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Try doing some basic maths questions in the Lean Theorem Prover. Functions, real numbers, equivalence relations and groups. Click on README.md and then on "Open in CoCalc with one click".

Project: Xena
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License: APACHE
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import tactic.linarith

example (e b c a v0 v1 : ℚ) (h1 : v0 = 5*a) (h2 : v1 = 3*b) (h3 : v0 + v1 + c = 10) :
  v0 + 5 + (v1 - 3) + (c - 2) = 10 :=
by linarith

example (ε : ℚ) (h1 : ε > 0) : ε / 2 + ε / 3 + ε / 7 < ε :=
 by linarith

example (x y z : ℚ) (h1 : 2*x  < 3*y) (h2 : -4*x + z/2 < 0)
        (h3 : 12*y - z < 0)  : false :=
by linarith

example (ε : ℚ) (h1 : ε > 0) : ε / 2 < ε :=
by linarith

example (ε : ℚ) (h1 : ε > 0) : ε / 3 + ε / 3 + ε / 3 = ε :=
by linarith

example (a b c : ℚ)  (h2 : b + 2 > 3 + b) : false :=
by linarith {discharger := `[ring SOP]}

example (a b c : ℚ)  (h2 : b + 2 > 3 + b) : false :=
by linarith

example (a b c : ℚ) (x y : ℤ) (h1 : x ≤ 3*y) (h2 : b + 2 > 3 + b) : false :=
by linarith {restrict_type := ℚ}

example (g v V c h : ℚ) (h1 : h = 0) (h2 : v = V) (h3 : V > 0) (h4 : g > 0)
        (h5 : 0 ≤ c) (h6 : c < 1) :
  v ≤ V :=
by linarith

example (x y z : ℚ) (h1 : 2*x + ((-3)*y) < 0) (h2 : (-4)*x + 2*z < 0)
       (h3 : 12*y + (-4)* z < 0) (h4 : nat.prime 7) : false :=
by linarith

example (x y z : ℚ) (h1 : 2*1*x + (3)*(y*(-1)) < 0) (h2 : (-2)*x*2 < -(z + z))
       (h3 : 12*y + (-4)* z < 0) (h4 : nat.prime 7) : false :=
by linarith

example (x y z : ℤ) (h1 : 2*x  < 3*y) (h2 : -4*x + 2*z < 0)
        (h3 : 12*y - 4* z < 0)  : false :=
by linarith

example (x y z : ℤ) (h1 : 2*x  < 3*y) (h2 : -4*x + 2*z < 0) (h3 : x*y < 5)
        (h3 : 12*y - 4* z < 0)  : false :=
by linarith

example (x y z : ℤ) (h1 : 2*x  < 3*y) (h2 : -4*x + 2*z < 0) (h3 : x*y < 5) :
        ¬ 12*y - 4* z < 0 :=
by linarith

example (w x y z : ℤ) (h1 : 4*x + (-3)*y + 6*w ≤ 0) (h2 : (-1)*x < 0)
        (h3 : y < 0) (h4 : w ≥ 0) (h5 : nat.prime x.nat_abs) : false :=
by linarith

example (a b c : ℚ) (h1 : a > 0) (h2 : b > 5) (h3 : c < -10)
        (h4 : a + b - c < 3)  : false :=
by linarith

example (a b c : ℚ) (h2 : b > 0) (h3 : ¬ b ≥ 0) : false :=
by linarith

example (a b c : ℚ) (h2 : (2 : ℚ) > 3)  : a + b - c ≥ 3 :=
by linarith {exfalso := ff}

example (x : ℚ) (hx : x > 0) (h : x.num < 0) : false :=
by linarith [rat.num_pos_iff_pos.mpr hx, h]

example (x : ℚ) (hx : x > 0) (h : x.num < 0) : false :=
by linarith only [rat.num_pos_iff_pos.mpr hx, h]

example (x y z : ℚ) (hx : x ≤ 3*y) (h2 : y ≤ 2*z) (h3 : x ≥ 6*z) : x = 3*y :=
by linarith

example (x y z : ℕ) (hx : x ≤ 3*y) (h2 : y ≤ 2*z) (h3 : x ≥ 6*z) : x = 3*y :=
by linarith

example (x y z : ℚ) (hx : ¬ x > 3*y) (h2 : ¬ y > 2*z) (h3 : x ≥ 6*z) : x = 3*y :=
by linarith

example (h1 : (1 : ℕ) < 1) : false :=
by linarith

example (a b c : ℚ) (h2 : b > 0) (h3 : b < 0) : nat.prime 10 :=
by linarith

example (a b c : ℕ) : a + b ≥ a :=
by linarith

example (a b c : ℕ) : ¬ a + b < a :=
by linarith

example (x y : ℚ) (h : 6 + ((x + 4) * x + (6 + 3 * y) * y) = 3) (h' : (x + 4) * x ≥ 0)
  (h'' : (6 + 3 * y) * y ≥ 0)  : false :=
by linarith

example (x y : ℕ) (h : 6 + ((x + 4) * x + (6 + 3 * y) * y) = 3) : false :=
by linarith

example (a b i : ℕ) (h1 :  ¬ a < i) (h2 : b < i) (h3 : a ≤ b) : false :=
by linarith

example (n : ℕ) (h1 : n ≤ 3) (h2 : n > 2) : n = 3 := by linarith

example (z : ℕ) (hz : ¬ z ≥ 2) (h2 : ¬ z + 1 ≤ 2) : false :=
by linarith

example (z : ℕ) (hz : ¬ z ≥ 2) : z + 1 ≤ 2 :=
by linarith

example (a b c : ℚ) (h1 : 1 / a < b) (h2 : b < c) : 1 / a < c :=
by linarith

example
(N : ℕ) (n : ℕ) (Hirrelevant : n > N)
(A : ℚ) (l : ℚ) (h : A - l ≤ -(A - l)) (h_1 : ¬A ≤ -A) (h_2 : ¬l ≤ -l)
(h_3 : -(A - l) < 1) :  A < l + 1 := by linarith

example (d : ℚ) (q n : ℕ) (h1 : ((q : ℚ) - 1)*n ≥ 0) (h2 : d = 2/3*(((q : ℚ) - 1)*n)) :
  d ≤ ((q : ℚ) - 1)*n :=
by linarith

example (d : ℚ) (q n : ℕ) (h1 : ((q : ℚ) - 1)*n ≥ 0) (h2 : d = 2/3*(((q : ℚ) - 1)*n)) :
  ((q : ℚ) - 1)*n - d = 1/3 * (((q : ℚ) - 1)*n) :=
by linarith

example (a : ℚ) (ha : 0 ≤ a) : 0 * 0 ≤ 2 * a :=
by linarith

example (x : ℚ) : id x ≥ x :=
by success_if_fail {linarith}; linarith!

example (x y z : ℚ) (hx : x < 5) (hx2 : x > 5) (hy : y < 5000000000) (hz : z > 34*y) : false :=
by linarith only [hx, hx2]

example (x y z : ℚ) (hx : x < 5) (hy : y < 5000000000) (hz : z > 34*y) : x ≤ 5 :=
by linarith only [hx]

example (x y : ℚ) (h : x < y) : x ≠ y := by linarith

example (x y : ℚ) (h : x < y) : ¬ x = y := by linarith