author | blanchet |
Wed, 23 Apr 2014 10:23:26 +0200 | |
changeset 56640 | 0a35354137a5 |
parent 56639 | c9d6b581bd3b |
child 56641 | 029997d3b5d8 |
permissions | -rw-r--r-- |
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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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||
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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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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 |
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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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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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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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(*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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* non-recursive 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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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 |