feat(library/algebra): add missing theorems to algebra library

library/algebra/group.lean

@@ -541,6 +541,7 @@ section add_group
 
   theorem add_eq_of_eq_sub {a b c : A} (H : a = c - b) : a + b = c :=
   add_eq_of_eq_add_neg H
+
 end add_group
 
 structure add_comm_group [class] (A : Type) extends add_group A, add_comm_monoid A
@@ -585,6 +586,8 @@ section add_comm_group
   theorem sub_eq_sub_add_sub (a b c : A) : a - b = c - b + (a - c) :=
     by rewrite [add_sub, sub_add_cancel] ⬝ !add.comm
 
+  theorem neg_neg_sub_neg (a b : A) : - (-a - -b) = a - b :=
+    by rewrite [neg_sub, sub_neg_eq_add, neg_add_eq_sub]
 end add_comm_group
 
 definition group_of_add_group (A : Type) [G : add_group A] : group A :=

library/algebra/order.lean

@@ -290,6 +290,17 @@ section
     (assume H : ¬ a ≤ b,
       (inr (assume H1 : a = b, H (H1 ▸ !le.refl))))
 
+  theorem eq_or_lt_of_not_lt {a b : A} (H : ¬ a < b) : a = b ∨ b < a :=
+    if Heq : a = b then or.inl Heq else or.inr (lt_of_not_ge (λ Hge, H (lt_of_le_of_ne Hge Heq)))
+
+  theorem eq_or_lt_of_le {a b : A} (H : a ≤ b) : a = b ∨ a < b :=
+    begin
+      cases eq_or_lt_of_not_lt (not_lt_of_ge H),
+      exact or.inl a_1⁻¹,
+      exact or.inr a_1
+    end
+
+
   -- testing equality first may result in more definitional equalities
   definition lt.cases {B : Type} (a b : A) (t_lt t_eq t_gt : B) : B :=
   if a = b then t_eq else (if a < b then t_lt else t_gt)

library/algebra/ordered_group.lean

@@ -493,6 +493,20 @@ section
   theorem sub_lt_right_of_lt_add : a < b + c → a - c < b :=
     iff.mp !lt_add_iff_sub_lt_right
 
+  theorem sub_lt_of_sub_lt : a - b < c → a - c < b :=
+    begin
+      intro H,
+      apply sub_lt_left_of_lt_add,
+      apply lt_add_of_sub_lt_right _ _ _ H
+    end
+
+  theorem sub_le_of_sub_le : a - b ≤ c → a - c ≤ b :=
+    begin
+      intro H,
+      apply sub_left_le_of_le_add,
+      apply le_add_of_sub_right_le _ _ _ H
+    end
+
   -- TODO: the Isabelle library has varations on a + b ≤ b ↔ a ≤ 0
   theorem le_iff_le_of_sub_eq_sub {a b c d : A} (H : a - b = c - d) : a ≤ b ↔ c ≤ d :=
   calc
@@ -553,6 +567,9 @@ section
   theorem sub_le_of_nonneg (H : b ≥ 0) : a - b ≤ a :=
    add_le_of_le_of_nonpos (le.refl a) (neg_nonpos_of_nonneg H)
 
+  theorem sub_lt_of_pos {b : A} (H : b > 0) : a - b < a :=
+   add_lt_of_le_of_neg (le.refl a) (neg_neg_of_pos H)
+
   theorem neg_add_neg_le_neg_of_pos {a : A} (H : a > 0) : -a + -a ≤ -a :=
     !neg_add ▸ neg_le_neg (le_add_of_nonneg_left (le_of_lt H))
 end

library/algebra/ordered_ring.lean

@@ -672,6 +672,19 @@ section
   theorem sub_le_of_abs_sub_le_right (H : abs (a - b) ≤ c) : a - c ≤ b :=
     sub_le_of_abs_sub_le_left (!abs_sub ▸ H)
 
+  theorem sub_lt_of_abs_sub_lt_left (H : abs (a - b) < c) : b - c < a :=
+    if Hz : 0 ≤ a - b then
+      (calc
+        a ≥ b : (iff.mp !sub_nonneg_iff_le) Hz
+      ... > b - c : sub_lt_of_pos _ (lt_of_le_of_lt !abs_nonneg H))
+    else
+      (have Habs : b - a < c, by rewrite [abs_of_neg (lt_of_not_ge Hz) at H, neg_sub at H]; apply H,
+       have Habs' : b < c + a, from lt_add_of_sub_lt_right _ _ _ Habs,
+       sub_lt_left_of_lt_add _ _ _ Habs')
+
+  theorem sub_lt_of_abs_sub_lt_right (H : abs (a - b) < c) : a - c < b :=
+    sub_lt_of_abs_sub_lt_left (!abs_sub ▸ H)
+
   theorem abs_sub_square (a b : A) : abs (a - b) * abs (a - b) = a * a + b * b - (1 + 1) * a * b :=
     by rewrite [abs_mul_abs_self, *mul_sub_left_distrib, *mul_sub_right_distrib,
              sub_add_eq_sub_sub, sub_neg_eq_add, *right_distrib, sub_add_eq_sub_sub, *one_mul,

library/data/real/complete.lean

@@ -7,9 +7,9 @@ This construction follows Bishop and Bridges (1985).
 
 At this point, we no longer proceed constructively: this file makes heavy use of decidability,
 excluded middle, and Hilbert choice. Two sets of definitions of Cauchy sequences, convergence,
-etc are available, one with rates and one without. The definitions with rates are amenable
-to be used constructively, if and when that development takes place. The second set of
-definitions are the usual classical ones.
+etc are available in the libray, one with rates and one without. The definitions here, with rates,
+ are amenable to be used constructively if and when that development takes place. The second set of
+definitions available in /library/theories/analysis/metric_space.lean are the usual classical ones.
 
 Here, we show that ℝ is complete. The proofs of Cauchy completeness and the supremum property
 are independent of each other.