Theory machine_ieee

⊢ ∀to_float from_float from_real_with_flags m w.
    convert to_float from_float from_real_with_flags m w =
    (let
       f = to_float w
     in
       case float_value f of
         Float r => from_real_with_flags m r
       | Infinity =>
         (clear_flags,
          from_float
            (if f.Sign = 0w then float_plus_infinity (:δ # ε)
             else float_minus_infinity (:δ # ε)))
       | NaN =>
         (check_for_signalling [f],from_float (@fp. float_is_nan fp)))

⊢ ∀x. float_to_fp16 x = x.Sign @@ x.Exponent @@ x.Significand

⊢ ∀x. float_to_fp32 x = x.Sign @@ x.Exponent @@ x.Significand

⊢ ∀x. float_to_fp64 x = x.Sign @@ x.Exponent @@ x.Significand

⊢ fp16_abs = float_to_fp16 ∘ float_abs ∘ fp16_to_float

⊢ ∀mode a b.
    fp16_add mode a b =
    float_to_fp16
      (SND (float_add mode (fp16_to_float a) (fp16_to_float b)))

⊢ ∀mode a b.
    fp16_add_with_flags mode a b =
    (I ## float_to_fp16)
      (float_add mode (fp16_to_float a) (fp16_to_float b))

⊢ fp16_bottom = float_to_fp16 (float_bottom (:10 # 5))

⊢ ∀a b.
    fp16_compare a b = float_compare (fp16_to_float a) (fp16_to_float b)

⊢ ∀mode a b.
    fp16_div mode a b =
    float_to_fp16
      (SND (float_div mode (fp16_to_float a) (fp16_to_float b)))

⊢ ∀mode a b.
    fp16_div_with_flags mode a b =
    (I ## float_to_fp16)
      (float_div mode (fp16_to_float a) (fp16_to_float b))

⊢ ∀a b. fp16_equal a b ⇔ float_equal (fp16_to_float a) (fp16_to_float b)

⊢ ∀a b.
    fp16_greaterEqual a b ⇔
    float_greater_equal (fp16_to_float a) (fp16_to_float b)

⊢ ∀a b.
    fp16_greaterThan a b ⇔
    float_greater_than (fp16_to_float a) (fp16_to_float b)

⊢ fp16_isFinite = float_is_finite ∘ fp16_to_float

⊢ fp16_isInfinite = float_is_infinite ∘ fp16_to_float

⊢ fp16_isIntegral = float_is_integral ∘ fp16_to_float

⊢ fp16_isNan = float_is_nan ∘ fp16_to_float

⊢ fp16_isNormal = float_is_normal ∘ fp16_to_float

⊢ fp16_isSignallingNan = float_is_signalling ∘ fp16_to_float

⊢ fp16_isSubnormal = float_is_subnormal ∘ fp16_to_float

⊢ fp16_isZero = float_is_zero ∘ fp16_to_float

⊢ ∀a b.
    fp16_lessEqual a b ⇔
    float_less_equal (fp16_to_float a) (fp16_to_float b)

⊢ ∀a b.
    fp16_lessThan a b ⇔ float_less_than (fp16_to_float a) (fp16_to_float b)

⊢ ∀mode a b c.
    fp16_mul_add mode a b c =
    float_to_fp16
      (SND
         (float_mul_add mode (fp16_to_float a) (fp16_to_float b)
            (fp16_to_float c)))

⊢ ∀mode a b c.
    fp16_mul_add_with_flags mode a b c =
    (I ## float_to_fp16)
      (float_mul_add mode (fp16_to_float a) (fp16_to_float b)
         (fp16_to_float c))

⊢ ∀mode a b.
    fp16_mul mode a b =
    float_to_fp16
      (SND (float_mul mode (fp16_to_float a) (fp16_to_float b)))

⊢ ∀mode a b c.
    fp16_mul_sub mode a b c =
    float_to_fp16
      (SND
         (float_mul_sub mode (fp16_to_float a) (fp16_to_float b)
            (fp16_to_float c)))

⊢ ∀mode a b c.
    fp16_mul_sub_with_flags mode a b c =
    (I ## float_to_fp16)
      (float_mul_sub mode (fp16_to_float a) (fp16_to_float b)
         (fp16_to_float c))

⊢ ∀mode a b.
    fp16_mul_with_flags mode a b =
    (I ## float_to_fp16)
      (float_mul mode (fp16_to_float a) (fp16_to_float b))

⊢ fp16_negInf = float_to_fp16 (float_minus_infinity (:10 # 5))

⊢ fp16_negMin = float_to_fp16 (float_minus_min (:10 # 5))

⊢ fp16_negZero = float_to_fp16 NEG0

⊢ fp16_negate = float_to_fp16 ∘ float_negate ∘ fp16_to_float

⊢ fp16_posInf = float_to_fp16 (float_plus_infinity (:10 # 5))

⊢ fp16_posMin = float_to_fp16 (float_plus_min (:10 # 5))

⊢ fp16_posZero = float_to_fp16 POS0

⊢ ∀mode.
    fp16_roundToIntegral mode =
    float_to_fp16 ∘ float_round_to_integral mode ∘ fp16_to_float

⊢ ∀mode.
    fp16_sqrt mode = float_to_fp16 ∘ SND ∘ float_sqrt mode ∘ fp16_to_float

⊢ ∀mode.
    fp16_sqrt_with_flags mode =
    (I ## float_to_fp16) ∘ float_sqrt mode ∘ fp16_to_float

⊢ ∀mode a b.
    fp16_sub mode a b =
    float_to_fp16
      (SND (float_sub mode (fp16_to_float a) (fp16_to_float b)))

⊢ ∀mode a b.
    fp16_sub_with_flags mode a b =
    (I ## float_to_fp16)
      (float_sub mode (fp16_to_float a) (fp16_to_float b))

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

⊢ fp16_to_fp32 = SND ∘ fp16_to_fp32_with_flags

⊢ fp16_to_fp32_with_flags =
  convert fp16_to_float float_to_fp32 real_to_fp32_with_flags
    roundTiesToEven

⊢ fp16_to_fp64 = SND ∘ fp16_to_fp64_with_flags

⊢ fp16_to_fp64_with_flags =
  convert fp16_to_float float_to_fp64 real_to_fp64_with_flags
    roundTiesToEven

⊢ ∀mode. fp16_to_int mode = float_to_int mode ∘ fp16_to_float

⊢ fp16_to_real = float_to_real ∘ fp16_to_float

⊢ fp16_to_value = float_value ∘ fp16_to_float

⊢ fp16_top = float_to_fp16 FLT_MAX

⊢ fp32_abs = float_to_fp32 ∘ float_abs ∘ fp32_to_float

⊢ ∀mode a b.
    fp32_add mode a b =
    float_to_fp32
      (SND (float_add mode (fp32_to_float a) (fp32_to_float b)))

⊢ ∀mode a b.
    fp32_add_with_flags mode a b =
    (I ## float_to_fp32)
      (float_add mode (fp32_to_float a) (fp32_to_float b))

⊢ fp32_bottom = float_to_fp32 (float_bottom (:23 # 8))

⊢ ∀a b.
    fp32_compare a b = float_compare (fp32_to_float a) (fp32_to_float b)

⊢ ∀mode a b.
    fp32_div mode a b =
    float_to_fp32
      (SND (float_div mode (fp32_to_float a) (fp32_to_float b)))

⊢ ∀mode a b.
    fp32_div_with_flags mode a b =
    (I ## float_to_fp32)
      (float_div mode (fp32_to_float a) (fp32_to_float b))

⊢ ∀a b. fp32_equal a b ⇔ float_equal (fp32_to_float a) (fp32_to_float b)

⊢ ∀a b.
    fp32_greaterEqual a b ⇔
    float_greater_equal (fp32_to_float a) (fp32_to_float b)

⊢ ∀a b.
    fp32_greaterThan a b ⇔
    float_greater_than (fp32_to_float a) (fp32_to_float b)

⊢ fp32_isFinite = float_is_finite ∘ fp32_to_float

⊢ fp32_isInfinite = float_is_infinite ∘ fp32_to_float

⊢ fp32_isIntegral = float_is_integral ∘ fp32_to_float

⊢ fp32_isNan = float_is_nan ∘ fp32_to_float

⊢ fp32_isNormal = float_is_normal ∘ fp32_to_float

⊢ fp32_isSignallingNan = float_is_signalling ∘ fp32_to_float

⊢ fp32_isSubnormal = float_is_subnormal ∘ fp32_to_float

⊢ fp32_isZero = float_is_zero ∘ fp32_to_float

⊢ ∀a b.
    fp32_lessEqual a b ⇔
    float_less_equal (fp32_to_float a) (fp32_to_float b)

⊢ ∀a b.
    fp32_lessThan a b ⇔ float_less_than (fp32_to_float a) (fp32_to_float b)

⊢ ∀mode a b c.
    fp32_mul_add mode a b c =
    float_to_fp32
      (SND
         (float_mul_add mode (fp32_to_float a) (fp32_to_float b)
            (fp32_to_float c)))

⊢ ∀mode a b c.
    fp32_mul_add_with_flags mode a b c =
    (I ## float_to_fp32)
      (float_mul_add mode (fp32_to_float a) (fp32_to_float b)
         (fp32_to_float c))

⊢ ∀mode a b.
    fp32_mul mode a b =
    float_to_fp32
      (SND (float_mul mode (fp32_to_float a) (fp32_to_float b)))

⊢ ∀mode a b c.
    fp32_mul_sub mode a b c =
    float_to_fp32
      (SND
         (float_mul_sub mode (fp32_to_float a) (fp32_to_float b)
            (fp32_to_float c)))

⊢ ∀mode a b c.
    fp32_mul_sub_with_flags mode a b c =
    (I ## float_to_fp32)
      (float_mul_sub mode (fp32_to_float a) (fp32_to_float b)
         (fp32_to_float c))

⊢ ∀mode a b.
    fp32_mul_with_flags mode a b =
    (I ## float_to_fp32)
      (float_mul mode (fp32_to_float a) (fp32_to_float b))

⊢ fp32_negInf = float_to_fp32 (float_minus_infinity (:23 # 8))

⊢ fp32_negMin = float_to_fp32 (float_minus_min (:23 # 8))

⊢ fp32_negZero = float_to_fp32 NEG0

⊢ fp32_negate = float_to_fp32 ∘ float_negate ∘ fp32_to_float

⊢ fp32_posInf = float_to_fp32 (float_plus_infinity (:23 # 8))

⊢ fp32_posMin = float_to_fp32 (float_plus_min (:23 # 8))

⊢ fp32_posZero = float_to_fp32 POS0

⊢ ∀mode.
    fp32_roundToIntegral mode =
    float_to_fp32 ∘ float_round_to_integral mode ∘ fp32_to_float

⊢ ∀mode.
    fp32_sqrt mode = float_to_fp32 ∘ SND ∘ float_sqrt mode ∘ fp32_to_float

⊢ ∀mode.
    fp32_sqrt_with_flags mode =
    (I ## float_to_fp32) ∘ float_sqrt mode ∘ fp32_to_float

⊢ ∀mode a b.
    fp32_sub mode a b =
    float_to_fp32
      (SND (float_sub mode (fp32_to_float a) (fp32_to_float b)))

⊢ ∀mode a b.
    fp32_sub_with_flags mode a b =
    (I ## float_to_fp32)
      (float_sub mode (fp32_to_float a) (fp32_to_float b))

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

⊢ ∀m. fp32_to_fp16 m = SND ∘ fp32_to_fp16_with_flags m

⊢ fp32_to_fp16_with_flags =
  convert fp32_to_float float_to_fp16 real_to_fp16_with_flags

⊢ fp32_to_fp64 = SND ∘ fp32_to_fp64_with_flags

⊢ fp32_to_fp64_with_flags =
  convert fp32_to_float float_to_fp64 real_to_fp64_with_flags
    roundTiesToEven

⊢ ∀mode. fp32_to_int mode = float_to_int mode ∘ fp32_to_float

⊢ fp32_to_real = float_to_real ∘ fp32_to_float

⊢ fp32_to_value = float_value ∘ fp32_to_float

⊢ fp32_top = float_to_fp32 FLT_MAX

⊢ fp64_abs = float_to_fp64 ∘ float_abs ∘ fp64_to_float

⊢ ∀mode a b.
    fp64_add mode a b =
    float_to_fp64
      (SND (float_add mode (fp64_to_float a) (fp64_to_float b)))

⊢ ∀mode a b.
    fp64_add_with_flags mode a b =
    (I ## float_to_fp64)
      (float_add mode (fp64_to_float a) (fp64_to_float b))

⊢ fp64_bottom = float_to_fp64 (float_bottom (:52 # 11))

⊢ ∀a b.
    fp64_compare a b = float_compare (fp64_to_float a) (fp64_to_float b)

⊢ ∀mode a b.
    fp64_div mode a b =
    float_to_fp64
      (SND (float_div mode (fp64_to_float a) (fp64_to_float b)))

⊢ ∀mode a b.
    fp64_div_with_flags mode a b =
    (I ## float_to_fp64)
      (float_div mode (fp64_to_float a) (fp64_to_float b))

⊢ ∀a b. fp64_equal a b ⇔ float_equal (fp64_to_float a) (fp64_to_float b)

⊢ ∀a b.
    fp64_greaterEqual a b ⇔
    float_greater_equal (fp64_to_float a) (fp64_to_float b)

⊢ ∀a b.
    fp64_greaterThan a b ⇔
    float_greater_than (fp64_to_float a) (fp64_to_float b)

⊢ fp64_isFinite = float_is_finite ∘ fp64_to_float

⊢ fp64_isInfinite = float_is_infinite ∘ fp64_to_float

⊢ fp64_isIntegral = float_is_integral ∘ fp64_to_float

⊢ fp64_isNan = float_is_nan ∘ fp64_to_float

⊢ fp64_isNormal = float_is_normal ∘ fp64_to_float

⊢ fp64_isSignallingNan = float_is_signalling ∘ fp64_to_float

⊢ fp64_isSubnormal = float_is_subnormal ∘ fp64_to_float

⊢ fp64_isZero = float_is_zero ∘ fp64_to_float

⊢ ∀a b.
    fp64_lessEqual a b ⇔
    float_less_equal (fp64_to_float a) (fp64_to_float b)

⊢ ∀a b.
    fp64_lessThan a b ⇔ float_less_than (fp64_to_float a) (fp64_to_float b)

⊢ ∀mode a b c.
    fp64_mul_add mode a b c =
    float_to_fp64
      (SND
         (float_mul_add mode (fp64_to_float a) (fp64_to_float b)
            (fp64_to_float c)))

⊢ ∀mode a b c.
    fp64_mul_add_with_flags mode a b c =
    (I ## float_to_fp64)
      (float_mul_add mode (fp64_to_float a) (fp64_to_float b)
         (fp64_to_float c))

⊢ ∀mode a b.
    fp64_mul mode a b =
    float_to_fp64
      (SND (float_mul mode (fp64_to_float a) (fp64_to_float b)))

⊢ ∀mode a b c.
    fp64_mul_sub mode a b c =
    float_to_fp64
      (SND
         (float_mul_sub mode (fp64_to_float a) (fp64_to_float b)
            (fp64_to_float c)))

⊢ ∀mode a b c.
    fp64_mul_sub_with_flags mode a b c =
    (I ## float_to_fp64)
      (float_mul_sub mode (fp64_to_float a) (fp64_to_float b)
         (fp64_to_float c))

⊢ ∀mode a b.
    fp64_mul_with_flags mode a b =
    (I ## float_to_fp64)
      (float_mul mode (fp64_to_float a) (fp64_to_float b))

⊢ fp64_negInf = float_to_fp64 (float_minus_infinity (:52 # 11))

⊢ fp64_negMin = float_to_fp64 (float_minus_min (:52 # 11))

⊢ fp64_negZero = float_to_fp64 NEG0

⊢ fp64_negate = float_to_fp64 ∘ float_negate ∘ fp64_to_float

⊢ fp64_posInf = float_to_fp64 (float_plus_infinity (:52 # 11))

⊢ fp64_posMin = float_to_fp64 (float_plus_min (:52 # 11))

⊢ fp64_posZero = float_to_fp64 POS0

⊢ ∀mode.
    fp64_roundToIntegral mode =
    float_to_fp64 ∘ float_round_to_integral mode ∘ fp64_to_float

⊢ ∀mode.
    fp64_sqrt mode = float_to_fp64 ∘ SND ∘ float_sqrt mode ∘ fp64_to_float

⊢ ∀mode.
    fp64_sqrt_with_flags mode =
    (I ## float_to_fp64) ∘ float_sqrt mode ∘ fp64_to_float

⊢ ∀mode a b.
    fp64_sub mode a b =
    float_to_fp64
      (SND (float_sub mode (fp64_to_float a) (fp64_to_float b)))

⊢ ∀mode a b.
    fp64_sub_with_flags mode a b =
    (I ## float_to_fp64)
      (float_sub mode (fp64_to_float a) (fp64_to_float b))

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

⊢ ∀m. fp64_to_fp16 m = SND ∘ fp64_to_fp16_with_flags m

⊢ fp64_to_fp16_with_flags =
  convert fp64_to_float float_to_fp16 real_to_fp16_with_flags

⊢ ∀m. fp64_to_fp32 m = SND ∘ fp64_to_fp32_with_flags m

⊢ fp64_to_fp32_with_flags =
  convert fp64_to_float float_to_fp32 real_to_fp32_with_flags

⊢ ∀mode. fp64_to_int mode = float_to_int mode ∘ fp64_to_float

⊢ fp64_to_real = float_to_real ∘ fp64_to_float

⊢ fp64_to_value = float_value ∘ fp64_to_float

⊢ fp64_top = float_to_fp64 FLT_MAX

⊢ ∀mode a. int_to_fp16 mode a = real_to_fp16 mode (real_of_int a)

⊢ ∀mode a. int_to_fp32 mode a = real_to_fp32 mode (real_of_int a)

⊢ ∀mode a. int_to_fp64 mode a = real_to_fp64 mode (real_of_int a)

⊢ ∀mode. real_to_fp16 mode = float_to_fp16 ∘ real_to_float mode

⊢ ∀mode.
    real_to_fp16_with_flags mode =
    (I ## float_to_fp16) ∘ real_to_float_with_flags mode

⊢ ∀mode. real_to_fp32 mode = float_to_fp32 ∘ real_to_float mode

⊢ ∀mode.
    real_to_fp32_with_flags mode =
    (I ## float_to_fp32) ∘ real_to_float_with_flags mode

⊢ ∀mode. real_to_fp64 mode = float_to_fp64 ∘ real_to_float mode

⊢ ∀mode.
    real_to_fp64_with_flags mode =
    (I ## float_to_fp64) ∘ real_to_float_with_flags mode

⊢ ∀x. ∃y. x = fp16_to_float y

⊢ ∀x. ∃y. x = fp32_to_float y

⊢ ∀x. ∃y. x = fp64_to_float y

⊢ ∀x y. float_to_fp16 x = float_to_fp16 y ⇔ x = y

⊢ ∀x. float_to_fp16 (fp16_to_float x) = x

⊢ ∀x y. float_to_fp32 x = float_to_fp32 y ⇔ x = y

⊢ ∀x. float_to_fp32 (fp32_to_float x) = x

⊢ ∀x y. float_to_fp64 x = float_to_fp64 y ⇔ x = y

⊢ ∀x. float_to_fp64 (fp64_to_float x) = x

⊢ (∀a. fp16_abs (float_to_fp16 a) = float_to_fp16 (float_abs a)) ∧
  ∀a. fp16_abs (n2w a) = float_to_fp16 (float_abs (fp16_to_float (n2w a)))

⊢ (∀mode b a.
     fp16_add mode (float_to_fp16 a) (float_to_fp16 b) =
     float_to_fp16 (SND (float_add mode a b))) ∧
  (∀mode b a.
     fp16_add mode (float_to_fp16 a) (n2w b) =
     float_to_fp16 (SND (float_add mode a (fp16_to_float (n2w b))))) ∧
  (∀mode b a.
     fp16_add mode (n2w a) (float_to_fp16 b) =
     float_to_fp16 (SND (float_add mode (fp16_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp16_add mode (n2w a) (n2w b) =
    float_to_fp16
      (SND (float_add mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))))

⊢ (∀mode b a.
     fp16_add_with_flags mode (float_to_fp16 a) (float_to_fp16 b) =
     (I ## float_to_fp16) (float_add mode a b)) ∧
  (∀mode b a.
     fp16_add_with_flags mode (float_to_fp16 a) (n2w b) =
     (I ## float_to_fp16) (float_add mode a (fp16_to_float (n2w b)))) ∧
  (∀mode b a.
     fp16_add_with_flags mode (n2w a) (float_to_fp16 b) =
     (I ## float_to_fp16) (float_add mode (fp16_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp16_add_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp16)
      (float_add mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b)))

⊢ (∀b a.
     fp16_compare (float_to_fp16 a) (float_to_fp16 b) = float_compare a b) ∧
  (∀b a.
     fp16_compare (float_to_fp16 a) (n2w b) =
     float_compare a (fp16_to_float (n2w b))) ∧
  (∀b a.
     fp16_compare (n2w a) (float_to_fp16 b) =
     float_compare (fp16_to_float (n2w a)) b) ∧
  ∀b a.
    fp16_compare (n2w a) (n2w b) =
    float_compare (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

⊢ (∀mode b a.
     fp16_div mode (float_to_fp16 a) (float_to_fp16 b) =
     float_to_fp16 (SND (float_div mode a b))) ∧
  (∀mode b a.
     fp16_div mode (float_to_fp16 a) (n2w b) =
     float_to_fp16 (SND (float_div mode a (fp16_to_float (n2w b))))) ∧
  (∀mode b a.
     fp16_div mode (n2w a) (float_to_fp16 b) =
     float_to_fp16 (SND (float_div mode (fp16_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp16_div mode (n2w a) (n2w b) =
    float_to_fp16
      (SND (float_div mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))))

⊢ (∀mode b a.
     fp16_div_with_flags mode (float_to_fp16 a) (float_to_fp16 b) =
     (I ## float_to_fp16) (float_div mode a b)) ∧
  (∀mode b a.
     fp16_div_with_flags mode (float_to_fp16 a) (n2w b) =
     (I ## float_to_fp16) (float_div mode a (fp16_to_float (n2w b)))) ∧
  (∀mode b a.
     fp16_div_with_flags mode (n2w a) (float_to_fp16 b) =
     (I ## float_to_fp16) (float_div mode (fp16_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp16_div_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp16)
      (float_div mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b)))

⊢ (∀b a. fp16_equal (float_to_fp16 a) (float_to_fp16 b) ⇔ float_equal a b) ∧
  (∀b a.
     fp16_equal (float_to_fp16 a) (n2w b) ⇔
     float_equal a (fp16_to_float (n2w b))) ∧
  (∀b a.
     fp16_equal (n2w a) (float_to_fp16 b) ⇔
     float_equal (fp16_to_float (n2w a)) b) ∧
  ∀b a.
    fp16_equal (n2w a) (n2w b) ⇔
    float_equal (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

⊢ (∀b a.
     fp16_greaterEqual (float_to_fp16 a) (float_to_fp16 b) ⇔
     float_greater_equal a b) ∧
  (∀b a.
     fp16_greaterEqual (float_to_fp16 a) (n2w b) ⇔
     float_greater_equal a (fp16_to_float (n2w b))) ∧
  (∀b a.
     fp16_greaterEqual (n2w a) (float_to_fp16 b) ⇔
     float_greater_equal (fp16_to_float (n2w a)) b) ∧
  ∀b a.
    fp16_greaterEqual (n2w a) (n2w b) ⇔
    float_greater_equal (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

⊢ (∀b a.
     fp16_greaterThan (float_to_fp16 a) (float_to_fp16 b) ⇔
     float_greater_than a b) ∧
  (∀b a.
     fp16_greaterThan (float_to_fp16 a) (n2w b) ⇔
     float_greater_than a (fp16_to_float (n2w b))) ∧
  (∀b a.
     fp16_greaterThan (n2w a) (float_to_fp16 b) ⇔
     float_greater_than (fp16_to_float (n2w a)) b) ∧
  ∀b a.
    fp16_greaterThan (n2w a) (n2w b) ⇔
    float_greater_than (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

⊢ (∀a. fp16_isFinite (float_to_fp16 a) ⇔ float_is_finite a) ∧
  ∀a. fp16_isFinite (n2w a) ⇔ float_is_finite (fp16_to_float (n2w a))

⊢ (∀a. fp16_isInfinite (float_to_fp16 a) ⇔ float_is_infinite a) ∧
  ∀a. fp16_isInfinite (n2w a) ⇔ float_is_infinite (fp16_to_float (n2w a))

⊢ (∀a. fp16_isIntegral (float_to_fp16 a) ⇔ float_is_integral a) ∧
  ∀a. fp16_isIntegral (n2w a) ⇔ float_is_integral (fp16_to_float (n2w a))

⊢ (∀a. fp16_isNan (float_to_fp16 a) ⇔ float_is_nan a) ∧
  ∀a. fp16_isNan (n2w a) ⇔ float_is_nan (fp16_to_float (n2w a))

⊢ (∀a. fp16_isNormal (float_to_fp16 a) ⇔ float_is_normal a) ∧
  ∀a. fp16_isNormal (n2w a) ⇔ float_is_normal (fp16_to_float (n2w a))

⊢ (∀a. fp16_isSignallingNan (float_to_fp16 a) ⇔ float_is_signalling a) ∧
  ∀a. fp16_isSignallingNan (n2w a) ⇔
      float_is_signalling (fp16_to_float (n2w a))

⊢ (∀a. fp16_isSubnormal (float_to_fp16 a) ⇔ float_is_subnormal a) ∧
  ∀a. fp16_isSubnormal (n2w a) ⇔ float_is_subnormal (fp16_to_float (n2w a))

⊢ (∀a. fp16_isZero (float_to_fp16 a) ⇔ float_is_zero a) ∧
  ∀a. fp16_isZero (n2w a) ⇔ float_is_zero (fp16_to_float (n2w a))

⊢ (∀b a.
     fp16_lessEqual (float_to_fp16 a) (float_to_fp16 b) ⇔
     float_less_equal a b) ∧
  (∀b a.
     fp16_lessEqual (float_to_fp16 a) (n2w b) ⇔
     float_less_equal a (fp16_to_float (n2w b))) ∧
  (∀b a.
     fp16_lessEqual (n2w a) (float_to_fp16 b) ⇔
     float_less_equal (fp16_to_float (n2w a)) b) ∧
  ∀b a.
    fp16_lessEqual (n2w a) (n2w b) ⇔
    float_less_equal (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

⊢ (∀b a.
     fp16_lessThan (float_to_fp16 a) (float_to_fp16 b) ⇔
     float_less_than a b) ∧
  (∀b a.
     fp16_lessThan (float_to_fp16 a) (n2w b) ⇔
     float_less_than a (fp16_to_float (n2w b))) ∧
  (∀b a.
     fp16_lessThan (n2w a) (float_to_fp16 b) ⇔
     float_less_than (fp16_to_float (n2w a)) b) ∧
  ∀b a.
    fp16_lessThan (n2w a) (n2w b) ⇔
    float_less_than (fp16_to_float (n2w a)) (fp16_to_float (n2w b))

⊢ (∀mode b a.
     fp16_mul mode (float_to_fp16 a) (float_to_fp16 b) =
     float_to_fp16 (SND (float_mul mode a b))) ∧
  (∀mode b a.
     fp16_mul mode (float_to_fp16 a) (n2w b) =
     float_to_fp16 (SND (float_mul mode a (fp16_to_float (n2w b))))) ∧
  (∀mode b a.
     fp16_mul mode (n2w a) (float_to_fp16 b) =
     float_to_fp16 (SND (float_mul mode (fp16_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp16_mul mode (n2w a) (n2w b) =
    float_to_fp16
      (SND (float_mul mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))))

⊢ (∀mode c b a.
     fp16_mul_add mode (float_to_fp16 a) (float_to_fp16 b)
       (float_to_fp16 c) =
     float_to_fp16 (SND (float_mul_add mode a b c))) ∧
  (∀mode c b a.
     fp16_mul_add mode (float_to_fp16 a) (float_to_fp16 b) (n2w c) =
     float_to_fp16 (SND (float_mul_add mode a b (fp16_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp16_mul_add mode (float_to_fp16 a) (n2w b) (float_to_fp16 c) =
     float_to_fp16 (SND (float_mul_add mode a (fp16_to_float (n2w b)) c))) ∧
  (∀mode c b a.
     fp16_mul_add mode (float_to_fp16 a) (n2w b) (n2w c) =
     float_to_fp16
       (SND
          (float_mul_add mode a (fp16_to_float (n2w b))
             (fp16_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp16_mul_add mode (n2w a) (float_to_fp16 b) (float_to_fp16 c) =
     float_to_fp16 (SND (float_mul_add mode (fp16_to_float (n2w a)) b c))) ∧
  (∀mode c b a.
     fp16_mul_add mode (n2w a) (float_to_fp16 b) (n2w c) =
     float_to_fp16
       (SND
          (float_mul_add mode (fp16_to_float (n2w a)) b
             (fp16_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp16_mul_add mode (n2w a) (n2w b) (float_to_fp16 c) =
     float_to_fp16
       (SND
          (float_mul_add mode (fp16_to_float (n2w a))
             (fp16_to_float (n2w b)) c))) ∧
  ∀mode c b a.
    fp16_mul_add mode (n2w a) (n2w b) (n2w c) =
    float_to_fp16
      (SND
         (float_mul_add mode (fp16_to_float (n2w a))
            (fp16_to_float (n2w b)) (fp16_to_float (n2w c))))

⊢ (∀mode c b a.
     fp16_mul_add_with_flags mode (float_to_fp16 a) (float_to_fp16 b)
       (float_to_fp16 c) =
     (I ## float_to_fp16) (float_mul_add mode a b c)) ∧
  (∀mode c b a.
     fp16_mul_add_with_flags mode (float_to_fp16 a) (float_to_fp16 b)
       (n2w c) =
     (I ## float_to_fp16) (float_mul_add mode a b (fp16_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp16_mul_add_with_flags mode (float_to_fp16 a) (n2w b)
       (float_to_fp16 c) =
     (I ## float_to_fp16) (float_mul_add mode a (fp16_to_float (n2w b)) c)) ∧
  (∀mode c b a.
     fp16_mul_add_with_flags mode (float_to_fp16 a) (n2w b) (n2w c) =
     (I ## float_to_fp16)
       (float_mul_add mode a (fp16_to_float (n2w b))
          (fp16_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp16_mul_add_with_flags mode (n2w a) (float_to_fp16 b)
       (float_to_fp16 c) =
     (I ## float_to_fp16) (float_mul_add mode (fp16_to_float (n2w a)) b c)) ∧
  (∀mode c b a.
     fp16_mul_add_with_flags mode (n2w a) (float_to_fp16 b) (n2w c) =
     (I ## float_to_fp16)
       (float_mul_add mode (fp16_to_float (n2w a)) b
          (fp16_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp16_mul_add_with_flags mode (n2w a) (n2w b) (float_to_fp16 c) =
     (I ## float_to_fp16)
       (float_mul_add mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))
          c)) ∧
  ∀mode c b a.
    fp16_mul_add_with_flags mode (n2w a) (n2w b) (n2w c) =
    (I ## float_to_fp16)
      (float_mul_add mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))
         (fp16_to_float (n2w c)))

⊢ (∀mode c b a.
     fp16_mul_sub mode (float_to_fp16 a) (float_to_fp16 b)
       (float_to_fp16 c) =
     float_to_fp16 (SND (float_mul_sub mode a b c))) ∧
  (∀mode c b a.
     fp16_mul_sub mode (float_to_fp16 a) (float_to_fp16 b) (n2w c) =
     float_to_fp16 (SND (float_mul_sub mode a b (fp16_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp16_mul_sub mode (float_to_fp16 a) (n2w b) (float_to_fp16 c) =
     float_to_fp16 (SND (float_mul_sub mode a (fp16_to_float (n2w b)) c))) ∧
  (∀mode c b a.
     fp16_mul_sub mode (float_to_fp16 a) (n2w b) (n2w c) =
     float_to_fp16
       (SND
          (float_mul_sub mode a (fp16_to_float (n2w b))
             (fp16_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp16_mul_sub mode (n2w a) (float_to_fp16 b) (float_to_fp16 c) =
     float_to_fp16 (SND (float_mul_sub mode (fp16_to_float (n2w a)) b c))) ∧
  (∀mode c b a.
     fp16_mul_sub mode (n2w a) (float_to_fp16 b) (n2w c) =
     float_to_fp16
       (SND
          (float_mul_sub mode (fp16_to_float (n2w a)) b
             (fp16_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp16_mul_sub mode (n2w a) (n2w b) (float_to_fp16 c) =
     float_to_fp16
       (SND
          (float_mul_sub mode (fp16_to_float (n2w a))
             (fp16_to_float (n2w b)) c))) ∧
  ∀mode c b a.
    fp16_mul_sub mode (n2w a) (n2w b) (n2w c) =
    float_to_fp16
      (SND
         (float_mul_sub mode (fp16_to_float (n2w a))
            (fp16_to_float (n2w b)) (fp16_to_float (n2w c))))

⊢ (∀mode c b a.
     fp16_mul_sub_with_flags mode (float_to_fp16 a) (float_to_fp16 b)
       (float_to_fp16 c) =
     (I ## float_to_fp16) (float_mul_sub mode a b c)) ∧
  (∀mode c b a.
     fp16_mul_sub_with_flags mode (float_to_fp16 a) (float_to_fp16 b)
       (n2w c) =
     (I ## float_to_fp16) (float_mul_sub mode a b (fp16_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp16_mul_sub_with_flags mode (float_to_fp16 a) (n2w b)
       (float_to_fp16 c) =
     (I ## float_to_fp16) (float_mul_sub mode a (fp16_to_float (n2w b)) c)) ∧
  (∀mode c b a.
     fp16_mul_sub_with_flags mode (float_to_fp16 a) (n2w b) (n2w c) =
     (I ## float_to_fp16)
       (float_mul_sub mode a (fp16_to_float (n2w b))
          (fp16_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp16_mul_sub_with_flags mode (n2w a) (float_to_fp16 b)
       (float_to_fp16 c) =
     (I ## float_to_fp16) (float_mul_sub mode (fp16_to_float (n2w a)) b c)) ∧
  (∀mode c b a.
     fp16_mul_sub_with_flags mode (n2w a) (float_to_fp16 b) (n2w c) =
     (I ## float_to_fp16)
       (float_mul_sub mode (fp16_to_float (n2w a)) b
          (fp16_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp16_mul_sub_with_flags mode (n2w a) (n2w b) (float_to_fp16 c) =
     (I ## float_to_fp16)
       (float_mul_sub mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))
          c)) ∧
  ∀mode c b a.
    fp16_mul_sub_with_flags mode (n2w a) (n2w b) (n2w c) =
    (I ## float_to_fp16)
      (float_mul_sub mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))
         (fp16_to_float (n2w c)))

⊢ (∀mode b a.
     fp16_mul_with_flags mode (float_to_fp16 a) (float_to_fp16 b) =
     (I ## float_to_fp16) (float_mul mode a b)) ∧
  (∀mode b a.
     fp16_mul_with_flags mode (float_to_fp16 a) (n2w b) =
     (I ## float_to_fp16) (float_mul mode a (fp16_to_float (n2w b)))) ∧
  (∀mode b a.
     fp16_mul_with_flags mode (n2w a) (float_to_fp16 b) =
     (I ## float_to_fp16) (float_mul mode (fp16_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp16_mul_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp16)
      (float_mul mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b)))

⊢ ∀x. ∃y. x = float_to_fp16 y

⊢ (∀a. fp16_negate (float_to_fp16 a) = float_to_fp16 (float_negate a)) ∧
  ∀a. fp16_negate (n2w a) =
      float_to_fp16 (float_negate (fp16_to_float (n2w a)))

⊢ (∀mode a.
     fp16_roundToIntegral mode (float_to_fp16 a) =
     float_to_fp16 (float_round_to_integral mode a)) ∧
  ∀mode a.
    fp16_roundToIntegral mode (n2w a) =
    float_to_fp16 (float_round_to_integral mode (fp16_to_float (n2w a)))

⊢ (∀mode a.
     fp16_sqrt mode (float_to_fp16 a) =
     float_to_fp16 (SND (float_sqrt mode a))) ∧
  ∀mode a.
    fp16_sqrt mode (n2w a) =
    float_to_fp16 (SND (float_sqrt mode (fp16_to_float (n2w a))))

⊢ (∀mode a.
     fp16_sqrt_with_flags mode (float_to_fp16 a) =
     (I ## float_to_fp16) (float_sqrt mode a)) ∧
  ∀mode a.
    fp16_sqrt_with_flags mode (n2w a) =
    (I ## float_to_fp16) (float_sqrt mode (fp16_to_float (n2w a)))

⊢ (∀mode b a.
     fp16_sub mode (float_to_fp16 a) (float_to_fp16 b) =
     float_to_fp16 (SND (float_sub mode a b))) ∧
  (∀mode b a.
     fp16_sub mode (float_to_fp16 a) (n2w b) =
     float_to_fp16 (SND (float_sub mode a (fp16_to_float (n2w b))))) ∧
  (∀mode b a.
     fp16_sub mode (n2w a) (float_to_fp16 b) =
     float_to_fp16 (SND (float_sub mode (fp16_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp16_sub mode (n2w a) (n2w b) =
    float_to_fp16
      (SND (float_sub mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b))))

⊢ (∀mode b a.
     fp16_sub_with_flags mode (float_to_fp16 a) (float_to_fp16 b) =
     (I ## float_to_fp16) (float_sub mode a b)) ∧
  (∀mode b a.
     fp16_sub_with_flags mode (float_to_fp16 a) (n2w b) =
     (I ## float_to_fp16) (float_sub mode a (fp16_to_float (n2w b)))) ∧
  (∀mode b a.
     fp16_sub_with_flags mode (n2w a) (float_to_fp16 b) =
     (I ## float_to_fp16) (float_sub mode (fp16_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp16_sub_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp16)
      (float_sub mode (fp16_to_float (n2w a)) (fp16_to_float (n2w b)))

⊢ ∀x y. fp16_to_float x = fp16_to_float y ⇔ x = y

⊢ ∀x. fp16_to_float (float_to_fp16 x) = x

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

⊢ (∀mode a. fp16_to_int mode (float_to_fp16 a) = float_to_int mode a) ∧
  ∀mode a.
    fp16_to_int mode (n2w a) = float_to_int mode (fp16_to_float (n2w a))

⊢ (∀a. fp16_to_real (float_to_fp16 a) = float_to_real a) ∧
  ∀a. fp16_to_real (n2w a) = float_to_real (fp16_to_float (n2w a))

⊢ (∀a. fp16_to_value (float_to_fp16 a) = float_value a) ∧
  ∀a. fp16_to_value (n2w a) = float_value (fp16_to_float (n2w a))

⊢ (∀a. fp32_abs (float_to_fp32 a) = float_to_fp32 (float_abs a)) ∧
  ∀a. fp32_abs (n2w a) = float_to_fp32 (float_abs (fp32_to_float (n2w a)))

⊢ (∀mode b a.
     fp32_add mode (float_to_fp32 a) (float_to_fp32 b) =
     float_to_fp32 (SND (float_add mode a b))) ∧
  (∀mode b a.
     fp32_add mode (float_to_fp32 a) (n2w b) =
     float_to_fp32 (SND (float_add mode a (fp32_to_float (n2w b))))) ∧
  (∀mode b a.
     fp32_add mode (n2w a) (float_to_fp32 b) =
     float_to_fp32 (SND (float_add mode (fp32_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp32_add mode (n2w a) (n2w b) =
    float_to_fp32
      (SND (float_add mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))))

⊢ (∀mode b a.
     fp32_add_with_flags mode (float_to_fp32 a) (float_to_fp32 b) =
     (I ## float_to_fp32) (float_add mode a b)) ∧
  (∀mode b a.
     fp32_add_with_flags mode (float_to_fp32 a) (n2w b) =
     (I ## float_to_fp32) (float_add mode a (fp32_to_float (n2w b)))) ∧
  (∀mode b a.
     fp32_add_with_flags mode (n2w a) (float_to_fp32 b) =
     (I ## float_to_fp32) (float_add mode (fp32_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp32_add_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp32)
      (float_add mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b)))

⊢ (∀b a.
     fp32_compare (float_to_fp32 a) (float_to_fp32 b) = float_compare a b) ∧
  (∀b a.
     fp32_compare (float_to_fp32 a) (n2w b) =
     float_compare a (fp32_to_float (n2w b))) ∧
  (∀b a.
     fp32_compare (n2w a) (float_to_fp32 b) =
     float_compare (fp32_to_float (n2w a)) b) ∧
  ∀b a.
    fp32_compare (n2w a) (n2w b) =
    float_compare (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

⊢ (∀mode b a.
     fp32_div mode (float_to_fp32 a) (float_to_fp32 b) =
     float_to_fp32 (SND (float_div mode a b))) ∧
  (∀mode b a.
     fp32_div mode (float_to_fp32 a) (n2w b) =
     float_to_fp32 (SND (float_div mode a (fp32_to_float (n2w b))))) ∧
  (∀mode b a.
     fp32_div mode (n2w a) (float_to_fp32 b) =
     float_to_fp32 (SND (float_div mode (fp32_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp32_div mode (n2w a) (n2w b) =
    float_to_fp32
      (SND (float_div mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))))

⊢ (∀mode b a.
     fp32_div_with_flags mode (float_to_fp32 a) (float_to_fp32 b) =
     (I ## float_to_fp32) (float_div mode a b)) ∧
  (∀mode b a.
     fp32_div_with_flags mode (float_to_fp32 a) (n2w b) =
     (I ## float_to_fp32) (float_div mode a (fp32_to_float (n2w b)))) ∧
  (∀mode b a.
     fp32_div_with_flags mode (n2w a) (float_to_fp32 b) =
     (I ## float_to_fp32) (float_div mode (fp32_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp32_div_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp32)
      (float_div mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b)))

⊢ (∀b a. fp32_equal (float_to_fp32 a) (float_to_fp32 b) ⇔ float_equal a b) ∧
  (∀b a.
     fp32_equal (float_to_fp32 a) (n2w b) ⇔
     float_equal a (fp32_to_float (n2w b))) ∧
  (∀b a.
     fp32_equal (n2w a) (float_to_fp32 b) ⇔
     float_equal (fp32_to_float (n2w a)) b) ∧
  ∀b a.
    fp32_equal (n2w a) (n2w b) ⇔
    float_equal (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

⊢ (∀b a.
     fp32_greaterEqual (float_to_fp32 a) (float_to_fp32 b) ⇔
     float_greater_equal a b) ∧
  (∀b a.
     fp32_greaterEqual (float_to_fp32 a) (n2w b) ⇔
     float_greater_equal a (fp32_to_float (n2w b))) ∧
  (∀b a.
     fp32_greaterEqual (n2w a) (float_to_fp32 b) ⇔
     float_greater_equal (fp32_to_float (n2w a)) b) ∧
  ∀b a.
    fp32_greaterEqual (n2w a) (n2w b) ⇔
    float_greater_equal (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

⊢ (∀b a.
     fp32_greaterThan (float_to_fp32 a) (float_to_fp32 b) ⇔
     float_greater_than a b) ∧
  (∀b a.
     fp32_greaterThan (float_to_fp32 a) (n2w b) ⇔
     float_greater_than a (fp32_to_float (n2w b))) ∧
  (∀b a.
     fp32_greaterThan (n2w a) (float_to_fp32 b) ⇔
     float_greater_than (fp32_to_float (n2w a)) b) ∧
  ∀b a.
    fp32_greaterThan (n2w a) (n2w b) ⇔
    float_greater_than (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

⊢ (∀a. fp32_isFinite (float_to_fp32 a) ⇔ float_is_finite a) ∧
  ∀a. fp32_isFinite (n2w a) ⇔ float_is_finite (fp32_to_float (n2w a))

⊢ (∀a. fp32_isInfinite (float_to_fp32 a) ⇔ float_is_infinite a) ∧
  ∀a. fp32_isInfinite (n2w a) ⇔ float_is_infinite (fp32_to_float (n2w a))

⊢ (∀a. fp32_isIntegral (float_to_fp32 a) ⇔ float_is_integral a) ∧
  ∀a. fp32_isIntegral (n2w a) ⇔ float_is_integral (fp32_to_float (n2w a))

⊢ (∀a. fp32_isNan (float_to_fp32 a) ⇔ float_is_nan a) ∧
  ∀a. fp32_isNan (n2w a) ⇔ float_is_nan (fp32_to_float (n2w a))

⊢ (∀a. fp32_isNormal (float_to_fp32 a) ⇔ float_is_normal a) ∧
  ∀a. fp32_isNormal (n2w a) ⇔ float_is_normal (fp32_to_float (n2w a))

⊢ (∀a. fp32_isSignallingNan (float_to_fp32 a) ⇔ float_is_signalling a) ∧
  ∀a. fp32_isSignallingNan (n2w a) ⇔
      float_is_signalling (fp32_to_float (n2w a))

⊢ (∀a. fp32_isSubnormal (float_to_fp32 a) ⇔ float_is_subnormal a) ∧
  ∀a. fp32_isSubnormal (n2w a) ⇔ float_is_subnormal (fp32_to_float (n2w a))

⊢ (∀a. fp32_isZero (float_to_fp32 a) ⇔ float_is_zero a) ∧
  ∀a. fp32_isZero (n2w a) ⇔ float_is_zero (fp32_to_float (n2w a))

⊢ (∀b a.
     fp32_lessEqual (float_to_fp32 a) (float_to_fp32 b) ⇔
     float_less_equal a b) ∧
  (∀b a.
     fp32_lessEqual (float_to_fp32 a) (n2w b) ⇔
     float_less_equal a (fp32_to_float (n2w b))) ∧
  (∀b a.
     fp32_lessEqual (n2w a) (float_to_fp32 b) ⇔
     float_less_equal (fp32_to_float (n2w a)) b) ∧
  ∀b a.
    fp32_lessEqual (n2w a) (n2w b) ⇔
    float_less_equal (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

⊢ (∀b a.
     fp32_lessThan (float_to_fp32 a) (float_to_fp32 b) ⇔
     float_less_than a b) ∧
  (∀b a.
     fp32_lessThan (float_to_fp32 a) (n2w b) ⇔
     float_less_than a (fp32_to_float (n2w b))) ∧
  (∀b a.
     fp32_lessThan (n2w a) (float_to_fp32 b) ⇔
     float_less_than (fp32_to_float (n2w a)) b) ∧
  ∀b a.
    fp32_lessThan (n2w a) (n2w b) ⇔
    float_less_than (fp32_to_float (n2w a)) (fp32_to_float (n2w b))

⊢ (∀mode b a.
     fp32_mul mode (float_to_fp32 a) (float_to_fp32 b) =
     float_to_fp32 (SND (float_mul mode a b))) ∧
  (∀mode b a.
     fp32_mul mode (float_to_fp32 a) (n2w b) =
     float_to_fp32 (SND (float_mul mode a (fp32_to_float (n2w b))))) ∧
  (∀mode b a.
     fp32_mul mode (n2w a) (float_to_fp32 b) =
     float_to_fp32 (SND (float_mul mode (fp32_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp32_mul mode (n2w a) (n2w b) =
    float_to_fp32
      (SND (float_mul mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))))

⊢ (∀mode c b a.
     fp32_mul_add mode (float_to_fp32 a) (float_to_fp32 b)
       (float_to_fp32 c) =
     float_to_fp32 (SND (float_mul_add mode a b c))) ∧
  (∀mode c b a.
     fp32_mul_add mode (float_to_fp32 a) (float_to_fp32 b) (n2w c) =
     float_to_fp32 (SND (float_mul_add mode a b (fp32_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp32_mul_add mode (float_to_fp32 a) (n2w b) (float_to_fp32 c) =
     float_to_fp32 (SND (float_mul_add mode a (fp32_to_float (n2w b)) c))) ∧
  (∀mode c b a.
     fp32_mul_add mode (float_to_fp32 a) (n2w b) (n2w c) =
     float_to_fp32
       (SND
          (float_mul_add mode a (fp32_to_float (n2w b))
             (fp32_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp32_mul_add mode (n2w a) (float_to_fp32 b) (float_to_fp32 c) =
     float_to_fp32 (SND (float_mul_add mode (fp32_to_float (n2w a)) b c))) ∧
  (∀mode c b a.
     fp32_mul_add mode (n2w a) (float_to_fp32 b) (n2w c) =
     float_to_fp32
       (SND
          (float_mul_add mode (fp32_to_float (n2w a)) b
             (fp32_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp32_mul_add mode (n2w a) (n2w b) (float_to_fp32 c) =
     float_to_fp32
       (SND
          (float_mul_add mode (fp32_to_float (n2w a))
             (fp32_to_float (n2w b)) c))) ∧
  ∀mode c b a.
    fp32_mul_add mode (n2w a) (n2w b) (n2w c) =
    float_to_fp32
      (SND
         (float_mul_add mode (fp32_to_float (n2w a))
            (fp32_to_float (n2w b)) (fp32_to_float (n2w c))))

⊢ (∀mode c b a.
     fp32_mul_add_with_flags mode (float_to_fp32 a) (float_to_fp32 b)
       (float_to_fp32 c) =
     (I ## float_to_fp32) (float_mul_add mode a b c)) ∧
  (∀mode c b a.
     fp32_mul_add_with_flags mode (float_to_fp32 a) (float_to_fp32 b)
       (n2w c) =
     (I ## float_to_fp32) (float_mul_add mode a b (fp32_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp32_mul_add_with_flags mode (float_to_fp32 a) (n2w b)
       (float_to_fp32 c) =
     (I ## float_to_fp32) (float_mul_add mode a (fp32_to_float (n2w b)) c)) ∧
  (∀mode c b a.
     fp32_mul_add_with_flags mode (float_to_fp32 a) (n2w b) (n2w c) =
     (I ## float_to_fp32)
       (float_mul_add mode a (fp32_to_float (n2w b))
          (fp32_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp32_mul_add_with_flags mode (n2w a) (float_to_fp32 b)
       (float_to_fp32 c) =
     (I ## float_to_fp32) (float_mul_add mode (fp32_to_float (n2w a)) b c)) ∧
  (∀mode c b a.
     fp32_mul_add_with_flags mode (n2w a) (float_to_fp32 b) (n2w c) =
     (I ## float_to_fp32)
       (float_mul_add mode (fp32_to_float (n2w a)) b
          (fp32_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp32_mul_add_with_flags mode (n2w a) (n2w b) (float_to_fp32 c) =
     (I ## float_to_fp32)
       (float_mul_add mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))
          c)) ∧
  ∀mode c b a.
    fp32_mul_add_with_flags mode (n2w a) (n2w b) (n2w c) =
    (I ## float_to_fp32)
      (float_mul_add mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))
         (fp32_to_float (n2w c)))

⊢ (∀mode c b a.
     fp32_mul_sub mode (float_to_fp32 a) (float_to_fp32 b)
       (float_to_fp32 c) =
     float_to_fp32 (SND (float_mul_sub mode a b c))) ∧
  (∀mode c b a.
     fp32_mul_sub mode (float_to_fp32 a) (float_to_fp32 b) (n2w c) =
     float_to_fp32 (SND (float_mul_sub mode a b (fp32_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp32_mul_sub mode (float_to_fp32 a) (n2w b) (float_to_fp32 c) =
     float_to_fp32 (SND (float_mul_sub mode a (fp32_to_float (n2w b)) c))) ∧
  (∀mode c b a.
     fp32_mul_sub mode (float_to_fp32 a) (n2w b) (n2w c) =
     float_to_fp32
       (SND
          (float_mul_sub mode a (fp32_to_float (n2w b))
             (fp32_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp32_mul_sub mode (n2w a) (float_to_fp32 b) (float_to_fp32 c) =
     float_to_fp32 (SND (float_mul_sub mode (fp32_to_float (n2w a)) b c))) ∧
  (∀mode c b a.
     fp32_mul_sub mode (n2w a) (float_to_fp32 b) (n2w c) =
     float_to_fp32
       (SND
          (float_mul_sub mode (fp32_to_float (n2w a)) b
             (fp32_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp32_mul_sub mode (n2w a) (n2w b) (float_to_fp32 c) =
     float_to_fp32
       (SND
          (float_mul_sub mode (fp32_to_float (n2w a))
             (fp32_to_float (n2w b)) c))) ∧
  ∀mode c b a.
    fp32_mul_sub mode (n2w a) (n2w b) (n2w c) =
    float_to_fp32
      (SND
         (float_mul_sub mode (fp32_to_float (n2w a))
            (fp32_to_float (n2w b)) (fp32_to_float (n2w c))))

⊢ (∀mode c b a.
     fp32_mul_sub_with_flags mode (float_to_fp32 a) (float_to_fp32 b)
       (float_to_fp32 c) =
     (I ## float_to_fp32) (float_mul_sub mode a b c)) ∧
  (∀mode c b a.
     fp32_mul_sub_with_flags mode (float_to_fp32 a) (float_to_fp32 b)
       (n2w c) =
     (I ## float_to_fp32) (float_mul_sub mode a b (fp32_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp32_mul_sub_with_flags mode (float_to_fp32 a) (n2w b)
       (float_to_fp32 c) =
     (I ## float_to_fp32) (float_mul_sub mode a (fp32_to_float (n2w b)) c)) ∧
  (∀mode c b a.
     fp32_mul_sub_with_flags mode (float_to_fp32 a) (n2w b) (n2w c) =
     (I ## float_to_fp32)
       (float_mul_sub mode a (fp32_to_float (n2w b))
          (fp32_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp32_mul_sub_with_flags mode (n2w a) (float_to_fp32 b)
       (float_to_fp32 c) =
     (I ## float_to_fp32) (float_mul_sub mode (fp32_to_float (n2w a)) b c)) ∧
  (∀mode c b a.
     fp32_mul_sub_with_flags mode (n2w a) (float_to_fp32 b) (n2w c) =
     (I ## float_to_fp32)
       (float_mul_sub mode (fp32_to_float (n2w a)) b
          (fp32_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp32_mul_sub_with_flags mode (n2w a) (n2w b) (float_to_fp32 c) =
     (I ## float_to_fp32)
       (float_mul_sub mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))
          c)) ∧
  ∀mode c b a.
    fp32_mul_sub_with_flags mode (n2w a) (n2w b) (n2w c) =
    (I ## float_to_fp32)
      (float_mul_sub mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))
         (fp32_to_float (n2w c)))

⊢ (∀mode b a.
     fp32_mul_with_flags mode (float_to_fp32 a) (float_to_fp32 b) =
     (I ## float_to_fp32) (float_mul mode a b)) ∧
  (∀mode b a.
     fp32_mul_with_flags mode (float_to_fp32 a) (n2w b) =
     (I ## float_to_fp32) (float_mul mode a (fp32_to_float (n2w b)))) ∧
  (∀mode b a.
     fp32_mul_with_flags mode (n2w a) (float_to_fp32 b) =
     (I ## float_to_fp32) (float_mul mode (fp32_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp32_mul_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp32)
      (float_mul mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b)))

⊢ ∀x. ∃y. x = float_to_fp32 y

⊢ (∀a. fp32_negate (float_to_fp32 a) = float_to_fp32 (float_negate a)) ∧
  ∀a. fp32_negate (n2w a) =
      float_to_fp32 (float_negate (fp32_to_float (n2w a)))

⊢ (∀mode a.
     fp32_roundToIntegral mode (float_to_fp32 a) =
     float_to_fp32 (float_round_to_integral mode a)) ∧
  ∀mode a.
    fp32_roundToIntegral mode (n2w a) =
    float_to_fp32 (float_round_to_integral mode (fp32_to_float (n2w a)))

⊢ (∀mode a.
     fp32_sqrt mode (float_to_fp32 a) =
     float_to_fp32 (SND (float_sqrt mode a))) ∧
  ∀mode a.
    fp32_sqrt mode (n2w a) =
    float_to_fp32 (SND (float_sqrt mode (fp32_to_float (n2w a))))

⊢ (∀mode a.
     fp32_sqrt_with_flags mode (float_to_fp32 a) =
     (I ## float_to_fp32) (float_sqrt mode a)) ∧
  ∀mode a.
    fp32_sqrt_with_flags mode (n2w a) =
    (I ## float_to_fp32) (float_sqrt mode (fp32_to_float (n2w a)))

⊢ (∀mode b a.
     fp32_sub mode (float_to_fp32 a) (float_to_fp32 b) =
     float_to_fp32 (SND (float_sub mode a b))) ∧
  (∀mode b a.
     fp32_sub mode (float_to_fp32 a) (n2w b) =
     float_to_fp32 (SND (float_sub mode a (fp32_to_float (n2w b))))) ∧
  (∀mode b a.
     fp32_sub mode (n2w a) (float_to_fp32 b) =
     float_to_fp32 (SND (float_sub mode (fp32_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp32_sub mode (n2w a) (n2w b) =
    float_to_fp32
      (SND (float_sub mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b))))

⊢ (∀mode b a.
     fp32_sub_with_flags mode (float_to_fp32 a) (float_to_fp32 b) =
     (I ## float_to_fp32) (float_sub mode a b)) ∧
  (∀mode b a.
     fp32_sub_with_flags mode (float_to_fp32 a) (n2w b) =
     (I ## float_to_fp32) (float_sub mode a (fp32_to_float (n2w b)))) ∧
  (∀mode b a.
     fp32_sub_with_flags mode (n2w a) (float_to_fp32 b) =
     (I ## float_to_fp32) (float_sub mode (fp32_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp32_sub_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp32)
      (float_sub mode (fp32_to_float (n2w a)) (fp32_to_float (n2w b)))

⊢ ∀x y. fp32_to_float x = fp32_to_float y ⇔ x = y

⊢ ∀x. fp32_to_float (float_to_fp32 x) = x

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

⊢ (∀mode a. fp32_to_int mode (float_to_fp32 a) = float_to_int mode a) ∧
  ∀mode a.
    fp32_to_int mode (n2w a) = float_to_int mode (fp32_to_float (n2w a))

⊢ (∀a. fp32_to_real (float_to_fp32 a) = float_to_real a) ∧
  ∀a. fp32_to_real (n2w a) = float_to_real (fp32_to_float (n2w a))

⊢ (∀a. fp32_to_value (float_to_fp32 a) = float_value a) ∧
  ∀a. fp32_to_value (n2w a) = float_value (fp32_to_float (n2w a))

⊢ (∀a. fp64_abs (float_to_fp64 a) = float_to_fp64 (float_abs a)) ∧
  ∀a. fp64_abs (n2w a) = float_to_fp64 (float_abs (fp64_to_float (n2w a)))

⊢ (∀mode b a.
     fp64_add mode (float_to_fp64 a) (float_to_fp64 b) =
     float_to_fp64 (SND (float_add mode a b))) ∧
  (∀mode b a.
     fp64_add mode (float_to_fp64 a) (n2w b) =
     float_to_fp64 (SND (float_add mode a (fp64_to_float (n2w b))))) ∧
  (∀mode b a.
     fp64_add mode (n2w a) (float_to_fp64 b) =
     float_to_fp64 (SND (float_add mode (fp64_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp64_add mode (n2w a) (n2w b) =
    float_to_fp64
      (SND (float_add mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))))

⊢ (∀mode b a.
     fp64_add_with_flags mode (float_to_fp64 a) (float_to_fp64 b) =
     (I ## float_to_fp64) (float_add mode a b)) ∧
  (∀mode b a.
     fp64_add_with_flags mode (float_to_fp64 a) (n2w b) =
     (I ## float_to_fp64) (float_add mode a (fp64_to_float (n2w b)))) ∧
  (∀mode b a.
     fp64_add_with_flags mode (n2w a) (float_to_fp64 b) =
     (I ## float_to_fp64) (float_add mode (fp64_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp64_add_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp64)
      (float_add mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b)))

⊢ (∀b a.
     fp64_compare (float_to_fp64 a) (float_to_fp64 b) = float_compare a b) ∧
  (∀b a.
     fp64_compare (float_to_fp64 a) (n2w b) =
     float_compare a (fp64_to_float (n2w b))) ∧
  (∀b a.
     fp64_compare (n2w a) (float_to_fp64 b) =
     float_compare (fp64_to_float (n2w a)) b) ∧
  ∀b a.
    fp64_compare (n2w a) (n2w b) =
    float_compare (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

⊢ (∀mode b a.
     fp64_div mode (float_to_fp64 a) (float_to_fp64 b) =
     float_to_fp64 (SND (float_div mode a b))) ∧
  (∀mode b a.
     fp64_div mode (float_to_fp64 a) (n2w b) =
     float_to_fp64 (SND (float_div mode a (fp64_to_float (n2w b))))) ∧
  (∀mode b a.
     fp64_div mode (n2w a) (float_to_fp64 b) =
     float_to_fp64 (SND (float_div mode (fp64_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp64_div mode (n2w a) (n2w b) =
    float_to_fp64
      (SND (float_div mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))))

⊢ (∀mode b a.
     fp64_div_with_flags mode (float_to_fp64 a) (float_to_fp64 b) =
     (I ## float_to_fp64) (float_div mode a b)) ∧
  (∀mode b a.
     fp64_div_with_flags mode (float_to_fp64 a) (n2w b) =
     (I ## float_to_fp64) (float_div mode a (fp64_to_float (n2w b)))) ∧
  (∀mode b a.
     fp64_div_with_flags mode (n2w a) (float_to_fp64 b) =
     (I ## float_to_fp64) (float_div mode (fp64_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp64_div_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp64)
      (float_div mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b)))

⊢ (∀b a. fp64_equal (float_to_fp64 a) (float_to_fp64 b) ⇔ float_equal a b) ∧
  (∀b a.
     fp64_equal (float_to_fp64 a) (n2w b) ⇔
     float_equal a (fp64_to_float (n2w b))) ∧
  (∀b a.
     fp64_equal (n2w a) (float_to_fp64 b) ⇔
     float_equal (fp64_to_float (n2w a)) b) ∧
  ∀b a.
    fp64_equal (n2w a) (n2w b) ⇔
    float_equal (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

⊢ (∀b a.
     fp64_greaterEqual (float_to_fp64 a) (float_to_fp64 b) ⇔
     float_greater_equal a b) ∧
  (∀b a.
     fp64_greaterEqual (float_to_fp64 a) (n2w b) ⇔
     float_greater_equal a (fp64_to_float (n2w b))) ∧
  (∀b a.
     fp64_greaterEqual (n2w a) (float_to_fp64 b) ⇔
     float_greater_equal (fp64_to_float (n2w a)) b) ∧
  ∀b a.
    fp64_greaterEqual (n2w a) (n2w b) ⇔
    float_greater_equal (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

⊢ (∀b a.
     fp64_greaterThan (float_to_fp64 a) (float_to_fp64 b) ⇔
     float_greater_than a b) ∧
  (∀b a.
     fp64_greaterThan (float_to_fp64 a) (n2w b) ⇔
     float_greater_than a (fp64_to_float (n2w b))) ∧
  (∀b a.
     fp64_greaterThan (n2w a) (float_to_fp64 b) ⇔
     float_greater_than (fp64_to_float (n2w a)) b) ∧
  ∀b a.
    fp64_greaterThan (n2w a) (n2w b) ⇔
    float_greater_than (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

⊢ (∀a. fp64_isFinite (float_to_fp64 a) ⇔ float_is_finite a) ∧
  ∀a. fp64_isFinite (n2w a) ⇔ float_is_finite (fp64_to_float (n2w a))

⊢ (∀a. fp64_isInfinite (float_to_fp64 a) ⇔ float_is_infinite a) ∧
  ∀a. fp64_isInfinite (n2w a) ⇔ float_is_infinite (fp64_to_float (n2w a))

⊢ (∀a. fp64_isIntegral (float_to_fp64 a) ⇔ float_is_integral a) ∧
  ∀a. fp64_isIntegral (n2w a) ⇔ float_is_integral (fp64_to_float (n2w a))

⊢ (∀a. fp64_isNan (float_to_fp64 a) ⇔ float_is_nan a) ∧
  ∀a. fp64_isNan (n2w a) ⇔ float_is_nan (fp64_to_float (n2w a))

⊢ (∀a. fp64_isNormal (float_to_fp64 a) ⇔ float_is_normal a) ∧
  ∀a. fp64_isNormal (n2w a) ⇔ float_is_normal (fp64_to_float (n2w a))

⊢ (∀a. fp64_isSignallingNan (float_to_fp64 a) ⇔ float_is_signalling a) ∧
  ∀a. fp64_isSignallingNan (n2w a) ⇔
      float_is_signalling (fp64_to_float (n2w a))

⊢ (∀a. fp64_isSubnormal (float_to_fp64 a) ⇔ float_is_subnormal a) ∧
  ∀a. fp64_isSubnormal (n2w a) ⇔ float_is_subnormal (fp64_to_float (n2w a))

⊢ (∀a. fp64_isZero (float_to_fp64 a) ⇔ float_is_zero a) ∧
  ∀a. fp64_isZero (n2w a) ⇔ float_is_zero (fp64_to_float (n2w a))

⊢ (∀b a.
     fp64_lessEqual (float_to_fp64 a) (float_to_fp64 b) ⇔
     float_less_equal a b) ∧
  (∀b a.
     fp64_lessEqual (float_to_fp64 a) (n2w b) ⇔
     float_less_equal a (fp64_to_float (n2w b))) ∧
  (∀b a.
     fp64_lessEqual (n2w a) (float_to_fp64 b) ⇔
     float_less_equal (fp64_to_float (n2w a)) b) ∧
  ∀b a.
    fp64_lessEqual (n2w a) (n2w b) ⇔
    float_less_equal (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

⊢ (∀b a.
     fp64_lessThan (float_to_fp64 a) (float_to_fp64 b) ⇔
     float_less_than a b) ∧
  (∀b a.
     fp64_lessThan (float_to_fp64 a) (n2w b) ⇔
     float_less_than a (fp64_to_float (n2w b))) ∧
  (∀b a.
     fp64_lessThan (n2w a) (float_to_fp64 b) ⇔
     float_less_than (fp64_to_float (n2w a)) b) ∧
  ∀b a.
    fp64_lessThan (n2w a) (n2w b) ⇔
    float_less_than (fp64_to_float (n2w a)) (fp64_to_float (n2w b))

⊢ (∀mode b a.
     fp64_mul mode (float_to_fp64 a) (float_to_fp64 b) =
     float_to_fp64 (SND (float_mul mode a b))) ∧
  (∀mode b a.
     fp64_mul mode (float_to_fp64 a) (n2w b) =
     float_to_fp64 (SND (float_mul mode a (fp64_to_float (n2w b))))) ∧
  (∀mode b a.
     fp64_mul mode (n2w a) (float_to_fp64 b) =
     float_to_fp64 (SND (float_mul mode (fp64_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp64_mul mode (n2w a) (n2w b) =
    float_to_fp64
      (SND (float_mul mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))))

⊢ (∀mode c b a.
     fp64_mul_add mode (float_to_fp64 a) (float_to_fp64 b)
       (float_to_fp64 c) =
     float_to_fp64 (SND (float_mul_add mode a b c))) ∧
  (∀mode c b a.
     fp64_mul_add mode (float_to_fp64 a) (float_to_fp64 b) (n2w c) =
     float_to_fp64 (SND (float_mul_add mode a b (fp64_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp64_mul_add mode (float_to_fp64 a) (n2w b) (float_to_fp64 c) =
     float_to_fp64 (SND (float_mul_add mode a (fp64_to_float (n2w b)) c))) ∧
  (∀mode c b a.
     fp64_mul_add mode (float_to_fp64 a) (n2w b) (n2w c) =
     float_to_fp64
       (SND
          (float_mul_add mode a (fp64_to_float (n2w b))
             (fp64_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp64_mul_add mode (n2w a) (float_to_fp64 b) (float_to_fp64 c) =
     float_to_fp64 (SND (float_mul_add mode (fp64_to_float (n2w a)) b c))) ∧
  (∀mode c b a.
     fp64_mul_add mode (n2w a) (float_to_fp64 b) (n2w c) =
     float_to_fp64
       (SND
          (float_mul_add mode (fp64_to_float (n2w a)) b
             (fp64_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp64_mul_add mode (n2w a) (n2w b) (float_to_fp64 c) =
     float_to_fp64
       (SND
          (float_mul_add mode (fp64_to_float (n2w a))
             (fp64_to_float (n2w b)) c))) ∧
  ∀mode c b a.
    fp64_mul_add mode (n2w a) (n2w b) (n2w c) =
    float_to_fp64
      (SND
         (float_mul_add mode (fp64_to_float (n2w a))
            (fp64_to_float (n2w b)) (fp64_to_float (n2w c))))

⊢ (∀mode c b a.
     fp64_mul_add_with_flags mode (float_to_fp64 a) (float_to_fp64 b)
       (float_to_fp64 c) =
     (I ## float_to_fp64) (float_mul_add mode a b c)) ∧
  (∀mode c b a.
     fp64_mul_add_with_flags mode (float_to_fp64 a) (float_to_fp64 b)
       (n2w c) =
     (I ## float_to_fp64) (float_mul_add mode a b (fp64_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp64_mul_add_with_flags mode (float_to_fp64 a) (n2w b)
       (float_to_fp64 c) =
     (I ## float_to_fp64) (float_mul_add mode a (fp64_to_float (n2w b)) c)) ∧
  (∀mode c b a.
     fp64_mul_add_with_flags mode (float_to_fp64 a) (n2w b) (n2w c) =
     (I ## float_to_fp64)
       (float_mul_add mode a (fp64_to_float (n2w b))
          (fp64_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp64_mul_add_with_flags mode (n2w a) (float_to_fp64 b)
       (float_to_fp64 c) =
     (I ## float_to_fp64) (float_mul_add mode (fp64_to_float (n2w a)) b c)) ∧
  (∀mode c b a.
     fp64_mul_add_with_flags mode (n2w a) (float_to_fp64 b) (n2w c) =
     (I ## float_to_fp64)
       (float_mul_add mode (fp64_to_float (n2w a)) b
          (fp64_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp64_mul_add_with_flags mode (n2w a) (n2w b) (float_to_fp64 c) =
     (I ## float_to_fp64)
       (float_mul_add mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))
          c)) ∧
  ∀mode c b a.
    fp64_mul_add_with_flags mode (n2w a) (n2w b) (n2w c) =
    (I ## float_to_fp64)
      (float_mul_add mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))
         (fp64_to_float (n2w c)))

⊢ (∀mode c b a.
     fp64_mul_sub mode (float_to_fp64 a) (float_to_fp64 b)
       (float_to_fp64 c) =
     float_to_fp64 (SND (float_mul_sub mode a b c))) ∧
  (∀mode c b a.
     fp64_mul_sub mode (float_to_fp64 a) (float_to_fp64 b) (n2w c) =
     float_to_fp64 (SND (float_mul_sub mode a b (fp64_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp64_mul_sub mode (float_to_fp64 a) (n2w b) (float_to_fp64 c) =
     float_to_fp64 (SND (float_mul_sub mode a (fp64_to_float (n2w b)) c))) ∧
  (∀mode c b a.
     fp64_mul_sub mode (float_to_fp64 a) (n2w b) (n2w c) =
     float_to_fp64
       (SND
          (float_mul_sub mode a (fp64_to_float (n2w b))
             (fp64_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp64_mul_sub mode (n2w a) (float_to_fp64 b) (float_to_fp64 c) =
     float_to_fp64 (SND (float_mul_sub mode (fp64_to_float (n2w a)) b c))) ∧
  (∀mode c b a.
     fp64_mul_sub mode (n2w a) (float_to_fp64 b) (n2w c) =
     float_to_fp64
       (SND
          (float_mul_sub mode (fp64_to_float (n2w a)) b
             (fp64_to_float (n2w c))))) ∧
  (∀mode c b a.
     fp64_mul_sub mode (n2w a) (n2w b) (float_to_fp64 c) =
     float_to_fp64
       (SND
          (float_mul_sub mode (fp64_to_float (n2w a))
             (fp64_to_float (n2w b)) c))) ∧
  ∀mode c b a.
    fp64_mul_sub mode (n2w a) (n2w b) (n2w c) =
    float_to_fp64
      (SND
         (float_mul_sub mode (fp64_to_float (n2w a))
            (fp64_to_float (n2w b)) (fp64_to_float (n2w c))))

⊢ (∀mode c b a.
     fp64_mul_sub_with_flags mode (float_to_fp64 a) (float_to_fp64 b)
       (float_to_fp64 c) =
     (I ## float_to_fp64) (float_mul_sub mode a b c)) ∧
  (∀mode c b a.
     fp64_mul_sub_with_flags mode (float_to_fp64 a) (float_to_fp64 b)
       (n2w c) =
     (I ## float_to_fp64) (float_mul_sub mode a b (fp64_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp64_mul_sub_with_flags mode (float_to_fp64 a) (n2w b)
       (float_to_fp64 c) =
     (I ## float_to_fp64) (float_mul_sub mode a (fp64_to_float (n2w b)) c)) ∧
  (∀mode c b a.
     fp64_mul_sub_with_flags mode (float_to_fp64 a) (n2w b) (n2w c) =
     (I ## float_to_fp64)
       (float_mul_sub mode a (fp64_to_float (n2w b))
          (fp64_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp64_mul_sub_with_flags mode (n2w a) (float_to_fp64 b)
       (float_to_fp64 c) =
     (I ## float_to_fp64) (float_mul_sub mode (fp64_to_float (n2w a)) b c)) ∧
  (∀mode c b a.
     fp64_mul_sub_with_flags mode (n2w a) (float_to_fp64 b) (n2w c) =
     (I ## float_to_fp64)
       (float_mul_sub mode (fp64_to_float (n2w a)) b
          (fp64_to_float (n2w c)))) ∧
  (∀mode c b a.
     fp64_mul_sub_with_flags mode (n2w a) (n2w b) (float_to_fp64 c) =
     (I ## float_to_fp64)
       (float_mul_sub mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))
          c)) ∧
  ∀mode c b a.
    fp64_mul_sub_with_flags mode (n2w a) (n2w b) (n2w c) =
    (I ## float_to_fp64)
      (float_mul_sub mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))
         (fp64_to_float (n2w c)))

⊢ (∀mode b a.
     fp64_mul_with_flags mode (float_to_fp64 a) (float_to_fp64 b) =
     (I ## float_to_fp64) (float_mul mode a b)) ∧
  (∀mode b a.
     fp64_mul_with_flags mode (float_to_fp64 a) (n2w b) =
     (I ## float_to_fp64) (float_mul mode a (fp64_to_float (n2w b)))) ∧
  (∀mode b a.
     fp64_mul_with_flags mode (n2w a) (float_to_fp64 b) =
     (I ## float_to_fp64) (float_mul mode (fp64_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp64_mul_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp64)
      (float_mul mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b)))

⊢ ∀x. ∃y. x = float_to_fp64 y

⊢ (∀a. fp64_negate (float_to_fp64 a) = float_to_fp64 (float_negate a)) ∧
  ∀a. fp64_negate (n2w a) =
      float_to_fp64 (float_negate (fp64_to_float (n2w a)))

⊢ (∀mode a.
     fp64_roundToIntegral mode (float_to_fp64 a) =
     float_to_fp64 (float_round_to_integral mode a)) ∧
  ∀mode a.
    fp64_roundToIntegral mode (n2w a) =
    float_to_fp64 (float_round_to_integral mode (fp64_to_float (n2w a)))

⊢ (∀mode a.
     fp64_sqrt mode (float_to_fp64 a) =
     float_to_fp64 (SND (float_sqrt mode a))) ∧
  ∀mode a.
    fp64_sqrt mode (n2w a) =
    float_to_fp64 (SND (float_sqrt mode (fp64_to_float (n2w a))))

⊢ (∀mode a.
     fp64_sqrt_with_flags mode (float_to_fp64 a) =
     (I ## float_to_fp64) (float_sqrt mode a)) ∧
  ∀mode a.
    fp64_sqrt_with_flags mode (n2w a) =
    (I ## float_to_fp64) (float_sqrt mode (fp64_to_float (n2w a)))

⊢ (∀mode b a.
     fp64_sub mode (float_to_fp64 a) (float_to_fp64 b) =
     float_to_fp64 (SND (float_sub mode a b))) ∧
  (∀mode b a.
     fp64_sub mode (float_to_fp64 a) (n2w b) =
     float_to_fp64 (SND (float_sub mode a (fp64_to_float (n2w b))))) ∧
  (∀mode b a.
     fp64_sub mode (n2w a) (float_to_fp64 b) =
     float_to_fp64 (SND (float_sub mode (fp64_to_float (n2w a)) b))) ∧
  ∀mode b a.
    fp64_sub mode (n2w a) (n2w b) =
    float_to_fp64
      (SND (float_sub mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b))))

⊢ (∀mode b a.
     fp64_sub_with_flags mode (float_to_fp64 a) (float_to_fp64 b) =
     (I ## float_to_fp64) (float_sub mode a b)) ∧
  (∀mode b a.
     fp64_sub_with_flags mode (float_to_fp64 a) (n2w b) =
     (I ## float_to_fp64) (float_sub mode a (fp64_to_float (n2w b)))) ∧
  (∀mode b a.
     fp64_sub_with_flags mode (n2w a) (float_to_fp64 b) =
     (I ## float_to_fp64) (float_sub mode (fp64_to_float (n2w a)) b)) ∧
  ∀mode b a.
    fp64_sub_with_flags mode (n2w a) (n2w b) =
    (I ## float_to_fp64)
      (float_sub mode (fp64_to_float (n2w a)) (fp64_to_float (n2w b)))

⊢ ∀x y. fp64_to_float x = fp64_to_float y ⇔ x = y

⊢ ∀x. fp64_to_float (float_to_fp64 x) = x

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

⊢ (∀mode a. fp64_to_int mode (float_to_fp64 a) = float_to_int mode a) ∧
  ∀mode a.
    fp64_to_int mode (n2w a) = float_to_int mode (fp64_to_float (n2w a))

⊢ (∀a. fp64_to_real (float_to_fp64 a) = float_to_real a) ∧
  ∀a. fp64_to_real (n2w a) = float_to_real (fp64_to_float (n2w a))

⊢ (∀a. fp64_to_value (float_to_fp64 a) = float_value a) ∧
  ∀a. fp64_to_value (n2w a) = float_value (fp64_to_float (n2w a))