author | huffman |
Tue, 12 Dec 2006 07:46:40 +0100 | |
changeset 21787 | 9edd495b6330 |
parent 21404 | eb85850d3eb7 |
child 21847 | 59a68ed9f2f2 |
permissions | -rw-r--r-- |
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(* Title : 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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header{*Hypernatural numbers*} |
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theory HyperNat |
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imports Star |
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begin |
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types 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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subsection{*Properties Transferred from Naturals*} |
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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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||
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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 (insert add_strict_left_mono [OF zero_less_one], auto) |
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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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apply safe |
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apply (rule_tac x = "n - (1::hypnat) " in exI) |
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apply (simp add: hypnat_gt_zero_iff) |
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apply (insert add_le_less_mono [OF _ zero_less_one, of 0], auto) |
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108 |
done |
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109 |
|
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lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))" |
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111 |
by (simp add: linorder_not_le [symmetric] add_commute [of x]) |
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|
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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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-- {* elimination of @{text -} on @{text hypnat} *} |
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116 |
proof (cases "a<b" rule: case_split) |
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117 |
case True |
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118 |
thus ?thesis |
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119 |
by (auto simp add: hypnat_add_self_not_less order_less_imp_le |
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120 |
hypnat_diff_is_0_eq [THEN iffD2]) |
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121 |
next |
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122 |
case False |
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123 |
thus ?thesis |
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by (auto simp add: linorder_not_less dest: order_le_less_trans) |
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125 |
qed |
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126 |
|
17433 | 127 |
subsection{*Properties of the set of embedded natural numbers*} |
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lemma of_nat_eq_star_of [simp]: "of_nat = star_of" |
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130 |
proof |
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131 |
fix n :: nat |
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show "of_nat n = star_of n" by transfer simp |
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133 |
qed |
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134 |
|
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lemma Nats_eq_Standard: "(Nats :: nat star set) = Standard" |
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136 |
by (auto simp add: Nats_def Standard_def) |
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137 |
|
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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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|
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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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143 |
|
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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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|
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lemma of_nat_eq_add [rule_format]: |
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149 |
"\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat" |
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150 |
apply (induct n) |
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151 |
apply (auto simp add: add_assoc) |
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152 |
apply (case_tac x) |
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153 |
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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|
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|
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subsection{*Infinite Hypernatural Numbers -- @{term HNatInfinite}*} |
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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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subsubsection {* Closure Rules *} |
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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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234 |
done |
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|
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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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|
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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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|
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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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250 |
done |
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|
252 |
||
253 |
subsection{*Existence of an infinite hypernatural number*} |
|
254 |
||
19765 | 255 |
definition |
17433 | 256 |
(* 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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|
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lemma hypnat_of_nat_neq_whn: "hypnat_of_nat n \<noteq> whn" |
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by (simp add: hypnat_omega_def star_of_def star_n_eq_iff |
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262 |
FreeUltrafilterNat.finite) |
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263 |
|
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lemma whn_neq_hypnat_of_nat: "whn \<noteq> hypnat_of_nat n" |
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by (simp add: hypnat_omega_def star_of_def star_n_eq_iff |
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266 |
FreeUltrafilterNat.finite) |
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267 |
|
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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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|
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lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite" |
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by (simp add: HNatInfinite_def) |
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|
20695 | 274 |
text{* Example of an hypersequence (i.e. an extended standard sequence) |
275 |
whose term with an hypernatural suffix is an infinitesimal i.e. |
|
276 |
the whn'nth term of the hypersequence is a member of Infinitesimal*} |
|
277 |
||
278 |
lemma SEQ_Infinitesimal: |
|
279 |
"( *f* (%n::nat. inverse(real(Suc n)))) whn : Infinitesimal" |
|
280 |
apply (simp add: hypnat_omega_def starfun star_n_inverse) |
|
281 |
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff) |
|
282 |
apply (simp add: real_of_nat_Suc_gt_zero FreeUltrafilterNat_inverse_real_of_posnat) |
|
283 |
done |
|
17433 | 284 |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
285 |
lemma lemma_unbounded_set [simp]: "{n::nat. m < n} \<in> FreeUltrafilterNat" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
286 |
apply (insert finite_atMost [of m]) |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
287 |
apply (simp add: atMost_def) |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
288 |
apply (drule FreeUltrafilterNat_finite) |
14468 | 289 |
apply (drule FreeUltrafilterNat_Compl_mem, ultra) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
290 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
291 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
292 |
lemma Compl_Collect_le: "- {n::nat. N \<le> n} = {n. n < N}" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
293 |
by (simp add: Collect_neg_eq [symmetric] linorder_not_le) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
294 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
295 |
lemma hypnat_of_nat_eq: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
296 |
"hypnat_of_nat m = star_n (%n::nat. m)" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
17332
diff
changeset
|
297 |
by (simp add: star_of_def) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
298 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
299 |
lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}" |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
300 |
by (simp add: Nats_def image_def) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
301 |
|
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
302 |
lemma Nats_less_whn: "n \<in> Nats \<Longrightarrow> n < whn" |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
303 |
by (simp add: Nats_less_HNatInfinite) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
304 |
|
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
305 |
lemma Nats_le_whn: "n \<in> Nats \<Longrightarrow> n \<le> whn" |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
306 |
by (simp add: Nats_le_HNatInfinite) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
307 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
308 |
lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn" |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
309 |
by (simp add: Nats_less_whn) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
310 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
311 |
lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn" |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
312 |
by (simp add: Nats_le_whn) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
313 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
314 |
lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn" |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
315 |
by (simp add: Nats_less_whn) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
316 |
|
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
317 |
lemma hypnat_one_less_hypnat_omega [simp]: "1 < whn" |
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
318 |
by (simp add: Nats_less_whn) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
319 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
320 |
|
17433 | 321 |
subsubsection{*Alternative characterization of the set of infinite hypernaturals*} |
15070 | 322 |
|
323 |
text{* @{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}*} |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
324 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
325 |
(*??delete? similar reasoning in hypnat_omega_gt_SHNat above*) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
326 |
lemma HNatInfinite_FreeUltrafilterNat_lemma: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
327 |
"\<forall>N::nat. {n. f n \<noteq> N} \<in> FreeUltrafilterNat |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
328 |
==> {n. N < f n} \<in> FreeUltrafilterNat" |
15251 | 329 |
apply (induct_tac N) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
330 |
apply (drule_tac x = 0 in spec) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
331 |
apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem, drule FreeUltrafilterNat_Int, assumption, simp) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
332 |
apply (drule_tac x = "Suc n" in spec, ultra) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
333 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
334 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
335 |
lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
336 |
apply (auto simp add: HNatInfinite_def SHNat_eq hypnat_of_nat_eq) |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
337 |
apply (rule_tac x = x in star_cases) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
338 |
apply (auto elim: HNatInfinite_FreeUltrafilterNat_lemma |
20740
5a103b43da5a
reorganized HNatInfinite proofs; simplified and renamed some lemmas
huffman
parents:
20730
diff
changeset
|
339 |
simp add: star_n_less FreeUltrafilterNat_Compl_iff1 |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
340 |
star_n_eq_iff Collect_neg_eq [symmetric]) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
341 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
342 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
343 |
|
17433 | 344 |
subsubsection{*Alternative Characterization of @{term HNatInfinite} using |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
345 |
Free Ultrafilter*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
346 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
347 |
lemma HNatInfinite_FreeUltrafilterNat: |
20552
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
348 |
"star_n X \<in> HNatInfinite ==> \<forall>u. {n. u < X n}: FreeUltrafilterNat" |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
349 |
apply (auto simp add: HNatInfinite_iff SHNat_eq) |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
350 |
apply (drule_tac x="star_of u" in spec, simp) |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
351 |
apply (simp add: star_of_def star_n_less) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
352 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
353 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
354 |
lemma FreeUltrafilterNat_HNatInfinite: |
20552
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
355 |
"\<forall>u. {n. u < X n}: FreeUltrafilterNat ==> star_n X \<in> HNatInfinite" |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
356 |
by (auto simp add: star_n_less HNatInfinite_iff SHNat_eq hypnat_of_nat_eq) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
357 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
358 |
lemma HNatInfinite_FreeUltrafilterNat_iff: |
20552
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
359 |
"(star_n X \<in> HNatInfinite) = (\<forall>u. {n. u < X n}: FreeUltrafilterNat)" |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
360 |
by (rule iffI [OF HNatInfinite_FreeUltrafilterNat |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
361 |
FreeUltrafilterNat_HNatInfinite]) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
362 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
363 |
subsection{*Embedding of the Hypernaturals into the Hyperreals*} |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
364 |
text{*Obtained using the nonstandard extension of the naturals*} |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
365 |
|
19765 | 366 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20740
diff
changeset
|
367 |
hypreal_of_hypnat :: "hypnat => hypreal" where |
19765 | 368 |
"hypreal_of_hypnat = *f* real" |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
369 |
|
17332
4910cf8c0cd2
added theorem attributes transfer_intro, transfer_unfold, transfer_refold; simplified some proofs; some rearranging
huffman
parents:
17318
diff
changeset
|
370 |
declare hypreal_of_hypnat_def [transfer_unfold] |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
371 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
372 |
lemma hypreal_of_hypnat: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
373 |
"hypreal_of_hypnat (star_n X) = star_n (%n. real (X n))" |
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
374 |
by (simp add: hypreal_of_hypnat_def starfun) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
375 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
376 |
lemma hypreal_of_hypnat_inject [simp]: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
377 |
"!!m n. (hypreal_of_hypnat m = hypreal_of_hypnat n) = (m=n)" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
378 |
by transfer (rule real_of_nat_inject) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
379 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
380 |
lemma hypreal_of_hypnat_zero: "hypreal_of_hypnat 0 = 0" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
381 |
by transfer (rule real_of_nat_zero) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
382 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
383 |
lemma hypreal_of_hypnat_one: "hypreal_of_hypnat (1::hypnat) = 1" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
384 |
by transfer simp |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
385 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
386 |
lemma hypreal_of_hypnat_add [simp]: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
387 |
"!!m n. hypreal_of_hypnat (m + n) = hypreal_of_hypnat m + hypreal_of_hypnat n" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
388 |
by transfer (rule real_of_nat_add) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
389 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
390 |
lemma hypreal_of_hypnat_mult [simp]: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
391 |
"!!m n. hypreal_of_hypnat (m * n) = hypreal_of_hypnat m * hypreal_of_hypnat n" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
392 |
by transfer (rule real_of_nat_mult) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
393 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
394 |
lemma hypreal_of_hypnat_less_iff [simp]: |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
395 |
"!!m n. (hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
396 |
by transfer (rule real_of_nat_less_iff) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
397 |
|
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
398 |
lemma hypreal_of_hypnat_eq_zero_iff: "(hypreal_of_hypnat N = 0) = (N = 0)" |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
399 |
by (simp add: hypreal_of_hypnat_zero [symmetric]) |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
400 |
declare hypreal_of_hypnat_eq_zero_iff [simp] |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
401 |
|
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
402 |
lemma hypreal_of_hypnat_ge_zero [simp]: "!!n. 0 \<le> hypreal_of_hypnat n" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
403 |
by transfer (rule real_of_nat_ge_zero) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
404 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
405 |
lemma HNatInfinite_inverse_Infinitesimal [simp]: |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14371
diff
changeset
|
406 |
"n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal" |
17318
bc1c75855f3d
starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents:
17299
diff
changeset
|
407 |
apply (cases n) |
20552
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
408 |
apply (auto simp add: hypreal_of_hypnat star_n_inverse real_norm_def |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
409 |
HNatInfinite_FreeUltrafilterNat_iff |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
410 |
Infinitesimal_FreeUltrafilterNat_iff2) |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
411 |
apply (drule_tac x="Suc m" in spec) |
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
19765
diff
changeset
|
412 |
apply (erule ultra, simp) |
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
413 |
done |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
414 |
|
14420
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
415 |
lemma HNatInfinite_hypreal_of_hypnat_gt_zero: |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
416 |
"N \<in> HNatInfinite ==> 0 < hypreal_of_hypnat N" |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
417 |
apply (rule ccontr) |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
418 |
apply (simp add: hypreal_of_hypnat_zero [symmetric] linorder_not_less) |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
419 |
done |
4e72cd222e0b
converted Hyperreal/HTranscendental to Isar script
paulson
parents:
14415
diff
changeset
|
420 |
|
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
13487
diff
changeset
|
421 |
end |