(* Title: HOL/HOLCF/Domain_Aux.thy

Author: Brian Huffman

*)

header {* Domain package support *}

theory Domain_Aux

imports Map_Functions Fixrec

begin

subsection {* Continuous isomorphisms *}

text {* A locale for continuous isomorphisms *}

locale iso =

fixes abs :: "'a -> 'b"

fixes rep :: "'b -> 'a"

assumes abs_iso [simp]: "rep·(abs·x) = x"

assumes rep_iso [simp]: "abs·(rep·y) = y"

begin

lemma swap: "iso rep abs"

by (rule iso.intro [OF rep_iso abs_iso])

lemma abs_below: "(abs·x \<sqsubseteq> abs·y) = (x \<sqsubseteq> y)"

proof

assume "abs·x \<sqsubseteq> abs·y"

then have "rep·(abs·x) \<sqsubseteq> rep·(abs·y)" by (rule monofun_cfun_arg)

then show "x \<sqsubseteq> y" by simp

next

assume "x \<sqsubseteq> y"

then show "abs·x \<sqsubseteq> abs·y" by (rule monofun_cfun_arg)

qed

lemma rep_below: "(rep·x \<sqsubseteq> rep·y) = (x \<sqsubseteq> y)"

by (rule iso.abs_below [OF swap])

lemma abs_eq: "(abs·x = abs·y) = (x = y)"

by (simp add: po_eq_conv abs_below)

lemma rep_eq: "(rep·x = rep·y) = (x = y)"

by (rule iso.abs_eq [OF swap])

lemma abs_strict: "abs·⊥ = ⊥"

proof -

have "⊥ \<sqsubseteq> rep·⊥" ..

then have "abs·⊥ \<sqsubseteq> abs·(rep·⊥)" by (rule monofun_cfun_arg)

then have "abs·⊥ \<sqsubseteq> ⊥" by simp

then show ?thesis by (rule bottomI)

qed

lemma rep_strict: "rep·⊥ = ⊥"

by (rule iso.abs_strict [OF swap])

lemma abs_defin': "abs·x = ⊥ ==> x = ⊥"

proof -

have "x = rep·(abs·x)" by simp

also assume "abs·x = ⊥"

also note rep_strict

finally show "x = ⊥" .

qed

lemma rep_defin': "rep·z = ⊥ ==> z = ⊥"

by (rule iso.abs_defin' [OF swap])

lemma abs_defined: "z ≠ ⊥ ==> abs·z ≠ ⊥"

by (erule contrapos_nn, erule abs_defin')

lemma rep_defined: "z ≠ ⊥ ==> rep·z ≠ ⊥"

by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)

lemma abs_bottom_iff: "(abs·x = ⊥) = (x = ⊥)"

by (auto elim: abs_defin' intro: abs_strict)

lemma rep_bottom_iff: "(rep·x = ⊥) = (x = ⊥)"

by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms)

lemma casedist_rule: "rep·x = ⊥ ∨ P ==> x = ⊥ ∨ P"

by (simp add: rep_bottom_iff)

lemma compact_abs_rev: "compact (abs·x) ==> compact x"

proof (unfold compact_def)

assume "adm (λy. abs·x \<notsqsubseteq> y)"

with cont_Rep_cfun2

have "adm (λy. abs·x \<notsqsubseteq> abs·y)" by (rule adm_subst)

then show "adm (λy. x \<notsqsubseteq> y)" using abs_below by simp

qed

lemma compact_rep_rev: "compact (rep·x) ==> compact x"

by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)

lemma compact_abs: "compact x ==> compact (abs·x)"

by (rule compact_rep_rev) simp

lemma compact_rep: "compact x ==> compact (rep·x)"

by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)

lemma iso_swap: "(x = abs·y) = (rep·x = y)"

proof

assume "x = abs·y"

then have "rep·x = rep·(abs·y)" by simp

then show "rep·x = y" by simp

next

assume "rep·x = y"

then have "abs·(rep·x) = abs·y" by simp

then show "x = abs·y" by simp

qed

end

subsection {* Proofs about take functions *}

text {*

This section contains lemmas that are used in a module that supports

the domain isomorphism package; the module contains proofs related

to take functions and the finiteness predicate.

*}

lemma deflation_abs_rep:

fixes abs and rep and d

assumes abs_iso: "!!x. rep·(abs·x) = x"

assumes rep_iso: "!!y. abs·(rep·y) = y"

shows "deflation d ==> deflation (abs oo d oo rep)"

by (rule ep_pair.deflation_e_d_p) (simp add: ep_pair.intro assms)

lemma deflation_chain_min:

assumes chain: "chain d"

assumes defl: "!!n. deflation (d n)"

shows "d m·(d n·x) = d (min m n)·x"

proof (rule linorder_le_cases)

assume "m ≤ n"

with chain have "d m \<sqsubseteq> d n" by (rule chain_mono)

then have "d m·(d n·x) = d m·x"

by (rule deflation_below_comp1 [OF defl defl])

moreover from `m ≤ n` have "min m n = m" by simp

ultimately show ?thesis by simp

next

assume "n ≤ m"

with chain have "d n \<sqsubseteq> d m" by (rule chain_mono)

then have "d m·(d n·x) = d n·x"

by (rule deflation_below_comp2 [OF defl defl])

moreover from `n ≤ m` have "min m n = n" by simp

ultimately show ?thesis by simp

qed

lemma lub_ID_take_lemma:

assumes "chain t" and "(\<Squnion>n. t n) = ID"

assumes "!!n. t n·x = t n·y" shows "x = y"

proof -

have "(\<Squnion>n. t n·x) = (\<Squnion>n. t n·y)"

using assms(3) by simp

then have "(\<Squnion>n. t n)·x = (\<Squnion>n. t n)·y"

using assms(1) by (simp add: lub_distribs)

then show "x = y"

using assms(2) by simp

qed

lemma lub_ID_reach:

assumes "chain t" and "(\<Squnion>n. t n) = ID"

shows "(\<Squnion>n. t n·x) = x"

using assms by (simp add: lub_distribs)

lemma lub_ID_take_induct:

assumes "chain t" and "(\<Squnion>n. t n) = ID"

assumes "adm P" and "!!n. P (t n·x)" shows "P x"

proof -

from `chain t` have "chain (λn. t n·x)" by simp

from `adm P` this `!!n. P (t n·x)` have "P (\<Squnion>n. t n·x)" by (rule admD)

with `chain t` `(\<Squnion>n. t n) = ID` show "P x" by (simp add: lub_distribs)

qed

subsection {* Finiteness *}

text {*

Let a ``decisive'' function be a deflation that maps every input to

either itself or bottom. Then if a domain's take functions are all

decisive, then all values in the domain are finite.

*}

definition

decisive :: "('a::pcpo -> 'a) => bool"

where

"decisive d <-> (∀x. d·x = x ∨ d·x = ⊥)"

lemma decisiveI: "(!!x. d·x = x ∨ d·x = ⊥) ==> decisive d"

unfolding decisive_def by simp

lemma decisive_cases:

assumes "decisive d" obtains "d·x = x" | "d·x = ⊥"

using assms unfolding decisive_def by auto

lemma decisive_bottom: "decisive ⊥"

unfolding decisive_def by simp

lemma decisive_ID: "decisive ID"

unfolding decisive_def by simp

lemma decisive_ssum_map:

assumes f: "decisive f"

assumes g: "decisive g"

shows "decisive (ssum_map·f·g)"

apply (rule decisiveI, rename_tac s)

apply (case_tac s, simp_all)

apply (rule_tac x=x in decisive_cases [OF f], simp_all)

apply (rule_tac x=y in decisive_cases [OF g], simp_all)

done

lemma decisive_sprod_map:

assumes f: "decisive f"

assumes g: "decisive g"

shows "decisive (sprod_map·f·g)"

apply (rule decisiveI, rename_tac s)

apply (case_tac s, simp_all)

apply (rule_tac x=x in decisive_cases [OF f], simp_all)

apply (rule_tac x=y in decisive_cases [OF g], simp_all)

done

lemma decisive_abs_rep:

fixes abs rep

assumes iso: "iso abs rep"

assumes d: "decisive d"

shows "decisive (abs oo d oo rep)"

apply (rule decisiveI)

apply (rule_tac x="rep·x" in decisive_cases [OF d])

apply (simp add: iso.rep_iso [OF iso])

apply (simp add: iso.abs_strict [OF iso])

done

lemma lub_ID_finite:

assumes chain: "chain d"

assumes lub: "(\<Squnion>n. d n) = ID"

assumes decisive: "!!n. decisive (d n)"

shows "∃n. d n·x = x"

proof -

have 1: "chain (λn. d n·x)" using chain by simp

have 2: "(\<Squnion>n. d n·x) = x" using chain lub by (rule lub_ID_reach)

have "∀n. d n·x = x ∨ d n·x = ⊥"

using decisive unfolding decisive_def by simp

hence "range (λn. d n·x) ⊆ {x, ⊥}"

by auto

hence "finite (range (λn. d n·x))"

by (rule finite_subset, simp)

with 1 have "finite_chain (λn. d n·x)"

by (rule finite_range_imp_finch)

then have "∃n. (\<Squnion>n. d n·x) = d n·x"

unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)

with 2 show "∃n. d n·x = x" by (auto elim: sym)

qed

lemma lub_ID_finite_take_induct:

assumes "chain d" and "(\<Squnion>n. d n) = ID" and "!!n. decisive (d n)"

shows "(!!n. P (d n·x)) ==> P x"

using lub_ID_finite [OF assms] by metis

subsection {* Proofs about constructor functions *}

text {* Lemmas for proving nchotomy rule: *}

lemma ex_one_bottom_iff:

"(∃x. P x ∧ x ≠ ⊥) = P ONE"

by simp

lemma ex_up_bottom_iff:

"(∃x. P x ∧ x ≠ ⊥) = (∃x. P (up·x))"

by (safe, case_tac x, auto)

lemma ex_sprod_bottom_iff:

"(∃y. P y ∧ y ≠ ⊥) =

(∃x y. (P (:x, y:) ∧ x ≠ ⊥) ∧ y ≠ ⊥)"

by (safe, case_tac y, auto)

lemma ex_sprod_up_bottom_iff:

"(∃y. P y ∧ y ≠ ⊥) =

(∃x y. P (:up·x, y:) ∧ y ≠ ⊥)"

by (safe, case_tac y, simp, case_tac x, auto)

lemma ex_ssum_bottom_iff:

"(∃x. P x ∧ x ≠ ⊥) =

((∃x. P (sinl·x) ∧ x ≠ ⊥) ∨

(∃x. P (sinr·x) ∧ x ≠ ⊥))"

by (safe, case_tac x, auto)

lemma exh_start: "p = ⊥ ∨ (∃x. p = x ∧ x ≠ ⊥)"

by auto

lemmas ex_bottom_iffs =

ex_ssum_bottom_iff

ex_sprod_up_bottom_iff

ex_sprod_bottom_iff

ex_up_bottom_iff

ex_one_bottom_iff

text {* Rules for turning nchotomy into exhaust: *}

lemma exh_casedist0: "[|R; R ==> P|] ==> P" (* like make_elim *)

by auto

lemma exh_casedist1: "((P ∨ Q ==> R) ==> S) ≡ ([|P ==> R; Q ==> R|] ==> S)"

by rule auto

lemma exh_casedist2: "(∃x. P x ==> Q) ≡ (!!x. P x ==> Q)"

by rule auto

lemma exh_casedist3: "(P ∧ Q ==> R) ≡ (P ==> Q ==> R)"

by rule auto

lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3

text {* Rules for proving constructor properties *}

lemmas con_strict_rules =

sinl_strict sinr_strict spair_strict1 spair_strict2

lemmas con_bottom_iff_rules =

sinl_bottom_iff sinr_bottom_iff spair_bottom_iff up_defined ONE_defined

lemmas con_below_iff_rules =

sinl_below sinr_below sinl_below_sinr sinr_below_sinl con_bottom_iff_rules

lemmas con_eq_iff_rules =

sinl_eq sinr_eq sinl_eq_sinr sinr_eq_sinl con_bottom_iff_rules

lemmas sel_strict_rules =

cfcomp2 sscase1 sfst_strict ssnd_strict fup1

lemma sel_app_extra_rules:

"sscase·ID·⊥·(sinr·x) = ⊥"

"sscase·ID·⊥·(sinl·x) = x"

"sscase·⊥·ID·(sinl·x) = ⊥"

"sscase·⊥·ID·(sinr·x) = x"

"fup·ID·(up·x) = x"

by (cases "x = ⊥", simp, simp)+

lemmas sel_app_rules =

sel_strict_rules sel_app_extra_rules

ssnd_spair sfst_spair up_defined spair_defined

lemmas sel_bottom_iff_rules =

cfcomp2 sfst_bottom_iff ssnd_bottom_iff

lemmas take_con_rules =

ssum_map_sinl' ssum_map_sinr' sprod_map_spair' u_map_up

deflation_strict deflation_ID ID1 cfcomp2

subsection {* ML setup *}

ML_file "Tools/Domain/domain_take_proofs.ML"

ML_file "Tools/cont_consts.ML"

ML_file "Tools/cont_proc.ML"

ML_file "Tools/Domain/domain_constructors.ML"

ML_file "Tools/Domain/domain_induction.ML"

setup Domain_Take_Proofs.setup

end