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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | import OAI.Algebra.DepthFive.Basic
namespace OAI
namespace Problem335
@[simp] theorem ceilSqrt_zero : ceilSqrt 0 = 0 := by
simp [ceilSqrt]
@[simp] theorem ceilSqrt_one : ceilSqrt 1 = 1 := by
simp [ceilSqrt]
@[simp] theorem ceilSqrt_two : ceilSqrt 2 = 2 := by
norm_num [ceilSqrt, Nat.sqrt_two]
theorem sqrt_le_ceilSqrt (n : ℕ) : Nat.sqrt n ≤ ceilSqrt n := by
unfold ceilSqrt
omega
theorem ceilSqrt_le_sqrt_add_one (n : ℕ) : ceilSqrt n ≤ Nat.sqrt n + 1 := by
unfold ceilSqrt
split <;> omega
theorem self_le_ceilSqrt_sq (n : ℕ) : n ≤ ceilSqrt n ^ 2 := by
unfold ceilSqrt
split
· rename_i h
simpa using h.ge
· simpa [Nat.succ_eq_add_one] using (Nat.lt_succ_sqrt' n).le
theorem ceilSqrt_pos {n : ℕ} (hn : 0 < n) : 0 < ceilSqrt n :=
lt_of_lt_of_le (Nat.sqrt_pos.mpr hn) (sqrt_le_ceilSqrt n)
theorem ceilSqrt_le_self (n : ℕ) : ceilSqrt n ≤ n := by
rcases Nat.eq_zero_or_pos n with rfl | hn
· simp
by_cases h : n = 1
· subst n
simp
have hn' : 1 < n := by omega
exact (ceilSqrt_le_sqrt_add_one n).trans (Nat.succ_le_of_lt (Nat.sqrt_lt_self hn'))
theorem two_le_ceilSqrt {n : ℕ} (hn : 2 ≤ n) : 2 ≤ ceilSqrt n := by
have h := self_le_ceilSqrt_sq n
by_contra hh
have hh' : ceilSqrt n ≤ 1 := by omega
have : ceilSqrt n ^ 2 ≤ 1 ^ 2 := Nat.pow_le_pow_left hh' 2
omega
theorem ceilSqrt_cast_le_real_sqrt_add_one (n : ℕ) :
(ceilSqrt n : ℝ) ≤ Real.sqrt (n : ℝ) + 1 := by
have hnat : (Nat.sqrt n : ℝ) ≤ Real.sqrt (n : ℝ) := by
apply Real.le_sqrt_of_sq_le
exact_mod_cast Nat.sqrt_le' n
have hceil : (ceilSqrt n : ℝ) ≤ (Nat.sqrt n : ℝ) + 1 := by
exact_mod_cast ceilSqrt_le_sqrt_add_one n
linarith
theorem real_sqrt_le_ceilSqrt_cast (n : ℕ) :
Real.sqrt (n : ℝ) ≤ (ceilSqrt n : ℝ) := by
apply (Real.sqrt_le_left (Nat.cast_nonneg _)).mpr
exact_mod_cast self_le_ceilSqrt_sq n
theorem ceilDiv_le_iff {n t r : ℕ} (ht : 0 < t) :
ceilDiv n t ≤ r ↔ n ≤ r * t := by
unfold ceilDiv
rw [Nat.div_le_iff_le_mul_add_pred ht]
rw [Nat.mul_comm t r]
omega
theorem self_le_ceilDiv_mul (n : ℕ) {t : ℕ} (ht : 0 < t) :
n ≤ ceilDiv n t * t :=
(ceilDiv_le_iff ht).mp le_rfl
theorem ceilDiv_le_self (n : ℕ) {t : ℕ} (ht : 0 < t) :
ceilDiv n t ≤ n := by
apply (ceilDiv_le_iff ht).mpr
exact Nat.le_mul_of_pos_right n ht
theorem ceilDiv_pos {n t : ℕ} (hn : 0 < n) (ht : 0 < t) :
0 < ceilDiv n t := by
by_contra h
have hh : ceilDiv n t ≤ 0 := by omega
have := (ceilDiv_le_iff ht).mp hh
omega
theorem ceilDiv_ceilSqrt_le (n : ℕ) : ceilDiv n (ceilSqrt n) ≤ ceilSqrt n := by
rcases Nat.eq_zero_or_pos n with rfl | hn
· simp [ceilDiv]
apply (ceilDiv_le_iff (ceilSqrt_pos hn)).mpr
simpa [pow_two] using self_le_ceilSqrt_sq n
end Problem335
end OAI
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