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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.apply_fun
open function

example (X Y Z : Type) (f : X → Y) (g : Y → Z) (H : injective $ g ∘ f) :
  injective f :=
begin
  intros x x' h,
  apply_fun g at h,
  exact H h
end

example (f : ℕ → ℕ) (a b : ℕ) (monof : monotone f)  (h : a ≤ b) : f a ≤ f b :=
begin
  apply_fun f at h,
  assumption,
  assumption
end

example (a b : ℤ) (h : a = b) : a + 1 = b + 1 :=
begin
  apply_fun (λ n, n+1) at h,
  -- check that `h` was β-reduced
  guard_hyp' h := a + 1 = b + 1,
  exact h
end

example (f : ℕ → ℕ) (a b : ℕ) (monof : monotone f)  (h : a ≤ b) : f a ≤ f b :=
begin
  apply_fun f at h using monof,
  assumption
end

-- monotonicity will be proved by `mono` in the next example
example (a b : ℕ) (h : a ≤ b) : a + 1 ≤ b + 1 :=
begin
  apply_fun (λ n, n+1) at h,
  exact h
end