author  blanchet 
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permissions  rwrr 
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(* Title: HOL/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_compat" :: 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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56346  22 
lemma image_Collect_subsetI: "(\<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> underS R j" 
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unfolding underS_def by simp 
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lemma underS_E: "i \<in> underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R" 
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unfolding underS_def by simp 
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lemma underS_Field: "i \<in> underS R j \<Longrightarrow> i \<in> Field R" 
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unfolding 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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55811  52 
lemma convol_expand_snd': 
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assumes "(fst o f = g)" 

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shows "h = snd o f \<longleftrightarrow> <g, h> = f" 

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proof  

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from assms have *: "<g, snd o f> = f" by (rule convol_expand_snd) 

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then have "h = snd o f \<longleftrightarrow> h = snd o <g, snd o f>" by simp 

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moreover have "\<dots> \<longleftrightarrow> h = snd o f" by (simp add: snd_convol) 

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moreover have "\<dots> \<longleftrightarrow> <g, h> = f" by (subst (2) *[symmetric]) (auto simp: convol_def fun_eq_iff) 

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ultimately show ?thesis by simp 

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qed 

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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 f_the_inv_into_f_bij_betw: "bij_betw f A B \<Longrightarrow> 
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(bij_betw f A B \<Longrightarrow> x \<in> B) \<Longrightarrow> f (the_inv_into A f x) = x" 

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unfolding bij_betw_def by (blast intro: f_the_inv_into_f) 

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lemma ex_bij_betw: "A \<le>o (r :: 'b rel) \<Longrightarrow> \<exists>f B :: 'b set. bij_betw f B A" 
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by (subst (asm) internalize_card_of_ordLeq) 

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(auto dest!: iffD2[OF card_of_ordIso ordIso_symmetric]) 

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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 assms) 
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lemma nchotomy_relcomppE: 
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assumes "\<And>y. \<exists>x. y = f x" "(r OO s) a c" "\<And>b. r a (f b) \<Longrightarrow> s (f b) c \<Longrightarrow> P" 
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shows P 

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proof (rule relcompp.cases[OF assms(2)], hypsubst) 

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fix b assume "r a b" "s b c" 

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moreover from assms(1) obtain b' where "b = f b'" by blast 

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ultimately show P by (blast intro: assms(3)) 

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qed 

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lemma vimage2p_rel_fun: "rel_fun (vimage2p f g R) R f g" 
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unfolding rel_fun_def vimage2p_def by auto 

52731  179 

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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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55945  183 
lemma id_transfer: "rel_fun A A id id" 
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unfolding rel_fun_def by simp 

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lemma ssubst_Pair_rhs: "\<lbrakk>(r, s) \<in> R; s' = s\<rbrakk> \<Longrightarrow> (r, s') \<in> R" 
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by (rule ssubst) 
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lemma fun_cong_unused_0: "f = (\<lambda>x. g) \<Longrightarrow> f (\<lambda>x. 0) = g" 
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by (erule arg_cong) 
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lemma snd_o_convol: "(snd \<circ> (\<lambda>x. (f x, g x))) = g" 
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by (rule ext) simp 

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lemma inj_on_convol_id: "inj_on (\<lambda>x. (x, f x)) X" 

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unfolding inj_on_def by simp 

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lemma case_prod_app: "case_prod f x y = case_prod (\<lambda>l r. f l r y) x" 
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by (case_tac x) simp 
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lemma case_sum_o_map_sum: "case_sum l r (map_sum f g x) = case_sum (l \<circ> f) (r \<circ> g) x" 
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by (case_tac x) simp+ 
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lemma case_prod_o_map_prod: "case_prod h (map_prod f g x) = case_prod (\<lambda>l r. h (f l) (g r)) x" 
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by (case_tac x) simp+ 
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lemma prod_inj_map: "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (map_prod f g)" 
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by (simp add: inj_on_def) 
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declare [[ML_print_depth = 10000]] (*###*) 
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ML_file "Tools/BNF/bnf_lfp_util.ML" 
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ML_file "Tools/BNF/bnf_lfp_tactics.ML" 

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ML_file "Tools/BNF/bnf_lfp.ML" 

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ML_file "Tools/BNF/bnf_lfp_compat.ML" 

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ML_file "Tools/BNF/bnf_lfp_rec_sugar_more.ML" 
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ML_file "Tools/BNF/bnf_lfp_size.ML" 
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hide_fact (open) id_transfer 
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datatype_new ('a, 'b) j = J0  J 'a "('a, 'b) j" 
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thm j.size j.rec_o_map j.size_o_map 
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datatype_new 'a l = N nat nat  C 'a "'a l" 
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thm l.size l.rec_o_map l.size_o_map 
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datatype_new ('a, 'b) x = XN 'b  XC 'a "('a, 'b) x" 
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thm x.size x.rec_o_map x.size_o_map 
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datatype_new 
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'a tl = TN  TC "'a mt" "'a tl" and 
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'a mt = MT 'a "'a tl" 
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thm tl.size tl.rec_o_map tl.size_o_map 
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thm mt.size mt.rec_o_map mt.size_o_map 
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datatype_new 'a t = T nat 'a "'a t l" 
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thm t.size t.rec_o_map t.size_o_map 
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datatype_new 'a fset = FSet0  FSet 'a "'a fset" 
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thm fset.size fset.rec_o_map fset.size_o_map 
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datatype_new 'a u = U 'a "'a u fset" 
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thm u.size u.rec_o_map u.size_o_map 
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244 

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datatype_new 
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('a, 'b) v = V "nat l"  V' 'a "('a, 'b) w" and 
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('a, 'b) w = W 'b "('a, 'b) v fset l" 
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thm v.size v.rec_o_map v.size_o_map 
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thm w.size w.rec_o_map w.size_o_map 
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250 

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(*TODO: 
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* deal with *unused* dead variables and other odd cases (e.g. recursion through fun) 
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* what happens if recursion through arbitrary bnf, like 'fsize'? 
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* by default 
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* offer possibility to register size function and theorems 
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* nonrecursive types use 'case' instead of 'rec', causes trouble (revert?) 
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* compat with old size? 
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* recursion of old through new (e.g. through list)? 
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* recursion of new through old? 
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* should they share theory data? 
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* code generator setup? 
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*) 
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48975
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added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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end 
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datatype_new 'a x = X0  X 'a (*###*) 
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thm x.size 
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thm x.size_o_map 
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datatype_new 'a x = X0  X 'a "'a x" (*###*) 
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thm x.size 
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thm x.size_o_map 
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datatype_new 'a l = N  C 'a "'a l" 
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datatype_new ('a, 'b) tl = TN 'b  TC 'a "'a l" 
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end 