Require Import ZArith Int63 Floats.

Open Scope float_scope.

Definition two := Eval compute in of_int63 2%int63.
Definition three := Eval compute in of_int63 3%int63.
Definition six := Eval compute in of_int63 6%int63.

Check (eq_refl : three * two = six).
Check (eq_refl six <: three * two = six).
Check (eq_refl six <<: three * two = six).
Definition compute1 := Eval compute in three * two.
Check (eq_refl compute1 : six = six).

Definition huge := Eval compute in ldexp one 1023%Z.
Definition tiny := Eval compute in ldexp one (-1023)%Z.

Check (eq_refl : huge * tiny = one).
Check (eq_refl one <: huge * tiny = one).
Check (eq_refl one <<: huge * tiny = one).
Definition compute2 := Eval compute in huge * tiny.
Check (eq_refl compute2 : one = one).

Check (eq_refl : huge * huge = infinity).
Check (eq_refl infinity <: huge * huge = infinity).
Check (eq_refl infinity <<: huge * huge = infinity).
Definition compute3 := Eval compute in huge * huge.
Check (eq_refl compute3 : infinity = infinity).

Check (eq_refl : one * nan = nan).
Check (eq_refl nan <: one * nan = nan).
Check (eq_refl nan <<: one * nan = nan).
Definition compute4 := Eval compute in one * nan.
Check (eq_refl compute4 : nan = nan).

Check (eq_refl : infinity * infinity = infinity).
Check (eq_refl infinity <: infinity * infinity = infinity).
Check (eq_refl infinity <<: infinity * infinity = infinity).
Definition compute5 := Eval compute in infinity * infinity.
Check (eq_refl compute5 : infinity = infinity).

Check (eq_refl : infinity * neg_infinity = neg_infinity).
Check (eq_refl neg_infinity <: infinity * neg_infinity = neg_infinity).
Check (eq_refl neg_infinity <<: infinity * neg_infinity = neg_infinity).
Definition compute6 := Eval compute in infinity * neg_infinity.
Check (eq_refl compute6 : neg_infinity = neg_infinity).

Check (eq_refl : zero * zero = zero).
Check (eq_refl zero <: zero * zero = zero).
Check (eq_refl zero <<: zero * zero = zero).
Check (eq_refl : neg_zero * zero = neg_zero).
Check (eq_refl neg_zero <: neg_zero * zero = neg_zero).
Check (eq_refl neg_zero <<: neg_zero * zero = neg_zero).
Check (eq_refl : neg_zero * neg_zero = zero).
Check (eq_refl zero <: neg_zero * neg_zero = zero).
Check (eq_refl zero <<: neg_zero * neg_zero = zero).
Check (eq_refl : zero * neg_zero = neg_zero).
Check (eq_refl neg_zero <: zero * neg_zero = neg_zero).
Check (eq_refl neg_zero <<: zero * neg_zero = neg_zero).

Check (eq_refl : huge * neg_infinity = neg_infinity).
Check (eq_refl neg_infinity <: huge * neg_infinity = neg_infinity).
Check (eq_refl neg_infinity <<: huge * neg_infinity = neg_infinity).

Check (eq_refl : one * tiny = tiny).
Check (eq_refl tiny <: one * tiny = tiny).
Check (eq_refl tiny <<: one * tiny = tiny).
Check (eq_refl : one * huge = huge).
Check (eq_refl huge <: one * huge = huge).
Check (eq_refl huge <<: one * huge = huge).
Check (eq_refl : zero * huge = zero).
Check (eq_refl zero <: zero * huge = zero).
Check (eq_refl zero <<: zero * huge = zero).
Check (eq_refl : zero * (-huge) = neg_zero).
Check (eq_refl neg_zero <: zero * (-huge) = neg_zero).
Check (eq_refl neg_zero <<: zero * (-huge) = neg_zero).

Check (eq_refl : zero * infinity = nan).
Check (eq_refl nan <: zero * infinity = nan).
Check (eq_refl nan <<: zero * infinity = nan).
Check (eq_refl : neg_infinity * zero = nan).
Check (eq_refl nan <: neg_infinity * zero = nan).
Check (eq_refl nan <<: neg_infinity * zero = nan).