--- a/src/HOLCF/Algebraic.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Algebraic.thy Wed Oct 20 21:26:51 2010 -0700
@@ -211,43 +211,4 @@
lemma cast_strict2 [simp]: "cast\<cdot>A\<cdot>\<bottom> = \<bottom>"
by (rule cast.below [THEN UU_I])
-subsection {* Deflation membership relation *}
-
-definition
- in_defl :: "udom \<Rightarrow> defl \<Rightarrow> bool" (infixl ":::" 50)
-where
- "x ::: A \<longleftrightarrow> cast\<cdot>A\<cdot>x = x"
-
-lemma adm_in_defl: "adm (\<lambda>x. x ::: A)"
-unfolding in_defl_def by simp
-
-lemma in_deflI: "cast\<cdot>A\<cdot>x = x \<Longrightarrow> x ::: A"
-unfolding in_defl_def .
-
-lemma cast_fixed: "x ::: A \<Longrightarrow> cast\<cdot>A\<cdot>x = x"
-unfolding in_defl_def .
-
-lemma cast_in_defl [simp]: "cast\<cdot>A\<cdot>x ::: A"
-unfolding in_defl_def by (rule cast.idem)
-
-lemma bottom_in_defl [simp]: "\<bottom> ::: A"
-unfolding in_defl_def by (rule cast_strict2)
-
-lemma defl_belowD: "A \<sqsubseteq> B \<Longrightarrow> x ::: A \<Longrightarrow> x ::: B"
-unfolding in_defl_def
- apply (rule deflation.belowD)
- apply (rule deflation_cast)
- apply (erule monofun_cfun_arg)
- apply assumption
-done
-
-lemma rev_defl_belowD: "x ::: A \<Longrightarrow> A \<sqsubseteq> B \<Longrightarrow> x ::: B"
-by (rule defl_belowD)
-
-lemma defl_belowI: "(\<And>x. x ::: A \<Longrightarrow> x ::: B) \<Longrightarrow> A \<sqsubseteq> B"
-apply (rule cast_below_imp_below)
-apply (rule cast.belowI)
-apply (simp add: in_defl_def)
-done
-
end
--- a/src/HOLCF/Cfun.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Cfun.thy Wed Oct 20 21:26:51 2010 -0700
@@ -534,32 +534,28 @@
default_sort pcpo
definition
- strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where
- "strictify = (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
+ strict :: "'a \<rightarrow> 'b \<rightarrow> 'b" where
+ "strict = (\<Lambda> x. if x = \<bottom> then \<bottom> else ID)"
-text {* results about strictify *}
+lemma cont_strict: "cont (\<lambda>x. if x = \<bottom> then \<bottom> else y)"
+unfolding cont_def is_lub_def is_ub_def ball_simps
+by (simp add: lub_eq_bottom_iff)
-lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)"
-by simp
+lemma strict_conv_if: "strict\<cdot>x = (if x = \<bottom> then \<bottom> else ID)"
+unfolding strict_def by (simp add: cont_strict)
-lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
-apply (rule monofunI)
-apply (auto simp add: monofun_cfun_arg)
-done
+lemma strict1 [simp]: "strict\<cdot>\<bottom> = \<bottom>"
+by (simp add: strict_conv_if)
-lemma cont_strictify2: "cont (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
-apply (rule contI2)
-apply (rule monofun_strictify2)
-apply (case_tac "(\<Squnion>i. Y i) = \<bottom>", simp)
-apply (simp add: contlub_cfun_arg del: if_image_distrib)
-apply (drule chain_UU_I_inverse2, clarify, rename_tac j)
-apply (rule lub_mono2, rule_tac x=j in exI, simp_all)
-apply (auto dest!: chain_mono_less)
-done
+lemma strict2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strict\<cdot>x = ID"
+by (simp add: strict_conv_if)
+
+ definition
+ strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where
+ "strictify = (\<Lambda> f x. strict\<cdot>x\<cdot>(f\<cdot>x))"
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
- unfolding strictify_def
- by (simp add: cont_strictify1 cont_strictify2 cont2cont_LAM)
+unfolding strictify_def by simp
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
by (simp add: strictify_conv_if)
--- a/src/HOLCF/Domain.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Domain.thy Wed Oct 20 21:26:51 2010 -0700
@@ -11,8 +11,8 @@
("Tools/cont_proc.ML")
("Tools/Domain/domain_constructors.ML")
("Tools/Domain/domain_axioms.ML")
- ("Tools/Domain/domain_theorems.ML")
- ("Tools/Domain/domain_extender.ML")
+ ("Tools/Domain/domain_induction.ML")
+ ("Tools/Domain/domain.ML")
begin
default_sort pcpo
@@ -20,61 +20,31 @@
subsection {* Casedist *}
+text {* Lemmas for proving nchotomy rule: *}
+
lemma ex_one_defined_iff:
"(\<exists>x. P x \<and> x \<noteq> \<bottom>) = P ONE"
- apply safe
- apply (rule_tac p=x in oneE)
- apply simp
- apply simp
- apply force
- done
+by simp
lemma ex_up_defined_iff:
"(\<exists>x. P x \<and> x \<noteq> \<bottom>) = (\<exists>x. P (up\<cdot>x))"
- apply safe
- apply (rule_tac p=x in upE)
- apply simp
- apply fast
- apply (force intro!: up_defined)
- done
+by (safe, case_tac x, auto)
lemma ex_sprod_defined_iff:
"(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
(\<exists>x y. (P (:x, y:) \<and> x \<noteq> \<bottom>) \<and> y \<noteq> \<bottom>)"
- apply safe
- apply (rule_tac p=y in sprodE)
- apply simp
- apply fast
- apply (force intro!: spair_defined)
- done
+by (safe, case_tac y, auto)
lemma ex_sprod_up_defined_iff:
"(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
(\<exists>x y. P (:up\<cdot>x, y:) \<and> y \<noteq> \<bottom>)"
- apply safe
- apply (rule_tac p=y in sprodE)
- apply simp
- apply (rule_tac p=x in upE)
- apply simp
- apply fast
- apply (force intro!: spair_defined)
- done
+by (safe, case_tac y, simp, case_tac x, auto)
lemma ex_ssum_defined_iff:
"(\<exists>x. P x \<and> x \<noteq> \<bottom>) =
((\<exists>x. P (sinl\<cdot>x) \<and> x \<noteq> \<bottom>) \<or>
(\<exists>x. P (sinr\<cdot>x) \<and> x \<noteq> \<bottom>))"
- apply (rule iffI)
- apply (erule exE)
- apply (erule conjE)
- apply (rule_tac p=x in ssumE)
- apply simp
- apply (rule disjI1, fast)
- apply (rule disjI2, fast)
- apply (erule disjE)
- apply force
- apply force
- done
+by (safe, case_tac x, auto)
lemma exh_start: "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)"
by auto
@@ -86,7 +56,7 @@
ex_up_defined_iff
ex_one_defined_iff
-text {* Rules for turning exh into casedist *}
+text {* Rules for turning nchotomy into exhaust: *}
lemma exh_casedist0: "\<lbrakk>R; R \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" (* like make_elim *)
by auto
@@ -103,23 +73,11 @@
lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3
-subsection {* Combinators for building copy functions *}
-
-lemmas domain_map_stricts =
- ssum_map_strict sprod_map_strict u_map_strict
-
-lemmas domain_map_simps =
- ssum_map_sinl ssum_map_sinr sprod_map_spair u_map_up
-
-
subsection {* Installing the domain package *}
lemmas con_strict_rules =
sinl_strict sinr_strict spair_strict1 spair_strict2
-lemmas con_defin_rules =
- sinl_defined sinr_defined spair_defined up_defined ONE_defined
-
lemmas con_defined_iff_rules =
sinl_defined_iff sinr_defined_iff spair_strict_iff up_defined ONE_defined
@@ -155,7 +113,7 @@
use "Tools/cont_proc.ML"
use "Tools/Domain/domain_axioms.ML"
use "Tools/Domain/domain_constructors.ML"
-use "Tools/Domain/domain_theorems.ML"
-use "Tools/Domain/domain_extender.ML"
+use "Tools/Domain/domain_induction.ML"
+use "Tools/Domain/domain.ML"
end
--- a/src/HOLCF/Fixrec.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Fixrec.thy Wed Oct 20 21:26:51 2010 -0700
@@ -115,7 +115,7 @@
definition
match_UU :: "'a \<rightarrow> 'c match \<rightarrow> 'c match"
where
- "match_UU = strictify\<cdot>(\<Lambda> x k. fail)"
+ "match_UU = (\<Lambda> x k. strict\<cdot>x\<cdot>fail)"
definition
match_Pair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match"
--- a/src/HOLCF/IsaMakefile Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/IsaMakefile Wed Oct 20 21:26:51 2010 -0700
@@ -70,12 +70,12 @@
Tools/cont_consts.ML \
Tools/cont_proc.ML \
Tools/holcf_library.ML \
- Tools/Domain/domain_extender.ML \
+ Tools/Domain/domain.ML \
Tools/Domain/domain_axioms.ML \
Tools/Domain/domain_constructors.ML \
+ Tools/Domain/domain_induction.ML \
Tools/Domain/domain_isomorphism.ML \
Tools/Domain/domain_take_proofs.ML \
- Tools/Domain/domain_theorems.ML \
Tools/fixrec.ML \
Tools/pcpodef.ML \
Tools/repdef.ML \
--- a/src/HOLCF/One.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/One.thy Wed Oct 20 21:26:51 2010 -0700
@@ -54,7 +54,7 @@
definition
one_when :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a" where
- "one_when = (\<Lambda> a. strictify\<cdot>(\<Lambda> _. a))"
+ "one_when = (\<Lambda> a x. strict\<cdot>x\<cdot>a)"
translations
"case x of XCONST ONE \<Rightarrow> t" == "CONST one_when\<cdot>t\<cdot>x"
--- a/src/HOLCF/Pcpo.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Pcpo.thy Wed Oct 20 21:26:51 2010 -0700
@@ -223,18 +223,14 @@
lemma UU_I: "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>"
by (subst eq_UU_iff)
+lemma lub_eq_bottom_iff: "chain Y \<Longrightarrow> (\<Squnion>i. Y i) = \<bottom> \<longleftrightarrow> (\<forall>i. Y i = \<bottom>)"
+by (simp only: eq_UU_iff lub_below_iff)
+
lemma chain_UU_I: "\<lbrakk>chain Y; (\<Squnion>i. Y i) = \<bottom>\<rbrakk> \<Longrightarrow> \<forall>i. Y i = \<bottom>"
-apply (rule allI)
-apply (rule UU_I)
-apply (erule subst)
-apply (erule is_ub_thelub)
-done
+by (simp add: lub_eq_bottom_iff)
lemma chain_UU_I_inverse: "\<forall>i::nat. Y i = \<bottom> \<Longrightarrow> (\<Squnion>i. Y i) = \<bottom>"
-apply (rule lub_chain_maxelem)
-apply (erule spec)
-apply simp
-done
+by simp
lemma chain_UU_I_inverse2: "(\<Squnion>i. Y i) \<noteq> \<bottom> \<Longrightarrow> \<exists>i::nat. Y i \<noteq> \<bottom>"
by (blast intro: chain_UU_I_inverse)
--- a/src/HOLCF/Pcpodef.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Pcpodef.thy Wed Oct 20 21:26:51 2010 -0700
@@ -57,13 +57,10 @@
subsection {* Proving a subtype is chain-finite *}
-lemma monofun_Rep:
+lemma ch2ch_Rep:
assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
- shows "monofun Rep"
-by (rule monofunI, unfold below)
-
-lemmas ch2ch_Rep = ch2ch_monofun [OF monofun_Rep]
-lemmas ub2ub_Rep = ub2ub_monofun [OF monofun_Rep]
+ shows "chain S \<Longrightarrow> chain (\<lambda>i. Rep (S i))"
+unfolding chain_def below .
theorem typedef_chfin:
fixes Abs :: "'a::chfin \<Rightarrow> 'b::po"
@@ -87,6 +84,11 @@
admissible predicate.
*}
+lemma typedef_is_lubI:
+ assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+ shows "range (\<lambda>i. Rep (S i)) <<| Rep x \<Longrightarrow> range S <<| x"
+unfolding is_lub_def is_ub_def below by simp
+
lemma Abs_inverse_lub_Rep:
fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
assumes type: "type_definition Rep Abs A"
@@ -104,15 +106,15 @@
and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
and adm: "adm (\<lambda>x. x \<in> A)"
shows "chain S \<Longrightarrow> range S <<| Abs (\<Squnion>i. Rep (S i))"
- apply (frule ch2ch_Rep [OF below])
- apply (rule is_lubI)
- apply (rule ub_rangeI)
- apply (simp only: below Abs_inverse_lub_Rep [OF type below adm])
- apply (erule is_ub_thelub)
- apply (simp only: below Abs_inverse_lub_Rep [OF type below adm])
- apply (erule is_lub_thelub)
- apply (erule ub2ub_Rep [OF below])
-done
+proof -
+ assume S: "chain S"
+ hence "chain (\<lambda>i. Rep (S i))" by (rule ch2ch_Rep [OF below])
+ hence "range (\<lambda>i. Rep (S i)) <<| (\<Squnion>i. Rep (S i))" by (rule cpo_lubI)
+ hence "range (\<lambda>i. Rep (S i)) <<| Rep (Abs (\<Squnion>i. Rep (S i)))"
+ by (simp only: Abs_inverse_lub_Rep [OF type below adm S])
+ thus "range S <<| Abs (\<Squnion>i. Rep (S i))"
+ by (rule typedef_is_lubI [OF below])
+qed
lemmas typedef_thelub = typedef_lub [THEN thelubI, standard]
@@ -152,18 +154,6 @@
composing it with another continuous function.
*}
-theorem typedef_is_lubI:
- assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
- shows "range (\<lambda>i. Rep (S i)) <<| Rep x \<Longrightarrow> range S <<| x"
- apply (rule is_lubI)
- apply (rule ub_rangeI)
- apply (subst below)
- apply (erule is_ub_lub)
- apply (subst below)
- apply (erule is_lub_lub)
- apply (erule ub2ub_Rep [OF below])
-done
-
theorem typedef_cont_Abs:
fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
fixes f :: "'c::cpo \<Rightarrow> 'a::cpo"
--- a/src/HOLCF/Representable.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Representable.thy Wed Oct 20 21:26:51 2010 -0700
@@ -18,80 +18,25 @@
lemma emb_prj: "emb\<cdot>((prj\<cdot>x)::'a) = cast\<cdot>DEFL('a)\<cdot>x"
by (simp add: cast_DEFL)
-lemma in_DEFL_iff:
- "x ::: DEFL('a) \<longleftrightarrow> emb\<cdot>((prj\<cdot>x)::'a) = x"
-by (simp add: in_defl_def cast_DEFL)
-
-lemma prj_inverse:
- "x ::: DEFL('a) \<Longrightarrow> emb\<cdot>((prj\<cdot>x)::'a) = x"
-by (simp only: in_DEFL_iff)
-
-lemma emb_in_DEFL [simp]:
- "emb\<cdot>(x::'a) ::: DEFL('a)"
-by (simp add: in_DEFL_iff)
-
-subsection {* Coerce operator *}
-
-definition coerce :: "'a \<rightarrow> 'b"
-where "coerce = prj oo emb"
-
-lemma beta_coerce: "coerce\<cdot>x = prj\<cdot>(emb\<cdot>x)"
-by (simp add: coerce_def)
-
-lemma prj_emb: "prj\<cdot>(emb\<cdot>x) = coerce\<cdot>x"
-by (simp add: coerce_def)
-
-lemma coerce_strict [simp]: "coerce\<cdot>\<bottom> = \<bottom>"
-by (simp add: coerce_def)
-
-lemma coerce_eq_ID [simp]: "(coerce :: 'a \<rightarrow> 'a) = ID"
-by (rule cfun_eqI, simp add: beta_coerce)
-
-lemma emb_coerce:
- "DEFL('a) \<sqsubseteq> DEFL('b)
- \<Longrightarrow> emb\<cdot>((coerce::'a \<rightarrow> 'b)\<cdot>x) = emb\<cdot>x"
- apply (simp add: beta_coerce)
- apply (rule prj_inverse)
- apply (erule defl_belowD)
- apply (rule emb_in_DEFL)
+lemma emb_prj_emb:
+ fixes x :: "'a"
+ assumes "DEFL('a) \<sqsubseteq> DEFL('b)"
+ shows "emb\<cdot>(prj\<cdot>(emb\<cdot>x) :: 'b) = emb\<cdot>x"
+unfolding emb_prj
+apply (rule cast.belowD)
+apply (rule monofun_cfun_arg [OF assms])
+apply (simp add: cast_DEFL)
done
-lemma coerce_prj:
- "DEFL('a) \<sqsubseteq> DEFL('b)
- \<Longrightarrow> (coerce::'b \<rightarrow> 'a)\<cdot>(prj\<cdot>x) = prj\<cdot>x"
- apply (simp add: coerce_def)
+lemma prj_emb_prj:
+ assumes "DEFL('a) \<sqsubseteq> DEFL('b)"
+ shows "prj\<cdot>(emb\<cdot>(prj\<cdot>x :: 'b)) = (prj\<cdot>x :: 'a)"
apply (rule emb_eq_iff [THEN iffD1])
apply (simp only: emb_prj)
apply (rule deflation_below_comp1)
apply (rule deflation_cast)
apply (rule deflation_cast)
- apply (erule monofun_cfun_arg)
-done
-
-lemma coerce_coerce [simp]:
- "DEFL('a) \<sqsubseteq> DEFL('b)
- \<Longrightarrow> coerce\<cdot>((coerce::'a \<rightarrow> 'b)\<cdot>x) = coerce\<cdot>x"
-by (simp add: beta_coerce prj_inverse defl_belowD)
-
-lemma coerce_inverse:
- "emb\<cdot>(x::'a) ::: DEFL('b) \<Longrightarrow> coerce\<cdot>(coerce\<cdot>x :: 'b) = x"
-by (simp only: beta_coerce prj_inverse emb_inverse)
-
-lemma coerce_type:
- "DEFL('a) \<sqsubseteq> DEFL('b)
- \<Longrightarrow> emb\<cdot>((coerce::'a \<rightarrow> 'b)\<cdot>x) ::: DEFL('a)"
-by (simp add: beta_coerce prj_inverse defl_belowD)
-
-lemma ep_pair_coerce:
- "DEFL('a) \<sqsubseteq> DEFL('b)
- \<Longrightarrow> ep_pair (coerce::'a \<rightarrow> 'b) (coerce::'b \<rightarrow> 'a)"
- apply (rule ep_pair.intro)
- apply simp
- apply (simp only: beta_coerce)
- apply (rule below_trans)
- apply (rule monofun_cfun_arg)
- apply (rule emb_prj_below)
- apply simp
+ apply (rule monofun_cfun_arg [OF assms])
done
text {* Isomorphism lemmas used internally by the domain package: *}
@@ -99,67 +44,79 @@
lemma domain_abs_iso:
fixes abs and rep
assumes DEFL: "DEFL('b) = DEFL('a)"
- assumes abs_def: "abs \<equiv> (coerce :: 'a \<rightarrow> 'b)"
- assumes rep_def: "rep \<equiv> (coerce :: 'b \<rightarrow> 'a)"
+ assumes abs_def: "(abs :: 'a \<rightarrow> 'b) \<equiv> prj oo emb"
+ assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
shows "rep\<cdot>(abs\<cdot>x) = x"
-unfolding abs_def rep_def by (simp add: DEFL)
+unfolding abs_def rep_def
+by (simp add: emb_prj_emb DEFL)
lemma domain_rep_iso:
fixes abs and rep
assumes DEFL: "DEFL('b) = DEFL('a)"
- assumes abs_def: "abs \<equiv> (coerce :: 'a \<rightarrow> 'b)"
- assumes rep_def: "rep \<equiv> (coerce :: 'b \<rightarrow> 'a)"
+ assumes abs_def: "(abs :: 'a \<rightarrow> 'b) \<equiv> prj oo emb"
+ assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
shows "abs\<cdot>(rep\<cdot>x) = x"
-unfolding abs_def rep_def by (simp add: DEFL [symmetric])
+unfolding abs_def rep_def
+by (simp add: emb_prj_emb DEFL)
+
+subsection {* Deflations as sets *}
+
+definition defl_set :: "defl \<Rightarrow> udom set"
+where "defl_set A = {x. cast\<cdot>A\<cdot>x = x}"
+
+lemma adm_defl_set: "adm (\<lambda>x. x \<in> defl_set A)"
+unfolding defl_set_def by simp
+lemma defl_set_bottom: "\<bottom> \<in> defl_set A"
+unfolding defl_set_def by simp
+
+lemma defl_set_cast [simp]: "cast\<cdot>A\<cdot>x \<in> defl_set A"
+unfolding defl_set_def by simp
+
+lemma defl_set_subset_iff: "defl_set A \<subseteq> defl_set B \<longleftrightarrow> A \<sqsubseteq> B"
+apply (simp add: defl_set_def subset_eq cast_below_cast [symmetric])
+apply (auto simp add: cast.belowI cast.belowD)
+done
subsection {* Proving a subtype is representable *}
text {*
- Temporarily relax type constraints for @{term emb}, and @{term prj}.
+ Temporarily relax type constraints for @{term emb} and @{term prj}.
*}
-setup {* Sign.add_const_constraint
- (@{const_name defl}, SOME @{typ "'a::pcpo itself \<Rightarrow> defl"}) *}
-
-setup {* Sign.add_const_constraint
- (@{const_name emb}, SOME @{typ "'a::pcpo \<rightarrow> udom"}) *}
-
-setup {* Sign.add_const_constraint
- (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::pcpo"}) *}
+setup {*
+ fold Sign.add_const_constraint
+ [ (@{const_name defl}, SOME @{typ "'a::pcpo itself \<Rightarrow> defl"})
+ , (@{const_name emb}, SOME @{typ "'a::pcpo \<rightarrow> udom"})
+ , (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::pcpo"}) ]
+*}
lemma typedef_rep_class:
fixes Rep :: "'a::pcpo \<Rightarrow> udom"
fixes Abs :: "udom \<Rightarrow> 'a::pcpo"
fixes t :: defl
- assumes type: "type_definition Rep Abs {x. x ::: t}"
+ assumes type: "type_definition Rep Abs (defl_set t)"
assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
assumes emb: "emb \<equiv> (\<Lambda> x. Rep x)"
assumes prj: "prj \<equiv> (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
assumes defl: "defl \<equiv> (\<lambda> a::'a itself. t)"
shows "OFCLASS('a, bifinite_class)"
proof
- have adm: "adm (\<lambda>x. x \<in> {x. x ::: t})"
- by (simp add: adm_in_defl)
have emb_beta: "\<And>x. emb\<cdot>x = Rep x"
unfolding emb
apply (rule beta_cfun)
- apply (rule typedef_cont_Rep [OF type below adm])
+ apply (rule typedef_cont_Rep [OF type below adm_defl_set])
done
have prj_beta: "\<And>y. prj\<cdot>y = Abs (cast\<cdot>t\<cdot>y)"
unfolding prj
apply (rule beta_cfun)
- apply (rule typedef_cont_Abs [OF type below adm])
+ apply (rule typedef_cont_Abs [OF type below adm_defl_set])
apply simp_all
done
- have emb_in_defl: "\<And>x::'a. emb\<cdot>x ::: t"
+ have prj_emb: "\<And>x::'a. prj\<cdot>(emb\<cdot>x) = x"
using type_definition.Rep [OF type]
- by (simp add: emb_beta)
- have prj_emb: "\<And>x::'a. prj\<cdot>(emb\<cdot>x) = x"
- unfolding prj_beta
- apply (simp add: cast_fixed [OF emb_in_defl])
- apply (simp add: emb_beta type_definition.Rep_inverse [OF type])
- done
+ unfolding prj_beta emb_beta defl_set_def
+ by (simp add: type_definition.Rep_inverse [OF type])
have emb_prj: "\<And>y. emb\<cdot>(prj\<cdot>y :: 'a) = cast\<cdot>t\<cdot>y"
unfolding prj_beta emb_beta
by (simp add: type_definition.Abs_inverse [OF type])
@@ -177,19 +134,14 @@
shows "DEFL('a::pcpo) = t"
unfolding assms ..
-text {* Restore original typing constraints *}
-
-setup {* Sign.add_const_constraint
- (@{const_name defl}, SOME @{typ "'a::bifinite itself \<Rightarrow> defl"}) *}
+text {* Restore original typing constraints. *}
-setup {* Sign.add_const_constraint
- (@{const_name emb}, SOME @{typ "'a::bifinite \<rightarrow> udom"}) *}
-
-setup {* Sign.add_const_constraint
- (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::bifinite"}) *}
-
-lemma adm_mem_Collect_in_defl: "adm (\<lambda>x. x \<in> {x. x ::: A})"
-unfolding mem_Collect_eq by (rule adm_in_defl)
+setup {*
+ fold Sign.add_const_constraint
+ [ (@{const_name defl}, SOME @{typ "'a::bifinite itself \<Rightarrow> defl"})
+ , (@{const_name emb}, SOME @{typ "'a::bifinite \<rightarrow> udom"})
+ , (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::bifinite"}) ]
+*}
use "Tools/repdef.ML"
@@ -258,22 +210,14 @@
apply (simp add: assms)
done
-lemma isodefl_coerce:
- fixes d :: "'a \<rightarrow> 'a"
- assumes DEFL: "DEFL('b) = DEFL('a)"
- shows "isodefl d t \<Longrightarrow> isodefl (coerce oo d oo coerce :: 'b \<rightarrow> 'b) t"
-unfolding isodefl_def
-apply (simp add: cfun_eq_iff)
-apply (simp add: emb_coerce coerce_prj DEFL)
-done
-
lemma isodefl_abs_rep:
fixes abs and rep and d
assumes DEFL: "DEFL('b) = DEFL('a)"
- assumes abs_def: "abs \<equiv> (coerce :: 'a \<rightarrow> 'b)"
- assumes rep_def: "rep \<equiv> (coerce :: 'b \<rightarrow> 'a)"
+ assumes abs_def: "(abs :: 'a \<rightarrow> 'b) \<equiv> prj oo emb"
+ assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
shows "isodefl d t \<Longrightarrow> isodefl (abs oo d oo rep) t"
-unfolding abs_def rep_def using DEFL by (rule isodefl_coerce)
+unfolding isodefl_def
+by (simp add: cfun_eq_iff assms prj_emb_prj emb_prj_emb)
lemma isodefl_cfun:
"isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
--- a/src/HOLCF/Sprod.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Sprod.thy Wed Oct 20 21:26:51 2010 -0700
@@ -27,9 +27,8 @@
type_notation (HTML output)
sprod ("(_ \<otimes>/ _)" [21,20] 20)
-lemma spair_lemma:
- "(strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a) \<in> Sprod"
-by (simp add: Sprod_def strictify_conv_if)
+lemma spair_lemma: "(strict\<cdot>b\<cdot>a, strict\<cdot>a\<cdot>b) \<in> Sprod"
+by (simp add: Sprod_def strict_conv_if)
subsection {* Definitions of constants *}
@@ -43,12 +42,11 @@
definition
spair :: "'a \<rightarrow> 'b \<rightarrow> ('a ** 'b)" where
- "spair = (\<Lambda> a b. Abs_Sprod
- (strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a))"
+ "spair = (\<Lambda> a b. Abs_Sprod (strict\<cdot>b\<cdot>a, strict\<cdot>a\<cdot>b))"
definition
ssplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a ** 'b) \<rightarrow> 'c" where
- "ssplit = (\<Lambda> f. strictify\<cdot>(\<Lambda> p. f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p)))"
+ "ssplit = (\<Lambda> f p. strict\<cdot>p\<cdot>(f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p)))"
syntax
"_stuple" :: "['a, args] => 'a ** 'b" ("(1'(:_,/ _:'))")
@@ -62,7 +60,7 @@
subsection {* Case analysis *}
lemma Rep_Sprod_spair:
- "Rep_Sprod (:a, b:) = (strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a)"
+ "Rep_Sprod (:a, b:) = (strict\<cdot>b\<cdot>a, strict\<cdot>a\<cdot>b)"
unfolding spair_def
by (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma)
@@ -76,7 +74,7 @@
apply (simp add: Rep_Sprod_simps Pair_fst_snd_eq)
apply (simp add: Sprod_def)
apply (erule disjE, simp)
-apply (simp add: strictify_conv_if)
+apply (simp add: strict_conv_if)
apply fast
done
@@ -91,22 +89,22 @@
subsection {* Properties of \emph{spair} *}
lemma spair_strict1 [simp]: "(:\<bottom>, y:) = \<bottom>"
-by (simp add: Rep_Sprod_simps strictify_conv_if)
+by (simp add: Rep_Sprod_simps strict_conv_if)
lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>"
-by (simp add: Rep_Sprod_simps strictify_conv_if)
+by (simp add: Rep_Sprod_simps strict_conv_if)
lemma spair_strict_iff [simp]: "((:x, y:) = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
-by (simp add: Rep_Sprod_simps strictify_conv_if)
+by (simp add: Rep_Sprod_simps strict_conv_if)
lemma spair_below_iff:
"((:a, b:) \<sqsubseteq> (:c, d:)) = (a = \<bottom> \<or> b = \<bottom> \<or> (a \<sqsubseteq> c \<and> b \<sqsubseteq> d))"
-by (simp add: Rep_Sprod_simps strictify_conv_if)
+by (simp add: Rep_Sprod_simps strict_conv_if)
lemma spair_eq_iff:
"((:a, b:) = (:c, d:)) =
(a = c \<and> b = d \<or> (a = \<bottom> \<or> b = \<bottom>) \<and> (c = \<bottom> \<or> d = \<bottom>))"
-by (simp add: Rep_Sprod_simps strictify_conv_if)
+by (simp add: Rep_Sprod_simps strict_conv_if)
lemma spair_strict: "x = \<bottom> \<or> y = \<bottom> \<Longrightarrow> (:x, y:) = \<bottom>"
by simp
@@ -197,7 +195,7 @@
by (rule compactI, simp add: ssnd_below_iff)
lemma compact_spair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (:x, y:)"
-by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if)
+by (rule compact_Sprod, simp add: Rep_Sprod_spair strict_conv_if)
lemma compact_spair_iff:
"compact (:x, y:) = (x = \<bottom> \<or> y = \<bottom> \<or> (compact x \<and> compact y))"
--- a/src/HOLCF/Ssum.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Ssum.thy Wed Oct 20 21:26:51 2010 -0700
@@ -34,28 +34,28 @@
definition
sinl :: "'a \<rightarrow> ('a ++ 'b)" where
- "sinl = (\<Lambda> a. Abs_Ssum (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>))"
+ "sinl = (\<Lambda> a. Abs_Ssum (strict\<cdot>a\<cdot>TT, a, \<bottom>))"
definition
sinr :: "'b \<rightarrow> ('a ++ 'b)" where
- "sinr = (\<Lambda> b. Abs_Ssum (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b))"
+ "sinr = (\<Lambda> b. Abs_Ssum (strict\<cdot>b\<cdot>FF, \<bottom>, b))"
-lemma sinl_Ssum: "(strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>) \<in> Ssum"
-by (simp add: Ssum_def strictify_conv_if)
+lemma sinl_Ssum: "(strict\<cdot>a\<cdot>TT, a, \<bottom>) \<in> Ssum"
+by (simp add: Ssum_def strict_conv_if)
-lemma sinr_Ssum: "(strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b) \<in> Ssum"
-by (simp add: Ssum_def strictify_conv_if)
+lemma sinr_Ssum: "(strict\<cdot>b\<cdot>FF, \<bottom>, b) \<in> Ssum"
+by (simp add: Ssum_def strict_conv_if)
-lemma sinl_Abs_Ssum: "sinl\<cdot>a = Abs_Ssum (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>)"
+lemma sinl_Abs_Ssum: "sinl\<cdot>a = Abs_Ssum (strict\<cdot>a\<cdot>TT, a, \<bottom>)"
by (unfold sinl_def, simp add: cont_Abs_Ssum sinl_Ssum)
-lemma sinr_Abs_Ssum: "sinr\<cdot>b = Abs_Ssum (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b)"
+lemma sinr_Abs_Ssum: "sinr\<cdot>b = Abs_Ssum (strict\<cdot>b\<cdot>FF, \<bottom>, b)"
by (unfold sinr_def, simp add: cont_Abs_Ssum sinr_Ssum)
-lemma Rep_Ssum_sinl: "Rep_Ssum (sinl\<cdot>a) = (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>)"
+lemma Rep_Ssum_sinl: "Rep_Ssum (sinl\<cdot>a) = (strict\<cdot>a\<cdot>TT, a, \<bottom>)"
by (simp add: sinl_Abs_Ssum Abs_Ssum_inverse sinl_Ssum)
-lemma Rep_Ssum_sinr: "Rep_Ssum (sinr\<cdot>b) = (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b)"
+lemma Rep_Ssum_sinr: "Rep_Ssum (sinr\<cdot>b) = (strict\<cdot>b\<cdot>FF, \<bottom>, b)"
by (simp add: sinr_Abs_Ssum Abs_Ssum_inverse sinr_Ssum)
subsection {* Properties of \emph{sinl} and \emph{sinr} *}
@@ -63,16 +63,16 @@
text {* Ordering *}
lemma sinl_below [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)"
-by (simp add: below_Ssum_def Rep_Ssum_sinl strictify_conv_if)
+by (simp add: below_Ssum_def Rep_Ssum_sinl strict_conv_if)
lemma sinr_below [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)"
-by (simp add: below_Ssum_def Rep_Ssum_sinr strictify_conv_if)
+by (simp add: below_Ssum_def Rep_Ssum_sinr strict_conv_if)
lemma sinl_below_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)"
-by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if)
+by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strict_conv_if)
lemma sinr_below_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)"
-by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if)
+by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strict_conv_if)
text {* Equality *}
@@ -117,10 +117,10 @@
text {* Compactness *}
lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)"
-by (rule compact_Ssum, simp add: Rep_Ssum_sinl strictify_conv_if)
+by (rule compact_Ssum, simp add: Rep_Ssum_sinl strict_conv_if)
lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)"
-by (rule compact_Ssum, simp add: Rep_Ssum_sinr strictify_conv_if)
+by (rule compact_Ssum, simp add: Rep_Ssum_sinr strict_conv_if)
lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x"
unfolding compact_def
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Tools/Domain/domain.ML Wed Oct 20 21:26:51 2010 -0700
@@ -0,0 +1,274 @@
+(* Title: HOLCF/Tools/Domain/domain.ML
+ Author: David von Oheimb
+ Author: Brian Huffman
+
+Theory extender for domain command, including theory syntax.
+*)
+
+signature DOMAIN =
+sig
+ val add_domain_cmd:
+ binding ->
+ ((string * string option) list * binding * mixfix *
+ (binding * (bool * binding option * string) list * mixfix) list) list
+ -> theory -> theory
+
+ val add_domain:
+ binding ->
+ ((string * string option) list * binding * mixfix *
+ (binding * (bool * binding option * typ) list * mixfix) list) list
+ -> theory -> theory
+
+ val add_new_domain_cmd:
+ binding ->
+ ((string * string option) list * binding * mixfix *
+ (binding * (bool * binding option * string) list * mixfix) list) list
+ -> theory -> theory
+
+ val add_new_domain:
+ binding ->
+ ((string * string option) list * binding * mixfix *
+ (binding * (bool * binding option * typ) list * mixfix) list) list
+ -> theory -> theory
+end;
+
+structure Domain :> DOMAIN =
+struct
+
+open HOLCF_Library;
+
+fun first (x,_,_) = x;
+fun second (_,x,_) = x;
+fun third (_,_,x) = x;
+
+(* ----- calls for building new thy and thms -------------------------------- *)
+
+type info =
+ Domain_Take_Proofs.iso_info list * Domain_Take_Proofs.take_induct_info;
+
+fun gen_add_domain
+ (prep_typ : theory -> (string * sort) list -> 'a -> typ)
+ (add_isos : (binding * mixfix * (typ * typ)) list -> theory -> info * theory)
+ (arg_sort : bool -> sort)
+ (comp_dbind : binding)
+ (raw_specs : ((string * string option) list * binding * mixfix *
+ (binding * (bool * binding option * 'a) list * mixfix) list) list)
+ (thy : theory) =
+ let
+ val dtnvs : (binding * typ list * mixfix) list =
+ let
+ fun readS (SOME s) = Syntax.read_sort_global thy s
+ | readS NONE = Sign.defaultS thy;
+ fun readTFree (a, s) = TFree (a, readS s);
+ in
+ map (fn (vs, dbind, mx, _) =>
+ (dbind, map readTFree vs, mx)) raw_specs
+ end;
+
+ fun thy_type (dbind, tvars, mx) = (dbind, length tvars, mx);
+ fun thy_arity (dbind, tvars, mx) =
+ (Sign.full_name thy dbind, map (snd o dest_TFree) tvars, arg_sort false);
+
+ (* this theory is used just for parsing and error checking *)
+ val tmp_thy = thy
+ |> Theory.copy
+ |> Sign.add_types (map thy_type dtnvs)
+ |> fold (AxClass.axiomatize_arity o thy_arity) dtnvs;
+
+ val dbinds : binding list =
+ map (fn (_,dbind,_,_) => dbind) raw_specs;
+ val raw_rhss :
+ (binding * (bool * binding option * 'a) list * mixfix) list list =
+ map (fn (_,_,_,cons) => cons) raw_specs;
+ val dtnvs' : (string * typ list) list =
+ map (fn (dbind, vs, mx) => (Sign.full_name thy dbind, vs)) dtnvs;
+
+ val all_cons = map (Binding.name_of o first) (flat raw_rhss);
+ val test_dupl_cons =
+ case duplicates (op =) all_cons of
+ [] => false | dups => error ("Duplicate constructors: "
+ ^ commas_quote dups);
+ val all_sels =
+ (map Binding.name_of o map_filter second o maps second) (flat raw_rhss);
+ val test_dupl_sels =
+ case duplicates (op =) all_sels of
+ [] => false | dups => error("Duplicate selectors: "^commas_quote dups);
+
+ fun test_dupl_tvars s =
+ case duplicates (op =) (map(fst o dest_TFree)s) of
+ [] => false | dups => error("Duplicate type arguments: "
+ ^commas_quote dups);
+ val test_dupl_tvars' = exists test_dupl_tvars (map snd dtnvs');
+
+ val sorts : (string * sort) list =
+ let val all_sorts = map (map dest_TFree o snd) dtnvs';
+ in
+ case distinct (eq_set (op =)) all_sorts of
+ [sorts] => sorts
+ | _ => error "Mutually recursive domains must have same type parameters"
+ end;
+
+ (* a lazy argument may have an unpointed type *)
+ (* unless the argument has a selector function *)
+ fun check_pcpo (lazy, sel, T) =
+ let val sort = arg_sort (lazy andalso is_none sel) in
+ if Sign.of_sort tmp_thy (T, sort) then ()
+ else error ("Constructor argument type is not of sort " ^
+ Syntax.string_of_sort_global tmp_thy sort ^ ": " ^
+ Syntax.string_of_typ_global tmp_thy T)
+ end;
+
+ (* test for free type variables, illegal sort constraints on rhs,
+ non-pcpo-types and invalid use of recursive type;
+ replace sorts in type variables on rhs *)
+ val map_tab = Domain_Take_Proofs.get_map_tab thy;
+ fun check_rec rec_ok (T as TFree (v,_)) =
+ if AList.defined (op =) sorts v then T
+ else error ("Free type variable " ^ quote v ^ " on rhs.")
+ | check_rec rec_ok (T as Type (s, Ts)) =
+ (case AList.lookup (op =) dtnvs' s of
+ NONE =>
+ let val rec_ok' = rec_ok andalso Symtab.defined map_tab s;
+ in Type (s, map (check_rec rec_ok') Ts) end
+ | SOME typevars =>
+ if typevars <> Ts
+ then error ("Recursion of type " ^
+ quote (Syntax.string_of_typ_global tmp_thy T) ^
+ " with different arguments")
+ else if rec_ok then T
+ else error ("Illegal indirect recursion of type " ^
+ quote (Syntax.string_of_typ_global tmp_thy T)))
+ | check_rec rec_ok (TVar _) = error "extender:check_rec";
+
+ fun prep_arg (lazy, sel, raw_T) =
+ let
+ val T = prep_typ tmp_thy sorts raw_T;
+ val _ = check_rec true T;
+ val _ = check_pcpo (lazy, sel, T);
+ in (lazy, sel, T) end;
+ fun prep_con (b, args, mx) = (b, map prep_arg args, mx);
+ fun prep_rhs cons = map prep_con cons;
+ val rhss : (binding * (bool * binding option * typ) list * mixfix) list list =
+ map prep_rhs raw_rhss;
+
+ fun mk_arg_typ (lazy, dest_opt, T) = if lazy then mk_upT T else T;
+ fun mk_con_typ (bind, args, mx) =
+ if null args then oneT else foldr1 mk_sprodT (map mk_arg_typ args);
+ fun mk_rhs_typ cons = foldr1 mk_ssumT (map mk_con_typ cons);
+
+ val absTs : typ list = map Type dtnvs';
+ val repTs : typ list = map mk_rhs_typ rhss;
+
+ val iso_spec : (binding * mixfix * (typ * typ)) list =
+ map (fn ((dbind, _, mx), eq) => (dbind, mx, eq))
+ (dtnvs ~~ (absTs ~~ repTs));
+
+ val ((iso_infos, take_info), thy) = add_isos iso_spec thy;
+
+ val (constr_infos, thy) =
+ thy
+ |> fold_map (fn ((dbind, cons), info) =>
+ Domain_Constructors.add_domain_constructors dbind cons info)
+ (dbinds ~~ rhss ~~ iso_infos);
+
+ val (take_rews, thy) =
+ Domain_Induction.comp_theorems comp_dbind
+ dbinds take_info constr_infos thy;
+ in
+ thy
+ end;
+
+fun define_isos (spec : (binding * mixfix * (typ * typ)) list) =
+ let
+ fun prep (dbind, mx, (lhsT, rhsT)) =
+ let val (dname, vs) = dest_Type lhsT;
+ in (map (fst o dest_TFree) vs, dbind, mx, rhsT, NONE) end;
+ in
+ Domain_Isomorphism.domain_isomorphism (map prep spec)
+ end;
+
+fun pcpo_arg lazy = if lazy then @{sort cpo} else @{sort pcpo};
+fun rep_arg lazy = @{sort bifinite};
+
+(* Adapted from src/HOL/Tools/Datatype/datatype_data.ML *)
+fun read_typ thy sorts str =
+ let
+ val ctxt = ProofContext.init_global thy
+ |> fold (Variable.declare_typ o TFree) sorts;
+ in Syntax.read_typ ctxt str end;
+
+fun cert_typ sign sorts raw_T =
+ let
+ val T = Type.no_tvars (Sign.certify_typ sign raw_T)
+ handle TYPE (msg, _, _) => error msg;
+ val sorts' = Term.add_tfreesT T sorts;
+ val _ =
+ case duplicates (op =) (map fst sorts') of
+ [] => ()
+ | dups => error ("Inconsistent sort constraints for " ^ commas dups)
+ in T end;
+
+val add_domain =
+ gen_add_domain cert_typ Domain_Axioms.add_axioms pcpo_arg;
+
+val add_new_domain =
+ gen_add_domain cert_typ define_isos rep_arg;
+
+val add_domain_cmd =
+ gen_add_domain read_typ Domain_Axioms.add_axioms pcpo_arg;
+
+val add_new_domain_cmd =
+ gen_add_domain read_typ define_isos rep_arg;
+
+
+(** outer syntax **)
+
+val _ = Keyword.keyword "lazy";
+
+val dest_decl : (bool * binding option * string) parser =
+ Parse.$$$ "(" |-- Scan.optional (Parse.$$$ "lazy" >> K true) false --
+ (Parse.binding >> SOME) -- (Parse.$$$ "::" |-- Parse.typ) --| Parse.$$$ ")" >> Parse.triple1
+ || Parse.$$$ "(" |-- Parse.$$$ "lazy" |-- Parse.typ --| Parse.$$$ ")"
+ >> (fn t => (true,NONE,t))
+ || Parse.typ >> (fn t => (false,NONE,t));
+
+val cons_decl =
+ Parse.binding -- Scan.repeat dest_decl -- Parse.opt_mixfix;
+
+val domain_decl =
+ (Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix) --
+ (Parse.$$$ "=" |-- Parse.enum1 "|" cons_decl);
+
+val domains_decl =
+ Scan.option (Parse.$$$ "(" |-- Parse.binding --| Parse.$$$ ")") --
+ Parse.and_list1 domain_decl;
+
+fun mk_domain
+ (definitional : bool)
+ (opt_name : binding option,
+ doms : ((((string * string option) list * binding) * mixfix) *
+ ((binding * (bool * binding option * string) list) * mixfix) list) list ) =
+ let
+ val names = map (fn (((_, t), _), _) => Binding.name_of t) doms;
+ val specs : ((string * string option) list * binding * mixfix *
+ (binding * (bool * binding option * string) list * mixfix) list) list =
+ map (fn (((vs, t), mx), cons) =>
+ (vs, t, mx, map (fn ((c, ds), mx) => (c, ds, mx)) cons)) doms;
+ val comp_dbind =
+ case opt_name of NONE => Binding.name (space_implode "_" names)
+ | SOME s => s;
+ in
+ if definitional
+ then add_new_domain_cmd comp_dbind specs
+ else add_domain_cmd comp_dbind specs
+ end;
+
+val _ =
+ Outer_Syntax.command "domain" "define recursive domains (HOLCF)"
+ Keyword.thy_decl (domains_decl >> (Toplevel.theory o mk_domain false));
+
+val _ =
+ Outer_Syntax.command "new_domain" "define recursive domains (HOLCF)"
+ Keyword.thy_decl (domains_decl >> (Toplevel.theory o mk_domain true));
+
+end;
--- a/src/HOLCF/Tools/Domain/domain_extender.ML Tue Oct 19 15:13:35 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,278 +0,0 @@
-(* Title: HOLCF/Tools/Domain/domain_extender.ML
- Author: David von Oheimb
- Author: Brian Huffman
-
-Theory extender for domain command, including theory syntax.
-*)
-
-signature DOMAIN_EXTENDER =
-sig
- val add_domain_cmd:
- binding ->
- ((string * string option) list * binding * mixfix *
- (binding * (bool * binding option * string) list * mixfix) list) list
- -> theory -> theory
-
- val add_domain:
- binding ->
- ((string * string option) list * binding * mixfix *
- (binding * (bool * binding option * typ) list * mixfix) list) list
- -> theory -> theory
-
- val add_new_domain_cmd:
- binding ->
- ((string * string option) list * binding * mixfix *
- (binding * (bool * binding option * string) list * mixfix) list) list
- -> theory -> theory
-
- val add_new_domain:
- binding ->
- ((string * string option) list * binding * mixfix *
- (binding * (bool * binding option * typ) list * mixfix) list) list
- -> theory -> theory
-end;
-
-structure Domain_Extender :> DOMAIN_EXTENDER =
-struct
-
-open HOLCF_Library;
-
-fun first (x,_,_) = x;
-fun second (_,x,_) = x;
-fun third (_,_,x) = x;
-
-fun upd_first f (x,y,z) = (f x, y, z);
-fun upd_second f (x,y,z) = ( x, f y, z);
-fun upd_third f (x,y,z) = ( x, y, f z);
-
-(* ----- general testing and preprocessing of constructor list -------------- *)
-fun check_and_sort_domain
- (arg_sort : bool -> sort)
- (dtnvs : (string * typ list) list)
- (cons'' : (binding * (bool * binding option * typ) list * mixfix) list list)
- (thy : theory)
- : ((string * typ list) *
- (binding * (bool * binding option * typ) list * mixfix) list) list =
- let
- val defaultS = Sign.defaultS thy;
-
- val test_dupl_typs =
- case duplicates (op =) (map fst dtnvs) of
- [] => false | dups => error ("Duplicate types: " ^ commas_quote dups);
-
- val all_cons = map (Binding.name_of o first) (flat cons'');
- val test_dupl_cons =
- case duplicates (op =) all_cons of
- [] => false | dups => error ("Duplicate constructors: "
- ^ commas_quote dups);
- val all_sels =
- (map Binding.name_of o map_filter second o maps second) (flat cons'');
- val test_dupl_sels =
- case duplicates (op =) all_sels of
- [] => false | dups => error("Duplicate selectors: "^commas_quote dups);
-
- fun test_dupl_tvars s =
- case duplicates (op =) (map(fst o dest_TFree)s) of
- [] => false | dups => error("Duplicate type arguments: "
- ^commas_quote dups);
- val test_dupl_tvars' = exists test_dupl_tvars (map snd dtnvs);
-
- (* test for free type variables, illegal sort constraints on rhs,
- non-pcpo-types and invalid use of recursive type;
- replace sorts in type variables on rhs *)
- fun analyse_equation ((dname,typevars),cons') =
- let
- val tvars = map dest_TFree typevars;
- val distinct_typevars = map TFree tvars;
- fun rm_sorts (TFree(s,_)) = TFree(s,[])
- | rm_sorts (Type(s,ts)) = Type(s,remove_sorts ts)
- | rm_sorts (TVar(s,_)) = TVar(s,[])
- and remove_sorts l = map rm_sorts l;
- fun analyse indirect (TFree(v,s)) =
- (case AList.lookup (op =) tvars v of
- NONE => error ("Free type variable " ^ quote v ^ " on rhs.")
- | SOME sort => if eq_set (op =) (s, defaultS) orelse
- eq_set (op =) (s, sort)
- then TFree(v,sort)
- else error ("Inconsistent sort constraint" ^
- " for type variable " ^ quote v))
- | analyse indirect (t as Type(s,typl)) =
- (case AList.lookup (op =) dtnvs s of
- NONE => Type (s, map (analyse false) typl)
- | SOME typevars =>
- if indirect
- then error ("Indirect recursion of type " ^
- quote (Syntax.string_of_typ_global thy t))
- else if dname <> s orelse
- (** BUG OR FEATURE?:
- mutual recursion may use different arguments **)
- remove_sorts typevars = remove_sorts typl
- then Type(s,map (analyse true) typl)
- else error ("Direct recursion of type " ^
- quote (Syntax.string_of_typ_global thy t) ^
- " with different arguments"))
- | analyse indirect (TVar _) = error "extender:analyse";
- fun check_pcpo lazy T =
- let val sort = arg_sort lazy in
- if Sign.of_sort thy (T, sort) then T
- else error ("Constructor argument type is not of sort " ^
- Syntax.string_of_sort_global thy sort ^ ": " ^
- Syntax.string_of_typ_global thy T)
- end;
- fun analyse_arg (lazy, sel, T) =
- (lazy, sel, check_pcpo lazy (analyse false T));
- fun analyse_con (b, args, mx) = (b, map analyse_arg args, mx);
- in ((dname,distinct_typevars), map analyse_con cons') end;
- in ListPair.map analyse_equation (dtnvs,cons'')
- end; (* let *)
-
-(* ----- calls for building new thy and thms -------------------------------- *)
-
-type info =
- Domain_Take_Proofs.iso_info list * Domain_Take_Proofs.take_induct_info;
-
-fun gen_add_domain
- (prep_typ : theory -> 'a -> typ)
- (add_isos : (binding * mixfix * (typ * typ)) list -> theory -> info * theory)
- (arg_sort : bool -> sort)
- (comp_dbind : binding)
- (eqs''' : ((string * string option) list * binding * mixfix *
- (binding * (bool * binding option * 'a) list * mixfix) list) list)
- (thy : theory) =
- let
- val dtnvs : (binding * typ list * mixfix) list =
- let
- fun readS (SOME s) = Syntax.read_sort_global thy s
- | readS NONE = Sign.defaultS thy;
- fun readTFree (a, s) = TFree (a, readS s);
- in
- map (fn (vs,dname:binding,mx,_) =>
- (dname, map readTFree vs, mx)) eqs'''
- end;
-
- fun thy_type (dname,tvars,mx) = (dname, length tvars, mx);
- fun thy_arity (dname,tvars,mx) =
- (Sign.full_name thy dname, map (snd o dest_TFree) tvars, arg_sort false);
-
- (* this theory is used just for parsing and error checking *)
- val tmp_thy = thy
- |> Theory.copy
- |> Sign.add_types (map thy_type dtnvs)
- |> fold (AxClass.axiomatize_arity o thy_arity) dtnvs;
-
- val dbinds : binding list =
- map (fn (_,dbind,_,_) => dbind) eqs''';
- val cons''' :
- (binding * (bool * binding option * 'a) list * mixfix) list list =
- map (fn (_,_,_,cons) => cons) eqs''';
- val cons'' :
- (binding * (bool * binding option * typ) list * mixfix) list list =
- map (map (upd_second (map (upd_third (prep_typ tmp_thy))))) cons''';
- val dtnvs' : (string * typ list) list =
- map (fn (dname,vs,mx) => (Sign.full_name thy dname,vs)) dtnvs;
- val eqs' : ((string * typ list) *
- (binding * (bool * binding option * typ) list * mixfix) list) list =
- check_and_sort_domain arg_sort dtnvs' cons'' tmp_thy;
- val dts : typ list = map (Type o fst) eqs';
-
- fun mk_arg_typ (lazy, dest_opt, T) = if lazy then mk_upT T else T;
- fun mk_con_typ (bind, args, mx) =
- if null args then oneT else foldr1 mk_sprodT (map mk_arg_typ args);
- fun mk_eq_typ (_, cons) = foldr1 mk_ssumT (map mk_con_typ cons);
- val repTs : typ list = map mk_eq_typ eqs';
-
- val iso_spec : (binding * mixfix * (typ * typ)) list =
- map (fn ((dbind, _, mx), eq) => (dbind, mx, eq))
- (dtnvs ~~ (dts ~~ repTs));
-
- val ((iso_infos, take_info), thy) = add_isos iso_spec thy;
-
- val (constr_infos, thy) =
- thy
- |> fold_map (fn ((dbind, (_,cs)), info) =>
- Domain_Constructors.add_domain_constructors dbind cs info)
- (dbinds ~~ eqs' ~~ iso_infos);
-
- val (take_rews, thy) =
- Domain_Theorems.comp_theorems comp_dbind
- dbinds take_info constr_infos thy;
- in
- thy
- end;
-
-fun define_isos (spec : (binding * mixfix * (typ * typ)) list) =
- let
- fun prep (dbind, mx, (lhsT, rhsT)) =
- let val (dname, vs) = dest_Type lhsT;
- in (map (fst o dest_TFree) vs, dbind, mx, rhsT, NONE) end;
- in
- Domain_Isomorphism.domain_isomorphism (map prep spec)
- end;
-
-fun pcpo_arg lazy = if lazy then @{sort cpo} else @{sort pcpo};
-fun rep_arg lazy = @{sort bifinite};
-
-val add_domain =
- gen_add_domain Sign.certify_typ Domain_Axioms.add_axioms pcpo_arg;
-
-val add_new_domain =
- gen_add_domain Sign.certify_typ define_isos rep_arg;
-
-val add_domain_cmd =
- gen_add_domain Syntax.read_typ_global Domain_Axioms.add_axioms pcpo_arg;
-
-val add_new_domain_cmd =
- gen_add_domain Syntax.read_typ_global define_isos rep_arg;
-
-
-(** outer syntax **)
-
-val _ = Keyword.keyword "lazy";
-
-val dest_decl : (bool * binding option * string) parser =
- Parse.$$$ "(" |-- Scan.optional (Parse.$$$ "lazy" >> K true) false --
- (Parse.binding >> SOME) -- (Parse.$$$ "::" |-- Parse.typ) --| Parse.$$$ ")" >> Parse.triple1
- || Parse.$$$ "(" |-- Parse.$$$ "lazy" |-- Parse.typ --| Parse.$$$ ")"
- >> (fn t => (true,NONE,t))
- || Parse.typ >> (fn t => (false,NONE,t));
-
-val cons_decl =
- Parse.binding -- Scan.repeat dest_decl -- Parse.opt_mixfix;
-
-val domain_decl =
- (Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix) --
- (Parse.$$$ "=" |-- Parse.enum1 "|" cons_decl);
-
-val domains_decl =
- Scan.option (Parse.$$$ "(" |-- Parse.binding --| Parse.$$$ ")") --
- Parse.and_list1 domain_decl;
-
-fun mk_domain
- (definitional : bool)
- (opt_name : binding option,
- doms : ((((string * string option) list * binding) * mixfix) *
- ((binding * (bool * binding option * string) list) * mixfix) list) list ) =
- let
- val names = map (fn (((_, t), _), _) => Binding.name_of t) doms;
- val specs : ((string * string option) list * binding * mixfix *
- (binding * (bool * binding option * string) list * mixfix) list) list =
- map (fn (((vs, t), mx), cons) =>
- (vs, t, mx, map (fn ((c, ds), mx) => (c, ds, mx)) cons)) doms;
- val comp_dbind =
- case opt_name of NONE => Binding.name (space_implode "_" names)
- | SOME s => s;
- in
- if definitional
- then add_new_domain_cmd comp_dbind specs
- else add_domain_cmd comp_dbind specs
- end;
-
-val _ =
- Outer_Syntax.command "domain" "define recursive domains (HOLCF)"
- Keyword.thy_decl (domains_decl >> (Toplevel.theory o mk_domain false));
-
-val _ =
- Outer_Syntax.command "new_domain" "define recursive domains (HOLCF)"
- Keyword.thy_decl (domains_decl >> (Toplevel.theory o mk_domain true));
-
-end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Tools/Domain/domain_induction.ML Wed Oct 20 21:26:51 2010 -0700
@@ -0,0 +1,438 @@
+(* Title: HOLCF/Tools/Domain/domain_induction.ML
+ Author: David von Oheimb
+ Author: Brian Huffman
+
+Proofs of high-level (co)induction rules for domain command.
+*)
+
+signature DOMAIN_INDUCTION =
+sig
+ val comp_theorems :
+ binding -> binding list ->
+ Domain_Take_Proofs.take_induct_info ->
+ Domain_Constructors.constr_info list ->
+ theory -> thm list * theory
+
+ val quiet_mode: bool Unsynchronized.ref;
+ val trace_domain: bool Unsynchronized.ref;
+end;
+
+structure Domain_Induction :> DOMAIN_INDUCTION =
+struct
+
+val quiet_mode = Unsynchronized.ref false;
+val trace_domain = Unsynchronized.ref false;
+
+fun message s = if !quiet_mode then () else writeln s;
+fun trace s = if !trace_domain then tracing s else ();
+
+open HOLCF_Library;
+
+(******************************************************************************)
+(***************************** proofs about take ******************************)
+(******************************************************************************)
+
+fun take_theorems
+ (dbinds : binding list)
+ (take_info : Domain_Take_Proofs.take_induct_info)
+ (constr_infos : Domain_Constructors.constr_info list)
+ (thy : theory) : thm list list * theory =
+let
+ val {take_consts, take_Suc_thms, deflation_take_thms, ...} = take_info;
+ val deflation_thms = Domain_Take_Proofs.get_deflation_thms thy;
+
+ val n = Free ("n", @{typ nat});
+ val n' = @{const Suc} $ n;
+
+ local
+ val newTs = map (#absT o #iso_info) constr_infos;
+ val subs = newTs ~~ map (fn t => t $ n) take_consts;
+ fun is_ID (Const (c, _)) = (c = @{const_name ID})
+ | is_ID _ = false;
+ in
+ fun map_of_arg v T =
+ let val m = Domain_Take_Proofs.map_of_typ thy subs T;
+ in if is_ID m then v else mk_capply (m, v) end;
+ end
+
+ fun prove_take_apps
+ ((dbind, take_const), constr_info) thy =
+ let
+ val {iso_info, con_specs, con_betas, ...} = constr_info;
+ val {abs_inverse, ...} = iso_info;
+ fun prove_take_app (con_const, args) =
+ let
+ val Ts = map snd args;
+ val ns = Name.variant_list ["n"] (Datatype_Prop.make_tnames Ts);
+ val vs = map Free (ns ~~ Ts);
+ val lhs = mk_capply (take_const $ n', list_ccomb (con_const, vs));
+ val rhs = list_ccomb (con_const, map2 map_of_arg vs Ts);
+ val goal = mk_trp (mk_eq (lhs, rhs));
+ val rules =
+ [abs_inverse] @ con_betas @ @{thms take_con_rules}
+ @ take_Suc_thms @ deflation_thms @ deflation_take_thms;
+ val tac = simp_tac (HOL_basic_ss addsimps rules) 1;
+ in
+ Goal.prove_global thy [] [] goal (K tac)
+ end;
+ val take_apps = map prove_take_app con_specs;
+ in
+ yield_singleton Global_Theory.add_thmss
+ ((Binding.qualified true "take_rews" dbind, take_apps),
+ [Simplifier.simp_add]) thy
+ end;
+in
+ fold_map prove_take_apps
+ (dbinds ~~ take_consts ~~ constr_infos) thy
+end;
+
+(******************************************************************************)
+(****************************** induction rules *******************************)
+(******************************************************************************)
+
+val case_UU_allI =
+ @{lemma "(!!x. x ~= UU ==> P x) ==> P UU ==> ALL x. P x" by metis};
+
+fun prove_induction
+ (comp_dbind : binding)
+ (constr_infos : Domain_Constructors.constr_info list)
+ (take_info : Domain_Take_Proofs.take_induct_info)
+ (take_rews : thm list)
+ (thy : theory) =
+let
+ val comp_dname = Binding.name_of comp_dbind;
+
+ val iso_infos = map #iso_info constr_infos;
+ val exhausts = map #exhaust constr_infos;
+ val con_rews = maps #con_rews constr_infos;
+ val {take_consts, take_induct_thms, ...} = take_info;
+
+ val newTs = map #absT iso_infos;
+ val P_names = Datatype_Prop.indexify_names (map (K "P") newTs);
+ val x_names = Datatype_Prop.indexify_names (map (K "x") newTs);
+ val P_types = map (fn T => T --> HOLogic.boolT) newTs;
+ val Ps = map Free (P_names ~~ P_types);
+ val xs = map Free (x_names ~~ newTs);
+ val n = Free ("n", HOLogic.natT);
+
+ fun con_assm defined p (con, args) =
+ let
+ val Ts = map snd args;
+ val ns = Name.variant_list P_names (Datatype_Prop.make_tnames Ts);
+ val vs = map Free (ns ~~ Ts);
+ val nonlazy = map snd (filter_out (fst o fst) (args ~~ vs));
+ fun ind_hyp (v, T) t =
+ case AList.lookup (op =) (newTs ~~ Ps) T of NONE => t
+ | SOME p' => Logic.mk_implies (mk_trp (p' $ v), t);
+ val t1 = mk_trp (p $ list_ccomb (con, vs));
+ val t2 = fold_rev ind_hyp (vs ~~ Ts) t1;
+ val t3 = Logic.list_implies (map (mk_trp o mk_defined) nonlazy, t2);
+ in fold_rev Logic.all vs (if defined then t3 else t2) end;
+ fun eq_assms ((p, T), cons) =
+ mk_trp (p $ HOLCF_Library.mk_bottom T) :: map (con_assm true p) cons;
+ val assms = maps eq_assms (Ps ~~ newTs ~~ map #con_specs constr_infos);
+
+ val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews);
+ fun quant_tac ctxt i = EVERY
+ (map (fn name => res_inst_tac ctxt [(("x", 0), name)] spec i) x_names);
+
+ (* FIXME: move this message to domain_take_proofs.ML *)
+ val is_finite = #is_finite take_info;
+ val _ = if is_finite
+ then message ("Proving finiteness rule for domain "^comp_dname^" ...")
+ else ();
+
+ val _ = trace " Proving finite_ind...";
+ val finite_ind =
+ let
+ val concls =
+ map (fn ((P, t), x) => P $ mk_capply (t $ n, x))
+ (Ps ~~ take_consts ~~ xs);
+ val goal = mk_trp (foldr1 mk_conj concls);
+
+ fun tacf {prems, context} =
+ let
+ (* Prove stronger prems, without definedness side conditions *)
+ fun con_thm p (con, args) =
+ let
+ val subgoal = con_assm false p (con, args);
+ val rules = prems @ con_rews @ simp_thms;
+ val simplify = asm_simp_tac (HOL_basic_ss addsimps rules);
+ fun arg_tac (lazy, _) =
+ rtac (if lazy then allI else case_UU_allI) 1;
+ val tacs =
+ rewrite_goals_tac @{thms atomize_all atomize_imp} ::
+ map arg_tac args @
+ [REPEAT (rtac impI 1), ALLGOALS simplify];
+ in
+ Goal.prove context [] [] subgoal (K (EVERY tacs))
+ end;
+ fun eq_thms (p, cons) = map (con_thm p) cons;
+ val conss = map #con_specs constr_infos;
+ val prems' = maps eq_thms (Ps ~~ conss);
+
+ val tacs1 = [
+ quant_tac context 1,
+ simp_tac HOL_ss 1,
+ InductTacs.induct_tac context [[SOME "n"]] 1,
+ simp_tac (take_ss addsimps prems) 1,
+ TRY (safe_tac HOL_cs)];
+ fun con_tac _ =
+ asm_simp_tac take_ss 1 THEN
+ (resolve_tac prems' THEN_ALL_NEW etac spec) 1;
+ fun cases_tacs (cons, exhaust) =
+ res_inst_tac context [(("y", 0), "x")] exhaust 1 ::
+ asm_simp_tac (take_ss addsimps prems) 1 ::
+ map con_tac cons;
+ val tacs = tacs1 @ maps cases_tacs (conss ~~ exhausts)
+ in
+ EVERY (map DETERM tacs)
+ end;
+ in Goal.prove_global thy [] assms goal tacf end;
+
+ val _ = trace " Proving ind...";
+ val ind =
+ let
+ val concls = map (op $) (Ps ~~ xs);
+ val goal = mk_trp (foldr1 mk_conj concls);
+ val adms = if is_finite then [] else map (mk_trp o mk_adm) Ps;
+ fun tacf {prems, context} =
+ let
+ fun finite_tac (take_induct, fin_ind) =
+ rtac take_induct 1 THEN
+ (if is_finite then all_tac else resolve_tac prems 1) THEN
+ (rtac fin_ind THEN_ALL_NEW solve_tac prems) 1;
+ val fin_inds = Project_Rule.projections context finite_ind;
+ in
+ TRY (safe_tac HOL_cs) THEN
+ EVERY (map finite_tac (take_induct_thms ~~ fin_inds))
+ end;
+ in Goal.prove_global thy [] (adms @ assms) goal tacf end
+
+ (* case names for induction rules *)
+ val dnames = map (fst o dest_Type) newTs;
+ val case_ns =
+ let
+ val adms =
+ if is_finite then [] else
+ if length dnames = 1 then ["adm"] else
+ map (fn s => "adm_" ^ Long_Name.base_name s) dnames;
+ val bottoms =
+ if length dnames = 1 then ["bottom"] else
+ map (fn s => "bottom_" ^ Long_Name.base_name s) dnames;
+ fun one_eq bot constr_info =
+ let fun name_of (c, args) = Long_Name.base_name (fst (dest_Const c));
+ in bot :: map name_of (#con_specs constr_info) end;
+ in adms @ flat (map2 one_eq bottoms constr_infos) end;
+
+ val inducts = Project_Rule.projections (ProofContext.init_global thy) ind;
+ fun ind_rule (dname, rule) =
+ ((Binding.empty, rule),
+ [Rule_Cases.case_names case_ns, Induct.induct_type dname]);
+
+in
+ thy
+ |> snd o Global_Theory.add_thms [
+ ((Binding.qualified true "finite_induct" comp_dbind, finite_ind), []),
+ ((Binding.qualified true "induct" comp_dbind, ind ), [])]
+ |> (snd o Global_Theory.add_thms (map ind_rule (dnames ~~ inducts)))
+end; (* prove_induction *)
+
+(******************************************************************************)
+(************************ bisimulation and coinduction ************************)
+(******************************************************************************)
+
+fun prove_coinduction
+ (comp_dbind : binding, dbinds : binding list)
+ (constr_infos : Domain_Constructors.constr_info list)
+ (take_info : Domain_Take_Proofs.take_induct_info)
+ (take_rews : thm list list)
+ (thy : theory) : theory =
+let
+ val iso_infos = map #iso_info constr_infos;
+ val newTs = map #absT iso_infos;
+
+ val {take_consts, take_0_thms, take_lemma_thms, ...} = take_info;
+
+ val R_names = Datatype_Prop.indexify_names (map (K "R") newTs);
+ val R_types = map (fn T => T --> T --> boolT) newTs;
+ val Rs = map Free (R_names ~~ R_types);
+ val n = Free ("n", natT);
+ val reserved = "x" :: "y" :: R_names;
+
+ (* declare bisimulation predicate *)
+ val bisim_bind = Binding.suffix_name "_bisim" comp_dbind;
+ val bisim_type = R_types ---> boolT;
+ val (bisim_const, thy) =
+ Sign.declare_const ((bisim_bind, bisim_type), NoSyn) thy;
+
+ (* define bisimulation predicate *)
+ local
+ fun one_con T (con, args) =
+ let
+ val Ts = map snd args;
+ val ns1 = Name.variant_list reserved (Datatype_Prop.make_tnames Ts);
+ val ns2 = map (fn n => n^"'") ns1;
+ val vs1 = map Free (ns1 ~~ Ts);
+ val vs2 = map Free (ns2 ~~ Ts);
+ val eq1 = mk_eq (Free ("x", T), list_ccomb (con, vs1));
+ val eq2 = mk_eq (Free ("y", T), list_ccomb (con, vs2));
+ fun rel ((v1, v2), T) =
+ case AList.lookup (op =) (newTs ~~ Rs) T of
+ NONE => mk_eq (v1, v2) | SOME r => r $ v1 $ v2;
+ val eqs = foldr1 mk_conj (map rel (vs1 ~~ vs2 ~~ Ts) @ [eq1, eq2]);
+ in
+ Library.foldr mk_ex (vs1 @ vs2, eqs)
+ end;
+ fun one_eq ((T, R), cons) =
+ let
+ val x = Free ("x", T);
+ val y = Free ("y", T);
+ val disj1 = mk_conj (mk_eq (x, mk_bottom T), mk_eq (y, mk_bottom T));
+ val disjs = disj1 :: map (one_con T) cons;
+ in
+ mk_all (x, mk_all (y, mk_imp (R $ x $ y, foldr1 mk_disj disjs)))
+ end;
+ val conjs = map one_eq (newTs ~~ Rs ~~ map #con_specs constr_infos);
+ val bisim_rhs = lambdas Rs (Library.foldr1 mk_conj conjs);
+ val bisim_eqn = Logic.mk_equals (bisim_const, bisim_rhs);
+ in
+ val (bisim_def_thm, thy) = thy |>
+ yield_singleton (Global_Theory.add_defs false)
+ ((Binding.qualified true "bisim_def" comp_dbind, bisim_eqn), []);
+ end (* local *)
+
+ (* prove coinduction lemma *)
+ val coind_lemma =
+ let
+ val assm = mk_trp (list_comb (bisim_const, Rs));
+ fun one ((T, R), take_const) =
+ let
+ val x = Free ("x", T);
+ val y = Free ("y", T);
+ val lhs = mk_capply (take_const $ n, x);
+ val rhs = mk_capply (take_const $ n, y);
+ in
+ mk_all (x, mk_all (y, mk_imp (R $ x $ y, mk_eq (lhs, rhs))))
+ end;
+ val goal =
+ mk_trp (foldr1 mk_conj (map one (newTs ~~ Rs ~~ take_consts)));
+ val rules = @{thm Rep_CFun_strict1} :: take_0_thms;
+ fun tacf {prems, context} =
+ let
+ val prem' = rewrite_rule [bisim_def_thm] (hd prems);
+ val prems' = Project_Rule.projections context prem';
+ val dests = map (fn th => th RS spec RS spec RS mp) prems';
+ fun one_tac (dest, rews) =
+ dtac dest 1 THEN safe_tac HOL_cs THEN
+ ALLGOALS (asm_simp_tac (HOL_basic_ss addsimps rews));
+ in
+ rtac @{thm nat.induct} 1 THEN
+ simp_tac (HOL_ss addsimps rules) 1 THEN
+ safe_tac HOL_cs THEN
+ EVERY (map one_tac (dests ~~ take_rews))
+ end
+ in
+ Goal.prove_global thy [] [assm] goal tacf
+ end;
+
+ (* prove individual coinduction rules *)
+ fun prove_coind ((T, R), take_lemma) =
+ let
+ val x = Free ("x", T);
+ val y = Free ("y", T);
+ val assm1 = mk_trp (list_comb (bisim_const, Rs));
+ val assm2 = mk_trp (R $ x $ y);
+ val goal = mk_trp (mk_eq (x, y));
+ fun tacf {prems, context} =
+ let
+ val rule = hd prems RS coind_lemma;
+ in
+ rtac take_lemma 1 THEN
+ asm_simp_tac (HOL_basic_ss addsimps (rule :: prems)) 1
+ end;
+ in
+ Goal.prove_global thy [] [assm1, assm2] goal tacf
+ end;
+ val coinds = map prove_coind (newTs ~~ Rs ~~ take_lemma_thms);
+ val coind_binds = map (Binding.qualified true "coinduct") dbinds;
+
+in
+ thy |> snd o Global_Theory.add_thms
+ (map Thm.no_attributes (coind_binds ~~ coinds))
+end; (* let *)
+
+(******************************************************************************)
+(******************************* main function ********************************)
+(******************************************************************************)
+
+fun comp_theorems
+ (comp_dbind : binding)
+ (dbinds : binding list)
+ (take_info : Domain_Take_Proofs.take_induct_info)
+ (constr_infos : Domain_Constructors.constr_info list)
+ (thy : theory) =
+let
+val comp_dname = Binding.name_of comp_dbind;
+
+(* Test for emptiness *)
+(* FIXME: reimplement emptiness test
+local
+ open Domain_Library;
+ val dnames = map (fst o fst) eqs;
+ val conss = map snd eqs;
+ fun rec_to ns lazy_rec (n,cons) = forall (exists (fn arg =>
+ is_rec arg andalso not (member (op =) ns (rec_of arg)) andalso
+ ((rec_of arg = n andalso not (lazy_rec orelse is_lazy arg)) orelse
+ rec_of arg <> n andalso rec_to (rec_of arg::ns)
+ (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
+ ) o snd) cons;
+ fun warn (n,cons) =
+ if rec_to [] false (n,cons)
+ then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
+ else false;
+in
+ val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
+ val is_emptys = map warn n__eqs;
+end;
+*)
+
+(* Test for indirect recursion *)
+local
+ val newTs = map (#absT o #iso_info) constr_infos;
+ fun indirect_typ (Type (_, Ts)) =
+ exists (fn T => member (op =) newTs T orelse indirect_typ T) Ts
+ | indirect_typ _ = false;
+ fun indirect_arg (_, T) = indirect_typ T;
+ fun indirect_con (_, args) = exists indirect_arg args;
+ fun indirect_eq cons = exists indirect_con cons;
+in
+ val is_indirect = exists indirect_eq (map #con_specs constr_infos);
+ val _ =
+ if is_indirect
+ then message "Indirect recursion detected, skipping proofs of (co)induction rules"
+ else message ("Proving induction properties of domain "^comp_dname^" ...");
+end;
+
+(* theorems about take *)
+
+val (take_rewss, thy) =
+ take_theorems dbinds take_info constr_infos thy;
+
+val {take_lemma_thms, take_0_thms, take_strict_thms, ...} = take_info;
+
+val take_rews = take_0_thms @ take_strict_thms @ flat take_rewss;
+
+(* prove induction rules, unless definition is indirect recursive *)
+val thy =
+ if is_indirect then thy else
+ prove_induction comp_dbind constr_infos take_info take_rews thy;
+
+val thy =
+ if is_indirect then thy else
+ prove_coinduction (comp_dbind, dbinds) constr_infos take_info take_rewss thy;
+
+in
+ (take_rews, thy)
+end; (* let *)
+end; (* struct *)
--- a/src/HOLCF/Tools/Domain/domain_isomorphism.ML Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Tools/Domain/domain_isomorphism.ML Wed Oct 20 21:26:51 2010 -0700
@@ -104,6 +104,7 @@
infixr 6 ->>;
infix -->>;
+val udomT = @{typ udom};
val deflT = @{typ "defl"};
fun mapT (T as Type (_, Ts)) =
@@ -113,7 +114,9 @@
fun mk_DEFL T =
Const (@{const_name defl}, Term.itselfT T --> deflT) $ Logic.mk_type T;
-fun coerce_const T = Const (@{const_name coerce}, T);
+fun emb_const T = Const (@{const_name emb}, T ->> udomT);
+fun prj_const T = Const (@{const_name prj}, udomT ->> T);
+fun coerce_const (T, U) = mk_cfcomp (prj_const U, emb_const T);
fun isodefl_const T =
Const (@{const_name isodefl}, (T ->> T) --> deflT --> HOLogic.boolT);
@@ -505,9 +508,9 @@
val rep_bind = Binding.suffix_name "_rep" tbind;
val abs_bind = Binding.suffix_name "_abs" tbind;
val ((rep_const, rep_def), thy) =
- define_const (rep_bind, coerce_const (lhsT ->> rhsT)) thy;
+ define_const (rep_bind, coerce_const (lhsT, rhsT)) thy;
val ((abs_const, abs_def), thy) =
- define_const (abs_bind, coerce_const (rhsT ->> lhsT)) thy;
+ define_const (abs_bind, coerce_const (rhsT, lhsT)) thy;
in
(((rep_const, abs_const), (rep_def, abs_def)), thy)
end;
--- a/src/HOLCF/Tools/Domain/domain_theorems.ML Tue Oct 19 15:13:35 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,438 +0,0 @@
-(* Title: HOLCF/Tools/Domain/domain_theorems.ML
- Author: David von Oheimb
- Author: Brian Huffman
-
-Proof generator for domain command.
-*)
-
-signature DOMAIN_THEOREMS =
-sig
- val comp_theorems :
- binding -> binding list ->
- Domain_Take_Proofs.take_induct_info ->
- Domain_Constructors.constr_info list ->
- theory -> thm list * theory
-
- val quiet_mode: bool Unsynchronized.ref;
- val trace_domain: bool Unsynchronized.ref;
-end;
-
-structure Domain_Theorems :> DOMAIN_THEOREMS =
-struct
-
-val quiet_mode = Unsynchronized.ref false;
-val trace_domain = Unsynchronized.ref false;
-
-fun message s = if !quiet_mode then () else writeln s;
-fun trace s = if !trace_domain then tracing s else ();
-
-open HOLCF_Library;
-
-(******************************************************************************)
-(***************************** proofs about take ******************************)
-(******************************************************************************)
-
-fun take_theorems
- (dbinds : binding list)
- (take_info : Domain_Take_Proofs.take_induct_info)
- (constr_infos : Domain_Constructors.constr_info list)
- (thy : theory) : thm list list * theory =
-let
- val {take_consts, take_Suc_thms, deflation_take_thms, ...} = take_info;
- val deflation_thms = Domain_Take_Proofs.get_deflation_thms thy;
-
- val n = Free ("n", @{typ nat});
- val n' = @{const Suc} $ n;
-
- local
- val newTs = map (#absT o #iso_info) constr_infos;
- val subs = newTs ~~ map (fn t => t $ n) take_consts;
- fun is_ID (Const (c, _)) = (c = @{const_name ID})
- | is_ID _ = false;
- in
- fun map_of_arg v T =
- let val m = Domain_Take_Proofs.map_of_typ thy subs T;
- in if is_ID m then v else mk_capply (m, v) end;
- end
-
- fun prove_take_apps
- ((dbind, take_const), constr_info) thy =
- let
- val {iso_info, con_specs, con_betas, ...} = constr_info;
- val {abs_inverse, ...} = iso_info;
- fun prove_take_app (con_const, args) =
- let
- val Ts = map snd args;
- val ns = Name.variant_list ["n"] (Datatype_Prop.make_tnames Ts);
- val vs = map Free (ns ~~ Ts);
- val lhs = mk_capply (take_const $ n', list_ccomb (con_const, vs));
- val rhs = list_ccomb (con_const, map2 map_of_arg vs Ts);
- val goal = mk_trp (mk_eq (lhs, rhs));
- val rules =
- [abs_inverse] @ con_betas @ @{thms take_con_rules}
- @ take_Suc_thms @ deflation_thms @ deflation_take_thms;
- val tac = simp_tac (HOL_basic_ss addsimps rules) 1;
- in
- Goal.prove_global thy [] [] goal (K tac)
- end;
- val take_apps = map prove_take_app con_specs;
- in
- yield_singleton Global_Theory.add_thmss
- ((Binding.qualified true "take_rews" dbind, take_apps),
- [Simplifier.simp_add]) thy
- end;
-in
- fold_map prove_take_apps
- (dbinds ~~ take_consts ~~ constr_infos) thy
-end;
-
-(******************************************************************************)
-(****************************** induction rules *******************************)
-(******************************************************************************)
-
-val case_UU_allI =
- @{lemma "(!!x. x ~= UU ==> P x) ==> P UU ==> ALL x. P x" by metis};
-
-fun prove_induction
- (comp_dbind : binding)
- (constr_infos : Domain_Constructors.constr_info list)
- (take_info : Domain_Take_Proofs.take_induct_info)
- (take_rews : thm list)
- (thy : theory) =
-let
- val comp_dname = Binding.name_of comp_dbind;
-
- val iso_infos = map #iso_info constr_infos;
- val exhausts = map #exhaust constr_infos;
- val con_rews = maps #con_rews constr_infos;
- val {take_consts, take_induct_thms, ...} = take_info;
-
- val newTs = map #absT iso_infos;
- val P_names = Datatype_Prop.indexify_names (map (K "P") newTs);
- val x_names = Datatype_Prop.indexify_names (map (K "x") newTs);
- val P_types = map (fn T => T --> HOLogic.boolT) newTs;
- val Ps = map Free (P_names ~~ P_types);
- val xs = map Free (x_names ~~ newTs);
- val n = Free ("n", HOLogic.natT);
-
- fun con_assm defined p (con, args) =
- let
- val Ts = map snd args;
- val ns = Name.variant_list P_names (Datatype_Prop.make_tnames Ts);
- val vs = map Free (ns ~~ Ts);
- val nonlazy = map snd (filter_out (fst o fst) (args ~~ vs));
- fun ind_hyp (v, T) t =
- case AList.lookup (op =) (newTs ~~ Ps) T of NONE => t
- | SOME p' => Logic.mk_implies (mk_trp (p' $ v), t);
- val t1 = mk_trp (p $ list_ccomb (con, vs));
- val t2 = fold_rev ind_hyp (vs ~~ Ts) t1;
- val t3 = Logic.list_implies (map (mk_trp o mk_defined) nonlazy, t2);
- in fold_rev Logic.all vs (if defined then t3 else t2) end;
- fun eq_assms ((p, T), cons) =
- mk_trp (p $ HOLCF_Library.mk_bottom T) :: map (con_assm true p) cons;
- val assms = maps eq_assms (Ps ~~ newTs ~~ map #con_specs constr_infos);
-
- val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews);
- fun quant_tac ctxt i = EVERY
- (map (fn name => res_inst_tac ctxt [(("x", 0), name)] spec i) x_names);
-
- (* FIXME: move this message to domain_take_proofs.ML *)
- val is_finite = #is_finite take_info;
- val _ = if is_finite
- then message ("Proving finiteness rule for domain "^comp_dname^" ...")
- else ();
-
- val _ = trace " Proving finite_ind...";
- val finite_ind =
- let
- val concls =
- map (fn ((P, t), x) => P $ mk_capply (t $ n, x))
- (Ps ~~ take_consts ~~ xs);
- val goal = mk_trp (foldr1 mk_conj concls);
-
- fun tacf {prems, context} =
- let
- (* Prove stronger prems, without definedness side conditions *)
- fun con_thm p (con, args) =
- let
- val subgoal = con_assm false p (con, args);
- val rules = prems @ con_rews @ simp_thms;
- val simplify = asm_simp_tac (HOL_basic_ss addsimps rules);
- fun arg_tac (lazy, _) =
- rtac (if lazy then allI else case_UU_allI) 1;
- val tacs =
- rewrite_goals_tac @{thms atomize_all atomize_imp} ::
- map arg_tac args @
- [REPEAT (rtac impI 1), ALLGOALS simplify];
- in
- Goal.prove context [] [] subgoal (K (EVERY tacs))
- end;
- fun eq_thms (p, cons) = map (con_thm p) cons;
- val conss = map #con_specs constr_infos;
- val prems' = maps eq_thms (Ps ~~ conss);
-
- val tacs1 = [
- quant_tac context 1,
- simp_tac HOL_ss 1,
- InductTacs.induct_tac context [[SOME "n"]] 1,
- simp_tac (take_ss addsimps prems) 1,
- TRY (safe_tac HOL_cs)];
- fun con_tac _ =
- asm_simp_tac take_ss 1 THEN
- (resolve_tac prems' THEN_ALL_NEW etac spec) 1;
- fun cases_tacs (cons, exhaust) =
- res_inst_tac context [(("y", 0), "x")] exhaust 1 ::
- asm_simp_tac (take_ss addsimps prems) 1 ::
- map con_tac cons;
- val tacs = tacs1 @ maps cases_tacs (conss ~~ exhausts)
- in
- EVERY (map DETERM tacs)
- end;
- in Goal.prove_global thy [] assms goal tacf end;
-
- val _ = trace " Proving ind...";
- val ind =
- let
- val concls = map (op $) (Ps ~~ xs);
- val goal = mk_trp (foldr1 mk_conj concls);
- val adms = if is_finite then [] else map (mk_trp o mk_adm) Ps;
- fun tacf {prems, context} =
- let
- fun finite_tac (take_induct, fin_ind) =
- rtac take_induct 1 THEN
- (if is_finite then all_tac else resolve_tac prems 1) THEN
- (rtac fin_ind THEN_ALL_NEW solve_tac prems) 1;
- val fin_inds = Project_Rule.projections context finite_ind;
- in
- TRY (safe_tac HOL_cs) THEN
- EVERY (map finite_tac (take_induct_thms ~~ fin_inds))
- end;
- in Goal.prove_global thy [] (adms @ assms) goal tacf end
-
- (* case names for induction rules *)
- val dnames = map (fst o dest_Type) newTs;
- val case_ns =
- let
- val adms =
- if is_finite then [] else
- if length dnames = 1 then ["adm"] else
- map (fn s => "adm_" ^ Long_Name.base_name s) dnames;
- val bottoms =
- if length dnames = 1 then ["bottom"] else
- map (fn s => "bottom_" ^ Long_Name.base_name s) dnames;
- fun one_eq bot constr_info =
- let fun name_of (c, args) = Long_Name.base_name (fst (dest_Const c));
- in bot :: map name_of (#con_specs constr_info) end;
- in adms @ flat (map2 one_eq bottoms constr_infos) end;
-
- val inducts = Project_Rule.projections (ProofContext.init_global thy) ind;
- fun ind_rule (dname, rule) =
- ((Binding.empty, rule),
- [Rule_Cases.case_names case_ns, Induct.induct_type dname]);
-
-in
- thy
- |> snd o Global_Theory.add_thms [
- ((Binding.qualified true "finite_induct" comp_dbind, finite_ind), []),
- ((Binding.qualified true "induct" comp_dbind, ind ), [])]
- |> (snd o Global_Theory.add_thms (map ind_rule (dnames ~~ inducts)))
-end; (* prove_induction *)
-
-(******************************************************************************)
-(************************ bisimulation and coinduction ************************)
-(******************************************************************************)
-
-fun prove_coinduction
- (comp_dbind : binding, dbinds : binding list)
- (constr_infos : Domain_Constructors.constr_info list)
- (take_info : Domain_Take_Proofs.take_induct_info)
- (take_rews : thm list list)
- (thy : theory) : theory =
-let
- val iso_infos = map #iso_info constr_infos;
- val newTs = map #absT iso_infos;
-
- val {take_consts, take_0_thms, take_lemma_thms, ...} = take_info;
-
- val R_names = Datatype_Prop.indexify_names (map (K "R") newTs);
- val R_types = map (fn T => T --> T --> boolT) newTs;
- val Rs = map Free (R_names ~~ R_types);
- val n = Free ("n", natT);
- val reserved = "x" :: "y" :: R_names;
-
- (* declare bisimulation predicate *)
- val bisim_bind = Binding.suffix_name "_bisim" comp_dbind;
- val bisim_type = R_types ---> boolT;
- val (bisim_const, thy) =
- Sign.declare_const ((bisim_bind, bisim_type), NoSyn) thy;
-
- (* define bisimulation predicate *)
- local
- fun one_con T (con, args) =
- let
- val Ts = map snd args;
- val ns1 = Name.variant_list reserved (Datatype_Prop.make_tnames Ts);
- val ns2 = map (fn n => n^"'") ns1;
- val vs1 = map Free (ns1 ~~ Ts);
- val vs2 = map Free (ns2 ~~ Ts);
- val eq1 = mk_eq (Free ("x", T), list_ccomb (con, vs1));
- val eq2 = mk_eq (Free ("y", T), list_ccomb (con, vs2));
- fun rel ((v1, v2), T) =
- case AList.lookup (op =) (newTs ~~ Rs) T of
- NONE => mk_eq (v1, v2) | SOME r => r $ v1 $ v2;
- val eqs = foldr1 mk_conj (map rel (vs1 ~~ vs2 ~~ Ts) @ [eq1, eq2]);
- in
- Library.foldr mk_ex (vs1 @ vs2, eqs)
- end;
- fun one_eq ((T, R), cons) =
- let
- val x = Free ("x", T);
- val y = Free ("y", T);
- val disj1 = mk_conj (mk_eq (x, mk_bottom T), mk_eq (y, mk_bottom T));
- val disjs = disj1 :: map (one_con T) cons;
- in
- mk_all (x, mk_all (y, mk_imp (R $ x $ y, foldr1 mk_disj disjs)))
- end;
- val conjs = map one_eq (newTs ~~ Rs ~~ map #con_specs constr_infos);
- val bisim_rhs = lambdas Rs (Library.foldr1 mk_conj conjs);
- val bisim_eqn = Logic.mk_equals (bisim_const, bisim_rhs);
- in
- val (bisim_def_thm, thy) = thy |>
- yield_singleton (Global_Theory.add_defs false)
- ((Binding.qualified true "bisim_def" comp_dbind, bisim_eqn), []);
- end (* local *)
-
- (* prove coinduction lemma *)
- val coind_lemma =
- let
- val assm = mk_trp (list_comb (bisim_const, Rs));
- fun one ((T, R), take_const) =
- let
- val x = Free ("x", T);
- val y = Free ("y", T);
- val lhs = mk_capply (take_const $ n, x);
- val rhs = mk_capply (take_const $ n, y);
- in
- mk_all (x, mk_all (y, mk_imp (R $ x $ y, mk_eq (lhs, rhs))))
- end;
- val goal =
- mk_trp (foldr1 mk_conj (map one (newTs ~~ Rs ~~ take_consts)));
- val rules = @{thm Rep_CFun_strict1} :: take_0_thms;
- fun tacf {prems, context} =
- let
- val prem' = rewrite_rule [bisim_def_thm] (hd prems);
- val prems' = Project_Rule.projections context prem';
- val dests = map (fn th => th RS spec RS spec RS mp) prems';
- fun one_tac (dest, rews) =
- dtac dest 1 THEN safe_tac HOL_cs THEN
- ALLGOALS (asm_simp_tac (HOL_basic_ss addsimps rews));
- in
- rtac @{thm nat.induct} 1 THEN
- simp_tac (HOL_ss addsimps rules) 1 THEN
- safe_tac HOL_cs THEN
- EVERY (map one_tac (dests ~~ take_rews))
- end
- in
- Goal.prove_global thy [] [assm] goal tacf
- end;
-
- (* prove individual coinduction rules *)
- fun prove_coind ((T, R), take_lemma) =
- let
- val x = Free ("x", T);
- val y = Free ("y", T);
- val assm1 = mk_trp (list_comb (bisim_const, Rs));
- val assm2 = mk_trp (R $ x $ y);
- val goal = mk_trp (mk_eq (x, y));
- fun tacf {prems, context} =
- let
- val rule = hd prems RS coind_lemma;
- in
- rtac take_lemma 1 THEN
- asm_simp_tac (HOL_basic_ss addsimps (rule :: prems)) 1
- end;
- in
- Goal.prove_global thy [] [assm1, assm2] goal tacf
- end;
- val coinds = map prove_coind (newTs ~~ Rs ~~ take_lemma_thms);
- val coind_binds = map (Binding.qualified true "coinduct") dbinds;
-
-in
- thy |> snd o Global_Theory.add_thms
- (map Thm.no_attributes (coind_binds ~~ coinds))
-end; (* let *)
-
-(******************************************************************************)
-(******************************* main function ********************************)
-(******************************************************************************)
-
-fun comp_theorems
- (comp_dbind : binding)
- (dbinds : binding list)
- (take_info : Domain_Take_Proofs.take_induct_info)
- (constr_infos : Domain_Constructors.constr_info list)
- (thy : theory) =
-let
-val comp_dname = Binding.name_of comp_dbind;
-
-(* Test for emptiness *)
-(* FIXME: reimplement emptiness test
-local
- open Domain_Library;
- val dnames = map (fst o fst) eqs;
- val conss = map snd eqs;
- fun rec_to ns lazy_rec (n,cons) = forall (exists (fn arg =>
- is_rec arg andalso not (member (op =) ns (rec_of arg)) andalso
- ((rec_of arg = n andalso not (lazy_rec orelse is_lazy arg)) orelse
- rec_of arg <> n andalso rec_to (rec_of arg::ns)
- (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
- ) o snd) cons;
- fun warn (n,cons) =
- if rec_to [] false (n,cons)
- then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
- else false;
-in
- val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
- val is_emptys = map warn n__eqs;
-end;
-*)
-
-(* Test for indirect recursion *)
-local
- val newTs = map (#absT o #iso_info) constr_infos;
- fun indirect_typ (Type (_, Ts)) =
- exists (fn T => member (op =) newTs T orelse indirect_typ T) Ts
- | indirect_typ _ = false;
- fun indirect_arg (_, T) = indirect_typ T;
- fun indirect_con (_, args) = exists indirect_arg args;
- fun indirect_eq cons = exists indirect_con cons;
-in
- val is_indirect = exists indirect_eq (map #con_specs constr_infos);
- val _ =
- if is_indirect
- then message "Indirect recursion detected, skipping proofs of (co)induction rules"
- else message ("Proving induction properties of domain "^comp_dname^" ...");
-end;
-
-(* theorems about take *)
-
-val (take_rewss, thy) =
- take_theorems dbinds take_info constr_infos thy;
-
-val {take_lemma_thms, take_0_thms, take_strict_thms, ...} = take_info;
-
-val take_rews = take_0_thms @ take_strict_thms @ flat take_rewss;
-
-(* prove induction rules, unless definition is indirect recursive *)
-val thy =
- if is_indirect then thy else
- prove_induction comp_dbind constr_infos take_info take_rews thy;
-
-val thy =
- if is_indirect then thy else
- prove_coinduction (comp_dbind, dbinds) constr_infos take_info take_rewss thy;
-
-in
- (take_rews, thy)
-end; (* let *)
-end; (* struct *)
--- a/src/HOLCF/Tools/fixrec.ML Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Tools/fixrec.ML Wed Oct 20 21:26:51 2010 -0700
@@ -6,10 +6,10 @@
signature FIXREC =
sig
- val add_fixrec: bool -> (binding * typ option * mixfix) list
- -> (Attrib.binding * term) list -> local_theory -> local_theory
- val add_fixrec_cmd: bool -> (binding * string option * mixfix) list
- -> (Attrib.binding * string) list -> local_theory -> local_theory
+ val add_fixrec: (binding * typ option * mixfix) list
+ -> (bool * (Attrib.binding * term)) list -> local_theory -> local_theory
+ val add_fixrec_cmd: (binding * string option * mixfix) list
+ -> (bool * (Attrib.binding * string)) list -> local_theory -> local_theory
val add_matchers: (string * string) list -> theory -> theory
val fixrec_simp_tac: Proof.context -> int -> tactic
val setup: theory -> theory
@@ -247,15 +247,15 @@
end;
(* builds a monadic term for matching a function definition pattern *)
-(* returns (name, arity, matcher) *)
+(* returns (constant, (vars, matcher)) *)
fun compile_lhs match_name pat rhs vs taken =
case pat of
Const(@{const_name Rep_CFun}, _) $ f $ x =>
let val (rhs', v, taken') = compile_pat match_name x rhs taken;
in compile_lhs match_name f rhs' (v::vs) taken' end
- | Free(_,_) => ((pat, length vs), big_lambdas vs rhs)
- | Const(_,_) => ((pat, length vs), big_lambdas vs rhs)
- | _ => fixrec_err ("function is not declared as constant in theory: "
+ | Free(_,_) => (pat, (vs, rhs))
+ | Const(_,_) => (pat, (vs, rhs))
+ | _ => fixrec_err ("invalid function pattern: "
^ ML_Syntax.print_term pat);
fun strip_alls t =
@@ -268,41 +268,24 @@
compile_lhs match_name lhs (mk_succeed rhs) [] (taken_names eq)
end;
-(* returns the sum (using +++) of the terms in ms *)
-(* also applies "run" to the result! *)
-fun fatbar arity ms =
- let
- fun LAM_Ts 0 t = ([], Term.fastype_of t)
- | LAM_Ts n (_ $ Abs(_,T,t)) =
- let val (Ts, U) = LAM_Ts (n-1) t in (T::Ts, U) end
- | LAM_Ts _ _ = fixrec_err "fatbar: internal error, not enough LAMs";
- fun unLAM 0 t = t
- | unLAM n (_$Abs(_,_,t)) = unLAM (n-1) t
- | unLAM _ _ = fixrec_err "fatbar: internal error, not enough LAMs";
- fun reLAM ([], U) t = t
- | reLAM (T::Ts, U) t = reLAM (Ts, T ->> U) (cabs_const(T,U)$Abs("",T,t));
- val msum = foldr1 mk_mplus (map (unLAM arity) ms);
- val (Ts, U) = LAM_Ts arity (hd ms)
- in
- reLAM (rev Ts, dest_matchT U) (mk_run msum)
- end;
-
(* this is the pattern-matching compiler function *)
fun compile_eqs match_name eqs =
let
- val ((names, arities), mats) =
- apfst ListPair.unzip (ListPair.unzip (map (compile_eq match_name) eqs));
- val cname =
- case distinct (op =) names of
+ val (consts, matchers) =
+ ListPair.unzip (map (compile_eq match_name) eqs);
+ val const =
+ case distinct (op =) consts of
[n] => n
| _ => fixrec_err "all equations in block must define the same function";
- val arity =
- case distinct (op =) arities of
- [a] => a
+ val vars =
+ case distinct (op = o pairself length) (map fst matchers) of
+ [vars] => vars
| _ => fixrec_err "all equations in block must have the same arity";
- val rhs = fatbar arity mats;
+ (* rename so all matchers use same free variables *)
+ fun rename (vs, t) = Term.subst_free (filter_out (op =) (vs ~~ vars)) t;
+ val rhs = big_lambdas vars (mk_run (foldr1 mk_mplus (map rename matchers)));
in
- mk_trp (cname === rhs)
+ mk_trp (const === rhs)
end;
(*************************************************************************)
@@ -352,11 +335,11 @@
fun gen_fixrec
prep_spec
- (strict : bool)
- raw_fixes
- raw_spec
+ (raw_fixes : (binding * 'a option * mixfix) list)
+ (raw_spec' : (bool * (Attrib.binding * 'b)) list)
(lthy : local_theory) =
let
+ val (skips, raw_spec) = ListPair.unzip raw_spec';
val (fixes : ((binding * typ) * mixfix) list,
spec : (Attrib.binding * term) list) =
fst (prep_spec raw_fixes raw_spec lthy);
@@ -369,7 +352,7 @@
fun block_of_name n =
map_filter
(fn (m,eq) => if m = n then SOME eq else NONE)
- (all_names ~~ spec);
+ (all_names ~~ (spec ~~ skips));
val blocks = map block_of_name names;
val matcher_tab = FixrecMatchData.get (ProofContext.theory_of lthy);
@@ -377,27 +360,25 @@
case Symtab.lookup matcher_tab c of SOME m => m
| NONE => fixrec_err ("unknown pattern constructor: " ^ c);
- val matches = map (compile_eqs match_name) (map (map snd) blocks);
+ val matches = map (compile_eqs match_name) (map (map (snd o fst)) blocks);
val spec' = map (pair Attrib.empty_binding) matches;
val (lthy, cnames, fixdef_thms, unfold_thms) =
add_fixdefs fixes spec' lthy;
+
+ val blocks' = map (map fst o filter_out snd) blocks;
+ val simps : (Attrib.binding * thm) list list =
+ map (make_simps lthy) (unfold_thms ~~ blocks');
+ fun mk_bind n : Attrib.binding =
+ (Binding.qualify true n (Binding.name "simps"),
+ [Attrib.internal (K Simplifier.simp_add)]);
+ val simps1 : (Attrib.binding * thm list) list =
+ map (fn (n,xs) => (mk_bind n, map snd xs)) (names ~~ simps);
+ val simps2 : (Attrib.binding * thm list) list =
+ map (apsnd (fn thm => [thm])) (flat simps);
+ val (_, lthy) = lthy
+ |> fold_map Local_Theory.note (simps1 @ simps2);
in
- if strict then let (* only prove simp rules if strict = true *)
- val simps : (Attrib.binding * thm) list list =
- map (make_simps lthy) (unfold_thms ~~ blocks);
- fun mk_bind n : Attrib.binding =
- (Binding.qualify true n (Binding.name "simps"),
- [Attrib.internal (K Simplifier.simp_add)]);
- val simps1 : (Attrib.binding * thm list) list =
- map (fn (n,xs) => (mk_bind n, map snd xs)) (names ~~ simps);
- val simps2 : (Attrib.binding * thm list) list =
- map (apsnd (fn thm => [thm])) (flat simps);
- val (_, lthy) = lthy
- |> fold_map Local_Theory.note (simps1 @ simps2);
- in
- lthy
- end
- else lthy
+ lthy
end;
in
@@ -412,10 +393,15 @@
(******************************** Parsers ********************************)
(*************************************************************************)
+val alt_specs' : (bool * (Attrib.binding * string)) list parser =
+ Parse.enum1 "|"
+ (Parse.opt_keyword "unchecked" -- Parse_Spec.spec --|
+ Scan.option (Scan.ahead (Parse.name || Parse.$$$ "[") -- Parse.!!! (Parse.$$$ "|")));
+
val _ =
Outer_Syntax.local_theory "fixrec" "define recursive functions (HOLCF)" Keyword.thy_decl
- ((Parse.opt_keyword "permissive" >> not) -- Parse.fixes -- Parse_Spec.where_alt_specs
- >> (fn ((strict, fixes), specs) => add_fixrec_cmd strict fixes specs));
+ (Parse.fixes -- (Parse.where_ |-- Parse.!!! alt_specs')
+ >> (fn (fixes, specs) => add_fixrec_cmd fixes specs));
val setup =
Method.setup @{binding fixrec_simp}
--- a/src/HOLCF/Tools/repdef.ML Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Tools/repdef.ML Wed Oct 20 21:26:51 2010 -0700
@@ -84,12 +84,11 @@
|> the_default (Binding.prefix_name "Rep_" name, Binding.prefix_name "Abs_" name);
(*set*)
- val in_defl = @{term "in_defl :: udom => defl => bool"};
- val set = HOLogic.Collect_const udomT $ Abs ("x", udomT, in_defl $ Bound 0 $ defl);
+ val set = @{const defl_set} $ defl;
(*pcpodef*)
- val tac1 = rtac @{thm CollectI} 1 THEN rtac @{thm bottom_in_defl} 1;
- val tac2 = rtac @{thm adm_mem_Collect_in_defl} 1;
+ val tac1 = rtac @{thm defl_set_bottom} 1;
+ val tac2 = rtac @{thm adm_defl_set} 1;
val ((info, cpo_info, pcpo_info), thy) = thy
|> Pcpodef.add_pcpodef def (SOME name) typ set (SOME morphs) (tac1, tac2);
--- a/src/HOLCF/Tutorial/Fixrec_ex.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/Tutorial/Fixrec_ex.thy Wed Oct 20 21:26:51 2010 -0700
@@ -115,22 +115,21 @@
because it only applies when the first pattern fails.
*}
-fixrec (permissive)
+fixrec
lzip2 :: "'a llist \<rightarrow> 'b llist \<rightarrow> ('a \<times> 'b) llist"
where
- "lzip2\<cdot>(lCons\<cdot>x\<cdot>xs)\<cdot>(lCons\<cdot>y\<cdot>ys) = lCons\<cdot>(x, y)\<cdot>(lzip\<cdot>xs\<cdot>ys)"
-| "lzip2\<cdot>xs\<cdot>ys = lNil"
+ "lzip2\<cdot>(lCons\<cdot>x\<cdot>xs)\<cdot>(lCons\<cdot>y\<cdot>ys) = lCons\<cdot>(x, y)\<cdot>(lzip2\<cdot>xs\<cdot>ys)"
+| (unchecked) "lzip2\<cdot>xs\<cdot>ys = lNil"
text {*
Usually fixrec tries to prove all equations as theorems.
- The "permissive" option overrides this behavior, so fixrec
- does not produce any simp rules.
+ The "unchecked" option overrides this behavior, so fixrec
+ does not attempt to prove that particular equation.
*}
text {* Simp rules can be generated later using @{text fixrec_simp}. *}
lemma lzip2_simps [simp]:
- "lzip2\<cdot>(lCons\<cdot>x\<cdot>xs)\<cdot>(lCons\<cdot>y\<cdot>ys) = lCons\<cdot>(x, y)\<cdot>(lzip\<cdot>xs\<cdot>ys)"
"lzip2\<cdot>(lCons\<cdot>x\<cdot>xs)\<cdot>lNil = lNil"
"lzip2\<cdot>lNil\<cdot>(lCons\<cdot>y\<cdot>ys) = lNil"
"lzip2\<cdot>lNil\<cdot>lNil = lNil"
--- a/src/HOLCF/ex/Domain_Proofs.thy Tue Oct 19 15:13:35 2010 +0100
+++ b/src/HOLCF/ex/Domain_Proofs.thy Wed Oct 20 21:26:51 2010 -0700
@@ -82,14 +82,14 @@
text {* Use @{text pcpodef} with the appropriate type combinator. *}
-pcpodef (open) 'a foo = "{x. x ::: foo_defl\<cdot>DEFL('a)}"
-by (simp_all add: adm_in_defl)
+pcpodef (open) 'a foo = "defl_set (foo_defl\<cdot>DEFL('a))"
+by (rule defl_set_bottom, rule adm_defl_set)
-pcpodef (open) 'a bar = "{x. x ::: bar_defl\<cdot>DEFL('a)}"
-by (simp_all add: adm_in_defl)
+pcpodef (open) 'a bar = "defl_set (bar_defl\<cdot>DEFL('a))"
+by (rule defl_set_bottom, rule adm_defl_set)
-pcpodef (open) 'a baz = "{x. x ::: baz_defl\<cdot>DEFL('a)}"
-by (simp_all add: adm_in_defl)
+pcpodef (open) 'a baz = "defl_set (baz_defl\<cdot>DEFL('a))"
+by (rule defl_set_bottom, rule adm_defl_set)
text {* Prove rep instance using lemma @{text typedef_rep_class}. *}
@@ -200,25 +200,25 @@
subsection {* Step 3: Define rep and abs functions *}
-text {* Define them all using @{text coerce}! *}
+text {* Define them all using @{text prj} and @{text emb}! *}
definition foo_rep :: "'a foo \<rightarrow> one \<oplus> ('a\<^sub>\<bottom> \<otimes> ('a bar)\<^sub>\<bottom>)"
-where "foo_rep \<equiv> coerce"
+where "foo_rep \<equiv> prj oo emb"
definition foo_abs :: "one \<oplus> ('a\<^sub>\<bottom> \<otimes> ('a bar)\<^sub>\<bottom>) \<rightarrow> 'a foo"
-where "foo_abs \<equiv> coerce"
+where "foo_abs \<equiv> prj oo emb"
definition bar_rep :: "'a bar \<rightarrow> ('a baz \<rightarrow> tr)\<^sub>\<bottom>"
-where "bar_rep \<equiv> coerce"
+where "bar_rep \<equiv> prj oo emb"
definition bar_abs :: "('a baz \<rightarrow> tr)\<^sub>\<bottom> \<rightarrow> 'a bar"
-where "bar_abs \<equiv> coerce"
+where "bar_abs \<equiv> prj oo emb"
definition baz_rep :: "'a baz \<rightarrow> ('a foo convex_pd \<rightarrow> tr)\<^sub>\<bottom>"
-where "baz_rep \<equiv> coerce"
+where "baz_rep \<equiv> prj oo emb"
definition baz_abs :: "('a foo convex_pd \<rightarrow> tr)\<^sub>\<bottom> \<rightarrow> 'a baz"
-where "baz_abs \<equiv> coerce"
+where "baz_abs \<equiv> prj oo emb"
text {* Prove isomorphism rules. *}