feat(rat/{basic,floor}) add floor lemmas (#5148)

Author: kappelmann

src/data/rat/basic.lean

@@ -586,6 +586,14 @@ begin
   simp [division_def, coe_int_eq_mk, mul_def one_ne_zero d0]
 end
 
+lemma exists_eq_mul_div_num_and_eq_mul_div_denom {n d : ℤ} (n_ne_zero : n ≠ 0)
+  (d_ne_zero : d ≠ 0) :
+  ∃ (c : ℤ), n = c * ((n : ℚ) / d).num ∧ (d : ℤ) = c * ((n : ℚ) / d).denom :=
+begin
+  have : ((n : ℚ) / d) = rat.mk n d, by rw [←rat.mk_eq_div],
+  exact rat.num_denom_mk n_ne_zero d_ne_zero this
+end
+
 theorem coe_int_eq_of_int (z : ℤ) : ↑z = of_int z :=
 (coe_int_eq_mk z).trans (of_int_eq_mk z).symm
 

src/data/rat/floor.lean

@@ -9,7 +9,8 @@ import algebra.floor
 
 ## Summary
 
-We define the `floor` function and the `floor_ring` instance on `ℚ`.
+We define the `floor` function and the `floor_ring` instance on `ℚ`. Some technical lemmas relating
+`floor` to integer division and modulo arithmetic are derived as well as some simple inequalities.
 
 ## Tags
 
@@ -37,4 +38,105 @@ instance : floor_ring ℚ :=
 
 protected lemma floor_def {q : ℚ} : ⌊q⌋ = q.num / q.denom := by { cases q, refl }
 
+lemma floor_int_div_nat_eq_div {n : ℤ} {d : ℕ} : ⌊(↑n : ℚ) / (↑d : ℚ)⌋ = n / (↑d : ℤ) :=
+begin
+  rw [rat.floor_def],
+  cases decidable.em (d = 0) with d_eq_zero d_ne_zero,
+  { simp [d_eq_zero] },
+  { cases decidable.em (n = 0) with n_eq_zero n_ne_zero,
+    { simp [n_eq_zero] },
+    { set q := (n : ℚ) / d with q_eq,
+      obtain ⟨c, n_eq_c_mul_num, d_eq_c_mul_denom⟩ : ∃ c, n = c * q.num ∧ (d : ℤ) = c * q.denom, by
+      { rw q_eq,
+        exact_mod_cast (@rat.exists_eq_mul_div_num_and_eq_mul_div_denom n d n_ne_zero
+          (by exact_mod_cast d_ne_zero)) },
+      suffices : q.num / q.denom = c * q.num / (c * q.denom),
+        by rwa [n_eq_c_mul_num, d_eq_c_mul_denom],
+      suffices : c > 0, by solve_by_elim [int.mul_div_mul_of_pos],
+      have q_denom_mul_c_pos : (0 : ℤ) < q.denom * c, by
+      { have : (d : ℤ) > 0, by exact_mod_cast (nat.pos_iff_ne_zero.elim_right d_ne_zero),
+        rwa [d_eq_c_mul_denom, mul_comm] at this },
+      suffices : (0 : ℤ) ≤ q.denom, from pos_of_mul_pos_left q_denom_mul_c_pos this,
+      exact_mod_cast (le_of_lt q.pos) } }
+end
+
+end rat
+
+
+lemma int.mod_nat_eq_sub_mul_floor_rat_div {n : ℤ} {d : ℕ} : n % d = n - d * ⌊(n : ℚ) / d⌋ :=
+by rw [(eq_sub_of_add_eq $ int.mod_add_div n d), rat.floor_int_div_nat_eq_div]
+
+lemma nat.coprime_sub_mul_floor_rat_div_of_coprime {n d : ℕ} (n_coprime_d : n.coprime d) :
+  ((n : ℤ) - d * ⌊(n : ℚ)/ d⌋).nat_abs.coprime d :=
+begin
+  have : (n : ℤ) % d = n - d * ⌊(n : ℚ)/ d⌋, from int.mod_nat_eq_sub_mul_floor_rat_div,
+  rw ←this,
+  have : d.coprime n, from n_coprime_d.symm,
+  rwa [nat.coprime, nat.gcd_rec] at this
+end
+
+
+namespace rat
+
+lemma num_lt_succ_floor_mul_denom (q : ℚ) : q.num < (⌊q⌋ + 1) * q.denom :=
+begin
+  suffices : (q.num : ℚ) < (⌊q⌋ + 1) * q.denom, by exact_mod_cast this,
+  suffices : (q.num : ℚ) < (q - fract q + 1) * q.denom, by
+  { have : (⌊q⌋ : ℚ) = q - fract q, from (eq_sub_of_add_eq $ floor_add_fract q),
+    rwa this },
+  suffices : (q.num : ℚ) < q.num + (1 - fract q) * q.denom, by
+  { have : (q - fract q + 1) * q.denom = q.num + (1 - fract q) * q.denom, calc
+      (q - fract q + 1) * q.denom = (q + (1 - fract q)) * q.denom            : by ring
+                              ... = q * q.denom + (1 - fract q) * q.denom    : by rw add_mul
+                              ... = q.num + (1 - fract q) * q.denom : by simp,
+    rwa this },
+  suffices : 0 < (1 - fract q) * q.denom, by { rw ←sub_lt_iff_lt_add', simpa },
+  have : 0 < 1 - fract q, by
+  { have : fract q < 1, from fract_lt_one q,
+    have : 0 + fract q < 1, by simp [this],
+    rwa lt_sub_iff_add_lt },
+  exact mul_pos this (by exact_mod_cast q.pos)
+end
+
+lemma fract_inv_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q): (fract q⁻¹).num < q.num :=
+begin
+  -- we know that the numerator must be positive
+  have q_num_pos : 0 < q.num, from rat.num_pos_iff_pos.elim_right q_pos,
+  -- we will work with the absolute value of the numerator, which is equal to the numerator
+  have q_num_abs_eq_q_num : (q.num.nat_abs : ℤ) = q.num, from
+    (int.nat_abs_of_nonneg $ le_of_lt q_num_pos),
+  set q_inv := (q.denom : ℚ) / q.num with q_inv_def,
+  have q_inv_eq : q⁻¹ = q_inv, from rat.inv_def',
+  suffices : (q_inv - ⌊q_inv⌋).num < q.num, by rwa q_inv_eq,
+  suffices : ((q.denom - q.num * ⌊q_inv⌋ : ℚ) / q.num).num < q.num, by
+    field_simp [this, (ne_of_gt q_num_pos)],
+  suffices : (q.denom : ℤ) - q.num * ⌊q_inv⌋ < q.num, by
+  { -- use that `q.num` and `q.denom` are coprime to show that the numerator stays unreduced
+    have : ((q.denom - q.num * ⌊q_inv⌋ : ℚ) / q.num).num = q.denom - q.num * ⌊q_inv⌋, by
+    { suffices : ((q.denom : ℤ) - q.num * ⌊q_inv⌋).nat_abs.coprime q.num.nat_abs, by
+        exact_mod_cast (rat.num_div_eq_of_coprime q_num_pos this),
+      have : (q.num.nat_abs : ℚ) = (q.num : ℚ), by exact_mod_cast q_num_abs_eq_q_num,
+      have tmp := nat.coprime_sub_mul_floor_rat_div_of_coprime q.cop.symm,
+      simpa only [this, q_num_abs_eq_q_num] using tmp },
+    rwa this },
+  -- to show the claim, start with the following inequality
+  have q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * q⁻¹.denom < q⁻¹.denom, by
+  { have : q⁻¹.num < (⌊q⁻¹⌋ + 1) * q⁻¹.denom, from rat.num_lt_succ_floor_mul_denom q⁻¹,
+    have : q⁻¹.num < ⌊q⁻¹⌋ * q⁻¹.denom + q⁻¹.denom, by rwa [right_distrib, one_mul] at this,
+    rwa [←sub_lt_iff_lt_add'] at this },
+  -- use that `q.num` and `q.denom` are coprime to show that q_inv is the unreduced reciprocal
+  -- of `q`
+  have : q_inv.num = q.denom ∧ q_inv.denom = q.num.nat_abs, by
+  { have coprime_q_denom_q_num : q.denom.coprime q.num.nat_abs, from q.cop.symm,
+    have : int.nat_abs q.denom = q.denom, by simp,
+    rw ←this at coprime_q_denom_q_num,
+    rw q_inv_def,
+    split,
+    { exact_mod_cast (rat.num_div_eq_of_coprime q_num_pos coprime_q_denom_q_num) },
+    { suffices : (((q.denom : ℚ) / q.num).denom : ℤ) = q.num.nat_abs, by exact_mod_cast this,
+      rw q_num_abs_eq_q_num,
+      exact_mod_cast (rat.denom_div_eq_of_coprime q_num_pos coprime_q_denom_q_num) } },
+  rwa [q_inv_eq, this.left, this.right, q_num_abs_eq_q_num, mul_comm] at q_inv_num_denom_ineq
+end
+
 end rat