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S05_Disjunction.lean
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S05_Disjunction.lean
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import LeanInVienna.Common
import Mathlib.Data.Real.Basic
namespace C03S05
section
variable {x y : ℝ}
example (h : y > x ^ 2) : y > 0 ∨ y < -1 := by
left
linarith [pow_two_nonneg x]
example (h : -y > x ^ 2 + 1) : y > 0 ∨ y < -1 := by
right
linarith [pow_two_nonneg x]
example (h : y > 0) : y > 0 ∨ y < -1 :=
Or.inl h
example (h : y < -1) : y > 0 ∨ y < -1 :=
Or.inr h
example : x < |y| → x < y ∨ x < -y := by
rcases le_or_gt 0 y with h | h
· rw [abs_of_nonneg h]
intro h; left; exact h
· rw [abs_of_neg h]
intro h; right; exact h
example : x < |y| → x < y ∨ x < -y := by
cases le_or_gt 0 y
case inl h =>
rw [abs_of_nonneg h]
intro h; left; exact h
case inr h =>
rw [abs_of_neg h]
intro h; right; exact h
example : x < |y| → x < y ∨ x < -y := by
cases le_or_gt 0 y
next h =>
rw [abs_of_nonneg h]
intro h; left; exact h
next h =>
rw [abs_of_neg h]
intro h; right; exact h
example : x < |y| → x < y ∨ x < -y := by
match le_or_gt 0 y with
| Or.inl h =>
rw [abs_of_nonneg h]
intro h; left; exact h
| Or.inr h =>
rw [abs_of_neg h]
intro h; right; exact h
namespace MyAbs
theorem le_abs_self (x : ℝ) : x ≤ |x| := by
sorry
theorem neg_le_abs_self (x : ℝ) : -x ≤ |x| := by
sorry
theorem abs_add (x y : ℝ) : |x + y| ≤ |x| + |y| := by
sorry
theorem lt_abs : x < |y| ↔ x < y ∨ x < -y := by
sorry
theorem abs_lt : |x| < y ↔ -y < x ∧ x < y := by
sorry
end MyAbs
end
example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := by
rcases lt_trichotomy x 0 with xlt | xeq | xgt
· left
exact xlt
· contradiction
· right; exact xgt
example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := by
rcases h with ⟨a, rfl⟩ | ⟨b, rfl⟩
· rw [mul_assoc]
apply dvd_mul_right
· rw [mul_comm, mul_assoc]
apply dvd_mul_right
example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by
sorry
example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
sorry
example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by
sorry
section
variable {R : Type*} [CommRing R] [IsDomain R]
variable (x y : R)
example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
sorry
example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by
sorry
end
example (P : Prop) : ¬¬P → P := by
intro h
cases em P
· assumption
· contradiction
example (P : Prop) : ¬¬P → P := by
intro h
by_cases h' : P
· assumption
contradiction
example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by
sorry