# Theory Code_Real_Approx_By_Float

theory Code_Real_Approx_By_Float
imports Complex_Main Code_Target_Int
```(*  Title:      HOL/Library/Code_Real_Approx_By_Float.thy
Author:     Florian Haftmann
Author:     Johannes Hölzl
Author:     Tobias Nipkow
*)

theory Code_Real_Approx_By_Float
imports Complex_Main Code_Target_Int
begin

text‹
❙‹WARNING!› This theory implements mathematical reals by machine reals
(floats). This is inconsistent. See the proof of False at the end of the
theory, where an equality on mathematical reals is (incorrectly) disproved
by mapping it to machine reals.

The ⬚‹value› command cannot display real results yet.

The only legitimate use of this theory is as a tool for code generation
purposes.
›

code_printing
type_constructor real ⇀
(SML) "real"
and (OCaml) "float"

code_printing
constant Ratreal ⇀

code_printing
constant "0 :: real" ⇀
(SML) "0.0"
and (OCaml) "0.0"
declare zero_real_code[code_unfold del]

code_printing
constant "1 :: real" ⇀
(SML) "1.0"
and (OCaml) "1.0"
declare one_real_code[code_unfold del]

code_printing
constant "HOL.equal :: real ⇒ real ⇒ bool" ⇀
(SML) "Real.== ((_), (_))"
and (OCaml) "Pervasives.(=)"

code_printing
constant "Orderings.less_eq :: real ⇒ real ⇒ bool" ⇀
(SML) "Real.<= ((_), (_))"
and (OCaml) "Pervasives.(<=)"

code_printing
constant "Orderings.less :: real ⇒ real ⇒ bool" ⇀
(SML) "Real.< ((_), (_))"
and (OCaml) "Pervasives.(<)"

code_printing
constant "op + :: real ⇒ real ⇒ real" ⇀
(SML) "Real.+ ((_), (_))"
and (OCaml) "Pervasives.( +. )"

code_printing
constant "op * :: real ⇒ real ⇒ real" ⇀
(SML) "Real.* ((_), (_))"
and (OCaml) "Pervasives.( *. )"

code_printing
constant "op - :: real ⇒ real ⇒ real" ⇀
(SML) "Real.- ((_), (_))"
and (OCaml) "Pervasives.( -. )"

code_printing
constant "uminus :: real ⇒ real" ⇀
(SML) "Real.~"
and (OCaml) "Pervasives.( ~-. )"

code_printing
constant "op / :: real ⇒ real ⇒ real" ⇀
(SML) "Real.'/ ((_), (_))"
and (OCaml) "Pervasives.( '/. )"

code_printing
constant "HOL.equal :: real ⇒ real ⇒ bool" ⇀
(SML) "Real.== ((_:real), (_))"

code_printing
constant "sqrt :: real ⇒ real" ⇀
(SML) "Math.sqrt"
and (OCaml) "Pervasives.sqrt"
declare sqrt_def[code del]

context
begin

qualified definition real_exp :: "real ⇒ real"
where "real_exp = exp"

lemma exp_eq_real_exp [code_unfold]: "exp = real_exp"
unfolding real_exp_def ..

end

code_printing
constant Code_Real_Approx_By_Float.real_exp ⇀
(SML) "Math.exp"
and (OCaml) "Pervasives.exp"
declare Code_Real_Approx_By_Float.real_exp_def[code del]
declare exp_def[code del]

code_printing
constant ln ⇀
(SML) "Math.ln"
and (OCaml) "Pervasives.ln"
declare ln_real_def[code del]

code_printing
constant cos ⇀
(SML) "Math.cos"
and (OCaml) "Pervasives.cos"
declare cos_def[code del]

code_printing
constant sin ⇀
(SML) "Math.sin"
and (OCaml) "Pervasives.sin"
declare sin_def[code del]

code_printing
constant pi ⇀
(SML) "Math.pi"
and (OCaml) "Pervasives.pi"
declare pi_def[code del]

code_printing
constant arctan ⇀
(SML) "Math.atan"
and (OCaml) "Pervasives.atan"
declare arctan_def[code del]

code_printing
constant arccos ⇀
(SML) "Math.scos"
and (OCaml) "Pervasives.acos"
declare arccos_def[code del]

code_printing
constant arcsin ⇀
(SML) "Math.asin"
and (OCaml) "Pervasives.asin"
declare arcsin_def[code del]

definition real_of_integer :: "integer ⇒ real"
where "real_of_integer = of_int ∘ int_of_integer"

code_printing
constant real_of_integer ⇀
(SML) "Real.fromInt"
and (OCaml) "Pervasives.float (Big'_int.int'_of'_big'_int (_))"

context
begin

qualified definition real_of_int :: "int ⇒ real"
where [code_abbrev]: "real_of_int = of_int"

lemma [code]: "real_of_int = real_of_integer ∘ integer_of_int"
by (simp add: fun_eq_iff real_of_integer_def real_of_int_def)

lemma [code_unfold del]: "0 ≡ (of_rat 0 :: real)"
by simp

lemma [code_unfold del]: "1 ≡ (of_rat 1 :: real)"
by simp

lemma [code_unfold del]: "numeral k ≡ (of_rat (numeral k) :: real)"
by simp

lemma [code_unfold del]: "- numeral k ≡ (of_rat (- numeral k) :: real)"
by simp

end

code_printing
constant Ratreal ⇀ (SML)

definition Realfract :: "int ⇒ int ⇒ real"
where "Realfract p q = of_int p / of_int q"

code_datatype Realfract

code_printing
constant Realfract ⇀ (SML) "Real.fromInt _/ '// Real.fromInt _"

lemma [code]: "Ratreal r = case_prod Realfract (quotient_of r)"
by (cases r) (simp add: Realfract_def quotient_of_Fract of_rat_rat)

lemma [code, code del]: "(HOL.equal :: real ⇒ real ⇒ bool) = (HOL.equal :: real ⇒ real ⇒ bool)"
..

lemma [code, code del]: "(plus :: real ⇒ real ⇒ real) = plus"
..

lemma [code, code del]: "(uminus :: real ⇒ real) = uminus"
..

lemma [code, code del]: "(minus :: real ⇒ real ⇒ real) = minus"
..

lemma [code, code del]: "(times :: real ⇒ real ⇒ real) = times"
..

lemma [code, code del]: "(divide :: real ⇒ real ⇒ real) = divide"
..

lemma [code]: "inverse r = 1 / r" for r :: real
by (fact inverse_eq_divide)