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| section "Relative Frequency LTL" | |
| theory RF_LTL | |
| imports Main "HOL-Library.Sublist" Auxiliary DynamicArchitectures.Dynamic_Architecture_Calculus | |
| begin | |
| type_synonym 's seq = "nat \<Rightarrow> 's" | |
| abbreviation "ccard n n' p \<equiv> card {i. i>n \<and> i \<le> n' \<and> p i}" | |
| lemma ccard_same: | |
| assumes "\<not> p (Suc n')" | |
| shows "ccard n n' p = ccard n (Suc n') p" | |
| proof - | |
| have "{i. i > n \<and> i \<le> Suc n' \<and> p i} = {i. i>n \<and> i \<le> n' \<and> p i}" | |
| proof | |
| show "{i. n < i \<and> i \<le> Suc n' \<and> p i} \<subseteq> {i. n < i \<and> i \<le> n' \<and> p i}" | |
| proof | |
| fix x assume "x \<in> {i. n < i \<and> i \<le> Suc n' \<and> p i}" | |
| hence "n<x" and "x\<le>Suc n'" and "p x" by auto | |
| with assms (1) have "x\<noteq>Suc n'" by auto | |
| with \<open>x\<le>Suc n'\<close> have "x \<le> n'" by simp | |
| with \<open>n<x\<close> \<open>p x\<close> show "x \<in> {i. n < i \<and> i \<le> n' \<and> p i}" by simp | |
| qed | |
| next | |
| show "{i. n < i \<and> i \<le> n' \<and> p i} \<subseteq> {i. n < i \<and> i \<le> Suc n' \<and> p i}" by auto | |
| qed | |
| thus ?thesis by simp | |
| qed | |
| lemma ccard_zero[simp]: | |
| fixes n::nat | |
| shows "ccard n n p = 0" | |
| by auto | |
| lemma ccard_inc: | |
| assumes "p (Suc n')" | |
| and "n' \<ge> n" | |
| shows "ccard n (Suc n') p = Suc (ccard n n' p)" | |
| proof - | |
| let ?A = "{i. i > n \<and> i \<le> n' \<and> p i}" | |
| have "finite ?A" by simp | |
| moreover have "Suc n' \<notin> ?A" by simp | |
| ultimately have "card (insert (Suc n') ?A) = Suc (card ?A)" using card_insert_disjoint[of ?A] by simp | |
| moreover have "insert (Suc n') ?A = {i. i>n \<and> i \<le> (Suc n') \<and> p i}" | |
| proof | |
| show "insert (Suc n') ?A \<subseteq> {i. n < i \<and> i \<le> Suc n' \<and> p i}" | |
| proof | |
| fix x assume "x \<in> insert (Suc n') {i. n < i \<and> i \<le> n' \<and> p i}" | |
| hence "x=Suc n' \<or> n < x \<and> x \<le> n' \<and> p x" by simp | |
| thus "x \<in> {i. n < i \<and> i \<le> Suc n' \<and> p i}" | |
| proof | |
| assume "x = Suc n'" | |
| with assms (1) assms (2) show ?thesis by simp | |
| next | |
| assume "n < x \<and> x \<le> n' \<and> p x" | |
| thus ?thesis by simp | |
| qed | |
| qed | |
| next | |
| show "{i. n < i \<and> i \<le> Suc n' \<and> p i} \<subseteq> insert (Suc n') ?A" by auto | |
| qed | |
| ultimately show ?thesis by simp | |
| qed | |
| lemma ccard_mono: | |
| assumes "n'\<ge>n" | |
| shows "n''\<ge>n' \<Longrightarrow> ccard n (n''::nat) p \<ge> ccard n n' p" | |
| proof (induction n'' rule: dec_induct) | |
| case base | |
| then show ?case .. | |
| next | |
| case (step n'') | |
| then show ?case | |
| proof cases | |
| assume "p (Suc n'')" | |
| moreover from step.hyps assms have "n\<le>n''" by simp | |
| ultimately have "ccard n (Suc n'') p = Suc (ccard n n'' p)" using ccard_inc[of p n'' n] by simp | |
| also have "\<dots> \<ge> ccard n n' p" using step.IH by simp | |
| finally show ?case . | |
| next | |
| assume "\<not> p (Suc n'')" | |
| moreover from step.hyps assms have "n\<le>n''" by simp | |
| ultimately have "ccard n (Suc n'') p = ccard n n'' p" using ccard_same[of p n'' n] by simp | |
| also have "\<dots> \<ge> ccard n n' p" using step.IH by simp | |
| finally show ?case by simp | |
| qed | |
| qed | |
| lemma ccard_ub[simp]: | |
| "ccard n n' p \<le> Suc n' - n" | |
| proof - | |
| have "{i. i>n \<and> i \<le> n' \<and> p i} \<subseteq> {i. i\<ge>n \<and> i \<le> n'}" by auto | |
| hence "ccard n n' p \<le> card {i. i\<ge>n \<and> i \<le> n'}" by (simp add: card_mono) | |
| moreover have "{i. i\<ge>n \<and> i \<le> n'} = {n..n'}" by auto | |
| hence "card {i. i\<ge>n \<and> i \<le> n'} = Suc n' - n" by simp | |
| ultimately show ?thesis by simp | |
| qed | |
| lemma ccard_sum: | |
| fixes n::nat | |
| assumes "n'\<ge>n''" | |
| and "n''\<ge>n" | |
| shows "ccard n n' P = ccard n n'' P + ccard n'' n' P" | |
| proof - | |
| have "ccard n n' P = card {i. i>n \<and> i \<le> n' \<and> P i}" by simp | |
| moreover have "{i. i>n \<and> i \<le> n' \<and> P i} = | |
| {i. i>n \<and> i \<le> n'' \<and> P i} \<union> {i. i>n'' \<and> i \<le> n' \<and> P i}" (is "?LHS = ?RHS") | |
| proof | |
| show "?LHS \<subseteq> ?RHS" by auto | |
| next | |
| show "?RHS \<subseteq> ?LHS" | |
| proof | |
| fix x | |
| assume "x\<in>?RHS" | |
| hence "x>n \<and> x \<le> n'' \<and> P x \<or> x>n'' \<and> x \<le> n' \<and> P x" by auto | |
| thus "x\<in>?LHS" | |
| proof | |
| assume "n < x \<and> x \<le> n'' \<and> P x" | |
| with assms show ?thesis by simp | |
| next | |
| assume "n'' < x \<and> x \<le> n' \<and> P x" | |
| with assms show ?thesis by simp | |
| qed | |
| qed | |
| qed | |
| hence "card ?LHS = card ?RHS" by simp | |
| ultimately have "ccard n n' P = card ?RHS" by simp | |
| moreover have "card ?RHS = card {i. i>n \<and> i \<le> n'' \<and> P i} + card {i. i>n'' \<and> i \<le> n' \<and> P i}" | |
| proof (rule card_Un_disjoint) | |
| show "finite {i. n < i \<and> i \<le> n'' \<and> P i}" by simp | |
| show "finite {i. n'' < i \<and> i \<le> n' \<and> P i}" by simp | |
| show "{i. n < i \<and> i \<le> n'' \<and> P i} \<inter> {i. n'' < i \<and> i \<le> n' \<and> P i} = {}" by auto | |
| qed | |
| moreover have "ccard n n'' P = card {i. i>n \<and> i \<le> n'' \<and> P i}" by simp | |
| moreover have "ccard n'' n' P= card {i. i>n'' \<and> i \<le> n' \<and> P i}" by simp | |
| ultimately show ?thesis by simp | |
| qed | |
| lemma ccard_ex: | |
| fixes n::nat | |
| shows "c\<ge>1 \<Longrightarrow> c < ccard n n'' P \<Longrightarrow> \<exists>n'<n''. n'>n \<and> ccard n n' P = c" | |
| proof (induction c rule: dec_induct) | |
| let ?l = "LEAST i::nat. n < i \<and> i < n'' \<and> P i" | |
| case base | |
| moreover have "ccard n n'' P \<le> Suc (card {i. n < i \<and> i < n'' \<and> P i})" | |
| proof - | |
| from \<open>ccard n n'' P > 1\<close> have "n''>n" using less_le_trans by force | |
| then obtain n' where "Suc n' = n''" and "Suc n' \<ge> n" by (metis lessE less_imp_le_nat) | |
| moreover have "{i. n < i \<and> i < Suc n' \<and> P i} = {i. n < i \<and> i \<le> n' \<and> P i}" by auto | |
| hence "card {i. n < i \<and> i < Suc n' \<and> P i} = card {i. n < i \<and> i \<le> n' \<and> P i}" by simp | |
| moreover have "card {i. n < i \<and> i \<le> Suc n' \<and> P i} \<le> Suc (card {i. n < i \<and> i \<le> n' \<and> P i})" | |
| proof cases | |
| assume "P (Suc n')" | |
| moreover from \<open>n''>n\<close> \<open>Suc n'=n''\<close> have "n'\<ge>n" by simp | |
| ultimately show ?thesis using ccard_inc[of P n' n] by simp | |
| next | |
| assume "\<not> P (Suc n')" | |
| moreover from \<open>n''>n\<close> \<open>Suc n'=n''\<close> have "n'\<ge>n" by simp | |
| ultimately show ?thesis using ccard_same[of P n' n] by simp | |
| qed | |
| ultimately show ?thesis by simp | |
| qed | |
| ultimately have "card {i. n < i \<and> i < n'' \<and> P i} \<ge> 1" by simp | |
| hence "{i. n < i \<and> i < n'' \<and> P i} \<noteq> {}" by fastforce | |
| hence "\<exists>i. n < i \<and> i < n'' \<and> P i" by auto | |
| hence "?l>n" and "?l<n''" and "P ?l" using LeastI_ex[of "\<lambda>i::nat. n < i \<and> i < n'' \<and> P i"] by auto | |
| moreover have "{i. n < i \<and> i \<le> ?l \<and> P i} = {?l}" | |
| proof | |
| show "{i. n < i \<and> i \<le> ?l \<and> P i} \<subseteq> {?l}" | |
| proof | |
| fix i | |
| assume "i\<in>{i. n < i \<and> i \<le> ?l \<and> P i}" | |
| hence "n < i" and "i \<le> ?l" and "P i" by auto | |
| with \<open>\<exists>i. n < i \<and> i < n'' \<and> P i\<close> have "i=?l" | |
| using Least_le[of "\<lambda>i. n < i \<and> i < n'' \<and> P i"] by (meson antisym le_less_trans) | |
| thus "i\<in>{?l}" by simp | |
| qed | |
| next | |
| show "{?l} \<subseteq> {i. n < i \<and> i \<le> ?l \<and> P i}" | |
| proof | |
| fix i | |
| assume "i\<in>{?l}" | |
| hence "i=?l" by simp | |
| with \<open>?l>n\<close> \<open>?l<n''\<close> \<open>P ?l\<close> show "i\<in>{i. n < i \<and> i \<le> ?l \<and> P i}" by simp | |
| qed | |
| qed | |
| hence "ccard n ?l P = 1" by simp | |
| ultimately show ?case by auto | |
| next | |
| case (step c) | |
| moreover from step.prems have "Suc c<ccard n n'' P" by simp | |
| ultimately obtain n' where "n'<n''" and "n < n'" and "ccard n n' P = c" by auto | |
| hence "ccard n n'' P = ccard n n' P + ccard n' n'' P" using ccard_sum[of n' n'' n] by simp | |
| with \<open>Suc c<ccard n n'' P\<close> \<open>ccard n n' P = c\<close> have "ccard n' n'' P>1" by simp | |
| moreover have "ccard n' n'' P \<le> Suc (card {i. n' < i \<and> i < n'' \<and> P i})" | |
| proof - | |
| from \<open>ccard n' n'' P > 1\<close> have "n''>n'" using less_le_trans by force | |
| then obtain n''' where "Suc n''' = n''" and "Suc n''' \<ge> n'" by (metis lessE less_imp_le_nat) | |
| moreover have "{i. n' < i \<and> i < Suc n''' \<and> P i} = {i. n' < i \<and> i \<le> n''' \<and> P i}" by auto | |
| hence "card {i. n' < i \<and> i < Suc n''' \<and> P i} = card {i. n' < i \<and> i \<le> n''' \<and> P i}" by simp | |
| moreover have "card {i. n' < i \<and> i \<le> Suc n''' \<and> P i} \<le> Suc (card {i. n' < i \<and> i \<le> n''' \<and> P i})" | |
| proof cases | |
| assume "P (Suc n''')" | |
| moreover from \<open>n''>n'\<close> \<open>Suc n'''=n''\<close> have "n'''\<ge>n'" by simp | |
| ultimately show ?thesis using ccard_inc[of P n''' n'] by simp | |
| next | |
| assume "\<not> P (Suc n''')" | |
| moreover from \<open>n''>n'\<close> \<open>Suc n'''=n''\<close> have "n'''\<ge>n'" by simp | |
| ultimately show ?thesis using ccard_same[of P n''' n'] by simp | |
| qed | |
| ultimately show ?thesis by simp | |
| qed | |
| ultimately have "card {i. n' < i \<and> i < n'' \<and> P i} \<ge> 1" by simp | |
| hence "{i. n' < i \<and> i < n'' \<and> P i} \<noteq> {}" by fastforce | |
| hence "\<exists>i. n' < i \<and> i < n'' \<and> P i" by auto | |
| let ?l = "LEAST i::nat. n' < i \<and> i < n'' \<and> P i" | |
| from \<open>\<exists>i. n' < i \<and> i < n'' \<and> P i\<close> have "n' < ?l" | |
| using LeastI_ex[of "\<lambda>i::nat. n' < i \<and> i < n'' \<and> P i"] by auto | |
| with \<open>n < n'\<close> have "ccard n ?l P = ccard n n' P + ccard n' ?l P" using ccard_sum[of n' ?l n] by simp | |
| moreover have "{i. n' < i \<and> i \<le> ?l \<and> P i} = {?l}" | |
| proof | |
| show "{i. n' < i \<and> i \<le> ?l \<and> P i} \<subseteq> {?l}" | |
| proof | |
| fix i | |
| assume "i\<in>{i. n' < i \<and> i \<le> ?l \<and> P i}" | |
| hence "n' < i" and "i \<le> ?l" and "P i" by auto | |
| with \<open>\<exists>i. n' < i \<and> i < n'' \<and> P i\<close> have "i=?l" | |
| using Least_le[of "\<lambda>i. n' < i \<and> i < n'' \<and> P i"] by (meson antisym le_less_trans) | |
| thus "i\<in>{?l}" by simp | |
| qed | |
| next | |
| show "{?l} \<subseteq> {i. n' < i \<and> i \<le> ?l \<and> P i}" | |
| proof | |
| fix i | |
| assume "i\<in>{?l}" | |
| hence "i=?l" by simp | |
| moreover from \<open>\<exists>i. n' < i \<and> i < n'' \<and> P i\<close> have "?l<n''" and "P ?l" | |
| using LeastI_ex[of "\<lambda>i. n' < i \<and> i < n'' \<and> P i"] by auto | |
| ultimately show "i\<in>{i. n' < i \<and> i \<le> ?l \<and> P i}" using \<open>?l>n'\<close> by simp | |
| qed | |
| qed | |
| hence "ccard n' ?l P = 1" by simp | |
| ultimately have "card {i. n < i \<and> i \<le> ?l \<and> P i} = Suc c" using \<open>ccard n n' P = c\<close> by simp | |
| moreover from \<open>\<exists>i. n' < i \<and> i < n'' \<and> P i\<close> have "n' < ?l" and "?l < n''" and "P ?l" | |
| using LeastI_ex[of "\<lambda>i::nat. n' < i \<and> i < n'' \<and> P i"] by auto | |
| with \<open>n < n'\<close> have "n<?l" and "?l<n''" by auto | |
| ultimately show ?case by auto | |
| qed | |
| lemma ccard_freq: | |
| assumes "(n'::nat)\<ge>n" | |
| and "ccard n n' P > ccard n n' Q + cnf" | |
| shows "\<exists>n' n''. ccard n' n'' P > cnf \<and> ccard n' n'' Q \<le> cnf" | |
| proof cases | |
| assume "cnf = 0" | |
| with assms(2) have "ccard n n' P > ccard n n' Q" by simp | |
| hence "card {i. n < i \<and> i \<le> n' \<and> P i}>card {i. n < i \<and> i \<le> n' \<and> Q i}" (is "card ?LHS>card ?RHS") | |
| by simp | |
| then obtain i where "i\<in>?LHS" and "\<not> i \<in> ?RHS" and "i>0" using cardEx[of ?LHS ?RHS] by auto | |
| hence "P i" and "\<not> Q i" by auto | |
| with \<open>i>0\<close> obtain n'' where "P (Suc n'')" and "\<not>Q (Suc n'')" using gr0_implies_Suc by auto | |
| hence "ccard n'' (Suc n'') P = 1" using ccard_inc by auto | |
| with \<open>cnf = 0\<close> have "ccard n'' (Suc n'') P > cnf" by simp | |
| moreover from \<open>\<not>Q (Suc n'')\<close> have "ccard n'' (Suc n'') Q = 0" using ccard_same[of Q n'' n''] by auto | |
| with \<open>cnf = 0\<close> have "ccard n'' (Suc n'') Q \<le> cnf" by simp | |
| ultimately show ?thesis by auto | |
| next | |
| assume "\<not> cnf = 0" | |
| show ?thesis | |
| proof (rule ccontr) | |
| assume "\<not> (\<exists>n' n''. ccard n' n'' P > cnf \<and> ccard n' n'' Q \<le> cnf)" | |
| hence hyp: "\<forall>n' n''. ccard n' n'' Q \<le> cnf \<longrightarrow> ccard n' n'' P \<le> cnf" | |
| using leI less_imp_le_nat by blast | |
| show False | |
| proof cases | |
| assume "ccard n n' Q \<le> cnf" | |
| with hyp have "ccard n n' P \<le> cnf" by simp | |
| with assms show False by simp | |
| next | |
| let ?gcond="\<lambda>n''. n''\<ge>n \<and> n''\<le>n' \<and> (\<exists>x\<ge>1. ccard n n'' Q = x * cnf)" | |
| let ?g= "GREATEST n''. ?gcond n''" | |
| assume "\<not> ccard n n' Q \<le> cnf" | |
| hence "ccard n n' Q > cnf" by simp | |
| hence "\<exists>n''. ?gcond n''" | |
| proof - | |
| from \<open>ccard n n' Q > cnf\<close> \<open>\<not>cnf=0\<close> obtain n'' where "n''>n" and "n''\<le>n'" and "ccard n n'' Q = cnf" | |
| using ccard_ex[of cnf n n' Q ] by auto | |
| moreover from \<open>ccard n n'' Q = cnf\<close> have "\<exists>x\<ge>1. ccard n n'' Q = x * cnf" by auto | |
| ultimately show ?thesis using less_imp_le_nat by blast | |
| qed | |
| moreover have "\<forall>n''>n'. \<not> ?gcond n''" by simp | |
| ultimately have gex: "\<exists>n''. ?gcond n'' \<and> (\<forall>n'''. ?gcond n''' \<longrightarrow> n'''\<le>n'')" | |
| using boundedGreatest[of ?gcond _ n'] by blast | |
| hence "\<exists>x\<ge>1. ccard n ?g Q = x * cnf" and "?g \<ge> n" using GreatestI_ex_nat[of ?gcond] by auto | |
| moreover {fix n'' | |
| have "n''\<ge>n \<Longrightarrow> \<exists>x\<ge>1. ccard n n'' Q = x * cnf \<Longrightarrow> ccard n n'' P \<le> ccard n n'' Q" | |
| proof (induction n'' rule: ge_induct) | |
| case (step n') | |
| from step.prems obtain x where "x\<ge>1" and cas: "ccard n n' Q = x * cnf" by auto | |
| then show ?case | |
| proof cases | |
| assume "x=1" | |
| with cas have "ccard n n' Q = cnf" by simp | |
| with hyp have "ccard n n' P \<le> cnf" by simp | |
| with \<open>ccard n n' Q = cnf\<close> show ?thesis by simp | |
| next | |
| assume "\<not>x=1" | |
| with \<open>x\<ge>1\<close> have "x>1" by simp | |
| hence "x-1 \<ge> 1" by simp | |
| moreover from \<open>cnf\<noteq>0\<close> \<open>x-1 \<ge> 1\<close> have "(x-1) * cnf < x * cnf \<and> (x - 1) * cnf \<noteq> 0" by auto | |
| with \<open>x-1 \<ge> 1\<close> \<open>cnf\<noteq>0\<close>\<open>ccard n n' Q = x * cnf\<close> obtain n'' | |
| where "n''>n" and "n''<n'" and "ccard n n'' Q = (x-1) * cnf" | |
| using ccard_ex[of "(x-1)*cnf" n n' Q ] by auto | |
| ultimately have "\<exists>x\<ge>1. ccard n n'' Q = x * cnf" and "n''\<ge>n" by auto | |
| with \<open>n''\<ge>n\<close> \<open>n''<n'\<close> have "ccard n n'' P \<le> ccard n n'' Q" using step.IH by simp | |
| moreover have "ccard n'' n' Q = cnf" | |
| proof - | |
| from \<open>x-1 \<ge> 1\<close> have "x*cnf = ((x-1) * cnf) + cnf" | |
| using semiring_normalization_rules(2)[of "(x - 1)" cnf] by simp | |
| with \<open>ccard n n'' Q = (x-1) * cnf\<close> \<open>ccard n n' Q = x * cnf\<close> | |
| have "ccard n n' Q = ccard n n'' Q + cnf" by simp | |
| moreover from \<open>n''\<ge>n\<close> \<open>n''<n'\<close> have "ccard n n' Q = ccard n n'' Q + ccard n'' n' Q" | |
| using ccard_sum[of n'' n' n] by simp | |
| ultimately show ?thesis by simp | |
| qed | |
| moreover from \<open>ccard n'' n' Q = cnf\<close> have "ccard n'' n' P \<le> cnf" using hyp by simp | |
| ultimately show ?thesis using \<open>n''\<ge>n\<close> \<open>n''<n'\<close> ccard_sum[of n'' n' n] by simp | |
| qed | |
| qed } note geq = this | |
| ultimately have "ccard n ?g P \<le> ccard n ?g Q" by simp | |
| moreover have "ccard ?g n' P \<le> cnf" | |
| proof (rule ccontr) | |
| assume "\<not> ccard ?g n' P \<le> cnf" | |
| hence "ccard ?g n' P > cnf" by simp | |
| have "ccard ?g n' Q > cnf" | |
| proof (rule ccontr) | |
| assume "\<not>ccard ?g n' Q > cnf" | |
| hence "ccard ?g n' Q \<le> cnf" by simp | |
| with \<open>ccard ?g n' P > cnf\<close> show False | |
| using \<open>\<not> (\<exists>n' n''. ccard n' n'' P > cnf \<and> ccard n' n'' Q \<le> cnf)\<close> by simp | |
| qed | |
| with \<open>\<not> cnf=0\<close> obtain n'' where "n''>?g" and "n''<n'" and "ccard ?g n'' Q = cnf" | |
| using ccard_ex[of cnf ?g n' Q] by auto | |
| moreover have "\<exists>x\<ge>1. ccard n n'' Q = x * cnf" | |
| proof - | |
| from \<open>\<exists>x\<ge>1. ccard n ?g Q = x * cnf\<close> obtain x where "x\<ge>1" and "ccard n ?g Q = x * cnf" by auto | |
| from \<open>n''>?g\<close> \<open>?g\<ge>n\<close> have "ccard n n'' Q = ccard n ?g Q + ccard ?g n'' Q" | |
| using ccard_sum[of ?g n'' n Q] by simp | |
| with \<open>ccard n ?g Q = x * cnf\<close> have "ccard n n'' Q = x * cnf + ccard ?g n'' Q" by simp | |
| with \<open>ccard ?g n'' Q = cnf\<close> have "ccard n n'' Q = Suc x * cnf" by simp | |
| thus ?thesis by auto | |
| qed | |
| moreover from \<open>n''>?g\<close> \<open>?g\<ge>n\<close> have "n''\<ge>n" by simp | |
| ultimately have "\<exists>n''>?g. ?gcond n''" by auto | |
| moreover from gex have "\<forall>n'''. ?gcond n''' \<longrightarrow> n'''\<le>?g" using Greatest_le_nat[of ?gcond] by auto | |
| ultimately show False by auto | |
| qed | |
| moreover from gex have "n'\<ge>?g" using GreatestI_ex_nat[of ?gcond] by auto | |
| ultimately have "ccard n n' P\<le>ccard n n' Q + cnf" using ccard_sum[of ?g n' n] using \<open>?g \<ge> n\<close> by simp | |
| with assms show False by simp | |
| qed | |
| qed | |
| qed | |
| locale honest = | |
| fixes bc:: "('a list) seq" | |
| and n::nat | |
| assumes growth: "n'\<noteq>0 \<Longrightarrow> n'\<le>n \<Longrightarrow> bc n' = bc (n'-1) \<or> (\<exists>b. bc n' = bc (n' - 1) @ b)" | |
| begin | |
| end | |
| locale dishonest = | |
| fixes bc:: "('a list) seq" | |
| and mining::"bool seq" | |
| assumes growth: "\<And>n::nat. prefix (bc (Suc n)) (bc n) \<or> (\<exists>b::'a. bc (Suc n) = bc n @ [b]) \<and> mining (Suc n)" | |
| begin | |
| lemma prefix_save: | |
| assumes "prefix sbc (bc n')" | |
| and "\<forall>n'''>n'. n'''\<le>n'' \<longrightarrow> length (bc n''') \<ge> length sbc" | |
| shows "n''\<ge>n' \<Longrightarrow> prefix sbc (bc n'')" | |
| proof (induction n'' rule: dec_induct) | |
| case base | |
| with assms(1) show ?case by simp | |
| next | |
| case (step n) | |
| from growth[of n] show ?case | |
| proof | |
| assume "prefix (bc (Suc n)) (bc n)" | |
| moreover from step.hyps have "length (bc (Suc n)) \<ge> length sbc" using assms(2) by simp | |
| ultimately show ?thesis using step.IH using prefix_length_prefix by auto | |
| next | |
| assume "(\<exists>b. bc (Suc n) = bc n @ [b]) \<and> mining (Suc n)" | |
| with step.IH show ?thesis by auto | |
| qed | |
| qed | |
| theorem prefix_length: | |
| assumes "prefix sbc (bc n')" and "\<not> prefix sbc (bc n'')" and "n'\<le>n''" | |
| shows "\<exists>n'''>n'. n'''\<le>n'' \<and> length (bc n''') < length sbc" | |
| proof (rule ccontr) | |
| assume "\<not> (\<exists>n'''>n'. n'''\<le>n'' \<and> length (bc n''') < length sbc)" | |
| hence "\<forall>n'''>n'. n'''\<le>n'' \<longrightarrow> length (bc n''') \<ge> length sbc" by auto | |
| with assms have "prefix sbc (bc n'')" using prefix_save[of sbc n' n''] by simp | |
| with assms (2) show False by simp | |
| qed | |
| theorem grow_mining: | |
| assumes "length (bc n) < length (bc (Suc n))" | |
| shows "mining (Suc n)" | |
| using assms growth leD prefix_length_le by blast | |
| lemma length_suc_length: | |
| "length (bc (Suc n)) \<le> Suc (length (bc n))" | |
| by (metis eq_iff growth le_SucI length_append_singleton prefix_length_le) | |
| end | |
| locale dishonest_growth = | |
| fixes bc:: "nat seq" | |
| and mining:: "nat \<Rightarrow> bool" | |
| assumes as1: "\<And>n::nat. bc (Suc n) \<le> Suc (bc n)" | |
| and as2: "\<And>n::nat. bc (Suc n) > bc n \<Longrightarrow> mining (Suc n)" | |
| begin | |
| end | |
| sublocale dishonest \<subseteq> dishonest_growth "\<lambda>n. length (bc n)" using grow_mining length_suc_length by unfold_locales auto | |
| context dishonest_growth | |
| begin | |
| theorem ccard_diff_lgth: | |
| "n'\<ge>n \<Longrightarrow> ccard n n' (\<lambda>n. mining n) \<ge> (bc n' - bc n)" | |
| proof (induction n' rule: dec_induct) | |
| case base | |
| then show ?case by simp | |
| next | |
| case (step n') | |
| from as1 have "bc (Suc n') < Suc (bc n') \<or> bc (Suc n') = Suc (bc n')" | |
| using le_neq_implies_less by blast | |
| then show ?case | |
| proof | |
| assume "bc (Suc n') < Suc (bc n')" | |
| hence "bc (Suc n') - bc n \<le> bc n' - bc n" by simp | |
| moreover from step.hyps have "ccard n (Suc n') (\<lambda>n. mining n) \<ge> ccard n n' (\<lambda>n. mining n)" | |
| using ccard_mono[of n n' "Suc n'"] by simp | |
| ultimately show ?thesis using step.IH by simp | |
| next | |
| assume "bc (Suc n') = Suc (bc n')" | |
| hence "bc (Suc n') - bc n \<le> Suc (bc n' - bc n)" by simp | |
| moreover from \<open>bc (Suc n') = Suc (bc n')\<close> have "mining (Suc n')" using as2 by simp | |
| with step.hyps have "ccard n (Suc n') (\<lambda>n. mining n) \<ge> Suc (ccard n n' (\<lambda>n. mining n))" | |
| using ccard_inc by simp | |
| ultimately show ?thesis using step.IH by simp | |
| qed | |
| qed | |
| end | |
| locale honest_growth = | |
| fixes bc:: "nat seq" | |
| and mining:: "nat \<Rightarrow> bool" | |
| and init:: nat | |
| assumes as1: "\<And>n::nat. bc (Suc n) \<ge> bc n" | |
| and as2: "\<And>n::nat. mining (Suc n) \<Longrightarrow> bc (Suc n) > bc n" | |
| begin | |
| lemma grow_mono: "n'\<ge>n\<Longrightarrow>bc n'\<ge>bc n" | |
| proof (induction n' rule: dec_induct) | |
| case base | |
| then show ?case by simp | |
| next | |
| case (step n') | |
| then show ?case using as1[of n'] by simp | |
| qed | |
| theorem ccard_diff_lgth: | |
| shows "n'\<ge>n \<Longrightarrow> bc n' - bc n \<ge> ccard n n' (\<lambda>n. mining n)" | |
| proof (induction n' rule: dec_induct) | |
| case base | |
| then show ?case by simp | |
| next | |
| case (step n') | |
| then show ?case | |
| proof cases | |
| assume "mining (Suc n')" | |
| with as2 have "bc (Suc n') > bc n'" by simp | |
| moreover from step.hyps have "bc n'\<ge>bc n" using grow_mono by simp | |
| ultimately have "bc (Suc n') - bc n > bc n' - bc n" by simp | |
| moreover from as1 have "bc (Suc n') - bc n \<ge> bc n' - bc n" by (simp add: diff_le_mono) | |
| moreover from \<open>mining (Suc n')\<close> step.hyps | |
| have "ccard n (Suc n') (\<lambda>n. mining n) \<le> Suc (ccard n n' (\<lambda>n. mining n))" | |
| using ccard_inc by simp | |
| ultimately show ?thesis using step.IH by simp | |
| next | |
| assume "\<not> mining (Suc n')" | |
| hence "ccard n (Suc n') (\<lambda>n. mining n) \<le> (ccard n n' (\<lambda>n. mining n))" using ccard_same by simp | |
| moreover from as1 have "bc (Suc n') - bc n \<ge> bc n' - bc n" by (simp add: diff_le_mono) | |
| ultimately show ?thesis using step.IH by simp | |
| qed | |
| qed | |
| end | |
| locale bounded_growth = hg: honest_growth hbc hmining + dg: dishonest_growth dbc dmining | |
| for hbc:: "nat seq" | |
| and dbc:: "nat seq" | |
| and hmining:: "nat \<Rightarrow> bool" | |
| and dmining:: "nat \<Rightarrow> bool" | |
| and sbc::nat | |
| and cnf::nat + | |
| assumes fair: "\<And>n n'. ccard n n' (\<lambda>n. dmining n) > cnf \<Longrightarrow> ccard n n' (\<lambda>n. hmining n) > cnf" | |
| and a2: "hbc 0 \<ge> sbc+cnf" | |
| and a3: "dbc 0 < sbc" | |
| begin | |
| theorem hn_upper_bound: shows "dbc n < hbc n" | |
| proof (rule ccontr) | |
| assume "\<not> dbc n < hbc n" | |
| hence "dbc n \<ge> hbc n" by simp | |
| moreover from a2 a3 have "hbc 0 > dbc 0 + cnf" by simp | |
| moreover have "hbc n\<ge>hbc 0" using hg.grow_mono by simp | |
| ultimately have "dbc n - dbc 0 > hbc n - hbc 0 + cnf" by simp | |
| moreover have "ccard 0 n (\<lambda>n. hmining n) \<le> hbc n - hbc 0" using hg.ccard_diff_lgth by simp | |
| moreover have "dbc n - dbc 0 \<le> ccard 0 n (\<lambda>n. dmining n)" using dg.ccard_diff_lgth by simp | |
| ultimately have "ccard 0 n (\<lambda>n. dmining n) > ccard 0 n (\<lambda>n. hmining n) + cnf" by simp | |
| hence "\<exists>n' n''. ccard n' n'' (\<lambda>n. dmining n) > cnf \<and> ccard n' n'' (\<lambda>n. hmining n) \<le> cnf" | |
| using ccard_freq by blast | |
| with fair show False using leD by blast | |
| qed | |
| end | |
| end |