author | haftmann |
Thu, 23 Nov 2017 17:03:27 +0000 | |
changeset 67087 | 733017b19de9 |
parent 64438 | f91cae6c1d74 |
child 67091 | 1393c2340eec |
permissions | -rw-r--r-- |
62479 | 1 |
(* Title: HOL/Nonstandard_Analysis/HyperNat.thy |
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Author: Jacques D. Fleuriot |
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Copyright: 1998 University of Cambridge |
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Converted to Isar and polished by lcp |
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*) |
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section \<open>Hypernatural numbers\<close> |
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theory HyperNat |
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imports StarDef |
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begin |
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type_synonym hypnat = "nat star" |
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abbreviation hypnat_of_nat :: "nat \<Rightarrow> nat star" |
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where "hypnat_of_nat \<equiv> star_of" |
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definition hSuc :: "hypnat \<Rightarrow> hypnat" |
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where hSuc_def [transfer_unfold]: "hSuc = *f* Suc" |
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subsection \<open>Properties Transferred from Naturals\<close> |
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lemma hSuc_not_zero [iff]: "\<And>m. hSuc m \<noteq> 0" |
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by transfer (rule Suc_not_Zero) |
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lemma zero_not_hSuc [iff]: "\<And>m. 0 \<noteq> hSuc m" |
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by transfer (rule Zero_not_Suc) |
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lemma hSuc_hSuc_eq [iff]: "\<And>m n. hSuc m = hSuc n \<longleftrightarrow> m = n" |
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by transfer (rule nat.inject) |
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lemma zero_less_hSuc [iff]: "\<And>n. 0 < hSuc n" |
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by transfer (rule zero_less_Suc) |
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lemma hypnat_minus_zero [simp]: "\<And>z::hypnat. z - z = 0" |
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by transfer (rule diff_self_eq_0) |
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lemma hypnat_diff_0_eq_0 [simp]: "\<And>n::hypnat. 0 - n = 0" |
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by transfer (rule diff_0_eq_0) |
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lemma hypnat_add_is_0 [iff]: "\<And>m n::hypnat. m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0" |
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by transfer (rule add_is_0) |
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lemma hypnat_diff_diff_left: "\<And>i j k::hypnat. i - j - k = i - (j + k)" |
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by transfer (rule diff_diff_left) |
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lemma hypnat_diff_commute: "\<And>i j k::hypnat. i - j - k = i - k - j" |
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by transfer (rule diff_commute) |
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lemma hypnat_diff_add_inverse [simp]: "\<And>m n::hypnat. n + m - n = m" |
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by transfer (rule diff_add_inverse) |
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lemma hypnat_diff_add_inverse2 [simp]: "\<And>m n::hypnat. m + n - n = m" |
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by transfer (rule diff_add_inverse2) |
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lemma hypnat_diff_cancel [simp]: "\<And>k m n::hypnat. (k + m) - (k + n) = m - n" |
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by transfer (rule diff_cancel) |
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lemma hypnat_diff_cancel2 [simp]: "\<And>k m n::hypnat. (m + k) - (n + k) = m - n" |
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by transfer (rule diff_cancel2) |
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lemma hypnat_diff_add_0 [simp]: "\<And>m n::hypnat. n - (n + m) = 0" |
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by transfer (rule diff_add_0) |
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lemma hypnat_diff_mult_distrib: "\<And>k m n::hypnat. (m - n) * k = (m * k) - (n * k)" |
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by transfer (rule diff_mult_distrib) |
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lemma hypnat_diff_mult_distrib2: "\<And>k m n::hypnat. k * (m - n) = (k * m) - (k * n)" |
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by transfer (rule diff_mult_distrib2) |
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lemma hypnat_le_zero_cancel [iff]: "\<And>n::hypnat. n \<le> 0 \<longleftrightarrow> n = 0" |
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by transfer (rule le_0_eq) |
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lemma hypnat_mult_is_0 [simp]: "\<And>m n::hypnat. m * n = 0 \<longleftrightarrow> m = 0 \<or> n = 0" |
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by transfer (rule mult_is_0) |
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lemma hypnat_diff_is_0_eq [simp]: "\<And>m n::hypnat. m - n = 0 \<longleftrightarrow> m \<le> n" |
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by transfer (rule diff_is_0_eq) |
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lemma hypnat_not_less0 [iff]: "\<And>n::hypnat. \<not> n < 0" |
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by transfer (rule not_less0) |
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lemma hypnat_less_one [iff]: "\<And>n::hypnat. n < 1 \<longleftrightarrow> n = 0" |
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by transfer (rule less_one) |
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lemma hypnat_add_diff_inverse: "\<And>m n::hypnat. \<not> m < n \<Longrightarrow> n + (m - n) = m" |
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by transfer (rule add_diff_inverse) |
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lemma hypnat_le_add_diff_inverse [simp]: "\<And>m n::hypnat. n \<le> m \<Longrightarrow> n + (m - n) = m" |
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by transfer (rule le_add_diff_inverse) |
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lemma hypnat_le_add_diff_inverse2 [simp]: "\<And>m n::hypnat. n \<le> m \<Longrightarrow> (m - n) + n = m" |
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by transfer (rule le_add_diff_inverse2) |
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declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le] |
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lemma hypnat_le0 [iff]: "\<And>n::hypnat. 0 \<le> n" |
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by transfer (rule le0) |
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lemma hypnat_le_add1 [simp]: "\<And>x n::hypnat. x \<le> x + n" |
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by transfer (rule le_add1) |
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lemma hypnat_add_self_le [simp]: "\<And>x n::hypnat. x \<le> n + x" |
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by transfer (rule le_add2) |
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lemma hypnat_add_one_self_less [simp]: "x < x + 1" for x :: hypnat |
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by (fact less_add_one) |
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lemma hypnat_neq0_conv [iff]: "\<And>n::hypnat. n \<noteq> 0 \<longleftrightarrow> 0 < n" |
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by transfer (rule neq0_conv) |
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lemma hypnat_gt_zero_iff: "0 < n \<longleftrightarrow> 1 \<le> n" for n :: hypnat |
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by (auto simp add: linorder_not_less [symmetric]) |
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lemma hypnat_gt_zero_iff2: "0 < n \<longleftrightarrow> (\<exists>m. n = m + 1)" for n :: hypnat |
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by (auto intro!: add_nonneg_pos exI[of _ "n - 1"] simp: hypnat_gt_zero_iff) |
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lemma hypnat_add_self_not_less: "\<not> x + y < x" for x y :: hypnat |
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by (simp add: linorder_not_le [symmetric] add.commute [of x]) |
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lemma hypnat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)" |
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for a b :: hypnat |
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\<comment> \<open>elimination of \<open>-\<close> on \<open>hypnat\<close>\<close> |
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proof (cases "a < b" rule: case_split) |
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case True |
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then show ?thesis |
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by (auto simp add: hypnat_add_self_not_less order_less_imp_le hypnat_diff_is_0_eq [THEN iffD2]) |
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next |
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case False |
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then show ?thesis |
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by (auto simp add: linorder_not_less dest: order_le_less_trans) |
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qed |
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subsection \<open>Properties of the set of embedded natural numbers\<close> |
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lemma of_nat_eq_star_of [simp]: "of_nat = star_of" |
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proof |
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show "of_nat n = star_of n" for n |
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by transfer simp |
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qed |
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lemma Nats_eq_Standard: "(Nats :: nat star set) = Standard" |
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by (auto simp: Nats_def Standard_def) |
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lemma hypnat_of_nat_mem_Nats [simp]: "hypnat_of_nat n \<in> Nats" |
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by (simp add: Nats_eq_Standard) |
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lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = 1" |
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by transfer simp |
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lemma hypnat_of_nat_Suc [simp]: "hypnat_of_nat (Suc n) = hypnat_of_nat n + 1" |
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by transfer simp |
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lemma of_nat_eq_add [rule_format]: "\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat" |
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apply (induct n) |
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apply (auto simp add: add.assoc) |
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apply (case_tac x) |
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apply (auto simp add: add.commute [of 1]) |
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done |
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lemma Nats_diff [simp]: "a \<in> Nats \<Longrightarrow> b \<in> Nats \<Longrightarrow> a - b \<in> Nats" for a b :: hypnat |
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by (simp add: Nats_eq_Standard) |
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subsection \<open>Infinite Hypernatural Numbers -- @{term HNatInfinite}\<close> |
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text \<open>The set of infinite hypernatural numbers.\<close> |
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definition HNatInfinite :: "hypnat set" |
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where "HNatInfinite = {n. n \<notin> Nats}" |
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lemma Nats_not_HNatInfinite_iff: "x \<in> Nats \<longleftrightarrow> x \<notin> HNatInfinite" |
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by (simp add: HNatInfinite_def) |
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lemma HNatInfinite_not_Nats_iff: "x \<in> HNatInfinite \<longleftrightarrow> x \<notin> Nats" |
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by (simp add: HNatInfinite_def) |
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lemma star_of_neq_HNatInfinite: "N \<in> HNatInfinite \<Longrightarrow> star_of n \<noteq> N" |
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by (auto simp add: HNatInfinite_def Nats_eq_Standard) |
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lemma star_of_Suc_lessI: "\<And>N. star_of n < N \<Longrightarrow> star_of (Suc n) \<noteq> N \<Longrightarrow> star_of (Suc n) < N" |
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by transfer (rule Suc_lessI) |
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lemma star_of_less_HNatInfinite: |
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assumes N: "N \<in> HNatInfinite" |
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shows "star_of n < N" |
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proof (induct n) |
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case 0 |
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from N have "star_of 0 \<noteq> N" |
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by (rule star_of_neq_HNatInfinite) |
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then show ?case by simp |
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next |
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case (Suc n) |
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from N have "star_of (Suc n) \<noteq> N" |
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by (rule star_of_neq_HNatInfinite) |
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with Suc show ?case |
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by (rule star_of_Suc_lessI) |
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qed |
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lemma star_of_le_HNatInfinite: "N \<in> HNatInfinite \<Longrightarrow> star_of n \<le> N" |
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by (rule star_of_less_HNatInfinite [THEN order_less_imp_le]) |
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subsubsection \<open>Closure Rules\<close> |
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lemma Nats_less_HNatInfinite: "x \<in> Nats \<Longrightarrow> y \<in> HNatInfinite \<Longrightarrow> x < y" |
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by (auto simp add: Nats_def star_of_less_HNatInfinite) |
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lemma Nats_le_HNatInfinite: "x \<in> Nats \<Longrightarrow> y \<in> HNatInfinite \<Longrightarrow> x \<le> y" |
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by (rule Nats_less_HNatInfinite [THEN order_less_imp_le]) |
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lemma zero_less_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 0 < x" |
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by (simp add: Nats_less_HNatInfinite) |
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lemma one_less_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 1 < x" |
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by (simp add: Nats_less_HNatInfinite) |
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lemma one_le_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 1 \<le> x" |
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by (simp add: Nats_le_HNatInfinite) |
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lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite" |
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by (simp add: HNatInfinite_def) |
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lemma Nats_downward_closed: "x \<in> Nats \<Longrightarrow> y \<le> x \<Longrightarrow> y \<in> Nats" for x y :: hypnat |
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apply (simp only: linorder_not_less [symmetric]) |
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apply (erule contrapos_np) |
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apply (drule HNatInfinite_not_Nats_iff [THEN iffD2]) |
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apply (erule (1) Nats_less_HNatInfinite) |
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done |
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lemma HNatInfinite_upward_closed: "x \<in> HNatInfinite \<Longrightarrow> x \<le> y \<Longrightarrow> y \<in> HNatInfinite" |
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apply (simp only: HNatInfinite_not_Nats_iff) |
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apply (erule contrapos_nn) |
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apply (erule (1) Nats_downward_closed) |
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done |
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lemma HNatInfinite_add: "x \<in> HNatInfinite \<Longrightarrow> x + y \<in> HNatInfinite" |
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apply (erule HNatInfinite_upward_closed) |
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apply (rule hypnat_le_add1) |
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done |
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lemma HNatInfinite_add_one: "x \<in> HNatInfinite \<Longrightarrow> x + 1 \<in> HNatInfinite" |
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by (rule HNatInfinite_add) |
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lemma HNatInfinite_diff: "x \<in> HNatInfinite \<Longrightarrow> y \<in> Nats \<Longrightarrow> x - y \<in> HNatInfinite" |
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apply (frule (1) Nats_le_HNatInfinite) |
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apply (simp only: HNatInfinite_not_Nats_iff) |
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apply (erule contrapos_nn) |
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apply (drule (1) Nats_add, simp) |
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done |
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lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite \<Longrightarrow> \<exists>y. x = y + 1" for x :: hypnat |
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apply (rule_tac x = "x - (1::hypnat) " in exI) |
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apply (simp add: Nats_le_HNatInfinite) |
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done |
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subsection \<open>Existence of an infinite hypernatural number\<close> |
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text \<open>\<open>\<omega>\<close> is in fact an infinite hypernatural number = \<open>[<1, 2, 3, \<dots>>]\<close>\<close> |
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definition whn :: hypnat |
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where hypnat_omega_def: "whn = star_n (\<lambda>n::nat. n)" |
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lemma hypnat_of_nat_neq_whn: "hypnat_of_nat n \<noteq> whn" |
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by (simp add: FreeUltrafilterNat.singleton' hypnat_omega_def star_of_def star_n_eq_iff) |
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lemma whn_neq_hypnat_of_nat: "whn \<noteq> hypnat_of_nat n" |
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by (simp add: FreeUltrafilterNat.singleton hypnat_omega_def star_of_def star_n_eq_iff) |
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lemma whn_not_Nats [simp]: "whn \<notin> Nats" |
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by (simp add: Nats_def image_def whn_neq_hypnat_of_nat) |
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lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite" |
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by (simp add: HNatInfinite_def) |
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lemma lemma_unbounded_set [simp]: "eventually (\<lambda>n::nat. m < n) \<U>" |
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by (rule filter_leD[OF FreeUltrafilterNat.le_cofinite]) |
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(auto simp add: cofinite_eq_sequentially eventually_at_top_dense) |
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lemma hypnat_of_nat_eq: "hypnat_of_nat m = star_n (\<lambda>n::nat. m)" |
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by (simp add: star_of_def) |
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lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}" |
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by (simp add: Nats_def image_def) |
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lemma Nats_less_whn: "n \<in> Nats \<Longrightarrow> n < whn" |
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by (simp add: Nats_less_HNatInfinite) |
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lemma Nats_le_whn: "n \<in> Nats \<Longrightarrow> n \<le> whn" |
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by (simp add: Nats_le_HNatInfinite) |
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lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn" |
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by (simp add: Nats_less_whn) |
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lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn" |
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by (simp add: Nats_le_whn) |
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lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn" |
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by (simp add: Nats_less_whn) |
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lemma hypnat_one_less_hypnat_omega [simp]: "1 < whn" |
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by (simp add: Nats_less_whn) |
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subsubsection \<open>Alternative characterization of the set of infinite hypernaturals\<close> |
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text \<open>@{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}\<close> |
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(*??delete? similar reasoning in hypnat_omega_gt_SHNat above*) |
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lemma HNatInfinite_FreeUltrafilterNat_lemma: |
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assumes "\<forall>N::nat. eventually (\<lambda>n. f n \<noteq> N) \<U>" |
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shows "eventually (\<lambda>n. N < f n) \<U>" |
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apply (induct N) |
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using assms |
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apply (drule_tac x = 0 in spec, simp) |
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using assms |
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apply (drule_tac x = "Suc N" in spec) |
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apply (auto elim: eventually_elim2) |
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done |
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lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}" |
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apply (safe intro!: Nats_less_HNatInfinite) |
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apply (auto simp add: HNatInfinite_def) |
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done |
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subsubsection \<open>Alternative Characterization of @{term HNatInfinite} using Free Ultrafilter\<close> |
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lemma HNatInfinite_FreeUltrafilterNat: |
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"star_n X \<in> HNatInfinite \<Longrightarrow> \<forall>u. eventually (\<lambda>n. u < X n) \<U>" |
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apply (auto simp add: HNatInfinite_iff SHNat_eq) |
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apply (drule_tac x="star_of u" in spec, simp) |
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apply (simp add: star_of_def star_less_def starP2_star_n) |
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done |
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lemma FreeUltrafilterNat_HNatInfinite: |
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"\<forall>u. eventually (\<lambda>n. u < X n) \<U> \<Longrightarrow> star_n X \<in> HNatInfinite" |
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by (auto simp add: star_less_def starP2_star_n HNatInfinite_iff SHNat_eq hypnat_of_nat_eq) |
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|
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lemma HNatInfinite_FreeUltrafilterNat_iff: |
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"(star_n X \<in> HNatInfinite) = (\<forall>u. eventually (\<lambda>n. u < X n) \<U>)" |
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by (rule iffI [OF HNatInfinite_FreeUltrafilterNat FreeUltrafilterNat_HNatInfinite]) |
345 |
||
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|
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subsection \<open>Embedding of the Hypernaturals into other types\<close> |
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definition of_hypnat :: "hypnat \<Rightarrow> 'a::semiring_1_cancel star" |
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where of_hypnat_def [transfer_unfold]: "of_hypnat = *f* of_nat" |
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lemma of_hypnat_0 [simp]: "of_hypnat 0 = 0" |
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by transfer (rule of_nat_0) |
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lemma of_hypnat_1 [simp]: "of_hypnat 1 = 1" |
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by transfer (rule of_nat_1) |
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|
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lemma of_hypnat_hSuc: "\<And>m. of_hypnat (hSuc m) = 1 + of_hypnat m" |
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by transfer (rule of_nat_Suc) |
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|
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lemma of_hypnat_add [simp]: "\<And>m n. of_hypnat (m + n) = of_hypnat m + of_hypnat n" |
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by transfer (rule of_nat_add) |
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|
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lemma of_hypnat_mult [simp]: "\<And>m n. of_hypnat (m * n) = of_hypnat m * of_hypnat n" |
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by transfer (rule of_nat_mult) |
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|
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lemma of_hypnat_less_iff [simp]: |
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"\<And>m n. of_hypnat m < (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m < n" |
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by transfer (rule of_nat_less_iff) |
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|
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lemma of_hypnat_0_less_iff [simp]: |
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"\<And>n. 0 < (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> 0 < n" |
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by transfer (rule of_nat_0_less_iff) |
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lemma of_hypnat_less_0_iff [simp]: "\<And>m. \<not> (of_hypnat m::'a::linordered_semidom star) < 0" |
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by transfer (rule of_nat_less_0_iff) |
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|
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lemma of_hypnat_le_iff [simp]: |
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"\<And>m n. of_hypnat m \<le> (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m \<le> n" |
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by transfer (rule of_nat_le_iff) |
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lemma of_hypnat_0_le_iff [simp]: "\<And>n. 0 \<le> (of_hypnat n::'a::linordered_semidom star)" |
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by transfer (rule of_nat_0_le_iff) |
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lemma of_hypnat_le_0_iff [simp]: "\<And>m. (of_hypnat m::'a::linordered_semidom star) \<le> 0 \<longleftrightarrow> m = 0" |
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by transfer (rule of_nat_le_0_iff) |
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lemma of_hypnat_eq_iff [simp]: |
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"\<And>m n. of_hypnat m = (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m = n" |
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by transfer (rule of_nat_eq_iff) |
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|
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lemma of_hypnat_eq_0_iff [simp]: "\<And>m. (of_hypnat m::'a::linordered_semidom star) = 0 \<longleftrightarrow> m = 0" |
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by transfer (rule of_nat_eq_0_iff) |
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|
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lemma HNatInfinite_of_hypnat_gt_zero: |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
29920
diff
changeset
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"N \<in> HNatInfinite \<Longrightarrow> (0::'a::linordered_semidom star) < of_hypnat N" |
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by (rule ccontr) (simp add: linorder_not_less) |
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|
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end |