Theory "machine_ieee"

Signature

Definitions

fp16_to_float_def

|- ∀w.
     fp16_to_float w =
     <|Sign := (15 >< 15) w; Exponent := (14 >< 10) w;
       Significand := (9 >< 0) w|>

float_to_fp16_def

|- ∀x. float_to_fp16 x = x.Sign @@ x.Exponent @@ x.Significand

fp16_to_real_def

|- fp16_to_real = float_to_real o fp16_to_float

real_to_fp16_def

|- ∀mode. real_to_fp16 mode = float_to_fp16 o round mode

fp16_roundToIntegral_def

|- ∀mode.
     fp16_roundToIntegral mode =
     float_to_fp16 o float_round_to_integral mode o fp16_to_float

fp16_sqrt_def

|- ∀mode. fp16_sqrt mode = float_to_fp16 o float_sqrt mode o fp16_to_float

fp16_negate_def

|- fp16_negate = float_to_fp16 o float_negate o fp16_to_float

fp16_abs_def

|- fp16_abs = float_to_fp16 o float_abs o fp16_to_float

fp16_isNan_def

|- fp16_isNan = float_is_nan o fp16_to_float

fp16_isIntegral_def

|- fp16_isIntegral = float_is_integral o fp16_to_float

fp16_isZero_def

|- fp16_isZero = float_is_zero o fp16_to_float

fp16_isNormal_def

|- fp16_isNormal = float_is_normal o fp16_to_float

fp16_isSubnormal_def

|- fp16_isSubnormal = float_is_subnormal o fp16_to_float

fp16_isFinite_def

|- fp16_isFinite = float_is_finite o fp16_to_float

fp16_isInfinite_def

|- fp16_isInfinite = float_is_infinite o fp16_to_float

fp16_posInf_def

|- fp16_posInf = float_to_fp16 (float_plus_infinity (:10 # 5))

fp16_negInf_def

|- fp16_negInf = float_to_fp16 (float_minus_infinity (:10 # 5))

fp16_posZero_def

|- fp16_posZero = float_to_fp16 (float_plus_zero (:10 # 5))

fp16_negZero_def

|- fp16_negZero = float_to_fp16 (float_minus_zero (:10 # 5))

fp16_posMin_def

|- fp16_posMin = float_to_fp16 (float_plus_min (:10 # 5))

fp16_negMin_def

|- fp16_negMin = float_to_fp16 (float_minus_min (:10 # 5))

fp16_top_def

|- fp16_top = float_to_fp16 (float_top (:10 # 5))

fp16_bottom_def

|- fp16_bottom = float_to_fp16 (float_bottom (:10 # 5))

fp16_add_def

|- ∀mode a b.
     fp16_add mode a b =
     float_to_fp16 ((mode float_add fp16_to_float a) (fp16_to_float b))

fp16_sub_def

|- ∀mode a b.
     fp16_sub mode a b =
     float_to_fp16 ((mode float_sub fp16_to_float a) (fp16_to_float b))

fp16_mul_def

|- ∀mode a b.
     fp16_mul mode a b =
     float_to_fp16 ((mode float_mul fp16_to_float a) (fp16_to_float b))

fp16_div_def

|- ∀mode a b.
     fp16_div mode a b =
     float_to_fp16 ((mode float_div fp16_to_float a) (fp16_to_float b))

fp16_equal_def

|- ∀a b. fp16_equal a b ⇔ float_equal (fp16_to_float a) (fp16_to_float b)

fp16_lessThan_def

|- ∀a b.
     fp16_lessThan a b ⇔ float_less_than (fp16_to_float a) (fp16_to_float b)

fp16_lessEqual_def

|- ∀a b.
     fp16_lessEqual a b ⇔ float_less_equal (fp16_to_float a) (fp16_to_float b)

fp16_greaterThan_def

|- ∀a b.
     fp16_greaterThan a b ⇔
     float_greater_than (fp16_to_float a) (fp16_to_float b)

fp16_greaterEqual_def

|- ∀a b.
     fp16_greaterEqual a b ⇔
     float_greater_equal (fp16_to_float a) (fp16_to_float b)

fp16_mul_add_def

|- ∀mode a b c.
     fp16_mul_add mode a b c =
     float_to_fp16
       (float_mul_add mode (fp16_to_float a) (fp16_to_float b)
          (fp16_to_float c))

fp32_to_float_def

|- ∀w.
     fp32_to_float w =
     <|Sign := (31 >< 31) w; Exponent := (30 >< 23) w;
       Significand := (22 >< 0) w|>

float_to_fp32_def

|- ∀x. float_to_fp32 x = x.Sign @@ x.Exponent @@ x.Significand

fp32_to_real_def

|- fp32_to_real = float_to_real o fp32_to_float

real_to_fp32_def

|- ∀mode. real_to_fp32 mode = float_to_fp32 o round mode

fp32_roundToIntegral_def

|- ∀mode.
     fp32_roundToIntegral mode =
     float_to_fp32 o float_round_to_integral mode o fp32_to_float

fp32_sqrt_def

|- ∀mode. fp32_sqrt mode = float_to_fp32 o float_sqrt mode o fp32_to_float

fp32_negate_def

|- fp32_negate = float_to_fp32 o float_negate o fp32_to_float

fp32_abs_def

|- fp32_abs = float_to_fp32 o float_abs o fp32_to_float

fp32_isNan_def

|- fp32_isNan = float_is_nan o fp32_to_float

fp32_isIntegral_def

|- fp32_isIntegral = float_is_integral o fp32_to_float

fp32_isZero_def

|- fp32_isZero = float_is_zero o fp32_to_float

fp32_isNormal_def

|- fp32_isNormal = float_is_normal o fp32_to_float

fp32_isSubnormal_def

|- fp32_isSubnormal = float_is_subnormal o fp32_to_float

fp32_isFinite_def

|- fp32_isFinite = float_is_finite o fp32_to_float

fp32_isInfinite_def

|- fp32_isInfinite = float_is_infinite o fp32_to_float

fp32_posInf_def

|- fp32_posInf = float_to_fp32 (float_plus_infinity (:23 # 8))

fp32_negInf_def

|- fp32_negInf = float_to_fp32 (float_minus_infinity (:23 # 8))

fp32_posZero_def

|- fp32_posZero = float_to_fp32 (float_plus_zero (:23 # 8))

fp32_negZero_def

|- fp32_negZero = float_to_fp32 (float_minus_zero (:23 # 8))

fp32_posMin_def

|- fp32_posMin = float_to_fp32 (float_plus_min (:23 # 8))

fp32_negMin_def

|- fp32_negMin = float_to_fp32 (float_minus_min (:23 # 8))

fp32_top_def

|- fp32_top = float_to_fp32 (float_top (:23 # 8))

fp32_bottom_def

|- fp32_bottom = float_to_fp32 (float_bottom (:23 # 8))

fp32_add_def

|- ∀mode a b.
     fp32_add mode a b =
     float_to_fp32 ((mode float_add fp32_to_float a) (fp32_to_float b))

fp32_sub_def

|- ∀mode a b.
     fp32_sub mode a b =
     float_to_fp32 ((mode float_sub fp32_to_float a) (fp32_to_float b))

fp32_mul_def

|- ∀mode a b.
     fp32_mul mode a b =
     float_to_fp32 ((mode float_mul fp32_to_float a) (fp32_to_float b))

fp32_div_def

|- ∀mode a b.
     fp32_div mode a b =
     float_to_fp32 ((mode float_div fp32_to_float a) (fp32_to_float b))

fp32_equal_def

|- ∀a b. fp32_equal a b ⇔ float_equal (fp32_to_float a) (fp32_to_float b)

fp32_lessThan_def

|- ∀a b.
     fp32_lessThan a b ⇔ float_less_than (fp32_to_float a) (fp32_to_float b)

fp32_lessEqual_def

|- ∀a b.
     fp32_lessEqual a b ⇔ float_less_equal (fp32_to_float a) (fp32_to_float b)

fp32_greaterThan_def

|- ∀a b.
     fp32_greaterThan a b ⇔
     float_greater_than (fp32_to_float a) (fp32_to_float b)

fp32_greaterEqual_def

|- ∀a b.
     fp32_greaterEqual a b ⇔
     float_greater_equal (fp32_to_float a) (fp32_to_float b)

fp32_mul_add_def

|- ∀mode a b c.
     fp32_mul_add mode a b c =
     float_to_fp32
       (float_mul_add mode (fp32_to_float a) (fp32_to_float b)
          (fp32_to_float c))

fp64_to_float_def

|- ∀w.
     fp64_to_float w =
     <|Sign := (63 >< 63) w; Exponent := (62 >< 52) w;
       Significand := (51 >< 0) w|>

float_to_fp64_def

|- ∀x. float_to_fp64 x = x.Sign @@ x.Exponent @@ x.Significand

fp64_to_real_def

|- fp64_to_real = float_to_real o fp64_to_float

real_to_fp64_def

|- ∀mode. real_to_fp64 mode = float_to_fp64 o round mode

fp64_roundToIntegral_def

|- ∀mode.
     fp64_roundToIntegral mode =
     float_to_fp64 o float_round_to_integral mode o fp64_to_float

fp64_sqrt_def

|- ∀mode. fp64_sqrt mode = float_to_fp64 o float_sqrt mode o fp64_to_float

fp64_negate_def

|- fp64_negate = float_to_fp64 o float_negate o fp64_to_float

fp64_abs_def

|- fp64_abs = float_to_fp64 o float_abs o fp64_to_float

fp64_isNan_def

|- fp64_isNan = float_is_nan o fp64_to_float

fp64_isIntegral_def

|- fp64_isIntegral = float_is_integral o fp64_to_float

fp64_isZero_def

|- fp64_isZero = float_is_zero o fp64_to_float

fp64_isNormal_def

|- fp64_isNormal = float_is_normal o fp64_to_float

fp64_isSubnormal_def

|- fp64_isSubnormal = float_is_subnormal o fp64_to_float

fp64_isFinite_def

|- fp64_isFinite = float_is_finite o fp64_to_float

fp64_isInfinite_def

|- fp64_isInfinite = float_is_infinite o fp64_to_float

fp64_posInf_def

|- fp64_posInf = float_to_fp64 (float_plus_infinity (:52 # 11))

fp64_negInf_def

|- fp64_negInf = float_to_fp64 (float_minus_infinity (:52 # 11))

fp64_posZero_def

|- fp64_posZero = float_to_fp64 (float_plus_zero (:52 # 11))

fp64_negZero_def

|- fp64_negZero = float_to_fp64 (float_minus_zero (:52 # 11))

fp64_posMin_def

|- fp64_posMin = float_to_fp64 (float_plus_min (:52 # 11))

fp64_negMin_def

|- fp64_negMin = float_to_fp64 (float_minus_min (:52 # 11))

fp64_top_def

|- fp64_top = float_to_fp64 (float_top (:52 # 11))

fp64_bottom_def

|- fp64_bottom = float_to_fp64 (float_bottom (:52 # 11))

fp64_add_def

|- ∀mode a b.
     fp64_add mode a b =
     float_to_fp64 ((mode float_add fp64_to_float a) (fp64_to_float b))

fp64_sub_def

|- ∀mode a b.
     fp64_sub mode a b =
     float_to_fp64 ((mode float_sub fp64_to_float a) (fp64_to_float b))

fp64_mul_def

|- ∀mode a b.
     fp64_mul mode a b =
     float_to_fp64 ((mode float_mul fp64_to_float a) (fp64_to_float b))

fp64_div_def

|- ∀mode a b.
     fp64_div mode a b =
     float_to_fp64 ((mode float_div fp64_to_float a) (fp64_to_float b))

fp64_equal_def

|- ∀a b. fp64_equal a b ⇔ float_equal (fp64_to_float a) (fp64_to_float b)

fp64_lessThan_def

|- ∀a b.
     fp64_lessThan a b ⇔ float_less_than (fp64_to_float a) (fp64_to_float b)

fp64_lessEqual_def

|- ∀a b.
     fp64_lessEqual a b ⇔ float_less_equal (fp64_to_float a) (fp64_to_float b)

fp64_greaterThan_def

|- ∀a b.
     fp64_greaterThan a b ⇔
     float_greater_than (fp64_to_float a) (fp64_to_float b)

fp64_greaterEqual_def

|- ∀a b.
     fp64_greaterEqual a b ⇔
     float_greater_equal (fp64_to_float a) (fp64_to_float b)

fp64_mul_add_def

|- ∀mode a b c.
     fp64_mul_add mode a b c =
     float_to_fp64
       (float_mul_add mode (fp64_to_float a) (fp64_to_float b)
          (fp64_to_float c))

Theorems

fp16_to_float_11

|- ∀x y. (fp16_to_float x = fp16_to_float y) ⇔ (x = y)

float_to_fp16_11

|- ∀x y. (float_to_fp16 x = float_to_fp16 y) ⇔ (x = y)

float_to_fp16_fp16_to_float

|- ∀x. float_to_fp16 (fp16_to_float x) = x

fp16_to_float_float_to_fp16

|- ∀x. fp16_to_float (float_to_fp16 x) = x

fp16_nchotomy

|- ∀x. ∃y. x = float_to_fp16 y

float_fp16_nchotomy

|- ∀x. ∃y. x = fp16_to_float y

fp16_to_float_n2w

|- ∀n.
     fp16_to_float (n2w n) =
     (let (q,f) = DIVMOD_2EXP 10 n in
      let (s,e) = DIVMOD_2EXP 5 q
      in
        <|Sign := n2w (s MOD 2); Exponent := n2w e; Significand := n2w f|>)

fp16_to_real_float_to_fp

|- ∀a. fp16_to_real (float_to_fp16 a) = float_to_real a

fp16_to_real_n2w

|- ∀a. fp16_to_real (n2w a) = float_to_real (fp16_to_float (n2w a))

fp16_abs_float_to_fp

|- ∀a. fp16_abs (float_to_fp16 a) = float_to_fp16 (float_abs a)

fp16_abs_n2w

|- ∀a. fp16_abs (n2w a) = float_to_fp16 (float_abs (fp16_to_float (n2w a)))

fp16_negate_float_to_fp

|- ∀a. fp16_negate (float_to_fp16 a) = float_to_fp16 (float_negate a)

fp16_negate_n2w

|- ∀a.
     fp16_negate (n2w a) =
     float_to_fp16 (float_negate (fp16_to_float (n2w a)))

fp16_isNan_float_to_fp

|- ∀a. fp16_isNan (float_to_fp16 a) ⇔ float_is_nan a

fp16_isNan_n2w

|- ∀a. fp16_isNan (n2w a) ⇔ float_is_nan (fp16_to_float (n2w a))

fp16_isIntegral_float_to_fp

|- ∀a. fp16_isIntegral (float_to_fp16 a) ⇔ float_is_integral a

fp16_isIntegral_n2w

|- ∀a. fp16_isIntegral (n2w a) ⇔ float_is_integral (fp16_to_float (n2w a))

fp16_isZero_float_to_fp

|- ∀a. fp16_isZero (float_to_fp16 a) ⇔ float_is_zero a

fp16_isZero_n2w

|- ∀a. fp16_isZero (n2w a) ⇔ float_is_zero (fp16_to_float (n2w a))

fp16_isNormal_float_to_fp

|- ∀a. fp16_isNormal (float_to_fp16 a) ⇔ float_is_normal a

fp16_isNormal_n2w

|- ∀a. fp16_isNormal (n2w a) ⇔ float_is_normal (fp16_to_float (n2w a))

fp16_isSubnormal_float_to_fp

|- ∀a. fp16_isSubnormal (float_to_fp16 a) ⇔ float_is_subnormal a

fp16_isSubnormal_n2w

|- ∀a. fp16_isSubnormal (n2w a) ⇔ float_is_subnormal (fp16_to_float (n2w a))

fp16_isFinite_float_to_fp

|- ∀a. fp16_isFinite (float_to_fp16 a) ⇔ float_is_finite a

fp16_isFinite_n2w

|- ∀a. fp16_isFinite (n2w a) ⇔ float_is_finite (fp16_to_float (n2w a))

fp16_isInfinite_float_to_fp

|- ∀a. fp16_isInfinite (float_to_fp16 a) ⇔ float_is_infinite a

fp16_isInfinite_n2w

|- ∀a. fp16_isInfinite (n2w a) ⇔ float_is_infinite (fp16_to_float (n2w a))

fp16_roundToIntegral_float_to_fp

|- ∀mode a.
     fp16_roundToIntegral mode (float_to_fp16 a) =
     float_to_fp16 (float_round_to_integral mode a)

fp16_roundToIntegral_n2w

|- ∀mode a.
     fp16_roundToIntegral mode (n2w a) =
     float_to_fp16 (float_round_to_integral mode (fp16_to_float (n2w a)))

fp16_sqrt_float_to_fp

|- ∀mode a.
     fp16_sqrt mode (float_to_fp16 a) = float_to_fp16 (float_sqrt mode a)

fp16_sqrt_n2w

|- ∀mode a.
     fp16_sqrt mode (n2w a) =
     float_to_fp16 (float_sqrt mode (fp16_to_float (n2w a)))

fp16_add_float_to_fp

|- ∀mode b a.
     fp16_add mode (float_to_fp16 a) (float_to_fp16 b) =
     float_to_fp16 ((mode float_add a) b)

fp16_add_n2w

|- ∀mode b a.
     fp16_add mode (n2w a) (n2w b) =
     float_to_fp16
       ((mode float_add fp16_to_float (n2w a)) (fp16_to_float (n2w b)))

fp16_sub_float_to_fp

|- ∀mode b a.
     fp16_sub mode (float_to_fp16 a) (float_to_fp16 b) =
     float_to_fp16 ((mode float_sub a) b)

fp16_sub_n2w

|- ∀mode b a.
     fp16_sub mode (n2w a) (n2w b) =
     float_to_fp16
       ((mode float_sub fp16_to_float (n2w a)) (fp16_to_float (n2w b)))

fp16_mul_float_to_fp

|- ∀mode b a.
     fp16_mul mode (float_to_fp16 a) (float_to_fp16 b) =
     float_to_fp16 ((mode float_mul a) b)

fp16_mul_n2w

|- ∀mode b a.
     fp16_mul mode (n2w a) (n2w b) =
     float_to_fp16
       ((mode float_mul fp16_to_float (n2w a)) (fp16_to_float (n2w b)))

fp16_div_float_to_fp

|- ∀mode b a.
     fp16_div mode (float_to_fp16 a) (float_to_fp16 b) =
     float_to_fp16 ((mode float_div a) b)

fp16_div_n2w

|- ∀mode b a.
     fp16_div mode (n2w a) (n2w b) =
     float_to_fp16
       ((mode float_div fp16_to_float (n2w a)) (fp16_to_float (n2w b)))

fp16_equal_float_to_fp

|- ∀b a. fp16_equal (float_to_fp16 a) (float_to_fp16 b) ⇔ float_equal a b

fp16_equal_n2w

|- ∀b a.
     fp16_equal (n2w a) (n2w b) ⇔
     float_equal (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

fp16_lessThan_float_to_fp

|- ∀b a.
     fp16_lessThan (float_to_fp16 a) (float_to_fp16 b) ⇔ float_less_than a b

fp16_lessThan_n2w

|- ∀b a.
     fp16_lessThan (n2w a) (n2w b) ⇔
     float_less_than (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

fp16_lessEqual_float_to_fp

|- ∀b a.
     fp16_lessEqual (float_to_fp16 a) (float_to_fp16 b) ⇔ float_less_equal a b

fp16_lessEqual_n2w

|- ∀b a.
     fp16_lessEqual (n2w a) (n2w b) ⇔
     float_less_equal (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

fp16_greaterThan_float_to_fp

|- ∀b a.
     fp16_greaterThan (float_to_fp16 a) (float_to_fp16 b) ⇔
     float_greater_than a b

fp16_greaterThan_n2w

|- ∀b a.
     fp16_greaterThan (n2w a) (n2w b) ⇔
     float_greater_than (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

fp16_greaterEqual_float_to_fp

|- ∀b a.
     fp16_greaterEqual (float_to_fp16 a) (float_to_fp16 b) ⇔
     float_greater_equal a b

fp16_greaterEqual_n2w

|- ∀b a.
     fp16_greaterEqual (n2w a) (n2w b) ⇔
     float_greater_equal (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

fp16_mul_add_float_to_fp

|- ∀mode c b a.
     fp16_mul_add mode (float_to_fp16 a) (float_to_fp16 b) (float_to_fp16 c) =
     float_to_fp16 (float_mul_add mode a b c)

fp16_mul_add_n2w

|- ∀mode c b a.
     fp16_mul_add mode (n2w a) (n2w b) (n2w c) =
     float_to_fp16
       (float_mul_add mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))
          (fp16_to_float (n2w c)))

fp32_to_float_11

|- ∀x y. (fp32_to_float x = fp32_to_float y) ⇔ (x = y)

float_to_fp32_11

|- ∀x y. (float_to_fp32 x = float_to_fp32 y) ⇔ (x = y)

float_to_fp32_fp32_to_float

|- ∀x. float_to_fp32 (fp32_to_float x) = x

fp32_to_float_float_to_fp32

|- ∀x. fp32_to_float (float_to_fp32 x) = x

fp32_nchotomy

|- ∀x. ∃y. x = float_to_fp32 y

float_fp32_nchotomy

|- ∀x. ∃y. x = fp32_to_float y

fp32_to_float_n2w

|- ∀n.
     fp32_to_float (n2w n) =
     (let (q,f) = DIVMOD_2EXP 23 n in
      let (s,e) = DIVMOD_2EXP 8 q
      in
        <|Sign := n2w (s MOD 2); Exponent := n2w e; Significand := n2w f|>)

fp32_to_real_float_to_fp

|- ∀a. fp32_to_real (float_to_fp32 a) = float_to_real a

fp32_to_real_n2w

|- ∀a. fp32_to_real (n2w a) = float_to_real (fp32_to_float (n2w a))

fp32_abs_float_to_fp

|- ∀a. fp32_abs (float_to_fp32 a) = float_to_fp32 (float_abs a)

fp32_abs_n2w

|- ∀a. fp32_abs (n2w a) = float_to_fp32 (float_abs (fp32_to_float (n2w a)))

fp32_negate_float_to_fp

|- ∀a. fp32_negate (float_to_fp32 a) = float_to_fp32 (float_negate a)

fp32_negate_n2w

|- ∀a.
     fp32_negate (n2w a) =
     float_to_fp32 (float_negate (fp32_to_float (n2w a)))

fp32_isNan_float_to_fp

|- ∀a. fp32_isNan (float_to_fp32 a) ⇔ float_is_nan a

fp32_isNan_n2w

|- ∀a. fp32_isNan (n2w a) ⇔ float_is_nan (fp32_to_float (n2w a))

fp32_isIntegral_float_to_fp

|- ∀a. fp32_isIntegral (float_to_fp32 a) ⇔ float_is_integral a

fp32_isIntegral_n2w

|- ∀a. fp32_isIntegral (n2w a) ⇔ float_is_integral (fp32_to_float (n2w a))

fp32_isZero_float_to_fp

|- ∀a. fp32_isZero (float_to_fp32 a) ⇔ float_is_zero a

fp32_isZero_n2w

|- ∀a. fp32_isZero (n2w a) ⇔ float_is_zero (fp32_to_float (n2w a))

fp32_isNormal_float_to_fp

|- ∀a. fp32_isNormal (float_to_fp32 a) ⇔ float_is_normal a

fp32_isNormal_n2w

|- ∀a. fp32_isNormal (n2w a) ⇔ float_is_normal (fp32_to_float (n2w a))

fp32_isSubnormal_float_to_fp

|- ∀a. fp32_isSubnormal (float_to_fp32 a) ⇔ float_is_subnormal a

fp32_isSubnormal_n2w

|- ∀a. fp32_isSubnormal (n2w a) ⇔ float_is_subnormal (fp32_to_float (n2w a))

fp32_isFinite_float_to_fp

|- ∀a. fp32_isFinite (float_to_fp32 a) ⇔ float_is_finite a

fp32_isFinite_n2w

|- ∀a. fp32_isFinite (n2w a) ⇔ float_is_finite (fp32_to_float (n2w a))

fp32_isInfinite_float_to_fp

|- ∀a. fp32_isInfinite (float_to_fp32 a) ⇔ float_is_infinite a

fp32_isInfinite_n2w

|- ∀a. fp32_isInfinite (n2w a) ⇔ float_is_infinite (fp32_to_float (n2w a))

fp32_roundToIntegral_float_to_fp

|- ∀mode a.
     fp32_roundToIntegral mode (float_to_fp32 a) =
     float_to_fp32 (float_round_to_integral mode a)

fp32_roundToIntegral_n2w

|- ∀mode a.
     fp32_roundToIntegral mode (n2w a) =
     float_to_fp32 (float_round_to_integral mode (fp32_to_float (n2w a)))

fp32_sqrt_float_to_fp

|- ∀mode a.
     fp32_sqrt mode (float_to_fp32 a) = float_to_fp32 (float_sqrt mode a)

fp32_sqrt_n2w

|- ∀mode a.
     fp32_sqrt mode (n2w a) =
     float_to_fp32 (float_sqrt mode (fp32_to_float (n2w a)))

fp32_add_float_to_fp

|- ∀mode b a.
     fp32_add mode (float_to_fp32 a) (float_to_fp32 b) =
     float_to_fp32 ((mode float_add a) b)

fp32_add_n2w

|- ∀mode b a.
     fp32_add mode (n2w a) (n2w b) =
     float_to_fp32
       ((mode float_add fp32_to_float (n2w a)) (fp32_to_float (n2w b)))

fp32_sub_float_to_fp

|- ∀mode b a.
     fp32_sub mode (float_to_fp32 a) (float_to_fp32 b) =
     float_to_fp32 ((mode float_sub a) b)

fp32_sub_n2w

|- ∀mode b a.
     fp32_sub mode (n2w a) (n2w b) =
     float_to_fp32
       ((mode float_sub fp32_to_float (n2w a)) (fp32_to_float (n2w b)))

fp32_mul_float_to_fp

|- ∀mode b a.
     fp32_mul mode (float_to_fp32 a) (float_to_fp32 b) =
     float_to_fp32 ((mode float_mul a) b)

fp32_mul_n2w

|- ∀mode b a.
     fp32_mul mode (n2w a) (n2w b) =
     float_to_fp32
       ((mode float_mul fp32_to_float (n2w a)) (fp32_to_float (n2w b)))

fp32_div_float_to_fp

|- ∀mode b a.
     fp32_div mode (float_to_fp32 a) (float_to_fp32 b) =
     float_to_fp32 ((mode float_div a) b)

fp32_div_n2w

|- ∀mode b a.
     fp32_div mode (n2w a) (n2w b) =
     float_to_fp32
       ((mode float_div fp32_to_float (n2w a)) (fp32_to_float (n2w b)))

fp32_equal_float_to_fp

|- ∀b a. fp32_equal (float_to_fp32 a) (float_to_fp32 b) ⇔ float_equal a b

fp32_equal_n2w

|- ∀b a.
     fp32_equal (n2w a) (n2w b) ⇔
     float_equal (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

fp32_lessThan_float_to_fp

|- ∀b a.
     fp32_lessThan (float_to_fp32 a) (float_to_fp32 b) ⇔ float_less_than a b

fp32_lessThan_n2w

|- ∀b a.
     fp32_lessThan (n2w a) (n2w b) ⇔
     float_less_than (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

fp32_lessEqual_float_to_fp

|- ∀b a.
     fp32_lessEqual (float_to_fp32 a) (float_to_fp32 b) ⇔ float_less_equal a b

fp32_lessEqual_n2w

|- ∀b a.
     fp32_lessEqual (n2w a) (n2w b) ⇔
     float_less_equal (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

fp32_greaterThan_float_to_fp

|- ∀b a.
     fp32_greaterThan (float_to_fp32 a) (float_to_fp32 b) ⇔
     float_greater_than a b

fp32_greaterThan_n2w

|- ∀b a.
     fp32_greaterThan (n2w a) (n2w b) ⇔
     float_greater_than (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

fp32_greaterEqual_float_to_fp

|- ∀b a.
     fp32_greaterEqual (float_to_fp32 a) (float_to_fp32 b) ⇔
     float_greater_equal a b

fp32_greaterEqual_n2w

|- ∀b a.
     fp32_greaterEqual (n2w a) (n2w b) ⇔
     float_greater_equal (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

fp32_mul_add_float_to_fp

|- ∀mode c b a.
     fp32_mul_add mode (float_to_fp32 a) (float_to_fp32 b) (float_to_fp32 c) =
     float_to_fp32 (float_mul_add mode a b c)

fp32_mul_add_n2w

|- ∀mode c b a.
     fp32_mul_add mode (n2w a) (n2w b) (n2w c) =
     float_to_fp32
       (float_mul_add mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))
          (fp32_to_float (n2w c)))

fp64_to_float_11

|- ∀x y. (fp64_to_float x = fp64_to_float y) ⇔ (x = y)

float_to_fp64_11

|- ∀x y. (float_to_fp64 x = float_to_fp64 y) ⇔ (x = y)

float_to_fp64_fp64_to_float

|- ∀x. float_to_fp64 (fp64_to_float x) = x

fp64_to_float_float_to_fp64

|- ∀x. fp64_to_float (float_to_fp64 x) = x

fp64_nchotomy

|- ∀x. ∃y. x = float_to_fp64 y

float_fp64_nchotomy

|- ∀x. ∃y. x = fp64_to_float y

fp64_to_float_n2w

|- ∀n.
     fp64_to_float (n2w n) =
     (let (q,f) = DIVMOD_2EXP 52 n in
      let (s,e) = DIVMOD_2EXP 11 q
      in
        <|Sign := n2w (s MOD 2); Exponent := n2w e; Significand := n2w f|>)

fp64_to_real_float_to_fp

|- ∀a. fp64_to_real (float_to_fp64 a) = float_to_real a

fp64_to_real_n2w

|- ∀a. fp64_to_real (n2w a) = float_to_real (fp64_to_float (n2w a))

fp64_abs_float_to_fp

|- ∀a. fp64_abs (float_to_fp64 a) = float_to_fp64 (float_abs a)

fp64_abs_n2w

|- ∀a. fp64_abs (n2w a) = float_to_fp64 (float_abs (fp64_to_float (n2w a)))

fp64_negate_float_to_fp

|- ∀a. fp64_negate (float_to_fp64 a) = float_to_fp64 (float_negate a)

fp64_negate_n2w

|- ∀a.
     fp64_negate (n2w a) =
     float_to_fp64 (float_negate (fp64_to_float (n2w a)))

fp64_isNan_float_to_fp

|- ∀a. fp64_isNan (float_to_fp64 a) ⇔ float_is_nan a

fp64_isNan_n2w

|- ∀a. fp64_isNan (n2w a) ⇔ float_is_nan (fp64_to_float (n2w a))

fp64_isIntegral_float_to_fp

|- ∀a. fp64_isIntegral (float_to_fp64 a) ⇔ float_is_integral a

fp64_isIntegral_n2w

|- ∀a. fp64_isIntegral (n2w a) ⇔ float_is_integral (fp64_to_float (n2w a))

fp64_isZero_float_to_fp

|- ∀a. fp64_isZero (float_to_fp64 a) ⇔ float_is_zero a

fp64_isZero_n2w

|- ∀a. fp64_isZero (n2w a) ⇔ float_is_zero (fp64_to_float (n2w a))

fp64_isNormal_float_to_fp

|- ∀a. fp64_isNormal (float_to_fp64 a) ⇔ float_is_normal a

fp64_isNormal_n2w

|- ∀a. fp64_isNormal (n2w a) ⇔ float_is_normal (fp64_to_float (n2w a))

fp64_isSubnormal_float_to_fp

|- ∀a. fp64_isSubnormal (float_to_fp64 a) ⇔ float_is_subnormal a

fp64_isSubnormal_n2w

|- ∀a. fp64_isSubnormal (n2w a) ⇔ float_is_subnormal (fp64_to_float (n2w a))

fp64_isFinite_float_to_fp

|- ∀a. fp64_isFinite (float_to_fp64 a) ⇔ float_is_finite a

fp64_isFinite_n2w

|- ∀a. fp64_isFinite (n2w a) ⇔ float_is_finite (fp64_to_float (n2w a))

fp64_isInfinite_float_to_fp

|- ∀a. fp64_isInfinite (float_to_fp64 a) ⇔ float_is_infinite a

fp64_isInfinite_n2w

|- ∀a. fp64_isInfinite (n2w a) ⇔ float_is_infinite (fp64_to_float (n2w a))

fp64_roundToIntegral_float_to_fp

|- ∀mode a.
     fp64_roundToIntegral mode (float_to_fp64 a) =
     float_to_fp64 (float_round_to_integral mode a)

fp64_roundToIntegral_n2w

|- ∀mode a.
     fp64_roundToIntegral mode (n2w a) =
     float_to_fp64 (float_round_to_integral mode (fp64_to_float (n2w a)))

fp64_sqrt_float_to_fp

|- ∀mode a.
     fp64_sqrt mode (float_to_fp64 a) = float_to_fp64 (float_sqrt mode a)

fp64_sqrt_n2w

|- ∀mode a.
     fp64_sqrt mode (n2w a) =
     float_to_fp64 (float_sqrt mode (fp64_to_float (n2w a)))

fp64_add_float_to_fp

|- ∀mode b a.
     fp64_add mode (float_to_fp64 a) (float_to_fp64 b) =
     float_to_fp64 ((mode float_add a) b)

fp64_add_n2w

|- ∀mode b a.
     fp64_add mode (n2w a) (n2w b) =
     float_to_fp64
       ((mode float_add fp64_to_float (n2w a)) (fp64_to_float (n2w b)))

fp64_sub_float_to_fp

|- ∀mode b a.
     fp64_sub mode (float_to_fp64 a) (float_to_fp64 b) =
     float_to_fp64 ((mode float_sub a) b)

fp64_sub_n2w

|- ∀mode b a.
     fp64_sub mode (n2w a) (n2w b) =
     float_to_fp64
       ((mode float_sub fp64_to_float (n2w a)) (fp64_to_float (n2w b)))

fp64_mul_float_to_fp

|- ∀mode b a.
     fp64_mul mode (float_to_fp64 a) (float_to_fp64 b) =
     float_to_fp64 ((mode float_mul a) b)

fp64_mul_n2w

|- ∀mode b a.
     fp64_mul mode (n2w a) (n2w b) =
     float_to_fp64
       ((mode float_mul fp64_to_float (n2w a)) (fp64_to_float (n2w b)))

fp64_div_float_to_fp

|- ∀mode b a.
     fp64_div mode (float_to_fp64 a) (float_to_fp64 b) =
     float_to_fp64 ((mode float_div a) b)

fp64_div_n2w

|- ∀mode b a.
     fp64_div mode (n2w a) (n2w b) =
     float_to_fp64
       ((mode float_div fp64_to_float (n2w a)) (fp64_to_float (n2w b)))

fp64_equal_float_to_fp

|- ∀b a. fp64_equal (float_to_fp64 a) (float_to_fp64 b) ⇔ float_equal a b

fp64_equal_n2w

|- ∀b a.
     fp64_equal (n2w a) (n2w b) ⇔
     float_equal (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

fp64_lessThan_float_to_fp

|- ∀b a.
     fp64_lessThan (float_to_fp64 a) (float_to_fp64 b) ⇔ float_less_than a b

fp64_lessThan_n2w

|- ∀b a.
     fp64_lessThan (n2w a) (n2w b) ⇔
     float_less_than (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

fp64_lessEqual_float_to_fp

|- ∀b a.
     fp64_lessEqual (float_to_fp64 a) (float_to_fp64 b) ⇔ float_less_equal a b

fp64_lessEqual_n2w

|- ∀b a.
     fp64_lessEqual (n2w a) (n2w b) ⇔
     float_less_equal (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

fp64_greaterThan_float_to_fp

|- ∀b a.
     fp64_greaterThan (float_to_fp64 a) (float_to_fp64 b) ⇔
     float_greater_than a b

fp64_greaterThan_n2w

|- ∀b a.
     fp64_greaterThan (n2w a) (n2w b) ⇔
     float_greater_than (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

fp64_greaterEqual_float_to_fp

|- ∀b a.
     fp64_greaterEqual (float_to_fp64 a) (float_to_fp64 b) ⇔
     float_greater_equal a b

fp64_greaterEqual_n2w

|- ∀b a.
     fp64_greaterEqual (n2w a) (n2w b) ⇔
     float_greater_equal (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

fp64_mul_add_float_to_fp

|- ∀mode c b a.
     fp64_mul_add mode (float_to_fp64 a) (float_to_fp64 b) (float_to_fp64 c) =
     float_to_fp64 (float_mul_add mode a b c)

fp64_mul_add_n2w

|- ∀mode c b a.
     fp64_mul_add mode (n2w a) (n2w b) (n2w c) =
     float_to_fp64
       (float_mul_add mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))
          (fp64_to_float (n2w c)))