author | wenzelm |
Wed, 26 Oct 2016 15:14:17 +0200 | |
changeset 64406 | 492de9062cd2 |
parent 62479 | 716336f19aa9 |
child 64435 | c93b0e6131c3 |
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 |
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hypnat_of_nat :: "nat => nat star" where |
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"hypnat_of_nat == star_of" |
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definition |
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hSuc :: "hypnat => hypnat" where |
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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) = (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]: "!!z. z - z = (0::hypnat)" |
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by transfer (rule diff_self_eq_0) |
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lemma hypnat_diff_0_eq_0 [simp]: "!!n. (0::hypnat) - n = 0" |
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by transfer (rule diff_0_eq_0) |
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lemma hypnat_add_is_0 [iff]: "!!m n. (m+n = (0::hypnat)) = (m=0 & n=0)" |
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by transfer (rule add_is_0) |
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lemma hypnat_diff_diff_left: "!!i j k. (i::hypnat) - j - k = i - (j+k)" |
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by transfer (rule diff_diff_left) |
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lemma hypnat_diff_commute: "!!i j k. (i::hypnat) - j - k = i-k-j" |
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by transfer (rule diff_commute) |
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lemma hypnat_diff_add_inverse [simp]: "!!m n. ((n::hypnat) + m) - n = m" |
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by transfer (rule diff_add_inverse) |
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lemma hypnat_diff_add_inverse2 [simp]: "!!m n. ((m::hypnat) + n) - n = m" |
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by transfer (rule diff_add_inverse2) |
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lemma hypnat_diff_cancel [simp]: "!!k m n. ((k::hypnat) + m) - (k+n) = m - n" |
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by transfer (rule diff_cancel) |
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lemma hypnat_diff_cancel2 [simp]: "!!k m n. ((m::hypnat) + k) - (n+k) = m - n" |
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by transfer (rule diff_cancel2) |
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lemma hypnat_diff_add_0 [simp]: "!!m n. (n::hypnat) - (n+m) = (0::hypnat)" |
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by transfer (rule diff_add_0) |
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lemma hypnat_diff_mult_distrib: "!!k m n. ((m::hypnat) - 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: "!!k m n. (k::hypnat) * (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]: "!!n. (n \<le> (0::hypnat)) = (n = 0)" |
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by transfer (rule le_0_eq) |
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lemma hypnat_mult_is_0 [simp]: "!!m n. (m*n = (0::hypnat)) = (m=0 | n=0)" |
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by transfer (rule mult_is_0) |
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lemma hypnat_diff_is_0_eq [simp]: "!!m n. ((m::hypnat) - n = 0) = (m \<le> n)" |
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by transfer (rule diff_is_0_eq) |
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lemma hypnat_not_less0 [iff]: "!!n. ~ n < (0::hypnat)" |
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by transfer (rule not_less0) |
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lemma hypnat_less_one [iff]: |
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"!!n. (n < (1::hypnat)) = (n=0)" |
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by transfer (rule less_one) |
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lemma hypnat_add_diff_inverse: "!!m n. ~ m<n ==> n+(m-n) = (m::hypnat)" |
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by transfer (rule add_diff_inverse) |
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lemma hypnat_le_add_diff_inverse [simp]: "!!m n. n \<le> m ==> n+(m-n) = (m::hypnat)" |
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by transfer (rule le_add_diff_inverse) |
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lemma hypnat_le_add_diff_inverse2 [simp]: "!!m n. n\<le>m ==> (m-n)+n = (m::hypnat)" |
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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]: "!!n. (0::hypnat) \<le> n" |
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by transfer (rule le0) |
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lemma hypnat_le_add1 [simp]: "!!x n. (x::hypnat) \<le> x + n" |
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by transfer (rule le_add1) |
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lemma hypnat_add_self_le [simp]: "!!x n. (x::hypnat) \<le> n + x" |
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by transfer (rule le_add2) |
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lemma hypnat_add_one_self_less [simp]: "(x::hypnat) < x + (1::hypnat)" |
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by (fact less_add_one) |
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lemma hypnat_neq0_conv [iff]: "!!n. (n \<noteq> 0) = (0 < (n::hypnat))" |
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by transfer (rule neq0_conv) |
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lemma hypnat_gt_zero_iff: "((0::hypnat) < n) = ((1::hypnat) \<le> n)" |
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by (auto simp add: linorder_not_less [symmetric]) |
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lemma hypnat_gt_zero_iff2: "(0 < n) = (\<exists>m. n = m + (1::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: "~ (x + y < (x::hypnat))" |
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by (simp add: linorder_not_le [symmetric] add.commute [of x]) |
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lemma hypnat_diff_split: |
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"P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))" |
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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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thus ?thesis |
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by (auto simp add: hypnat_add_self_not_less order_less_imp_le |
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hypnat_diff_is_0_eq [THEN iffD2]) |
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next |
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case False |
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thus ?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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fix n :: nat |
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show "of_nat n = star_of n" 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 add: 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::hypnat)" |
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by transfer simp |
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lemma hypnat_of_nat_Suc [simp]: |
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"hypnat_of_nat (Suc n) = hypnat_of_nat n + (1::hypnat)" |
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by transfer simp |
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lemma of_nat_eq_add [rule_format]: |
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"\<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; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats" |
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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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definition |
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(* the set of infinite hypernatural numbers *) |
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HNatInfinite :: "hypnat set" where |
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"HNatInfinite = {n. n \<notin> Nats}" |
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lemma Nats_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)" |
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by (simp add: HNatInfinite_def) |
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lemma HNatInfinite_not_Nats_iff: "(x \<in> HNatInfinite) = (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: |
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"\<And>N. \<lbrakk>star_of n < N; star_of (Suc n) \<noteq> N\<rbrakk> \<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" by (rule star_of_neq_HNatInfinite) |
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thus "star_of 0 < N" 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" by (rule star_of_neq_HNatInfinite) |
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with Suc show "star_of (Suc n) < N" 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: "\<lbrakk>x \<in> Nats; y \<in> HNatInfinite\<rbrakk> \<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: "\<lbrakk>x \<in> Nats; y \<in> HNatInfinite\<rbrakk> \<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: |
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"\<lbrakk>x \<in> Nats; (y::hypnat) \<le> x\<rbrakk> \<Longrightarrow> y \<in> Nats" |
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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: |
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"\<lbrakk>x \<in> HNatInfinite; x \<le> y\<rbrakk> \<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: |
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"\<lbrakk>x \<in> HNatInfinite; y \<in> Nats\<rbrakk> \<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 ==> \<exists>y. x = y + (1::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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definition |
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(* \<omega> is in fact an infinite hypernatural number = [<1,2,3,...>] *) |
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whn :: hypnat where |
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hypnat_omega_def: "whn = star_n (%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: |
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"hypnat_of_nat m = star_n (%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 |
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Free Ultrafilter\<close> |
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lemma HNatInfinite_FreeUltrafilterNat: |
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"star_n X \<in> HNatInfinite ==> \<forall>u. eventually (\<lambda>n. u < X n) FreeUltrafilterNat" |
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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) FreeUltrafilterNat ==> 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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lemma HNatInfinite_FreeUltrafilterNat_iff: |
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"(star_n X \<in> HNatInfinite) = (\<forall>u. eventually (\<lambda>n. u < X n) FreeUltrafilterNat)" |
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by (rule iffI [OF HNatInfinite_FreeUltrafilterNat |
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FreeUltrafilterNat_HNatInfinite]) |
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subsection \<open>Embedding of the Hypernaturals into other types\<close> |
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definition |
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of_hypnat :: "hypnat \<Rightarrow> 'a::semiring_1_cancel star" where |
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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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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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lemma of_hypnat_add [simp]: |
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"\<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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lemma of_hypnat_mult [simp]: |
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"\<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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377 |
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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)) = (m < n)" |
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by transfer (rule of_nat_less_iff) |
381 |
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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)) = (0 < n)" |
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by transfer (rule of_nat_0_less_iff) |
385 |
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lemma of_hypnat_less_0_iff [simp]: |
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"\<And>m. \<not> (of_hypnat m::'a::linordered_semidom star) < 0" |
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by transfer (rule of_nat_less_0_iff) |
389 |
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lemma of_hypnat_le_iff [simp]: |
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391 |
"\<And>m n. (of_hypnat m \<le> (of_hypnat n::'a::linordered_semidom star)) = (m \<le> n)" |
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by transfer (rule of_nat_le_iff) |
393 |
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lemma of_hypnat_0_le_iff [simp]: |
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"\<And>n. 0 \<le> (of_hypnat n::'a::linordered_semidom star)" |
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by transfer (rule of_nat_0_le_iff) |
397 |
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lemma of_hypnat_le_0_iff [simp]: |
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"\<And>m. ((of_hypnat m::'a::linordered_semidom star) \<le> 0) = (m = 0)" |
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by transfer (rule of_nat_le_0_iff) |
401 |
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402 |
lemma of_hypnat_eq_iff [simp]: |
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403 |
"\<And>m n. (of_hypnat m = (of_hypnat n::'a::linordered_semidom star)) = (m = n)" |
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by transfer (rule of_nat_eq_iff) |
405 |
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lemma of_hypnat_eq_0_iff [simp]: |
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407 |
"\<And>m. ((of_hypnat m::'a::linordered_semidom star) = 0) = (m = 0)" |
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by transfer (rule of_nat_eq_0_iff) |
409 |
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lemma HNatInfinite_of_hypnat_gt_zero: |
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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) |
413 |
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414 |
end |