# Library Coq.Numbers.Natural.Abstract.NSqrt

Properties of Square Root Function
Require Import NAxioms NSub NZSqrt.
Module NSqrtProp (Import A : NAxiomsSig')(Import B : NSubProp A).
Module Import Private_NZSqrt := Nop <+ NZSqrtProp A A B.
Ltac auto' := trivial; try rewrite <- neq_0_lt_0; auto using le_0_l.
Ltac wrap l := intros; apply l; auto'.
We redefine NZSqrt's results, without the non-negative hyps
Lemma sqrt_spec' : forall a, √a*√a <= a < S (√a) * S (√a).
Definition sqrt_unique : forall a b, b*b<=a<(S b)*(S b) -> √a == b
:= sqrt_unique.
Lemma sqrt_square : forall a, √(a*a) == a.
Definition sqrt_le_mono : forall a b, a<=b -> √a <= √b
:= sqrt_le_mono.
Definition sqrt_lt_cancel : forall a b, √a < √b -> a < b
:= sqrt_lt_cancel.
Lemma sqrt_le_square : forall a b, b*b<=a <-> b <= √a.
Lemma sqrt_lt_square : forall a b, a<b*b <-> √a < b.
Definition sqrt_0 := sqrt_0.
Definition sqrt_1 := sqrt_1.
Definition sqrt_2 := sqrt_2.
Definition sqrt_lt_lin : forall a, 1<a -> √a<a
:= sqrt_lt_lin.
Lemma sqrt_le_lin : forall a, √a<=a.
Definition sqrt_mul_below : forall a b, √a * √b <= √(a*b)
:= sqrt_mul_below.
Lemma sqrt_mul_above : forall a b, √(a*b) < S (√a) * S (√b).
Lemma sqrt_succ_le : forall a, √(S a) <= S (√a).
Lemma sqrt_succ_or : forall a, √(S a) == S (√a) \/ √(S a) == √a.
Definition sqrt_add_le : forall a b, √(a+b) <= √a + √b
:= sqrt_add_le.
Lemma add_sqrt_le : forall a b, √a + √b <= √(2*(a+b)).
For the moment, we include stuff about sqrt_up with patching them.
Include NZSqrtUpProp A A B Private_NZSqrt.
End NSqrtProp.