Theory HLog

```(*  Title:      HOL/Nonstandard_Analysis/HLog.thy
Author:     Jacques D. Fleuriot
Copyright:  2000, 2001 University of Edinburgh
*)

section ‹Logarithms: Non-Standard Version›

theory HLog
imports HTranscendental
begin

definition powhr :: "hypreal ⇒ hypreal ⇒ hypreal"  (infixr "powhr" 80)
where [transfer_unfold]: "x powhr a = starfun2 (powr) x a"

definition hlog :: "hypreal ⇒ hypreal ⇒ hypreal"
where [transfer_unfold]: "hlog a x = starfun2 log a x"

lemma powhr: "(star_n X) powhr (star_n Y) = star_n (λn. (X n) powr (Y n))"

lemma powhr_one_eq_one [simp]: "⋀a. 1 powhr a = 1"
by transfer simp

lemma powhr_mult: "⋀a x y. 0 < x ⟹ 0 < y ⟹ (x * y) powhr a = (x powhr a) * (y powhr a)"

lemma powhr_gt_zero [simp]: "⋀a x. 0 < x powhr a ⟷ x ≠ 0"
by transfer simp

lemma powhr_not_zero [simp]: "⋀a x. x powhr a ≠ 0 ⟷ x ≠ 0"
by transfer simp

lemma powhr_divide: "⋀a x y. 0 ≤ x ⟹ 0 ≤ y ⟹ (x / y) powhr a = (x powhr a) / (y powhr a)"
by transfer (rule powr_divide)

lemma powhr_add: "⋀a b x. x powhr (a + b) = (x powhr a) * (x powhr b)"

lemma powhr_powhr: "⋀a b x. (x powhr a) powhr b = x powhr (a * b)"
by transfer (rule powr_powr)

lemma powhr_powhr_swap: "⋀a b x. (x powhr a) powhr b = (x powhr b) powhr a"
by transfer (rule powr_powr_swap)

lemma powhr_minus: "⋀a x. x powhr (- a) = inverse (x powhr a)"
by transfer (rule powr_minus)

lemma powhr_minus_divide: "x powhr (- a) = 1 / (x powhr a)"

lemma powhr_less_mono: "⋀a b x. a < b ⟹ 1 < x ⟹ x powhr a < x powhr b"
by transfer simp

lemma powhr_less_cancel: "⋀a b x. x powhr a < x powhr b ⟹ 1 < x ⟹ a < b"
by transfer simp

lemma powhr_less_cancel_iff [simp]: "1 < x ⟹ x powhr a < x powhr b ⟷ a < b"
by (blast intro: powhr_less_cancel powhr_less_mono)

lemma powhr_le_cancel_iff [simp]: "1 < x ⟹ x powhr a ≤ x powhr b ⟷ a ≤ b"

lemma hlog: "hlog (star_n X) (star_n Y) = star_n (λn. log (X n) (Y n))"

lemma hlog_starfun_ln: "⋀x. ( *f* ln) x = hlog (( *f* exp) 1) x"
by transfer (rule log_ln)

lemma powhr_hlog_cancel [simp]: "⋀a x. 0 < a ⟹ a ≠ 1 ⟹ 0 < x ⟹ a powhr (hlog a x) = x"
by transfer simp

lemma hlog_powhr_cancel [simp]: "⋀a y. 0 < a ⟹ a ≠ 1 ⟹ hlog a (a powhr y) = y"
by transfer simp

lemma hlog_mult:
"⋀a x y. 0 < a ⟹ a ≠ 1 ⟹ 0 < x ⟹ 0 < y ⟹ hlog a (x * y) = hlog a x + hlog a y"
by transfer (rule log_mult)

lemma hlog_as_starfun: "⋀a x. 0 < a ⟹ a ≠ 1 ⟹ hlog a x = ( *f* ln) x / ( *f* ln) a"

lemma hlog_eq_div_starfun_ln_mult_hlog:
"⋀a b x. 0 < a ⟹ a ≠ 1 ⟹ 0 < b ⟹ b ≠ 1 ⟹ 0 < x ⟹
hlog a x = (( *f* ln) b / ( *f* ln) a) * hlog b x"
by transfer (rule log_eq_div_ln_mult_log)

lemma powhr_as_starfun: "⋀a x. x powhr a = (if x = 0 then 0 else ( *f* exp) (a * ( *f* real_ln) x))"

lemma HInfinite_powhr:
"x ∈ HInfinite ⟹ 0 < x ⟹ a ∈ HFinite - Infinitesimal ⟹ 0 < a ⟹ x powhr a ∈ HInfinite"
by (auto intro!: starfun_ln_ge_zero starfun_ln_HInfinite
HInfinite_HFinite_not_Infinitesimal_mult2 starfun_exp_HInfinite
simp add: order_less_imp_le HInfinite_gt_zero_gt_one powhr_as_starfun zero_le_mult_iff)

lemma hlog_hrabs_HInfinite_Infinitesimal:
"x ∈ HFinite - Infinitesimal ⟹ a ∈ HInfinite ⟹ 0 < a ⟹ hlog a ¦x¦ ∈ Infinitesimal"
apply (frule HInfinite_gt_zero_gt_one)
apply (auto intro!: starfun_ln_HFinite_not_Infinitesimal
HInfinite_inverse_Infinitesimal Infinitesimal_HFinite_mult2
hlog_as_starfun divide_inverse)
done

lemma hlog_HInfinite_as_starfun: "a ∈ HInfinite ⟹ 0 < a ⟹ hlog a x = ( *f* ln) x / ( *f* ln) a"
by (rule hlog_as_starfun) auto

lemma hlog_one [simp]: "⋀a. hlog a 1 = 0"
by transfer simp

lemma hlog_eq_one [simp]: "⋀a. 0 < a ⟹ a ≠ 1 ⟹ hlog a a = 1"
by transfer (rule log_eq_one)

lemma hlog_inverse: "0 < a ⟹ a ≠ 1 ⟹ 0 < x ⟹ hlog a (inverse x) = - hlog a x"
by (rule add_left_cancel [of "hlog a x", THEN iffD1]) (simp add: hlog_mult [symmetric])

lemma hlog_divide: "0 < a ⟹ a ≠ 1 ⟹ 0 < x ⟹ 0 < y ⟹ hlog a (x / y) = hlog a x - hlog a y"
by (simp add: hlog_mult hlog_inverse divide_inverse)

lemma hlog_less_cancel_iff [simp]:
"⋀a x y. 1 < a ⟹ 0 < x ⟹ 0 < y ⟹ hlog a x < hlog a y ⟷ x < y"
by transfer simp

lemma hlog_le_cancel_iff [simp]: "1 < a ⟹ 0 < x ⟹ 0 < y ⟹ hlog a x ≤ hlog a y ⟷ x ≤ y"