# Theory HLog

theory HLog
imports HTranscendental
```(*  Title       : HLog.thy
Author      : Jacques D. Fleuriot
Copyright   : 2000,2001 University of Edinburgh
*)

section‹Logarithms: Non-Standard Version›

theory HLog
imports HTranscendental
begin

(* should be in NSA.ML *)
lemma epsilon_ge_zero [simp]: "0 ≤ ε"
by (simp add: epsilon_def star_n_zero_num star_n_le)

lemma hpfinite_witness: "ε : {x. 0 ≤ x & x : HFinite}"
by auto

definition
powhr  :: "[hypreal,hypreal] => hypreal"     (infixr "powhr" 80) where
[transfer_unfold]: "x powhr a = starfun2 (op 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 (%n. (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 (%n. 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"
apply (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)
done

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"
apply (rule add_left_cancel [of "hlog a x", THEN iffD1])
done

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)"