--- a/src/HOLCF/Universal.thy Fri Mar 13 13:06:36 2009 +0100
+++ b/src/HOLCF/Universal.thy Fri Mar 13 07:30:47 2009 -0700
@@ -13,35 +13,35 @@
definition
node :: "nat \<Rightarrow> ubasis \<Rightarrow> ubasis set \<Rightarrow> ubasis"
where
- "node i x A = Suc (prod2nat (i, prod2nat (x, set2nat A)))"
+ "node i a S = Suc (prod2nat (i, prod2nat (a, set2nat S)))"
-lemma node_not_0 [simp]: "node i x A \<noteq> 0"
+lemma node_not_0 [simp]: "node i a S \<noteq> 0"
unfolding node_def by simp
-lemma node_gt_0 [simp]: "0 < node i x A"
+lemma node_gt_0 [simp]: "0 < node i a S"
unfolding node_def by simp
lemma node_inject [simp]:
- "\<lbrakk>finite A; finite B\<rbrakk>
- \<Longrightarrow> node i x A = node j y B \<longleftrightarrow> i = j \<and> x = y \<and> A = B"
+ "\<lbrakk>finite S; finite T\<rbrakk>
+ \<Longrightarrow> node i a S = node j b T \<longleftrightarrow> i = j \<and> a = b \<and> S = T"
unfolding node_def by simp
-lemma node_gt0: "i < node i x A"
+lemma node_gt0: "i < node i a S"
unfolding node_def less_Suc_eq_le
by (rule le_prod2nat_1)
-lemma node_gt1: "x < node i x A"
+lemma node_gt1: "a < node i a S"
unfolding node_def less_Suc_eq_le
by (rule order_trans [OF le_prod2nat_1 le_prod2nat_2])
lemma nat_less_power2: "n < 2^n"
by (induct n) simp_all
-lemma node_gt2: "\<lbrakk>finite A; y \<in> A\<rbrakk> \<Longrightarrow> y < node i x A"
+lemma node_gt2: "\<lbrakk>finite S; b \<in> S\<rbrakk> \<Longrightarrow> b < node i a S"
unfolding node_def less_Suc_eq_le set2nat_def
apply (rule order_trans [OF _ le_prod2nat_2])
apply (rule order_trans [OF _ le_prod2nat_2])
-apply (rule order_trans [where y="setsum (op ^ 2) {y}"])
+apply (rule order_trans [where y="setsum (op ^ 2) {b}"])
apply (simp add: nat_less_power2 [THEN order_less_imp_le])
apply (erule setsum_mono2, simp, simp)
done
@@ -52,7 +52,7 @@
lemma node_cases:
assumes 1: "x = 0 \<Longrightarrow> P"
- assumes 2: "\<And>i y A. \<lbrakk>finite A; x = node i y A\<rbrakk> \<Longrightarrow> P"
+ assumes 2: "\<And>i a S. \<lbrakk>finite S; x = node i a S\<rbrakk> \<Longrightarrow> P"
shows "P"
apply (cases x)
apply (erule 1)
@@ -65,7 +65,7 @@
lemma node_induct:
assumes 1: "P 0"
- assumes 2: "\<And>i x A. \<lbrakk>P x; finite A; \<forall>y\<in>A. P y\<rbrakk> \<Longrightarrow> P (node i x A)"
+ assumes 2: "\<And>i a S. \<lbrakk>P a; finite S; \<forall>b\<in>S. P b\<rbrakk> \<Longrightarrow> P (node i a S)"
shows "P x"
apply (induct x rule: nat_less_induct)
apply (case_tac n rule: node_cases)
@@ -78,13 +78,13 @@
inductive
ubasis_le :: "nat \<Rightarrow> nat \<Rightarrow> bool"
where
- ubasis_le_refl: "ubasis_le x x"
+ ubasis_le_refl: "ubasis_le a a"
| ubasis_le_trans:
- "\<lbrakk>ubasis_le x y; ubasis_le y z\<rbrakk> \<Longrightarrow> ubasis_le x z"
+ "\<lbrakk>ubasis_le a b; ubasis_le b c\<rbrakk> \<Longrightarrow> ubasis_le a c"
| ubasis_le_lower:
- "finite A \<Longrightarrow> ubasis_le x (node i x A)"
+ "finite S \<Longrightarrow> ubasis_le a (node i a S)"
| ubasis_le_upper:
- "\<lbrakk>finite A; y \<in> A; ubasis_le x y\<rbrakk> \<Longrightarrow> ubasis_le (node i x A) y"
+ "\<lbrakk>finite S; b \<in> S; ubasis_le a b\<rbrakk> \<Longrightarrow> ubasis_le (node i a S) b"
lemma ubasis_le_minimal: "ubasis_le 0 x"
apply (induct x rule: node_induct)
@@ -99,8 +99,8 @@
ubasis_until :: "(ubasis \<Rightarrow> bool) \<Rightarrow> ubasis \<Rightarrow> ubasis"
where
"ubasis_until P 0 = 0"
-| "finite A \<Longrightarrow> ubasis_until P (node i x A) =
- (if P (node i x A) then node i x A else ubasis_until P x)"
+| "finite S \<Longrightarrow> ubasis_until P (node i a S) =
+ (if P (node i a S) then node i a S else ubasis_until P a)"
apply clarify
apply (rule_tac x=b in node_cases)
apply simp
@@ -157,8 +157,8 @@
done
lemma ubasis_until_mono:
- assumes "\<And>i x A y. \<lbrakk>finite A; P (node i x A); y \<in> A; ubasis_le x y\<rbrakk> \<Longrightarrow> P y"
- shows "ubasis_le x y \<Longrightarrow> ubasis_le (ubasis_until P x) (ubasis_until P y)"
+ assumes "\<And>i a S b. \<lbrakk>finite S; P (node i a S); b \<in> S; ubasis_le a b\<rbrakk> \<Longrightarrow> P b"
+ shows "ubasis_le a b \<Longrightarrow> ubasis_le (ubasis_until P a) (ubasis_until P b)"
apply (induct set: ubasis_le)
apply (rule ubasis_le_refl)
apply (erule (1) ubasis_le_trans)
@@ -510,6 +510,12 @@
lemma rank_le_iff: "rank x \<le> n \<longleftrightarrow> cb_take n x = x"
by (rule iffI [OF rank_leD rank_leI])
+lemma rank_compact_bot [simp]: "rank compact_bot = 0"
+using rank_leI [of 0 compact_bot] by simp
+
+lemma rank_eq_0_iff [simp]: "rank x = 0 \<longleftrightarrow> x = compact_bot"
+using rank_le_iff [of x 0] by auto
+
definition
rank_le :: "'a compact_basis \<Rightarrow> 'a compact_basis set"
where
@@ -558,15 +564,15 @@
lemma rank_lt_Un_rank_eq: "rank_lt x \<union> rank_eq x = rank_le x"
unfolding rank_lt_def rank_eq_def rank_le_def by auto
-subsubsection {* Reordering of basis elements *}
+subsubsection {* Sequencing basis elements *}
definition
- reorder :: "'a compact_basis \<Rightarrow> nat"
+ place :: "'a compact_basis \<Rightarrow> nat"
where
- "reorder x = card (rank_lt x) + choose_pos (rank_eq x) x"
+ "place x = card (rank_lt x) + choose_pos (rank_eq x) x"
-lemma reorder_bounded: "reorder x < card (rank_le x)"
-unfolding reorder_def
+lemma place_bounded: "place x < card (rank_le x)"
+unfolding place_def
apply (rule ord_less_eq_trans)
apply (rule add_strict_left_mono)
apply (rule choose_pos_bounded)
@@ -579,53 +585,77 @@
apply (simp add: rank_lt_Un_rank_eq)
done
-lemma reorder_ge: "card (rank_lt x) \<le> reorder x"
-unfolding reorder_def by simp
+lemma place_ge: "card (rank_lt x) \<le> place x"
+unfolding place_def by simp
-lemma reorder_rank_mono:
+lemma place_rank_mono:
fixes x y :: "'a compact_basis"
- shows "rank x < rank y \<Longrightarrow> reorder x < reorder y"
-apply (rule less_le_trans [OF reorder_bounded])
-apply (rule order_trans [OF _ reorder_ge])
+ shows "rank x < rank y \<Longrightarrow> place x < place y"
+apply (rule less_le_trans [OF place_bounded])
+apply (rule order_trans [OF _ place_ge])
apply (rule card_mono)
apply (rule finite_rank_lt)
apply (simp add: rank_le_def rank_lt_def subset_eq)
done
-lemma reorder_eqD: "reorder x = reorder y \<Longrightarrow> x = y"
+lemma place_eqD: "place x = place y \<Longrightarrow> x = y"
apply (rule linorder_cases [where x="rank x" and y="rank y"])
- apply (drule reorder_rank_mono, simp)
- apply (simp add: reorder_def)
+ apply (drule place_rank_mono, simp)
+ apply (simp add: place_def)
apply (rule inj_on_choose_pos [where A="rank_eq x", THEN inj_onD])
apply (rule finite_rank_eq)
apply (simp cong: rank_lt_cong rank_eq_cong)
apply (simp add: rank_eq_def)
apply (simp add: rank_eq_def)
- apply (drule reorder_rank_mono, simp)
+ apply (drule place_rank_mono, simp)
done
-lemma inj_reorder: "inj reorder"
-by (rule inj_onI, erule reorder_eqD)
+lemma inj_place: "inj place"
+by (rule inj_onI, erule place_eqD)
subsubsection {* Embedding and projection on basis elements *}
+definition
+ sub :: "'a compact_basis \<Rightarrow> 'a compact_basis"
+where
+ "sub x = (case rank x of 0 \<Rightarrow> compact_bot | Suc k \<Rightarrow> cb_take k x)"
+
+lemma rank_sub_less: "x \<noteq> compact_bot \<Longrightarrow> rank (sub x) < rank x"
+unfolding sub_def
+apply (cases "rank x", simp)
+apply (simp add: less_Suc_eq_le)
+apply (rule rank_leI)
+apply (rule cb_take_idem)
+done
+
+lemma place_sub_less: "x \<noteq> compact_bot \<Longrightarrow> place (sub x) < place x"
+apply (rule place_rank_mono)
+apply (erule rank_sub_less)
+done
+
+lemma sub_below: "sub x \<sqsubseteq> x"
+unfolding sub_def by (cases "rank x", simp_all add: cb_take_less)
+
+lemma rank_less_imp_below_sub: "\<lbrakk>x \<sqsubseteq> y; rank x < rank y\<rbrakk> \<Longrightarrow> x \<sqsubseteq> sub y"
+unfolding sub_def
+apply (cases "rank y", simp)
+apply (simp add: less_Suc_eq_le)
+apply (subgoal_tac "cb_take nat x \<sqsubseteq> cb_take nat y")
+apply (simp add: rank_leD)
+apply (erule cb_take_mono)
+done
+
function
basis_emb :: "'a compact_basis \<Rightarrow> ubasis"
where
"basis_emb x = (if x = compact_bot then 0 else
- node
- (reorder x)
- (case rank x of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> basis_emb (cb_take k x))
- (basis_emb ` {y. reorder y < reorder x \<and> x \<sqsubseteq> y}))"
+ node (place x) (basis_emb (sub x))
+ (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y}))"
by auto
termination basis_emb
-apply (relation "measure reorder", simp)
-apply simp
-apply (rule reorder_rank_mono)
-apply (simp add: less_Suc_eq_le)
-apply (rule rank_leI)
-apply (rule cb_take_idem)
+apply (relation "measure place", simp)
+apply (simp add: place_sub_less)
apply simp
done
@@ -634,101 +664,68 @@
lemma basis_emb_compact_bot [simp]: "basis_emb compact_bot = 0"
by (simp add: basis_emb.simps)
-lemma fin1: "finite {y. reorder y < reorder x \<and> x \<sqsubseteq> y}"
+lemma fin1: "finite {y. place y < place x \<and> x \<sqsubseteq> y}"
apply (subst Collect_conj_eq)
apply (rule finite_Int)
apply (rule disjI1)
-apply (subgoal_tac "finite (reorder -` {n. n < reorder x})", simp)
-apply (rule finite_vimageI [OF _ inj_reorder])
+apply (subgoal_tac "finite (place -` {n. n < place x})", simp)
+apply (rule finite_vimageI [OF _ inj_place])
apply (simp add: lessThan_def [symmetric])
done
-lemma fin2: "finite (basis_emb ` {y. reorder y < reorder x \<and> x \<sqsubseteq> y})"
+lemma fin2: "finite (basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y})"
by (rule finite_imageI [OF fin1])
-lemma basis_emb_mono [OF refl]:
- "\<lbrakk>n = max (reorder x) (reorder y); x \<sqsubseteq> y\<rbrakk>
- \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)"
-proof (induct n arbitrary: x y rule: less_induct)
+lemma rank_place_mono:
+ "\<lbrakk>place x < place y; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> rank x < rank y"
+apply (rule linorder_cases, assumption)
+apply (simp add: place_def cong: rank_lt_cong rank_eq_cong)
+apply (drule choose_pos_lessD)
+apply (rule finite_rank_eq)
+apply (simp add: rank_eq_def)
+apply (simp add: rank_eq_def)
+apply simp
+apply (drule place_rank_mono, simp)
+done
+
+lemma basis_emb_mono:
+ "x \<sqsubseteq> y \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)"
+proof (induct n \<equiv> "max (place x) (place y)" arbitrary: x y rule: less_induct)
case (less n)
- assume IH:
- "\<And>(m::nat) (x::'a compact_basis) y.
- \<lbrakk>m < n; m = max (reorder x) (reorder y); x \<sqsubseteq> y\<rbrakk>
- \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)"
- assume n: "n = max (reorder x) (reorder y)"
- assume less: "x \<sqsubseteq> y"
- show ?case
- proof (cases)
- assume "x = compact_bot"
- thus ?case by (simp add: ubasis_le_minimal)
- next
- assume x_neq [simp]: "x \<noteq> compact_bot"
- with less have y_neq [simp]: "y \<noteq> compact_bot"
- apply clarify
- apply (drule antisym_less [OF compact_bot_minimal])
- apply simp
+ hence IH:
+ "\<And>(a::'a compact_basis) b.
+ \<lbrakk>max (place a) (place b) < max (place x) (place y); a \<sqsubseteq> b\<rbrakk>
+ \<Longrightarrow> ubasis_le (basis_emb a) (basis_emb b)"
+ by simp
+ show ?case proof (rule linorder_cases)
+ assume "place x < place y"
+ then have "rank x < rank y"
+ using `x \<sqsubseteq> y` by (rule rank_place_mono)
+ with `place x < place y` show ?case
+ apply (case_tac "y = compact_bot", simp)
+ apply (simp add: basis_emb.simps [of y])
+ apply (rule ubasis_le_trans [OF _ ubasis_le_lower [OF fin2]])
+ apply (rule IH)
+ apply (simp add: less_max_iff_disj)
+ apply (erule place_sub_less)
+ apply (erule rank_less_imp_below_sub [OF `x \<sqsubseteq> y`])
done
- show ?case
- proof (rule linorder_cases)
- assume 1: "reorder x < reorder y"
- show ?case
- proof (rule linorder_cases)
- assume "rank x < rank y"
- with 1 show ?case
- apply (case_tac "rank y", simp)
- apply (subst basis_emb.simps [where x=y])
- apply simp
- apply (rule ubasis_le_trans [OF _ ubasis_le_lower [OF fin2]])
- apply (rule IH [OF _ refl, unfolded n])
- apply (simp add: less_max_iff_disj)
- apply (rule reorder_rank_mono)
- apply (simp add: less_Suc_eq_le)
- apply (rule rank_leI)
- apply (rule cb_take_idem)
- apply (simp add: less_Suc_eq_le)
- apply (subgoal_tac "cb_take nat x \<sqsubseteq> cb_take nat y")
- apply (simp add: rank_leD)
- apply (rule cb_take_mono [OF less])
- done
- next
- assume "rank x = rank y"
- with 1 show ?case
- apply (simp add: reorder_def)
- apply (simp cong: rank_lt_cong rank_eq_cong)
- apply (drule choose_pos_lessD)
- apply (rule finite_rank_eq)
- apply (simp add: rank_eq_def)
- apply (simp add: rank_eq_def)
- apply (simp add: less)
- done
- next
- assume "rank x > rank y"
- hence "reorder x > reorder y"
- by (rule reorder_rank_mono)
- with 1 show ?case by simp
- qed
- next
- assume "reorder x = reorder y"
- hence "x = y" by (rule reorder_eqD)
- thus ?case by (simp add: ubasis_le_refl)
- next
- assume "reorder x > reorder y"
- with less show ?case
- apply (simp add: basis_emb.simps [where x=x])
- apply (rule ubasis_le_upper [OF fin2], simp)
- apply (cases "rank x")
- apply (simp add: ubasis_le_minimal)
- apply simp
- apply (rule IH [OF _ refl, unfolded n])
- apply (simp add: less_max_iff_disj)
- apply (rule reorder_rank_mono)
- apply (simp add: less_Suc_eq_le)
- apply (rule rank_leI)
- apply (rule cb_take_idem)
- apply (erule rev_trans_less)
- apply (rule cb_take_less)
- done
- qed
+ next
+ assume "place x = place y"
+ hence "x = y" by (rule place_eqD)
+ thus ?case by (simp add: ubasis_le_refl)
+ next
+ assume "place x > place y"
+ with `x \<sqsubseteq> y` show ?case
+ apply (case_tac "x = compact_bot", simp add: ubasis_le_minimal)
+ apply (simp add: basis_emb.simps [of x])
+ apply (rule ubasis_le_upper [OF fin2], simp)
+ apply (rule IH)
+ apply (simp add: less_max_iff_disj)
+ apply (erule place_sub_less)
+ apply (erule rev_trans_less)
+ apply (rule sub_below)
+ done
qed
qed
@@ -740,14 +737,14 @@
apply (simp add: basis_emb.simps)
apply (simp add: basis_emb.simps)
apply (simp add: basis_emb.simps)
- apply (simp add: fin2 inj_eq [OF inj_reorder])
+ apply (simp add: fin2 inj_eq [OF inj_place])
done
definition
- basis_prj :: "nat \<Rightarrow> 'a compact_basis"
+ basis_prj :: "ubasis \<Rightarrow> 'a compact_basis"
where
"basis_prj x = inv basis_emb
- (ubasis_until (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)) x)"
+ (ubasis_until (\<lambda>x. x \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> ubasis)) x)"
lemma basis_prj_basis_emb: "\<And>x. basis_prj (basis_emb x) = x"
unfolding basis_prj_def
@@ -758,8 +755,8 @@
done
lemma basis_prj_node:
- "\<lbrakk>finite A; node i x A \<notin> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)\<rbrakk>
- \<Longrightarrow> basis_prj (node i x A) = (basis_prj x :: 'a compact_basis)"
+ "\<lbrakk>finite S; node i a S \<notin> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)\<rbrakk>
+ \<Longrightarrow> basis_prj (node i a S) = (basis_prj a :: 'a compact_basis)"
unfolding basis_prj_def by simp
lemma basis_prj_0: "basis_prj 0 = compact_bot"
@@ -767,32 +764,41 @@
apply (rule basis_prj_basis_emb)
done
-lemma basis_prj_mono: "ubasis_le x y \<Longrightarrow> basis_prj x \<sqsubseteq> basis_prj y"
- apply (erule ubasis_le.induct)
- apply (rule refl_less)
- apply (erule (1) trans_less)
- apply (case_tac "node i x A \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
- apply (erule rangeE, rename_tac a)
- apply (case_tac "a = compact_bot", simp)
- apply (simp add: basis_prj_basis_emb)
- apply (simp add: basis_emb.simps)
- apply (clarsimp simp add: fin2)
- apply (case_tac "rank a", simp)
- apply (simp add: basis_prj_0)
- apply (simp add: basis_prj_basis_emb)
- apply (rule cb_take_less)
- apply (simp add: basis_prj_node)
- apply (case_tac "node i x A \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
- apply (erule rangeE, rename_tac a)
- apply (case_tac "a = compact_bot", simp)
- apply (simp add: basis_prj_basis_emb)
- apply (simp add: basis_emb.simps)
- apply (clarsimp simp add: fin2)
- apply (case_tac "rank a", simp add: basis_prj_basis_emb)
- apply (simp add: basis_prj_basis_emb)
- apply (simp add: basis_prj_node)
+lemma node_eq_basis_emb_iff:
+ "finite S \<Longrightarrow> node i a S = basis_emb x \<longleftrightarrow>
+ x \<noteq> compact_bot \<and> i = place x \<and> a = basis_emb (sub x) \<and>
+ S = basis_emb ` {y. place y < place x \<and> x \<sqsubseteq> y}"
+apply (cases "x = compact_bot", simp)
+apply (simp add: basis_emb.simps [of x])
+apply (simp add: fin2)
done
+lemma basis_prj_mono: "ubasis_le a b \<Longrightarrow> basis_prj a \<sqsubseteq> basis_prj b"
+proof (induct a b rule: ubasis_le.induct)
+ case (ubasis_le_refl a) show ?case by (rule refl_less)
+next
+ case (ubasis_le_trans a b c) thus ?case by - (rule trans_less)
+next
+ case (ubasis_le_lower S a i) thus ?case
+ apply (case_tac "node i a S \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
+ apply (erule rangeE, rename_tac x)
+ apply (simp add: basis_prj_basis_emb)
+ apply (simp add: node_eq_basis_emb_iff)
+ apply (simp add: basis_prj_basis_emb)
+ apply (rule sub_below)
+ apply (simp add: basis_prj_node)
+ done
+next
+ case (ubasis_le_upper S b a i) thus ?case
+ apply (case_tac "node i a S \<in> range (basis_emb :: 'a compact_basis \<Rightarrow> nat)")
+ apply (erule rangeE, rename_tac x)
+ apply (simp add: basis_prj_basis_emb)
+ apply (clarsimp simp add: node_eq_basis_emb_iff)
+ apply (simp add: basis_prj_basis_emb)
+ apply (simp add: basis_prj_node)
+ done
+qed
+
lemma basis_emb_prj_less: "ubasis_le (basis_emb (basis_prj x)) x"
unfolding basis_prj_def
apply (subst f_inv_f [where f=basis_emb])
@@ -806,7 +812,8 @@
node
choose
choose_pos
- reorder
+ place
+ sub
subsubsection {* EP-pair from any bifinite domain into @{typ udom} *}