--- a/src/HOL/Library/Multiset.thy Wed Jul 11 11:23:24 2007 +0200
+++ b/src/HOL/Library/Multiset.thy Wed Jul 11 11:24:36 2007 +0200
@@ -386,28 +386,28 @@
subsubsection {* Well-foundedness *}
definition
- mult1 :: "('a => 'a => bool) => 'a multiset => 'a multiset => bool" where
+ mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
"mult1 r =
- (%N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
- (\<forall>b. b :# K --> r b a))"
+ {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
+ (\<forall>b. b :# K --> (b, a) \<in> r)}"
definition
- mult :: "('a => 'a => bool) => 'a multiset => 'a multiset => bool" where
- "mult r = (mult1 r)\<^sup>+\<^sup>+"
+ mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
+ "mult r = (mult1 r)\<^sup>+"
-lemma not_less_empty [iff]: "\<not> mult1 r M {#}"
+lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
by (simp add: mult1_def)
-lemma less_add: "mult1 r N (M0 + {#a#})==>
- (\<exists>M. mult1 r M M0 \<and> N = M + {#a#}) \<or>
- (\<exists>K. (\<forall>b. b :# K --> r b a) \<and> N = M0 + K)"
+lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
+ (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
+ (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
(is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
proof (unfold mult1_def)
- let ?r = "\<lambda>K a. \<forall>b. b :# K --> r b a"
+ let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
- let ?case1 = "?case1 ?R"
+ let ?case1 = "?case1 {(N, M). ?R N M}"
- assume "?R N (M0 + {#a#})"
+ assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
then have "\<exists>a' M0' K.
M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
then show "?case1 \<or> ?case2"
@@ -435,80 +435,80 @@
qed
qed
-lemma all_accessible: "wfP r ==> \<forall>M. acc (mult1 r) M"
+lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
proof
let ?R = "mult1 r"
let ?W = "acc ?R"
{
fix M M0 a
- assume M0: "?W M0"
- and wf_hyp: "!!b. r b a ==> \<forall>M \<triangleright> ?W. ?W (M + {#b#})"
- and acc_hyp: "\<forall>M. ?R M M0 --> ?W (M + {#a#})"
- have "?W (M0 + {#a#})"
- proof (rule accI [of _ "M0 + {#a#}"])
+ assume M0: "M0 \<in> ?W"
+ and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
+ and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
+ have "M0 + {#a#} \<in> ?W"
+ proof (rule accI [of "M0 + {#a#}"])
fix N
- assume "?R N (M0 + {#a#})"
- then have "((\<exists>M. ?R M M0 \<and> N = M + {#a#}) \<or>
- (\<exists>K. (\<forall>b. b :# K --> r b a) \<and> N = M0 + K))"
+ assume "(N, M0 + {#a#}) \<in> ?R"
+ then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
+ (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
by (rule less_add)
- then show "?W N"
+ then show "N \<in> ?W"
proof (elim exE disjE conjE)
- fix M assume "?R M M0" and N: "N = M + {#a#}"
- from acc_hyp have "?R M M0 --> ?W (M + {#a#})" ..
- from this and `?R M M0` have "?W (M + {#a#})" ..
- then show "?W N" by (simp only: N)
+ fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
+ from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
+ from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
+ then show "N \<in> ?W" by (simp only: N)
next
fix K
assume N: "N = M0 + K"
- assume "\<forall>b. b :# K --> r b a"
- then have "?W (M0 + K)"
+ assume "\<forall>b. b :# K --> (b, a) \<in> r"
+ then have "M0 + K \<in> ?W"
proof (induct K)
case empty
- from M0 show "?W (M0 + {#})" by simp
+ from M0 show "M0 + {#} \<in> ?W" by simp
next
case (add K x)
- from add.prems have "r x a" by simp
- with wf_hyp have "\<forall>M \<triangleright> ?W. ?W (M + {#x#})" by blast
- moreover from add have "?W (M0 + K)" by simp
- ultimately have "?W ((M0 + K) + {#x#})" ..
- then show "?W (M0 + (K + {#x#}))" by (simp only: union_assoc)
+ from add.prems have "(x, a) \<in> r" by simp
+ with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
+ moreover from add have "M0 + K \<in> ?W" by simp
+ ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
+ then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
qed
- then show "?W N" by (simp only: N)
+ then show "N \<in> ?W" by (simp only: N)
qed
qed
} note tedious_reasoning = this
- assume wf: "wfP r"
+ assume wf: "wf r"
fix M
- show "?W M"
+ show "M \<in> ?W"
proof (induct M)
- show "?W {#}"
+ show "{#} \<in> ?W"
proof (rule accI)
- fix b assume "?R b {#}"
- with not_less_empty show "?W b" by contradiction
+ fix b assume "(b, {#}) \<in> ?R"
+ with not_less_empty show "b \<in> ?W" by contradiction
qed
- fix M a assume "?W M"
- from wf have "\<forall>M \<triangleright> ?W. ?W (M + {#a#})"
+ fix M a assume "M \<in> ?W"
+ from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
proof induct
fix a
- assume r: "!!b. r b a ==> \<forall>M \<triangleright> ?W. ?W (M + {#b#})"
- show "\<forall>M \<triangleright> ?W. ?W (M + {#a#})"
+ assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
+ show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
proof
- fix M assume "?W M"
- then show "?W (M + {#a#})"
+ fix M assume "M \<in> ?W"
+ then show "M + {#a#} \<in> ?W"
by (rule acc_induct) (rule tedious_reasoning [OF _ r])
qed
qed
- from this and `?W M` show "?W (M + {#a#})" ..
+ from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
qed
qed
-theorem wf_mult1: "wfP r ==> wfP (mult1 r)"
+theorem wf_mult1: "wf r ==> wf (mult1 r)"
by (rule acc_wfI) (rule all_accessible)
-theorem wf_mult: "wfP r ==> wfP (mult r)"
- unfolding mult_def by (rule wfP_trancl) (rule wf_mult1)
+theorem wf_mult: "wf r ==> wf (mult r)"
+ unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
subsubsection {* Closure-free presentation *}
@@ -521,16 +521,16 @@
text {* One direction. *}
lemma mult_implies_one_step:
- "transP r ==> mult r M N ==>
+ "trans r ==> (M, N) \<in> mult r ==>
\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
- (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. r k j)"
+ (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
apply (unfold mult_def mult1_def set_of_def)
- apply (erule converse_trancl_induct', clarify)
+ apply (erule converse_trancl_induct, clarify)
apply (rule_tac x = M0 in exI, simp, clarify)
- apply (case_tac "a :# Ka")
+ apply (case_tac "a :# K")
apply (rule_tac x = I in exI)
apply (simp (no_asm))
- apply (rule_tac x = "(Ka - {#a#}) + K" in exI)
+ apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
apply (simp (no_asm_simp) add: union_assoc [symmetric])
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
apply (simp add: diff_union_single_conv)
@@ -561,29 +561,30 @@
done
lemma one_step_implies_mult_aux:
- "\<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. r k j))
- --> mult r (I + K) (I + J)"
+ "trans r ==>
+ \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
+ --> (I + K, I + J) \<in> mult r"
apply (induct_tac n, auto)
apply (frule size_eq_Suc_imp_eq_union, clarify)
apply (rename_tac "J'", simp)
apply (erule notE, auto)
apply (case_tac "J' = {#}")
apply (simp add: mult_def)
- apply (rule trancl.r_into_trancl)
+ apply (rule r_into_trancl)
apply (simp add: mult1_def set_of_def, blast)
txt {* Now we know @{term "J' \<noteq> {#}"}. *}
- apply (cut_tac M = K and P = "\<lambda>x. r x a" in multiset_partition)
+ apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
apply (erule ssubst)
apply (simp add: Ball_def, auto)
apply (subgoal_tac
- "mult r ((I + {# x : K. r x a #}) + {# x : K. \<not> r x a #})
- ((I + {# x : K. r x a #}) + J')")
+ "((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #},
+ (I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r")
prefer 2
apply force
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
- apply (erule trancl_trans')
- apply (rule trancl.r_into_trancl)
+ apply (erule trancl_trans)
+ apply (rule r_into_trancl)
apply (simp add: mult1_def set_of_def)
apply (rule_tac x = a in exI)
apply (rule_tac x = "I + J'" in exI)
@@ -591,8 +592,8 @@
done
lemma one_step_implies_mult:
- "J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. r k j
- ==> mult r (I + K) (I + J)"
+ "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
+ ==> (I + K, I + J) \<in> mult r"
using one_step_implies_mult_aux by blast
@@ -601,10 +602,10 @@
instance multiset :: (type) ord ..
defs (overloaded)
- less_multiset_def: "op < == mult op <"
+ less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}"
le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)"
-lemma trans_base_order: "transP (op < :: 'a::order => 'a => bool)"
+lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
unfolding trans_def by (blast intro: order_less_trans)
text {*
@@ -629,7 +630,7 @@
text {* Transitivity. *}
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
- unfolding less_multiset_def mult_def by (blast intro: trancl_trans')
+ unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
text {* Asymmetry. *}
@@ -665,17 +666,17 @@
instance multiset :: (order) order
apply intro_classes
- apply (rule mult_less_le)
- apply (rule mult_le_refl)
- apply (erule mult_le_trans, assumption)
- apply (erule mult_le_antisym, assumption)
+ apply (rule mult_less_le)
+ apply (rule mult_le_refl)
+ apply (erule mult_le_trans, assumption)
+ apply (erule mult_le_antisym, assumption)
done
subsubsection {* Monotonicity of multiset union *}
lemma mult1_union:
- "mult1 r B D ==> mult1 r (C + B) (C + D)"
+ "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
apply (unfold mult1_def, auto)
apply (rule_tac x = a in exI)
apply (rule_tac x = "C + M0" in exI)
@@ -684,9 +685,9 @@
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
apply (unfold less_multiset_def mult_def)
- apply (erule trancl_induct')
- apply (blast intro: mult1_union)
- apply (blast intro: mult1_union trancl.r_into_trancl trancl_trans')
+ apply (erule trancl_induct)
+ apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
+ apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
done
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
@@ -709,10 +710,10 @@
apply (unfold le_multiset_def less_multiset_def)
apply (case_tac "M = {#}")
prefer 2
- apply (subgoal_tac "mult op < ({#} + {#}) ({#} + M)")
+ apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
prefer 2
apply (rule one_step_implies_mult)
- apply auto
+ apply (simp only: trans_def, auto)
done
lemma union_upper1: "A <= A + (B::'a::order multiset)"