--- a/src/HOL/Cardinals/Wellfounded_More_Base.thy Mon Nov 18 18:04:44 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,194 +0,0 @@
-(* Title: HOL/Cardinals/Wellfounded_More_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-More on well-founded relations (base).
-*)
-
-header {* More on Well-Founded Relations (Base) *}
-
-theory Wellfounded_More_Base
-imports Order_Relation_More_Base "~~/src/HOL/Library/Wfrec"
-begin
-
-
-text {* This section contains some variations of results in the theory
-@{text "Wellfounded.thy"}:
-\begin{itemize}
-\item means for slightly more direct definitions by well-founded recursion;
-\item variations of well-founded induction;
-\item means for proving a linear order to be a well-order.
-\end{itemize} *}
-
-
-subsection {* Well-founded recursion via genuine fixpoints *}
-
-
-(*2*)lemma wfrec_fixpoint:
-fixes r :: "('a * 'a) set" and
- H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
-assumes WF: "wf r" and ADM: "adm_wf r H"
-shows "wfrec r H = H (wfrec r H)"
-proof(rule ext)
- fix x
- have "wfrec r H x = H (cut (wfrec r H) r x) x"
- using wfrec[of r H] WF by simp
- also
- {have "\<And> y. (y,x) : r \<Longrightarrow> (cut (wfrec r H) r x) y = (wfrec r H) y"
- by (auto simp add: cut_apply)
- hence "H (cut (wfrec r H) r x) x = H (wfrec r H) x"
- using ADM adm_wf_def[of r H] by auto
- }
- finally show "wfrec r H x = H (wfrec r H) x" .
-qed
-
-
-
-subsection {* Characterizations of well-founded-ness *}
-
-
-text {* A transitive relation is well-founded iff it is ``locally" well-founded,
-i.e., iff its restriction to the lower bounds of of any element is well-founded. *}
-
-(*3*)lemma trans_wf_iff:
-assumes "trans r"
-shows "wf r = (\<forall>a. wf(r Int (r^-1``{a} \<times> r^-1``{a})))"
-proof-
- obtain R where R_def: "R = (\<lambda> a. r Int (r^-1``{a} \<times> r^-1``{a}))" by blast
- {assume *: "wf r"
- {fix a
- have "wf(R a)"
- using * R_def wf_subset[of r "R a"] by auto
- }
- }
- (* *)
- moreover
- {assume *: "\<forall>a. wf(R a)"
- have "wf r"
- proof(unfold wf_def, clarify)
- fix phi a
- assume **: "\<forall>a. (\<forall>b. (b,a) \<in> r \<longrightarrow> phi b) \<longrightarrow> phi a"
- obtain chi where chi_def: "chi = (\<lambda>b. (b,a) \<in> r \<longrightarrow> phi b)" by blast
- with * have "wf (R a)" by auto
- hence "(\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b) \<longrightarrow> (\<forall>b. chi b)"
- unfolding wf_def by blast
- moreover
- have "\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b"
- proof(auto simp add: chi_def R_def)
- fix b
- assume 1: "(b,a) \<in> r" and 2: "\<forall>c. (c, b) \<in> r \<and> (c, a) \<in> r \<longrightarrow> phi c"
- hence "\<forall>c. (c, b) \<in> r \<longrightarrow> phi c"
- using assms trans_def[of r] by blast
- thus "phi b" using ** by blast
- qed
- ultimately have "\<forall>b. chi b" by (rule mp)
- with ** chi_def show "phi a" by blast
- qed
- }
- ultimately show ?thesis using R_def by blast
-qed
-
-
-text {* The next lemma is a variation of @{text "wf_eq_minimal"} from Wellfounded,
-allowing one to assume the set included in the field. *}
-
-(*2*)lemma wf_eq_minimal2:
-"wf r = (\<forall>A. A <= Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r))"
-proof-
- let ?phi = "\<lambda> A. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r)"
- have "wf r = (\<forall>A. ?phi A)"
- by (auto simp: ex_in_conv [THEN sym], erule wfE_min, assumption, blast)
- (rule wfI_min, metis)
- (* *)
- also have "(\<forall>A. ?phi A) = (\<forall>B \<le> Field r. ?phi B)"
- proof
- assume "\<forall>A. ?phi A"
- thus "\<forall>B \<le> Field r. ?phi B" by simp
- next
- assume *: "\<forall>B \<le> Field r. ?phi B"
- show "\<forall>A. ?phi A"
- proof(clarify)
- fix A::"'a set" assume **: "A \<noteq> {}"
- obtain B where B_def: "B = A Int (Field r)" by blast
- show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r"
- proof(cases "B = {}")
- assume Case1: "B = {}"
- obtain a where 1: "a \<in> A \<and> a \<notin> Field r"
- using ** Case1 unfolding B_def by blast
- hence "\<forall>a' \<in> A. (a',a) \<notin> r" using 1 unfolding Field_def by blast
- thus ?thesis using 1 by blast
- next
- assume Case2: "B \<noteq> {}" have 1: "B \<le> Field r" unfolding B_def by blast
- obtain a where 2: "a \<in> B \<and> (\<forall>a' \<in> B. (a',a) \<notin> r)"
- using Case2 1 * by blast
- have "\<forall>a' \<in> A. (a',a) \<notin> r"
- proof(clarify)
- fix a' assume "a' \<in> A" and **: "(a',a) \<in> r"
- hence "a' \<in> B" unfolding B_def Field_def by blast
- thus False using 2 ** by blast
- qed
- thus ?thesis using 2 unfolding B_def by blast
- qed
- qed
- qed
- finally show ?thesis by blast
-qed
-
-subsection {* Characterizations of well-founded-ness *}
-
-text {* The next lemma and its corollary enable one to prove that
-a linear order is a well-order in a way which is more standard than
-via well-founded-ness of the strict version of the relation. *}
-
-(*3*)
-lemma Linear_order_wf_diff_Id:
-assumes LI: "Linear_order r"
-shows "wf(r - Id) = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
-proof(cases "r \<le> Id")
- assume Case1: "r \<le> Id"
- hence temp: "r - Id = {}" by blast
- hence "wf(r - Id)" by (simp add: temp)
- moreover
- {fix A assume *: "A \<le> Field r" and **: "A \<noteq> {}"
- obtain a where 1: "r = {} \<or> r = {(a,a)}" using LI
- unfolding order_on_defs using Case1 rel.Total_subset_Id by metis
- hence "A = {a} \<and> r = {(a,a)}" using * ** unfolding Field_def by blast
- hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" using 1 by blast
- }
- ultimately show ?thesis by blast
-next
- assume Case2: "\<not> r \<le> Id"
- hence 1: "Field r = Field(r - Id)" using Total_Id_Field LI
- unfolding order_on_defs by blast
- show ?thesis
- proof
- assume *: "wf(r - Id)"
- show "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
- proof(clarify)
- fix A assume **: "A \<le> Field r" and ***: "A \<noteq> {}"
- hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id"
- using 1 * unfolding wf_eq_minimal2 by simp
- moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
- using rel.Linear_order_in_diff_Id[of r] ** LI by blast
- ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" by blast
- qed
- next
- assume *: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
- show "wf(r - Id)"
- proof(unfold wf_eq_minimal2, clarify)
- fix A assume **: "A \<le> Field(r - Id)" and ***: "A \<noteq> {}"
- hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r"
- using 1 * by simp
- moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
- using rel.Linear_order_in_diff_Id[of r] ** LI mono_Field[of "r - Id" r] by blast
- ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id" by blast
- qed
- qed
-qed
-
-(*3*)corollary Linear_order_Well_order_iff:
-assumes "Linear_order r"
-shows "Well_order r = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
-using assms unfolding well_order_on_def using Linear_order_wf_diff_Id[of r] by blast
-
-end