feat(library/data/real): add more theorems concerning rationals embedded in reals

library/data/real/basic.lean

@@ -1206,6 +1206,10 @@ theorem of_rat_neg (a : ℚ) : of_rat (-a) = -of_rat a :=
 theorem of_rat_mul (a b : ℚ) : of_rat (a * b) = of_rat a * of_rat b :=
   quot.sound (r_mul_consts a b)
 
+theorem of_rat_zero : of_rat 0 = 0 := rfl
+
+theorem of_rat_one : of_rat 1 = 1 := rfl
+
 open int
 
 theorem of_int_add (a b : ℤ) : of_int (a + b) = of_int a + of_int b :=

library/data/real/complete.lean

@@ -213,6 +213,56 @@ theorem approx_spec (x : ℝ) (n : ℕ+) : abs (x - (of_rat (approx x n))) ≤ o
 theorem approx_spec' (x : ℝ) (n : ℕ+) : abs ((of_rat (approx x n)) - x) ≤ of_rat n⁻¹ :=
   by rewrite abs_sub; apply approx_spec
 
+theorem ex_rat_pos_lower_bound_of_pos {x : ℝ} (H : x > 0) : ∃ q : ℚ, q > 0 ∧ of_rat q ≤ x :=
+  if Hgeo : x ≥ 1 then
+    exists.intro 1 (and.intro zero_lt_one Hgeo)
+  else
+    have Hdp : 1 / x > 0, from one_div_pos_of_pos H,
+    begin
+      cases rat_approx (1 / x) 2 with q Hq,
+      have Hqp : q > 0, begin
+        apply lt_of_not_ge,
+        intro Hq2,
+        note Hx' := one_div_lt_one_div_of_lt H (lt_of_not_ge Hgeo),
+        rewrite div_one at Hx',
+        have Horqn : of_rat q ≤ 0, begin
+          krewrite -of_rat_zero,
+          apply of_rat_le_of_rat_of_le Hq2
+        end,
+        have Hgt1 : 1 / x - of_rat q > 1, from calc
+          1 / x - of_rat q = 1 / x + -of_rat q : sub_eq_add_neg
+                       ... ≥ 1 / x             : le_add_of_nonneg_right (neg_nonneg_of_nonpos Horqn)
+                       ... > 1                 : Hx',
+        have Hpos : 1 / x - of_rat q > 0, from gt.trans Hgt1 zero_lt_one,
+        rewrite [abs_of_pos Hpos at Hq],
+        apply not_le_of_gt Hgt1,
+        apply le.trans,
+        apply Hq,
+        krewrite -of_rat_one,
+        apply of_rat_le_of_rat_of_le,
+        apply inv_le_one
+      end,
+      existsi 1 / (2⁻¹ + q),
+      split,
+      apply div_pos_of_pos_of_pos,
+      exact zero_lt_one,
+      apply add_pos,
+      apply pnat.inv_pos,
+      exact Hqp,
+      note Hle2 := sub_le_of_abs_sub_le_right Hq,
+      note Hle3 := le_add_of_sub_left_le Hle2,
+      note Hle4 := one_div_le_of_one_div_le_of_pos H Hle3,
+      rewrite [of_rat_divide, of_rat_add],
+      exact Hle4
+    end
+
+theorem ex_rat_neg_upper_bound_of_neg {x : ℝ} (H : x < 0) : ∃ q : ℚ, q < 0 ∧ x ≤ of_rat q :=
+  have H' : -x > 0, from neg_pos_of_neg H,
+  obtain q [Hq1 Hq2], from ex_rat_pos_lower_bound_of_pos H',
+  exists.intro (-q) (and.intro
+    (neg_neg_of_pos Hq1)
+    (le_neg_of_le_neg Hq2))
+
 notation `r_seq` := ℕ+ → ℝ
 
 noncomputable definition converges_to_with_rate (X : r_seq) (a : ℝ) (N : ℕ+ → ℕ+) :=

library/data/real/division.lean

@@ -649,10 +649,6 @@ protected noncomputable definition discrete_linear_ordered_field [reducible] [tr
     decidable_lt    := dec_lt
    ⦄
 
-theorem of_rat_zero : of_rat (0:rat) = (0:real) := rfl
-
-theorem of_rat_one : of_rat (1:rat) = (1:real) := rfl
-
 theorem of_rat_divide (x y : ℚ) : of_rat (x / y) = of_rat x / of_rat y :=
 by_cases
   (assume yz : y = 0, by krewrite [yz, div_zero, +of_rat_zero, div_zero])

library/data/real/order.lean

@@ -1165,9 +1165,24 @@ theorem of_nat_lt_of_nat_of_lt {a b : ℕ} : (a < b) → of_nat a < of_nat b :=
 theorem lt_of_of_nat_lt_of_nat {a b : ℕ} : of_nat a < of_nat b → (a < b) :=
   iff.mp !of_nat_lt_of_nat_iff
 
+theorem of_rat_pos_of_pos {q : ℚ} (Hq : q > 0) : of_rat q > 0 :=
+  of_rat_lt_of_rat_of_lt Hq
+
+theorem of_rat_nonneg_of_nonneg {q : ℚ} (Hq : q ≥ 0) : of_rat q ≥ 0 :=
+  of_rat_le_of_rat_of_le Hq
+
+theorem of_rat_neg_of_neg {q : ℚ} (Hq : q < 0) : of_rat q < 0 :=
+  of_rat_lt_of_rat_of_lt Hq
+
+theorem of_rat_nonpos_of_nonpos {q : ℚ} (Hq : q ≤ 0) : of_rat q ≤ 0 :=
+  of_rat_le_of_rat_of_le Hq
+
 theorem of_nat_nonneg (a : ℕ) : of_nat a ≥ 0 :=
 of_rat_le_of_rat_of_le !rat.of_nat_nonneg
 
+theorem of_nat_succ_pos (k : ℕ) : 0 < of_nat k + 1 :=
+  add_pos_of_nonneg_of_pos (of_nat_nonneg k) real.zero_lt_one
+
 theorem of_rat_pow (a : ℚ) (n : ℕ) : of_rat (a^n) = (of_rat a)^n :=
 begin
   induction n with n ih,