_Let:
  assumes f: "cont (λx. f x)"
  assumes g1: "⋀y. cont (λx. g x y)"
  assumes g2: "⋀x. cont (λy. g x y)"
  shows "cont (λx. let y = f x in g x y)"
  unfolding Let_def using f g2 g1 by (rule cont_apply)

lemma cont2cont_Let' [simp, cont2cont]:
  assumes f: "cont (λx. f x)"
  assumes g: "cont (λp. g (fst p) (snd p))"
  shows "cont (λx. let y = f x in g x y)"
  using f
proof (rule cont2cont_Let)
  from g show "cont (λy. g x y)" for x
    by (simp add: prod_cont_iff)
  from g show "cont (λx. g x y)" for y
    by (simp add: prod_cont_iff)
qed

text ‹The simple version (suggested by Joachim Breitner) is needed if
  the type of the defined term is not a cpo.›

lemma cont2cont_Let_simple [simp, cont2cont]:
  assumes "⋀y. cont (λx. g x y)"
  shows "cont (λx. let y = t in g x y)"
  unfolding Let_def using assms .

end