Theory Cfun

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

header {* 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 (infixr "->" 0) = "cfun :: ('a => 'b) set"
  unfolding cfun_def by (auto intro: cont_const adm_cont)

type_notation (xsymbols)
  cfun  ("(_ ->/ _)" [1, 0] 0)

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

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

notation (HTML output)
  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
  "_Lambda" :: "[cargs, logic] => logic"  ("(3LAM _./ _)" [1000, 10] 10)

syntax (xsymbols)
  "_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)

text {* 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 @{text beta_cfun}
  rule.

  The simproc does not solve any more goals that would be solved by
  using @{text 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 @{text Abs_cfun_inverse2} instead
  of @{text 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 dest = Thm.dest_comb;
      val f = (snd o dest o snd o dest) ct;
      val [T, U] = Thm.dest_ctyp (ctyp_of_term f);
      val tr = instantiate' [SOME T, SOME U] [SOME f]
          (mk_meta_eq @{thm Abs_cfun_inverse2});
      val rules = Cont2ContData.get ctxt;
      val tac = SOLVED' (REPEAT_ALL_NEW (match_tac 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 \<sqsubseteq> g <-> (∀x. f·x \<sqsubseteq> g·x)" 
by (simp add: below_cfun_def fun_below_iff)

lemma cfun_belowI: "(!!x. f·x \<sqsubseteq> g·x) ==> f \<sqsubseteq> 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)"
apply (cut_tac x=f in Rep_cfun)
apply (simp add: cfun_def)
done

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·(\<Squnion>i. Y i) = (\<Squnion>i. f·(Y i))"
by (rule cont_Rep_cfun2 [THEN cont2contlubE])

lemma contlub_cfun_fun: "chain F ==> (\<Squnion>i. F i)·x = (\<Squnion>i. F i·x)"
by (rule cont_Rep_cfun1 [THEN cont2contlubE])

text {* monotonicity of application *}

lemma monofun_cfun_fun: "f \<sqsubseteq> g ==> f·x \<sqsubseteq> g·x"
by (simp add: cfun_below_iff)

lemma monofun_cfun_arg: "x \<sqsubseteq> y ==> f·x \<sqsubseteq> f·y"
by (rule monofun_Rep_cfun2 [THEN monofunE])

lemma monofun_cfun: "[|f \<sqsubseteq> g; x \<sqsubseteq> y|] ==> f·x \<sqsubseteq> 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|] ==> (\<Squnion>i. F i·(Y i)) = (\<Squnion>i. F i)·(\<Squnion>i. Y i)"
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub)

lemma lub_LAM:
  "[|!!x. chain (λi. F i x); !!i. cont (λx. F i x)|]
    ==> (\<Squnion>i. Λ x. F i x) = (Λ x. \<Squnion>i. F i x)"
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 ==> (\<Squnion>i. F i) = (Λ x. \<Squnion>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 -
  have 1: "!!y. cont (λx. (f x)·y)"
    using cont_Rep_cfun1 f by (rule cont_compose)
  show "cont (λx. (f x)·(t x))"
    using t cont_Rep_cfun2 1 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 "%x. 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)"
  unfolding monofun_def cfun_below_iff by simp

text {* cont2cont Lemma for @{term "%x. 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)
  fix x
  from f1 show "f x ∈ cfun" 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 \<sqsubseteq> g x|] ==> (Λ x. f x) \<sqsubseteq> (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::nat => 'a::cpo->'b::chfin)  
      ==> !s. ? n. (LUB i. Y i)$s = Y n$s"
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 \<sqsubseteq> g·y) = (x \<sqsubseteq> 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::'a::flat) \<sqsubseteq> y; x ≠ ⊥|] ==> x = y"
by (drule ax_flat, simp)

lemma flat_codom:
  "f·x = (c::'b::flat) ==> f·⊥ = ⊥ ∨ (∀z. f·z = c)"
apply (case_tac "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,@{text "_->_"}) 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 @{text "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 "!!x. cont (λy. if y = ⊥ then ⊥ else g x)"
    unfolding cont_def is_lub_def is_ub_def ball_simps
    by (simp add: lub_eq_bottom_iff)
  show "!!y. cont (λx. if y = ⊥ then ⊥ else g x)"
    by (simp add: g)
qed

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

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)"
unfolding strictify_def by simp

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)
  fix x show "cont (λy. g x y)"
    using g by (simp add: prod_cont_iff)
next
  fix y show "cont (λx. g x y)"
    using g 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