author  traytel 
Sat, 13 Jul 2013 13:03:21 +0200  
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parent 52634  7c4b56bac189 
child 52731  dacd47a0633f 
permissions  rwrr 
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(* Title: HOL/BNF/BNF_LFP.thy 
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Author: Dmitriy Traytel, TU Muenchen 
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Copyright 2012 
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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_Basic 
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keywords 
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"datatype_new" :: thy_decl 
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begin 
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49312  16 
lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}" 
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by blast 

18 

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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_o: "<f, g> o h = <f o h, g o h>" 
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unfolding convol_def by auto 
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49312  50 
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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49312  56 
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 
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apply (rule conjI) 

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apply blast 

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by (erule thin_rl) 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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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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end 