src/HOL/BNF/BNF_LFP.thy
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(*  Title:      HOL/BNF/BNF_LFP.thy
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Lorenz Panny, TU Muenchen
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012, 2013
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Least fixed point operation on bounded natural functors.
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*)
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header {* Least Fixed Point Operation on Bounded Natural Functors *}
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theory BNF_LFP
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imports BNF_FP_Base
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keywords
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  "datatype_new" :: thy_decl and
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  "datatype_new_compat" :: thy_decl and
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  "primrec_new" :: thy_decl
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begin
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lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}"
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by blast
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lemma image_Collect_subsetI:
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  "(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B"
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by blast
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lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X"
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by auto
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lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x"
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by auto
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lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> rel.underS R j"
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unfolding rel.underS_def by simp
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lemma underS_E: "i \<in> rel.underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"
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unfolding rel.underS_def by simp
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lemma underS_Field: "i \<in> rel.underS R j \<Longrightarrow> i \<in> Field R"
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unfolding rel.underS_def Field_def by auto
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lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
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unfolding Field_def by auto
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lemma fst_convol': "fst (<f, g> x) = f x"
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using fst_convol unfolding convol_def by simp
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lemma snd_convol': "snd (<f, g> x) = g x"
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using snd_convol unfolding convol_def by simp
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lemma convol_expand_snd: "fst o f = g \<Longrightarrow>  <g, snd o f> = f"
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unfolding convol_def by auto
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lemma convol_expand_snd': "(fst o f = g) \<Longrightarrow> (h = snd o f) \<longleftrightarrow> (<g, h> = f)"
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  by (metis convol_expand_snd snd_convol)
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definition inver where
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  "inver g f A = (ALL a : A. g (f a) = a)"
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lemma bij_betw_iff_ex:
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  "bij_betw f A B = (EX g. g ` B = A \<and> inver g f A \<and> inver f g B)" (is "?L = ?R")
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proof (rule iffI)
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  assume ?L
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  hence f: "f ` A = B" and inj_f: "inj_on f A" unfolding bij_betw_def by auto
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  let ?phi = "% b a. a : A \<and> f a = b"
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  have "ALL b : B. EX a. ?phi b a" using f by blast
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  then obtain g where g: "ALL b : B. g b : A \<and> f (g b) = b"
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    using bchoice[of B ?phi] by blast
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  hence gg: "ALL b : f ` A. g b : A \<and> f (g b) = b" using f by blast
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  have gf: "inver g f A" unfolding inver_def
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    by (metis (no_types) gg imageI[of _ A f] the_inv_into_f_f[OF inj_f])
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  moreover have "g ` B \<le> A \<and> inver f g B" using g unfolding inver_def by blast
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  moreover have "A \<le> g ` B"
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  proof safe
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    fix a assume a: "a : A"
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    hence "f a : B" using f by auto
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    moreover have "a = g (f a)" using a gf unfolding inver_def by auto
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    ultimately show "a : g ` B" by blast
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  qed
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  ultimately show ?R by blast
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next
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  assume ?R
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  then obtain g where g: "g ` B = A \<and> inver g f A \<and> inver f g B" by blast
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  show ?L unfolding bij_betw_def
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  proof safe
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    show "inj_on f A" unfolding inj_on_def
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    proof safe
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      fix a1 a2 assume a: "a1 : A"  "a2 : A" and "f a1 = f a2"
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      hence "g (f a1) = g (f a2)" by simp
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      thus "a1 = a2" using a g unfolding inver_def by simp
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    qed
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  next
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    fix a assume "a : A"
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    then obtain b where b: "b : B" and a: "a = g b" using g by blast
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    hence "b = f (g b)" using g unfolding inver_def by auto
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    thus "f a : B" unfolding a using b by simp
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  next
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    fix b assume "b : B"
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    hence "g b : A \<and> b = f (g b)" using g unfolding inver_def by auto
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    thus "b : f ` A" by auto
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  qed
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qed
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lemma bij_betw_ex_weakE:
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  "\<lbrakk>bij_betw f A B\<rbrakk> \<Longrightarrow> \<exists>g. g ` B \<subseteq> A \<and> inver g f A \<and> inver f g B"
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by (auto simp only: bij_betw_iff_ex)
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lemma inver_surj: "\<lbrakk>g ` B \<subseteq> A; f ` A \<subseteq> B; inver g f A\<rbrakk> \<Longrightarrow> g ` B = A"
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unfolding inver_def by auto (rule rev_image_eqI, auto)
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lemma inver_mono: "\<lbrakk>A \<subseteq> B; inver f g B\<rbrakk> \<Longrightarrow> inver f g A"
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unfolding inver_def by auto
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lemma inver_pointfree: "inver f g A = (\<forall>a \<in> A. (f o g) a = a)"
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unfolding inver_def by simp
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lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
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unfolding bij_betw_def by auto
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lemma bij_betw_imageE: "bij_betw f A B \<Longrightarrow> f ` A = B"
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unfolding bij_betw_def by auto
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lemma inverE: "\<lbrakk>inver f f' A; x \<in> A\<rbrakk> \<Longrightarrow> f (f' x) = x"
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unfolding inver_def by auto
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lemma bij_betw_inver1: "bij_betw f A B \<Longrightarrow> inver (inv_into A f) f A"
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unfolding bij_betw_def inver_def by auto
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lemma bij_betw_inver2: "bij_betw f A B \<Longrightarrow> inver f (inv_into A f) B"
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unfolding bij_betw_def inver_def by auto
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lemma bij_betwI: "\<lbrakk>bij_betw g B A; inver g f A; inver f g B\<rbrakk> \<Longrightarrow> bij_betw f A B"
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by (drule bij_betw_imageE, unfold bij_betw_iff_ex) blast
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lemma bij_betwI':
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  "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
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    \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
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    \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
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unfolding bij_betw_def inj_on_def by blast
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lemma surj_fun_eq:
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  assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
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  shows "g1 = g2"
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proof (rule ext)
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  fix y
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  from surj_on obtain x where "x \<in> X" and "y = f x" by blast
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  thus "g1 y = g2 y" using eq_on by simp
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qed
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lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
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unfolding wo_rel_def card_order_on_def by blast
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lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow>
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  \<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r"
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unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
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lemma Card_order_trans:
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  "\<lbrakk>Card_order r; x \<noteq> y; (x, y) \<in> r; y \<noteq> z; (y, z) \<in> r\<rbrakk> \<Longrightarrow> x \<noteq> z \<and> (x, z) \<in> r"
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unfolding card_order_on_def well_order_on_def linear_order_on_def
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  partial_order_on_def preorder_on_def trans_def antisym_def by blast
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lemma Cinfinite_limit2:
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 assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
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 shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
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proof -
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  from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
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    unfolding card_order_on_def well_order_on_def linear_order_on_def
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      partial_order_on_def preorder_on_def by auto
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  obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
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    using Cinfinite_limit[OF x1 r] by blast
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  obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
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    using Cinfinite_limit[OF x2 r] by blast
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  show ?thesis
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  proof (cases "y1 = y2")
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    case True with y1 y2 show ?thesis by blast
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  next
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    case False
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    with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
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      unfolding total_on_def by auto
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    thus ?thesis
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    proof
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      assume *: "(y1, y2) \<in> r"
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      with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
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      with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
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    next
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      assume *: "(y2, y1) \<in> r"
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      with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
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      with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
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    qed
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  qed
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qed
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lemma Cinfinite_limit_finite: "\<lbrakk>finite X; X \<subseteq> Field r; Cinfinite r\<rbrakk>
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 \<Longrightarrow> \<exists>y \<in> Field r. \<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)"
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proof (induct X rule: finite_induct)
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  case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
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next
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  case (insert x X)
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  then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
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  then obtain z where z: "z \<in> Field r" "x \<noteq> z \<and> (x, z) \<in> r" "y \<noteq> z \<and> (y, z) \<in> r"
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    using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
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  show ?case
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    apply (intro bexI ballI)
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    apply (erule insertE)
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    apply hypsubst
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    apply (rule z(2))
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    using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
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    apply blast
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    apply (rule z(1))
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    done
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qed
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lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
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by auto
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(*helps resolution*)
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lemma well_order_induct_imp:
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  "wo_rel r \<Longrightarrow> (\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> y \<in> Field r \<longrightarrow> P y \<Longrightarrow> x \<in> Field r \<longrightarrow> P x) \<Longrightarrow>
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     x \<in> Field r \<longrightarrow> P x"
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by (erule wo_rel.well_order_induct)
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lemma meta_spec2:
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  assumes "(\<And>x y. PROP P x y)"
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  shows "PROP P x y"
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by (rule `(\<And>x y. PROP P x y)`)
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52731
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lemma vimage2p_fun_rel: "(fun_rel (vimage2p f g R) R) f g"
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  unfolding fun_rel_def vimage2p_def by auto
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lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)"
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  unfolding vimage2p_def by auto
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ML_file "Tools/bnf_lfp_util.ML"
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ML_file "Tools/bnf_lfp_tactics.ML"
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ML_file "Tools/bnf_lfp.ML"
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ML_file "Tools/bnf_lfp_compat.ML"
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7f79f94a432c added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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end