feat(library/algebra): add to developments of group, ordered_group, and ring

library/algebra/group.lean

@@ -249,6 +249,12 @@ section group
   theorem eq_mul_inv_imp_mul_eq {a b c : A} (H : a = b * c⁻¹) : a * c = b :=
   !inv_inv ▸ (eq_mul_imp_mul_inv_eq H)
 
+  theorem mul_eq_iff_eq_inv_mul (a b c : A) : a * b = c ↔ b = a⁻¹ * c :=
+  iff.intro mul_eq_imp_eq_inv_mul eq_inv_mul_imp_mul_eq
+
+  theorem mul_eq_iff_eq_mul_inv (a b c : A) : a * b = c ↔ a = c * b⁻¹ :=
+  iff.intro mul_eq_imp_eq_mul_inv eq_mul_inv_imp_mul_eq
+
   definition group.to_left_cancel_semigroup [instance] : left_cancel_semigroup A :=
   left_cancel_semigroup.mk (@group.mul A s) (@group.mul_assoc A s)
     (take a b c,
@@ -371,6 +377,12 @@ section add_group
   theorem eq_add_neg_imp_add_eq {a b c : A} (H : a = b + -c) : a + c = b :=
   !neg_neg ▸ (eq_add_imp_add_neg_eq H)
 
+  theorem add_eq_iff_eq_neg_add (a b c : A) : a + b = c ↔ b = -a + c :=
+  iff.intro add_eq_imp_eq_neg_add eq_neg_add_imp_add_eq
+
+  theorem add_eq_iff_eq_add_neg (a b c : A) : a + b = c ↔ a = c + -b :=
+  iff.intro add_eq_imp_eq_add_neg eq_add_neg_imp_add_eq
+
   definition add_group.to_left_cancel_semigroup [instance] :
     add_left_cancel_semigroup A :=
   add_left_cancel_semigroup.mk (@add_group.add A s) (@add_group.add_assoc A s)

library/algebra/ordered_group.lean

@@ -153,8 +153,8 @@ end
 
 -- TODO: there is more we can do if we have max and min (in order.lean as well)
 
--- TODO: there is more we can do if we assume a ≤ b ↔ ∃c. a + c = b, but it is not clear whether
--- this gives us any useful generality
+-- TODO: there is more we can do if we assume a ≤ b ↔ ∃c. a + c = b. This covers the natural numbers,
+-- but it is not clear whether it provides any further useful generality.
 
 structure ordered_comm_group [class] (A : Type) extends add_comm_group A, order_pair A :=
 (add_le_left : ∀a b c, le a b → le (add c a) (add c b))
@@ -171,4 +171,138 @@ proof
   !neg_add_cancel_left ▸ !neg_add_cancel_left ▸ H'
 qed
 
+section
+
+  variables [s : ordered_comm_group A] (a b c d e : A)
+  include s
+
+  theorem le_imp_neg_le_neg {a b : A} (H : a ≤ b) : -b ≤ -a :=
+  have H1 : 0 ≤ -a + b, from !add_left_inv ▸ !(add_le_left H),
+  !add_neg_cancel_right ▸ !add_left_id ▸ add_le_right H1 (-b)
+--  !add_left_id ▸ !add_neg_cancel_right ▸ add_le_right H1 (-b)  -- doesn't work?
+
+  theorem neg_le_neg_iff : -a ≤ -b ↔ b ≤ a :=
+  iff.intro (take H, neg_neg a ▸ neg_neg b ▸ le_imp_neg_le_neg H) le_imp_neg_le_neg
+
+  theorem neg_le_zero_iff : -a ≤ 0 ↔ 0 ≤ a :=
+  neg_zero ▸ neg_le_neg_iff a 0
+
+  theorem zero_le_neg_iff : 0 ≤ -a ↔ a ≤ 0 :=
+  neg_zero ▸ neg_le_neg_iff 0 a
+
+  theorem lt_imp_neg_lt_neg {a b : A} (H : a < b) : -b < -a :=
+  have H1 : 0 < -a + b, from !add_left_inv ▸ !(add_lt_left H),
+  !add_neg_cancel_right ▸ !add_left_id ▸ add_lt_right H1 (-b)
+
+  theorem neg_lt_neg_iff : -a < -b ↔ b < a :=
+  iff.intro (take H, neg_neg a ▸ neg_neg b ▸ lt_imp_neg_lt_neg H) lt_imp_neg_lt_neg
+
+  theorem neg_lt_zero_iff : -a < 0 ↔ 0 < a :=
+  neg_zero ▸ neg_lt_neg_iff a 0
+
+  theorem zero_lt_neg_iff : 0 < -a ↔ a < 0 :=
+  neg_zero ▸ neg_lt_neg_iff 0 a
+
+  theorem le_neg_iff : a ≤ -b ↔ b ≤ -a := !neg_neg ▸ !neg_le_neg_iff
+
+  theorem neg_le_iff : -a ≤ b ↔ -b ≤ a := !neg_neg ▸ !neg_le_neg_iff
+
+  theorem lt_neg_iff : a < -b ↔ b < -a := !neg_neg ▸ !neg_lt_neg_iff
+
+  theorem neg_lt_iff : -a < b ↔ -b < a := !neg_neg ▸ !neg_lt_neg_iff
+
+  theorem minus_nonneg_iff : 0 ≤ a - b ↔ b ≤ a := !minus_self ▸ !add_le_right_iff
+
+  theorem minus_nonpos_iff : a - b ≤ 0 ↔ a ≤ b := !minus_self ▸ !add_le_right_iff
+
+  theorem minus_pos_iff : 0 < a - b ↔ b < a := !minus_self ▸ !add_lt_right_iff
+
+  theorem minus_neg_iff : a - b < 0 ↔ a < b := !minus_self ▸ !add_lt_right_iff
+
+  theorem add_le_iff_le_neg_add : a + b ≤ c ↔ b ≤ -a + c :=
+  have H: a + b ≤ c ↔ -a + (a + b) ≤ -a + c, from iff.symm (!add_le_left_iff),
+  !neg_add_cancel_left ▸ H
+
+  theorem add_le_iff_le_minus_left : a + b ≤ c ↔ b ≤ c - a :=
+  !add_comm ▸ !add_le_iff_le_neg_add
+
+  theorem add_le_iff_le_minus_right : a + b ≤ c ↔ a ≤ c - b :=
+  have H: a + b ≤ c ↔ a + b - b ≤ c - b, from iff.symm (!add_le_right_iff),
+  !add_neg_cancel_right ▸ H
+
+  theorem le_add_iff_neg_add_le : a ≤ b + c ↔ -b + a ≤ c :=
+  have H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_left_iff),
+  !neg_add_cancel_left ▸ H
+
+  theorem le_add_iff_minus_left_le : a ≤ b + c ↔ a - b ≤ c :=
+  !add_comm ▸ !le_add_iff_neg_add_le
+
+  theorem le_add_iff_minus_right_le : a ≤ b + c ↔ a - c ≤ b :=
+  have H: a ≤ b + c ↔ a - c ≤ b + c - c, from iff.symm (!add_le_right_iff),
+  !add_neg_cancel_right ▸ H
+
+  theorem add_lt_iff_lt_neg_add : a + b < c ↔ b < -a + c :=
+  have H: a + b < c ↔ -a + (a + b) < -a + c, from iff.symm (!add_lt_left_iff),
+  !neg_add_cancel_left ▸ H
+
+  theorem add_lt_iff_lt_minus_left : a + b < c ↔ b < c - a :=
+  !add_comm ▸ !add_lt_iff_lt_neg_add
+
+  theorem add_lt_iff_lt_minus_right : a + b < c ↔ a < c - b :=
+  have H: a + b < c ↔ a + b - b < c - b, from iff.symm (!add_lt_right_iff),
+  !add_neg_cancel_right ▸ H
+
+  theorem lt_add_iff_neg_add_lt : a < b + c ↔ -b + a < c :=
+  have H: a < b + c ↔ -b + a < -b + (b + c), from iff.symm (!add_lt_left_iff),
+  !neg_add_cancel_left ▸ H
+
+  theorem lt_add_iff_minus_left_lt : a < b + c ↔ a - b < c :=
+  !add_comm ▸ !lt_add_iff_neg_add_lt
+
+  theorem lt_add_iff_minus_right_lt : a < b + c ↔ a - c < b :=
+  have H: a < b + c ↔ a - c < b + c - c, from iff.symm (!add_lt_right_iff),
+  !add_neg_cancel_right ▸ H
+
+  -- TODO: the Isabelle library has varations on a + b ≤ b ↔ a ≤ 0
+
+  theorem minus_eq_imp_le_iff {a b c d : A} (H : a - b = c - d) : a ≤ b ↔ c ≤ d :=
+  calc
+    a ≤ b ↔ a - b ≤ 0 : iff.symm (minus_nonpos_iff a b)
+      ... ↔ c - d ≤ 0 : H ▸ !iff.refl
+      ... ↔ c ≤ d : minus_nonpos_iff c d
+
+  theorem minus_eq_imp_lt_iff {a b c d : A} (H : a - b = c - d) : a < b ↔ c < d :=
+  calc
+    a < b ↔ a - b < 0 : iff.symm (minus_neg_iff a b)
+      ... ↔ c - d < 0 : H ▸ !iff.refl
+      ... ↔ c < d : minus_neg_iff c d
+
+  theorem minus_le_left {a b : A} (H : a ≤ b) (c : A) : c - b ≤ c - a :=
+  add_le_left (le_imp_neg_le_neg H) c
+
+  theorem minus_le_right {a b : A} (H : a ≤ b) (c : A) : a - c ≤ b - c := add_le_right H (-c)
+
+  theorem minus_le {a b c d : A} (Hab : a ≤ b) (Hcd : c ≤ d) : a - d ≤ b - c :=
+  add_le Hab (le_imp_neg_le_neg Hcd)
+
+  theorem minus_lt_left {a b : A} (H : a < b) (c : A) : c - b < c - a :=
+  add_lt_left (lt_imp_neg_lt_neg H) c
+
+  theorem minus_lt_right {a b : A} (H : a < b) (c : A) : a - c < b - c := add_lt_right H (-c)
+
+  theorem minus_lt {a b c d : A} (Hab : a < b) (Hcd : c < d) : a - d < b - c :=
+  add_lt Hab (lt_imp_neg_lt_neg Hcd)
+
+  theorem minus_le_lt {a b c d : A} (Hab : a ≤ b) (Hcd : c < d) : a - d < b - c :=
+  add_le_lt Hab (lt_imp_neg_lt_neg Hcd)
+
+  theorem minus_lt_le {a b c d : A} (Hab : a < b) (Hcd : c ≤ d) : a - d < b - c :=
+  add_lt_le Hab (le_imp_neg_le_neg Hcd)
+
+end
+
+-- TODO: additional facts if the ordering is a linear ordering (e.g. -a = a ↔ a = 0)
+
+-- TODO: structures with abs
+
 end algebra

library/algebra/ring.lean

@@ -44,14 +44,6 @@ theorem zero_ne_one [s: zero_ne_one_class A] : 0 ≠ 1 := !zero_ne_one_class.zer
 structure semiring [class] (A : Type) extends add_comm_monoid A, monoid A, distrib A, mul_zero A,
     zero_ne_one_class A
 
-section
-
-  variable [s : semiring A]
-  include s
-
-  /- -/
-end
-
 structure comm_semiring [class] (A : Type) extends semiring A, comm_semigroup A
 
 
@@ -292,7 +284,64 @@ section
 
 end
 
-/- ring no_zero_divisors -/
 
+/- integral domains -/
+
+-- TODO: some properties here may extend to cancellative semirings. It is worth the effort?
+
+structure no_zero_divisors [class] (A : Type) extends has_mul A, has_zero A :=
+(eq_zero_or_eq_zero_of_mul_eq_zero : ∀a b, mul a b = zero → a = zero ∨ b = zero)
+
+theorem eq_zero_or_eq_zero_of_mul_eq_zero {A : Type} [s : no_zero_divisors A] {a b : A} (H : a * b = 0) :
+  a = 0 ∨ b = 0 := !no_zero_divisors.eq_zero_or_eq_zero_of_mul_eq_zero H
+
+structure integral_domain [class] (A : Type) extends comm_ring_dvd A, no_zero_divisors A
+
+section
+
+  variables [s : integral_domain A] (a b c d e : A)
+  include s
+
+  theorem mul_ne_zero_of_ne_zero_ne_zero {a b : A} (H1 : a ≠ 0) (H2 : b ≠ 0) : a * b ≠ 0 :=
+  assume H : a * b = 0,
+    or.elim (eq_zero_or_eq_zero_of_mul_eq_zero H) (assume H3, H1 H3) (assume H4, H2 H4)
+
+  theorem mul.cancel_right {a b c : A} (Ha : a ≠ 0) (H : b * a = c * a) : b = c :=
+  have H1 : b * a - c * a = 0, from iff.mp !eq_iff_minus_eq_zero H,
+  have H2 : (b - c) * a = 0, from eq.trans !minus_distrib_right H1,
+  have H3 : b - c = 0, from or.resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
+  iff.elim_right !eq_iff_minus_eq_zero H3
+
+  theorem mul.cancel_left {a b c : A} (Ha : a ≠ 0) (H : a * b = a * c) : b = c :=
+  have H1 : a * b - a * c = 0, from iff.mp !eq_iff_minus_eq_zero H,
+  have H2 : a * (b - c) = 0, from eq.trans !minus_distrib_left H1,
+  have H3 : b - c = 0, from or.resolve_right (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
+  iff.elim_right !eq_iff_minus_eq_zero H3
+
+  -- TODO: do we want the iff versions?
+
+  -- TODO: wait for simplifier
+  theorem square_eq_square_iff (a b : A) : a * a = b * b ↔ a = b ∨ a = -b := sorry
+
+  theorem square_eq_one_iff (a : A) : a * a = 1 ↔ a = 1 ∨ a = -1 := sorry
+
+  -- TODO: c - b * c → c = 0 ∨ b = 1 and variants
+
+  theorem dvd_of_mul_dvd_mul_left {a b c : A} (Ha : a ≠ 0) (Hdvd : a * b | a * c) : b | c :=
+  dvd_elim Hdvd
+    (take d,
+      assume H : a * b * d = a * c,
+      have H1 : b * d = c, from mul.cancel_left Ha (!mul_assoc ▸ H),
+      dvd_intro H1)
+
+  theorem dvd_of_mul_dvd_mul_right {a b c : A} (Ha : a ≠ 0) (Hdvd : b * a | c * a) : b | c :=
+  dvd_elim Hdvd
+    (take d,
+      assume H : b * a * d = c * a,
+      have H1 : b * d * a = c * a, from eq.trans !mul_right_comm H,
+      have H2 : b * d = c, from mul.cancel_right Ha H1,
+      dvd_intro H2)
+
+end
 
 end algebra