Theory Affine_Arithmetic.Print

section ‹Target Language debug messages›
theory Print

hide_const (open) floatarith.Max

subsection ‹Printing›

text ‹Just for debugging purposes›

definition print::"String.literal  unit" where "print x = ()"

context includes integer.lifting begin


code_printing constant print  (SML) "TextIO.print"

subsection ‹Write to File›

definition file_output::"String.literal  ((String.literal  unit)  'a)  'a" where
  "file_output _ f = f (λ_. ())"
code_printing constant file_output  (SML) "(fn s => fn f => File''_output (fn os => f (File'_Stream.output os)) (Path.explode s))"

subsection ‹Show for Floats›

definition showsp_float :: "float showsp"
  "showsp_float p x = (
    let m = mantissa x; e = exponent x in
      if e = 0 then showsp_int p m else showsp_int p m o shows_string ''*2^'' o showsp_int p e)"

lemma show_law_float [show_law_intros]:
  "show_law showsp_float r"
  by (auto simp: showsp_float_def Let_def show_law_simps intro!: show_lawI)

lemma showsp_float_append [show_law_simps]:
  "showsp_float p r (x @ y) = showsp_float p r x @ y"
  by (intro show_lawD show_law_intros)

local_setup Show_Generator.register_foreign_showsp @{typ float} @{term "showsp_float"} @{thm show_law_float}

derive "show" float

subsection ‹Convert Float to Decimal number›

text ‹type for decimal floating point numbers
  (currently just for printing, TODO? generalize theory Float for arbitrary base)›

datatype float10 = Float10 (mantissa10: int) (exponent10: int)
notation Float10 (infix "𝖾" 999)

partial_function (tailrec) normalize_float10
  where [code]: "normalize_float10 f =
    (if mantissa10 f mod 10  0  mantissa10 f = 0 then f
    else normalize_float10 (Float10 (mantissa10 f div 20) (exponent10 f + 1)))"

subsubsection ‹Version that should be easy to prove correct, but slow!›

context includes floatarith_notation begin

definition "float_to_float10_approximation f = the
  (do {
    let (x, y) = (mantissa f * 1024, exponent f - 10);
    let p = nat (bitlen (abs x) + bitlen (abs y) + 80); ― ‹FIXME: are there guarantees?›
    y_log  approx p (Mult (Num (of_int y))
      ((Mult (Ln (Num 2))
        (Inverse (Ln (Num 10)))))) [];
    let e_fl = floor_fl (lower y_log);
    let e = int_floor_fl e_fl;
    m  approx p (Mult (Num (of_int x)) (Powr (Num 10) (Add(Var 0) (Minus (Num e_fl))))) [Some y_log];
    let ml = lower m;
    let mu = upper m;
    Some (normalize_float10 (Float10 (int_floor_fl ml) e), normalize_float10 (Float10 (- int_floor_fl (- mu)) e))


lemma compute_float_of[code]: "float_of (real_of_float f) = f" by simp

subsection ‹Trusted, but faster version›

text ‹TODO: this is the HOL version of the SML-code in Approximation.thy›

lemma prod_case_call_mono[partial_function_mono]:
  "mono_tailrec (λf. (let (d, e) = a in (λy. f (c d e y))) b)"
  by (simp add: split_beta' call_mono)

definition divmod_int::"int  int  int * int"
where "divmod_int a b = (a div b, a mod b)"

partial_function (tailrec) f2f10_frac where
 "f2f10_frac c p r digits cnt e =
    (if r = 0 then (digits, cnt, 0)
    else if p = 0 then (digits, cnt, r)
    else (let
      (d, r) = divmod_int (r * 10) (power_int 2 (-e))
    in f2f10_frac (c  d  0) (if d  0  c then p - 1 else p) r
      (digits * 10 + d) (cnt + 1)) e)"
declare f2f10_frac.simps[code]

definition float2_float10::"int  bool  int  int  (int * int)" where
  "float2_float10 prec rd m e = (
    (m, e) = (if e < 0 then (m,e) else (m * power_int 2 e, 0));
    sgn = sgn m;
    m = abs m;

    round_down = (sgn = 1  rd)  (sgn = -1  ¬ rd);

    (x, r) = divmod_int m ((power_int 2 (-e)));

    p = ((if x = 0 then prec else prec - (log2 x + 1)) * 3) div 10 + 1;

    (digits, e10, r) = if p > 0 then f2f10_frac (x  0) p r 0 0 e else (0,0,0);

    digits = if round_down  r = 0 then digits else digits + 1

  in (sgn * (digits + x * (power_int 10 e10)), -e10))"

definition "lfloat10 r = (let f = float_of r in case_prod Float10 (float2_float10 20 True (mantissa f) (exponent f)))"
definition "ufloat10 r = (let f = float_of r in case_prod Float10 (float2_float10 20 False (mantissa f) (exponent f)))"

partial_function (tailrec) digits
  where [code]: "digits m ds = (if m = 0 then ds else digits (m div 10) (m mod 10 # ds))"

primrec showsp_float10 :: "float10 showsp"
  "showsp_float10 p (Float10 m e) = (
      ds = digits (nat (abs m)) [];
      d = int (length ds);
      e = e + d - 1;
      mp = take 1 ds;
      ms = drop 1 ds;
      ms = rev (dropWhile ((=) 0) (rev ms));
      show_digits = shows_list_gen (showsp_nat p) ''0'' '''' '''' ''''
    in (if m < 0 then shows_string ''-'' else (λx. x)) o
        show_digits mp o
        (if ms = [] then (λx. x) else shows_string ''.'' o show_digits ms) o
        (if e = 0 then (λx. x) else shows_string ''e'' o showsp_int p e))"

lemma show_law_float10_aux:
  fixes m e
  shows "show_law showsp_float10 (Float10 m e)"
  apply (rule show_lawI)
  unfolding showsp_float10.simps Let_def
  apply (simp add: show_law_simps )

lemma show_law_float10 [show_law_intros]: "show_law showsp_float10 r"
  by (cases r) (auto simp: show_law_float10_aux)

lemma showsp_float10_append [show_law_simps]:
  "showsp_float10 p r (x @ y) = showsp_float10 p r x @ y"
  by (intro show_lawD show_law_intros)

local_setup Show_Generator.register_foreign_showsp @{typ float10} @{term "showsp_float10"} @{thm show_law_float10}

derive "show" float10

definition "showsp_real p x = showsp_float10 p (lfloat10 x)"

lemma show_law_real[show_law_intros]: "show_law showsp_real x"
  using show_law_float10[of "lfloat10 x"]
  by (auto simp: showsp_real_def[abs_def] Let_def show_law_def
      simp del: showsp_float10.simps intro!: show_law_intros)

local_setup Show_Generator.register_foreign_showsp @{typ real} @{term "showsp_real"} @{thm show_law_real}
derive "show" real

subsection ‹gnuplot output›

subsubsection ‹vector output of 2D zonotope›

fun polychain_of_segments::"((real × real) × (real × real)) list  (real × real) list"
    "polychain_of_segments [] = []"
  | "polychain_of_segments (((x0, y0), z)#segs) = (x0, y0)#z#map snd segs"

definition shows_segments_of_aform
  where "shows_segments_of_aform a b xs color =
  shows_list_gen id '''' '''' ''⏎'' ''⏎'' (map (λ(x0, y0).
      shows_words (map lfloat10 [x0, y0]) o shows_space o shows_string color)
    (polychain_of_segments (segments_of_aform (prod_of_aforms (xs ! a) (xs ! b)))))"
abbreviation "show_segments_of_aform a b x c  shows_segments_of_aform a b x c ''''"

definition shows_box_of_aforms― ‹box and some further information›
where "shows_box_of_aforms (XS::real aform list) = (let
    RS = map (Radius' 20) XS;
    l = map (Inf_aform' 20) XS;
    u = map (Sup_aform' 20) XS
    in shows_words
      (l @ u @ RS) o shows_space o
      shows (card (((λx. pdevs_domain (snd x)) ` (set XS))))
abbreviation "show_box_of_aforms x  shows_box_of_aforms x ''''"

definition "pdevs_domains ((XS::real aform list)) = (((λx. pdevs_domain (snd x)) ` (set XS)))"

definition "generators XS =
      is = sorted_list_of_set (pdevs_domains XS);
      rs = map (λi. (i, map (λx. pdevs_apply (snd x) i) XS)) is
      (map fst XS, rs))"

definition shows_box_of_aforms_hr― ‹human readable›
where "shows_box_of_aforms_hr XS = (let
    RS = map (Radius' 20) XS;
    l = map (Inf_aform' 20) XS;
    u = map (Sup_aform' 20) XS
    in shows_paren (shows_words l) o shows_string '' .. '' o shows_paren (shows_words u) o
      shows_string ''; devs: '' o shows (card (pdevs_domains XS)) o
      shows_string ''; tdev: '' o shows_paren (shows_words RS)
abbreviation "show_box_of_aforms_hr x  shows_box_of_aforms_hr x ''''"

definition shows_aforms_hr― ‹human readable›
where "shows_aforms_hr XS = shows (generators XS)"

abbreviation "show_aform_hr x  shows_aforms_hr x ''''"