# 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"

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 = ⊥"

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"

lemma beta_cfun: "cont f ⟹ (Λ x. f x)⋅u = f u"

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"

text ‹Extensionality wrt. ordering for continuous functions›

lemma cfun_below_iff: "f ⊑ g ⟷ (∀x. f⋅x ⊑ g⋅x)"

lemma cfun_belowI: "(⋀x. f⋅x ⊑ g⋅x) ⟹ f ⊑ g"

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"

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))"

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

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)"

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
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)"

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)"

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

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)
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"

lemma cfcomp1: "(f oo g) = (Λ x. f⋅(g⋅x))"

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

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

lemma cfcomp_strict [simp]: "⊥ oo f = ⊥"

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
show "cont (λx. if y = ⊥ then ⊥ else g x)" for y
qed

lemma seq_conv_if: "seq⋅x = (if x = ⊥ then ⊥ else ID)"

lemma seq_simps [simp]:
"seq⋅⊥ = ⊥"
"seq⋅x⋅⊥ = ⊥"
"x ≠ ⊥ ⟹ seq⋅x = ID"

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)"

lemma strictify1 [simp]: "strictify⋅f⋅⊥ = ⊥"

lemma strictify2 [simp]: "x ≠ ⊥ ⟹ strictify⋅f⋅x = f⋅x"

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
from g show "cont (λx. g x y)" for y