refactor(library/algebra): factor out proofs from coercions
Coercions/instances should be simple definitions
This commit is contained in:
Leonardo de Moura
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d0e9e0e21c
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7149c49553
4 changed files with 106 additions and 116 deletions
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@ -251,23 +251,23 @@ section group
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theorem mul_eq_iff_eq_mul_inv (a b c : A) : a * b = c ↔ a = c * b⁻¹ :=
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iff.intro eq_mul_inv_of_mul_eq mul_eq_of_eq_mul_inv
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theorem mul_left_cancel {a b c : A} (H : a * b = a * c) : b = c :=
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calc b = a⁻¹ * (a * b) : inv_mul_cancel_left
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... = a⁻¹ * (a * c) : H
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... = c : inv_mul_cancel_left
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theorem mul_right_cancel {a b c : A} (H : a * b = c * b) : a = c :=
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calc a = (a * b) * b⁻¹ : mul_inv_cancel_right
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... = (c * b) * b⁻¹ : H
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... = c : mul_inv_cancel_right
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definition group.to_left_cancel_semigroup [instance] [coercion] : left_cancel_semigroup A :=
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⦃ left_cancel_semigroup, s,
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mul_left_cancel := take a b c,
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assume H : a * b = a * c,
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calc
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b = a⁻¹ * (a * b) : inv_mul_cancel_left
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... = a⁻¹ * (a * c) : H
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... = c : inv_mul_cancel_left⦄
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mul_left_cancel := @mul_left_cancel A s ⦄
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definition group.to_right_cancel_semigroup [instance] [coercion] : right_cancel_semigroup A :=
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⦃ right_cancel_semigroup, s,
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mul_right_cancel := take a b c,
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assume H : a * b = c * b,
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calc
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a = (a * b) * b⁻¹ : mul_inv_cancel_right
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... = (c * b) * b⁻¹ : H
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... = c : mul_inv_cancel_right⦄
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mul_right_cancel := @mul_right_cancel A s ⦄
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end group
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@ -378,25 +378,25 @@ section add_group
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theorem add_eq_iff_eq_add_neg (a b c : A) : a + b = c ↔ a = c + -b :=
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iff.intro eq_add_neg_of_add_eq add_eq_of_eq_add_neg
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theorem add_left_cancel {a b c : A} (H : a + b = a + c) : b = c :=
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calc b = -a + (a + b) : !neg_add_cancel_left⁻¹
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... = -a + (a + c) : H
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... = c : neg_add_cancel_left
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theorem add_right_cancel {a b c : A} (H : a + b = c + b) : a = c :=
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calc a = (a + b) + -b : !add_neg_cancel_right⁻¹
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... = (c + b) + -b : H
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... = c : add_neg_cancel_right
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definition add_group.to_left_cancel_semigroup [instance] [coercion] :
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add_left_cancel_semigroup A :=
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⦃ add_left_cancel_semigroup, s,
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add_left_cancel := take a b c,
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assume H : a + b = a + c,
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calc
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b = -a + (a + b) : !neg_add_cancel_left⁻¹
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... = -a + (a + c) : H
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... = c : neg_add_cancel_left ⦄
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add_left_cancel := @add_left_cancel A s ⦄
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definition add_group.to_add_right_cancel_semigroup [instance] [coercion] :
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add_right_cancel_semigroup A :=
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⦃ add_right_cancel_semigroup, s,
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add_right_cancel := take a b c,
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assume H : a + b = c + b,
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calc
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a = (a + b) + -b : !add_neg_cancel_right⁻¹
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... = (c + b) + -b : H
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... = c : add_neg_cancel_right ⦄
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add_right_cancel := @add_right_cancel A s ⦄
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/- sub -/
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@ -199,18 +199,16 @@ end
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structure ordered_comm_group [class] (A : Type) extends add_comm_group A, order_pair A :=
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(add_le_add_left : ∀a b, le a b → ∀c, le (add c a) (add c b))
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theorem ordered_comm_group.le_of_add_le_add_left [s : ordered_comm_group A] {a b c : A} (H : a + b ≤ a + c) : b ≤ c :=
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have H' : -a + (a + b) ≤ -a + (a + c), from ordered_comm_group.add_le_add_left _ _ H _,
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!neg_add_cancel_left ▸ !neg_add_cancel_left ▸ H'
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definition ordered_comm_group.to_ordered_cancel_comm_monoid [instance] [coercion]
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[s : ordered_comm_group A] : ordered_cancel_comm_monoid A :=
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⦃ ordered_cancel_comm_monoid, s,
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add_left_cancel := @add.left_cancel _ _,
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add_right_cancel := @add.right_cancel _ _,
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le_of_add_le_add_left :=
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proof
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take a b c : A,
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assume H : a + b ≤ a + c,
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have H' : -a + (a + b) ≤ -a + (a + c), from ordered_comm_group.add_le_add_left _ _ H _,
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!neg_add_cancel_left ▸ !neg_add_cancel_left ▸ H'
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qed ⦄
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add_left_cancel := @add.left_cancel A s,
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add_right_cancel := @add.right_cancel A s,
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le_of_add_le_add_left := @ordered_comm_group.le_of_add_le_add_left A s ⦄
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section
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variables [s : ordered_comm_group A] (a b c d e : A)
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@ -144,51 +144,42 @@ structure ordered_ring [class] (A : Type) extends ring A, ordered_comm_group A :
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(mul_nonneg : ∀a b, le zero a → le zero b → le zero (mul a b))
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(mul_pos : ∀a b, lt zero a → lt zero b → lt zero (mul a b))
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theorem ordered_ring.mul_le_mul_of_nonneg_left [s : ordered_ring A] {a b c : A}
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(Hab : a ≤ b) (Hc : 0 ≤ c) : c * a ≤ c * b :=
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have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
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have H2 : 0 ≤ c * (b - a), from ordered_ring.mul_nonneg _ _ Hc H1,
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iff.mp !sub_nonneg_iff_le (!mul_sub_left_distrib ▸ H2)
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theorem ordered_ring.mul_le_mul_of_nonneg_right [s : ordered_ring A] {a b c : A}
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(Hab : a ≤ b) (Hc : 0 ≤ c) : a * c ≤ b * c :=
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have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
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have H2 : 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg _ _ H1 Hc,
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iff.mp !sub_nonneg_iff_le (!mul_sub_right_distrib ▸ H2)
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theorem ordered_ring.mul_lt_mul_of_pos_left [s : ordered_ring A] {a b c : A}
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(Hab : a < b) (Hc : 0 < c) : c * a < c * b :=
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have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
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have H2 : 0 < c * (b - a), from ordered_ring.mul_pos _ _ Hc H1,
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iff.mp !sub_pos_iff_lt (!mul_sub_left_distrib ▸ H2)
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theorem ordered_ring.mul_lt_mul_of_pos_right [s : ordered_ring A] {a b c : A}
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(Hab : a < b) (Hc : 0 < c) : a * c < b * c :=
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have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
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have H2 : 0 < (b - a) * c, from ordered_ring.mul_pos _ _ H1 Hc,
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iff.mp !sub_pos_iff_lt (!mul_sub_right_distrib ▸ H2)
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definition ordered_ring.to_ordered_semiring [instance] [coercion] [s : ordered_ring A] :
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ordered_semiring A :=
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⦃ ordered_semiring, s,
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mul_zero := mul_zero,
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zero_mul := zero_mul,
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add_left_cancel := @add.left_cancel A _,
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add_right_cancel := @add.right_cancel A _,
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le_of_add_le_add_left := @le_of_add_le_add_left A _,
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mul_le_mul_of_nonneg_left := take a b c,
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assume Hab : a ≤ b,
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assume Hc : 0 ≤ c,
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show c * a ≤ c * b,
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proof
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have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
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have H2 : 0 ≤ c * (b - a), from ordered_ring.mul_nonneg _ _ Hc H1,
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iff.mp !sub_nonneg_iff_le (!mul_sub_left_distrib ▸ H2)
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qed,
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mul_le_mul_of_nonneg_right := take a b c,
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assume Hab : a ≤ b,
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assume Hc : 0 ≤ c,
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show a * c ≤ b * c,
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proof
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have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab,
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have H2 : 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg _ _ H1 Hc,
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iff.mp !sub_nonneg_iff_le (!mul_sub_right_distrib ▸ H2)
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qed,
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mul_lt_mul_of_pos_left := take a b c,
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assume Hab : a < b,
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assume Hc : 0 < c,
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show c * a < c * b,
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proof
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have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
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have H2 : 0 < c * (b - a), from ordered_ring.mul_pos _ _ Hc H1,
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iff.mp !sub_pos_iff_lt (!mul_sub_left_distrib ▸ H2)
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qed,
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mul_lt_mul_of_pos_right := take a b c,
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assume Hab : a < b,
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assume Hc : 0 < c,
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show a * c < b * c,
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proof
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have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab,
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have H2 : 0 < (b - a) * c, from ordered_ring.mul_pos _ _ H1 Hc,
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iff.mp !sub_pos_iff_lt (!mul_sub_right_distrib ▸ H2)
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qed
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⦄
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mul_zero := mul_zero,
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zero_mul := zero_mul,
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add_left_cancel := @add.left_cancel A s,
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add_right_cancel := @add.right_cancel A s,
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le_of_add_le_add_left := @le_of_add_le_add_left A s,
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mul_le_mul_of_nonneg_left := @ordered_ring.mul_le_mul_of_nonneg_left A s,
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mul_le_mul_of_nonneg_right := @ordered_ring.mul_le_mul_of_nonneg_right A s,
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mul_lt_mul_of_pos_left := @ordered_ring.mul_lt_mul_of_pos_left A s,
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mul_lt_mul_of_pos_right := @ordered_ring.mul_lt_mul_of_pos_right A s ⦄
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section
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variable [s : ordered_ring A]
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@ -246,39 +237,38 @@ definition linear_ordered_ring.to_linear_ordered_semiring [instance] [coercion]
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⦃ linear_ordered_semiring, s,
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mul_zero := mul_zero,
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zero_mul := zero_mul,
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add_left_cancel := @add.left_cancel A _,
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add_right_cancel := @add.right_cancel A _,
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le_of_add_le_add_left := @le_of_add_le_add_left A _,
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mul_le_mul_of_nonneg_left := @mul_le_mul_of_nonneg_left A _,
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mul_le_mul_of_nonneg_right := @mul_le_mul_of_nonneg_right A _,
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mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left A _,
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mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right A _,
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le_total := linear_ordered_ring.le_total
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⦄
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add_left_cancel := @add.left_cancel A s,
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add_right_cancel := @add.right_cancel A s,
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le_of_add_le_add_left := @le_of_add_le_add_left A s,
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mul_le_mul_of_nonneg_left := @mul_le_mul_of_nonneg_left A s,
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mul_le_mul_of_nonneg_right := @mul_le_mul_of_nonneg_right A s,
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mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left A s,
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mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right A s,
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le_total := linear_ordered_ring.le_total ⦄
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structure linear_ordered_comm_ring [class] (A : Type) extends linear_ordered_ring A, comm_monoid A
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theorem linear_ordered_comm_ring.eq_zero_or_eq_zero_of_mul_eq_zero [s : linear_ordered_comm_ring A]
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{a b : A} (H : a * b = 0) : a = 0 ∨ b = 0 :=
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lt.by_cases
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(assume Ha : 0 < a,
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lt.by_cases
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(assume Hb : 0 < b, absurd (H ▸ show 0 < a * b, from mul_pos Ha Hb) (lt.irrefl 0))
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(assume Hb : 0 = b, or.inr (Hb⁻¹))
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(assume Hb : 0 > b, absurd (H ▸ show 0 > a * b, from mul_neg_of_pos_of_neg Ha Hb) (lt.irrefl 0)))
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(assume Ha : 0 = a, or.inl (Ha⁻¹))
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(assume Ha : 0 > a,
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lt.by_cases
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(assume Hb : 0 < b, absurd (H ▸ show 0 > a * b, from mul_neg_of_neg_of_pos Ha Hb) (lt.irrefl 0))
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(assume Hb : 0 = b, or.inr (Hb⁻¹))
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(assume Hb : 0 > b, absurd (H ▸ show 0 < a * b, from mul_pos_of_neg_of_neg Ha Hb) (lt.irrefl 0)))
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-- Linearity implies no zero divisors. Doesn't need commutativity.
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definition linear_ordered_comm_ring.to_integral_domain [instance] [coercion]
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[s: linear_ordered_comm_ring A] :
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integral_domain A :=
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[s: linear_ordered_comm_ring A] : integral_domain A :=
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⦃ integral_domain, s,
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eq_zero_or_eq_zero_of_mul_eq_zero := take a b,
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assume H : a * b = 0,
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show a = 0 ∨ b = 0, from
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lt.by_cases
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(assume Ha : 0 < a,
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lt.by_cases
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(assume Hb : 0 < b, absurd (H ▸ mul_pos Ha Hb) (lt.irrefl 0))
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(assume Hb : 0 = b, or.inr (Hb⁻¹))
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(assume Hb : 0 > b, absurd (H ▸ mul_neg_of_pos_of_neg Ha Hb) (lt.irrefl 0)))
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(assume Ha : 0 = a, or.inl (Ha⁻¹))
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(assume Ha : 0 > a,
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lt.by_cases
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(assume Hb : 0 < b, absurd (H ▸ mul_neg_of_neg_of_pos Ha Hb) (lt.irrefl 0))
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(assume Hb : 0 = b, or.inr (Hb⁻¹))
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(assume Hb : 0 > b, absurd (H ▸ mul_pos_of_neg_of_neg Ha Hb) (lt.irrefl 0)))
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⦄
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eq_zero_or_eq_zero_of_mul_eq_zero :=
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@linear_ordered_comm_ring.eq_zero_or_eq_zero_of_mul_eq_zero A s ⦄
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section
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variable [s : linear_ordered_ring A]
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@ -152,22 +152,24 @@ end comm_semiring
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structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A, zero_ne_one_class A
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theorem ring.mul_zero [s : ring A] (a : A) : a * 0 = 0 :=
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have H : a * 0 + 0 = a * 0 + a * 0, from calc
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a * 0 + 0 = a * 0 : add_zero
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... = a * (0 + 0) : {(add_zero 0)⁻¹}
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... = a * 0 + a * 0 : ring.left_distrib,
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show a * 0 = 0, from (add.left_cancel H)⁻¹
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theorem ring.zero_mul [s : ring A] (a : A) : 0 * a = 0 :=
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have H : 0 * a + 0 = 0 * a + 0 * a, from calc
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0 * a + 0 = 0 * a : add_zero
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... = (0 + 0) * a : {(add_zero 0)⁻¹}
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... = 0 * a + 0 * a : ring.right_distrib,
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show 0 * a = 0, from (add.left_cancel H)⁻¹
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definition ring.to_semiring [instance] [coercion] [s : ring A] : semiring A :=
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⦃ semiring, s,
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mul_zero := take a,
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have H : a * 0 + 0 = a * 0 + a * 0, from
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calc
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a * 0 + 0 = a * 0 : add_zero
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... = a * (0 + 0) : {(add_zero 0)⁻¹}
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... = a * 0 + a * 0 : ring.left_distrib,
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show a * 0 = 0, from (add.left_cancel H)⁻¹,
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zero_mul := take a,
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have H : 0 * a + 0 = 0 * a + 0 * a, from
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calc
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0 * a + 0 = 0 * a : add_zero
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... = (0 + 0) * a : {(add_zero 0)⁻¹}
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... = 0 * a + 0 * a : ring.right_distrib,
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show 0 * a = 0, from (add.left_cancel H)⁻¹ ⦄
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mul_zero := ring.mul_zero,
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zero_mul := ring.zero_mul ⦄
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section
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variables [s : ring A] (a b c d e : A)
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