FinMatrix.CoqExt.RExt

Require Export QcExt.
Require Export RExtBase RExtCvt RExtStruct RExtBool RExtLt.
Require Export RExtPlus RExtMult RExtPow RExtOpp RExtInv.
Require Export RExtSqr RExtSqrt RExtAbs RExtSign RExtExp RExtLog RExtRpower.
Require Export RExtApprox RExtTrigo RExtAtan2.

Section test.
  Goal 1 = R1. ra. Qed.
  Goal R1² = 1. ra. Qed.
  Goal 1² = R1. ra. Qed.
  Goal /R1 = R1. ra. Qed.
  Goal /R1 = 1. ra. Qed.
  Goal /1 = R1. ra. Qed.
  Goal 0 <= R1. ra. Qed.
End test.

Section test.

  Goal forall r, r * r = 0 <-> r = 0. ra. Qed.
  Goal forall r, r * r <> 0 <-> r <> 0. ra. Qed.
  Goal forall r1 r2 : R, 0 <= r1 * r1 + r2 * r2. ra. Qed.
  Goal forall r1 r2 : R, r1 <> 0 \/ r2 <> 0 <-> r1 * r1 + r2 * r2 <> 0. ra. Qed.
  Goal forall r1 r2 : R, r1 * r1 + r2 * r2 <> 0 <-> 0 < r1 * r1 + r2 * r2. ra. Qed.
  Goal forall r1 r2 r3 : R,
      r1 <> 0 \/ r2 <> 0 \/ r3 <> 0 <-> r1 * r1 + r2 * r2 + r3 * r3 <> 0. ra. Qed.
  Goal forall r1 r2 r3 r4 : R,
      r1 <> 0 \/ r2 <> 0 \/ r3 <> 0 \/ r4 <> 0 <->
        r1 * r1 + r2 * r2 + r3 * r3 + r4 * r4 <> 0. ra. Qed.
End test.

Section test.
  Goal forall x, cos x * cos x + sin x * sin x = R1. ra. Qed.
  Goal forall x, sin x * sin x + cos x * cos x = R1. ra. Qed.
End test.

Section test.
  Goal forall r, 0 <= r * r. ra. Qed.
  Goal forall r, 0 <= r². ra. Qed.
  Goal forall r1 r2, 0 <= r1 * r1 + r2 * r2. ra. Qed.
  Goal forall r1 r2, 0 <= r1² + r2². ra. Qed.
  Goal forall r1 r2, 0 <= r1 * r1 + r2². ra. Qed.
  Goal forall r1 r2, 0 <= r1² + r2 * r2. ra. Qed.
  Goal forall r, 0 <= r -> 0 <= r * 5. ra. Qed.
  Goal forall r, 0 <= r -> 0 <= r * 5 * 10. ra. Qed.
  Goal forall r, 0 <= r -> 0 <= r * (/5) * 10. ra. Qed.
End test.

Section test.
  Goal forall r, 0 <> r -> 0 < r * r. ra. Qed.
  Goal forall r, 0 <> r -> 0 < r². ra. Qed.
  Goal forall r, 0 < r -> 0 < r * r. ra. Qed.
  Goal forall r, 0 < r -> 0 < r². ra. Qed.
  Goal forall r1 r2, r1 <> 0 -> 0 < r1 * r1 + r2 * r2. ra. Qed.
  Goal forall r1 r2, r1 <> 0 -> 0 < r1² + r2 * r2. ra. Qed.
  Goal forall r1 r2, r1 <> 0 -> 0 < r1 * r1 + r2². ra. Qed.

  Goal forall r1 r2, r2 <> 0 -> 0 < r1 * r1 + r2 * r2. ra. Qed.
  Goal forall r1 r2, r2 <> 0 -> 0 < r1² + r2 * r2. ra. Qed.
  Goal forall r1 r2, r2 <> 0 -> 0 < r1 * r1 + r2². ra. Qed.

  Goal forall r1 r2, 0 < r1 -> 0 < r1 * r1 + r2 * r2. ra. Qed.
  Goal forall r1 r2, 0 < r2 -> 0 < r1 * r1 + r2 * r2. ra. Qed.

  Goal forall r, 0 < r -> 0 < r * 10. ra. Qed.
  Goal forall r, 0 < r -> 0 < r + 10. ra. Qed.
  Goal forall r, 0 < r -> 0 < r * 100 + 23732. ra. Qed.
End test.

Section test.
  Goal forall r, r² <> 0 <-> r <> 0. ra. Qed.
  Goal forall r, r² <> 0 -> r <> 0. ra. Qed.
  Goal forall r, r <> 0 -> r * r <> 0. ra. Qed.
  Goal forall r, r <> 0 -> r² <> 0. ra. Qed.

  Goal forall r, 0 <> r * r -> r <> 0. ra. Qed.
  Goal forall r, r * r <> 0 -> 0 <> r. ra. Qed.
  Goal forall r, 0 <> r * r -> 0 <> r. ra. Qed.

  Goal forall r1 r2 r3 r4 : R,
    r1 <> 0 \/ r2 <> 0 \/ r3 <> 0 \/ r4 <> 0 <->
    r1 * r1 + r2 * r2 + r3 * r3 + r4 * r4 <> 0.
  Proof. ra. Qed.
End test.