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author | haftmann |

Thu, 12 May 2016 09:34:07 +0200 | |

changeset 63088 | f2177f5d2aed |

parent 63087 | be252979cfe5 |

child 63089 | 40134ddec3bf |

a quasi-recursive characterization of the multiset order (by Christian Sternagel)

src/HOL/Library/Multiset.thy | file | annotate | diff | comparison | revisions | |

src/HOL/Wellfounded.thy | file | annotate | diff | comparison | revisions |

--- a/src/HOL/Library/Multiset.thy Thu May 12 11:34:19 2016 +0200 +++ b/src/HOL/Library/Multiset.thy Thu May 12 09:34:07 2016 +0200 @@ -2066,6 +2066,117 @@ \<Longrightarrow> (I + K, I + J) \<in> mult r" using one_step_implies_mult_aux by blast +subsection \<open>A quasi-recursive characterization\<close> + +text \<open> + The decreasing parts \<open>A\<close> and \<open>B\<close> of multisets in a multiset-comparison + \<open>(I + B, I + A) \<in> mult r\<close>, can always be made disjoint. +\<close> +lemma decreasing_parts_disj: + assumes "irrefl r" and "trans r" + and "A \<noteq> {#}" and *: "\<forall>b\<in>#B. \<exists>a\<in>#A. (b, a) \<in> r" + defines "Z \<equiv> A #\<inter> B" + defines "X \<equiv> A - Z" + defines "Y \<equiv> B - Z" + shows "X \<noteq> {#} \<and> X #\<inter> Y = {#} \<and> + A = X + Z \<and> B = Y + Z \<and> (\<forall>y\<in>#Y. \<exists>x\<in>#X. (y, x) \<in> r)" +proof - + define D + where "D = set_mset A \<union> set_mset B" + let ?r = "r \<inter> D \<times> D" + have "irrefl ?r" and "trans ?r" and "finite ?r" + using \<open>irrefl r\<close> and \<open>trans r\<close> by (auto simp: D_def irrefl_def trans_Restr) + note wf_converse_induct = wf_induct [OF wf_converse [OF this]] + { fix b assume "b \<in># B" + then have "\<exists>x. x \<in># X \<and> (b, x) \<in> r" + proof (induction rule: wf_converse_induct) + case (1 b) + then obtain a where "b \<in># B" and "a \<in># A" and "(b, a) \<in> r" + using * by blast + then show ?case + proof (cases "a \<in># X") + case False + then have "a \<in># B" using \<open>a \<in># A\<close> + by (simp add: X_def Z_def not_in_iff) + (metis le_zero_eq not_in_iff) + moreover have "(a, b) \<in> (r \<inter> D \<times> D)\<inverse>" + using \<open>(b, a) \<in> r\<close> using \<open>b \<in># B\<close> and \<open>a \<in># B\<close> + by (auto simp: D_def) + ultimately obtain x where "x \<in># X" and "(a, x) \<in> r" + using "1.IH" by blast + moreover then have "(b, x) \<in> r" + using \<open>trans r\<close> and \<open>(b, a) \<in> r\<close> by (auto dest: transD) + ultimately show ?thesis by blast + qed blast + qed } + note B_less = this + then have "\<forall>y\<in>#Y. \<exists>x\<in>#X. (y, x) \<in> r" + by (auto simp: Y_def Z_def dest: in_diffD) + moreover have "X \<noteq> {#}" + proof - + obtain a where "a \<in># A" using \<open>A \<noteq> {#}\<close> + by (auto simp: multiset_eq_iff) + show ?thesis + proof (cases "a \<in># X") + case False + then have "a \<in># B" using \<open>a \<in># A\<close> + by (simp add: X_def Z_def not_in_iff) + (metis le_zero_eq not_in_iff) + then show ?thesis by (auto dest: B_less) + qed auto + qed + moreover have "A = X + Z" and "B = Y + Z" and "X #\<inter> Y = {#}" + by (auto simp: X_def Y_def Z_def multiset_eq_iff) + ultimately show ?thesis by blast +qed + +text \<open> + A predicate variant of the reflexive closure of \<open>mult\<close>, which is + executable whenever the given predicate \<open>P\<close> is. Together with the + standard code equations for \<open>op #\<inter>\<close> and \<open>op -\<close> this should yield + a quadratic (with respect to calls to \<open>P\<close>) implementation of \<open>multeqp\<close>. +\<close> +definition multeqp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where + "multeqp P N M = + (let Z = M #\<inter> N; X = M - Z; Y = N - Z in + (\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x))" + +lemma multeqp_iff: + assumes "irrefl R" and "trans R" + and [simp]: "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R" + shows "multeqp P N M \<longleftrightarrow> (N, M) \<in> (mult R)\<^sup>=" +proof + assume "(N, M) \<in> (mult R)\<^sup>=" + then show "multeqp P N M" + proof + assume "(N, M) \<in> mult R" + from mult_implies_one_step [OF \<open>trans R\<close> this] obtain I J K + where *: "I \<noteq> {#}" "\<forall>j\<in>#J. \<exists>i\<in>#I. (j, i) \<in> R" + and [simp]: "M = K + I" "N = K + J" by blast + from decreasing_parts_disj [OF \<open>irrefl R\<close> \<open>trans R\<close> *] + show "multeqp P N M" + by (auto simp: multeqp_def split: if_splits) + next + assume "(N, M) \<in> Id" then show "multeqp P N M" by (auto simp: multeqp_def) + qed +next + assume "multeqp P N M" + then obtain X Y Z where *: "Z = M #\<inter> N" "X = M - Z" "Y = N - Z" + and **: "\<forall>y\<in>#Y. \<exists>x\<in>#X. (y, x) \<in> R" by (auto simp: multeqp_def Let_def) + then have M: "M = Z + X" and N: "N = Z + Y" by (auto simp: multiset_eq_iff) + show "(N, M) \<in> (mult R)\<^sup>=" + proof (cases "X \<noteq> {#}") + case True + with * and ** have "(Z + Y, Z + X) \<in> mult R" + by (auto intro: one_step_implies_mult) + then show ?thesis by (simp add: M N) + next + case False + then show ?thesis using ** + by (cases "Y = {#}") (auto simp: M N) + qed +qed + subsubsection \<open>Partial-order properties\<close>

--- a/src/HOL/Wellfounded.thy Thu May 12 11:34:19 2016 +0200 +++ b/src/HOL/Wellfounded.thy Thu May 12 09:34:07 2016 +0200 @@ -459,6 +459,20 @@ apply (erule acyclic_converse [THEN iffD2]) done +text \<open> + Observe that the converse of an irreflexive, transitive, + and finite relation is again well-founded. Thus, we may + employ it for well-founded induction. +\<close> +lemma wf_converse: + assumes "irrefl r" and "trans r" and "finite r" + shows "wf (r\<inverse>)" +proof - + have "acyclic r" + using \<open>irrefl r\<close> and \<open>trans r\<close> by (simp add: irrefl_def acyclic_irrefl) + with \<open>finite r\<close> show ?thesis by (rule finite_acyclic_wf_converse) +qed + lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r" by (blast intro: finite_acyclic_wf wf_acyclic)