1(* Title: HOL/Nonstandard_Analysis/HLog.thy 2 Author: Jacques D. Fleuriot 3 Copyright: 2000, 2001 University of Edinburgh 4*) 5 6section \<open>Logarithms: Non-Standard Version\<close> 7 8theory HLog 9 imports HTranscendental 10begin 11 12definition powhr :: "hypreal \<Rightarrow> hypreal \<Rightarrow> hypreal" (infixr "powhr" 80) 13 where [transfer_unfold]: "x powhr a = starfun2 (powr) x a" 14 15definition hlog :: "hypreal \<Rightarrow> hypreal \<Rightarrow> hypreal" 16 where [transfer_unfold]: "hlog a x = starfun2 log a x" 17 18lemma powhr: "(star_n X) powhr (star_n Y) = star_n (\<lambda>n. (X n) powr (Y n))" 19 by (simp add: powhr_def starfun2_star_n) 20 21lemma powhr_one_eq_one [simp]: "\<And>a. 1 powhr a = 1" 22 by transfer simp 23 24lemma powhr_mult: "\<And>a x y. 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x * y) powhr a = (x powhr a) * (y powhr a)" 25 by transfer (simp add: powr_mult) 26 27lemma powhr_gt_zero [simp]: "\<And>a x. 0 < x powhr a \<longleftrightarrow> x \<noteq> 0" 28 by transfer simp 29 30lemma powhr_not_zero [simp]: "\<And>a x. x powhr a \<noteq> 0 \<longleftrightarrow> x \<noteq> 0" 31 by transfer simp 32 33lemma powhr_divide: "\<And>a x y. 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x / y) powhr a = (x powhr a) / (y powhr a)" 34 by transfer (rule powr_divide) 35 36lemma powhr_add: "\<And>a b x. x powhr (a + b) = (x powhr a) * (x powhr b)" 37 by transfer (rule powr_add) 38 39lemma powhr_powhr: "\<And>a b x. (x powhr a) powhr b = x powhr (a * b)" 40 by transfer (rule powr_powr) 41 42lemma powhr_powhr_swap: "\<And>a b x. (x powhr a) powhr b = (x powhr b) powhr a" 43 by transfer (rule powr_powr_swap) 44 45lemma powhr_minus: "\<And>a x. x powhr (- a) = inverse (x powhr a)" 46 by transfer (rule powr_minus) 47 48lemma powhr_minus_divide: "x powhr (- a) = 1 / (x powhr a)" 49 by (simp add: divide_inverse powhr_minus) 50 51lemma powhr_less_mono: "\<And>a b x. a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powhr a < x powhr b" 52 by transfer simp 53 54lemma powhr_less_cancel: "\<And>a b x. x powhr a < x powhr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b" 55 by transfer simp 56 57lemma powhr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> x powhr a < x powhr b \<longleftrightarrow> a < b" 58 by (blast intro: powhr_less_cancel powhr_less_mono) 59 60lemma powhr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> x powhr a \<le> x powhr b \<longleftrightarrow> a \<le> b" 61 by (simp add: linorder_not_less [symmetric]) 62 63lemma hlog: "hlog (star_n X) (star_n Y) = star_n (\<lambda>n. log (X n) (Y n))" 64 by (simp add: hlog_def starfun2_star_n) 65 66lemma hlog_starfun_ln: "\<And>x. ( *f* ln) x = hlog (( *f* exp) 1) x" 67 by transfer (rule log_ln) 68 69lemma powhr_hlog_cancel [simp]: "\<And>a x. 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powhr (hlog a x) = x" 70 by transfer simp 71 72lemma hlog_powhr_cancel [simp]: "\<And>a y. 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> hlog a (a powhr y) = y" 73 by transfer simp 74 75lemma hlog_mult: 76 "\<And>a x y. 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> hlog a (x * y) = hlog a x + hlog a y" 77 by transfer (rule log_mult) 78 79lemma hlog_as_starfun: "\<And>a x. 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> hlog a x = ( *f* ln) x / ( *f* ln) a" 80 by transfer (simp add: log_def) 81 82lemma hlog_eq_div_starfun_ln_mult_hlog: 83 "\<And>a b x. 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 84 hlog a x = (( *f* ln) b / ( *f* ln) a) * hlog b x" 85 by transfer (rule log_eq_div_ln_mult_log) 86 87lemma powhr_as_starfun: "\<And>a x. x powhr a = (if x = 0 then 0 else ( *f* exp) (a * ( *f* real_ln) x))" 88 by transfer (simp add: powr_def) 89 90lemma HInfinite_powhr: 91 "x \<in> HInfinite \<Longrightarrow> 0 < x \<Longrightarrow> a \<in> HFinite - Infinitesimal \<Longrightarrow> 0 < a \<Longrightarrow> x powhr a \<in> HInfinite" 92 by (auto intro!: starfun_ln_ge_zero starfun_ln_HInfinite 93 HInfinite_HFinite_not_Infinitesimal_mult2 starfun_exp_HInfinite 94 simp add: order_less_imp_le HInfinite_gt_zero_gt_one powhr_as_starfun zero_le_mult_iff) 95 96lemma hlog_hrabs_HInfinite_Infinitesimal: 97 "x \<in> HFinite - Infinitesimal \<Longrightarrow> a \<in> HInfinite \<Longrightarrow> 0 < a \<Longrightarrow> hlog a \<bar>x\<bar> \<in> Infinitesimal" 98 apply (frule HInfinite_gt_zero_gt_one) 99 apply (auto intro!: starfun_ln_HFinite_not_Infinitesimal 100 HInfinite_inverse_Infinitesimal Infinitesimal_HFinite_mult2 101 simp add: starfun_ln_HInfinite not_Infinitesimal_not_zero 102 hlog_as_starfun divide_inverse) 103 done 104 105lemma hlog_HInfinite_as_starfun: "a \<in> HInfinite \<Longrightarrow> 0 < a \<Longrightarrow> hlog a x = ( *f* ln) x / ( *f* ln) a" 106 by (rule hlog_as_starfun) auto 107 108lemma hlog_one [simp]: "\<And>a. hlog a 1 = 0" 109 by transfer simp 110 111lemma hlog_eq_one [simp]: "\<And>a. 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> hlog a a = 1" 112 by transfer (rule log_eq_one) 113 114lemma hlog_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> hlog a (inverse x) = - hlog a x" 115 by (rule add_left_cancel [of "hlog a x", THEN iffD1]) (simp add: hlog_mult [symmetric]) 116 117lemma hlog_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> hlog a (x / y) = hlog a x - hlog a y" 118 by (simp add: hlog_mult hlog_inverse divide_inverse) 119 120lemma hlog_less_cancel_iff [simp]: 121 "\<And>a x y. 1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> hlog a x < hlog a y \<longleftrightarrow> x < y" 122 by transfer simp 123 124lemma hlog_le_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> hlog a x \<le> hlog a y \<longleftrightarrow> x \<le> y" 125 by (simp add: linorder_not_less [symmetric]) 126 127end 128