feat(library/algebra/group): add rules for sub equalities
This commit is contained in:
Jeremy Avigad
Leonardo de Moura
parent
96b54a8007
commit
a607e7dd57
2 changed files with 57 additions and 28 deletions
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@ -208,29 +208,28 @@ section group
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... = 1 * b : H
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... = b : one_mul
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-- TODO: better names for the next eight theorems? (Also for additive ones.)
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theorem eq_mul_inv_of_mul_eq {a b c : A} (H : a * b = c) : a = c * b⁻¹ :=
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theorem eq_mul_inv_of_mul_eq {a b c : A} (H : a * c = b) : a = b * c⁻¹ :=
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H ▸ !mul_inv_cancel_right⁻¹
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theorem eq_inv_mul_of_mul_eq {a b c : A} (H : a * b = c) : b = a⁻¹ * c :=
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theorem eq_inv_mul_of_mul_eq {a b c : A} (H : b * a = c) : a = b⁻¹ * c :=
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H ▸ !inv_mul_cancel_left⁻¹
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theorem inv_mul_eq_of_eq_mul {a b c : A} (H : a = b * c) : b⁻¹ * a = c :=
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theorem inv_mul_eq_of_eq_mul {a b c : A} (H : b = a * c) : a⁻¹ * b = c :=
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H⁻¹ ▸ !inv_mul_cancel_left
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theorem mul_inv_eq_of_eq_mul {a b c : A} (H : a = b * c) : a * c⁻¹ = b :=
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theorem mul_inv_eq_of_eq_mul {a b c : A} (H : a = c * b) : a * b⁻¹ = c :=
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H⁻¹ ▸ !mul_inv_cancel_right
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theorem eq_mul_of_mul_inv_eq {a b c : A} (H : a * b⁻¹ = c) : a = c * b :=
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theorem eq_mul_of_mul_inv_eq {a b c : A} (H : a * c⁻¹ = b) : a = b * c :=
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!inv_inv ▸ (eq_mul_inv_of_mul_eq H)
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theorem eq_mul_of_inv_mul_eq {a b c : A} (H : a⁻¹ * b = c) : b = a * c :=
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theorem eq_mul_of_inv_mul_eq {a b c : A} (H : b⁻¹ * a = c) : a = b * c :=
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!inv_inv ▸ (eq_inv_mul_of_mul_eq H)
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theorem mul_eq_of_eq_inv_mul {a b c : A} (H : a = b⁻¹ * c) : b * a = c :=
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theorem mul_eq_of_eq_inv_mul {a b c : A} (H : b = a⁻¹ * c) : a * b = c :=
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!inv_inv ▸ (inv_mul_eq_of_eq_mul H)
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theorem mul_eq_of_eq_mul_inv {a b c : A} (H : a = b * c⁻¹) : a * c = b :=
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theorem mul_eq_of_eq_mul_inv {a b c : A} (H : a = c * b⁻¹) : a * b = c :=
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!inv_inv ▸ (mul_inv_eq_of_eq_mul H)
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theorem mul_eq_iff_eq_inv_mul (a b c : A) : a * b = c ↔ b = a⁻¹ * c :=
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@ -316,28 +315,29 @@ section add_group
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rewrite ⟨add.assoc, add_neg_cancel_left, add.right_inv⟩
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end
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theorem eq_add_neg_of_add_eq {a b c : A} (H : a + b = c) : a = c + -b :=
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-- TODO: delete these in favor of sub rules?
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theorem eq_add_neg_of_add_eq {a b c : A} (H : a + c = b) : a = b + -c :=
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H ▸ !add_neg_cancel_right⁻¹
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theorem eq_neg_add_of_add_eq {a b c : A} (H : a + b = c) : b = -a + c :=
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theorem eq_neg_add_of_add_eq {a b c : A} (H : b + a = c) : a = -b + c :=
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H ▸ !neg_add_cancel_left⁻¹
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theorem neg_add_eq_of_eq_add {a b c : A} (H : a = b + c) : -b + a = c :=
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theorem neg_add_eq_of_eq_add {a b c : A} (H : b = a + c) : -a + b = c :=
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H⁻¹ ▸ !neg_add_cancel_left
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theorem add_neg_eq_of_eq_add {a b c : A} (H : a = b + c) : a + -c = b :=
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theorem add_neg_eq_of_eq_add {a b c : A} (H : a = c + b) : a + -b = c :=
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H⁻¹ ▸ !add_neg_cancel_right
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theorem eq_add_of_add_neg_eq {a b c : A} (H : a + -b = c) : a = c + b :=
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theorem eq_add_of_add_neg_eq {a b c : A} (H : a + -c = b) : a = b + c :=
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!neg_neg ▸ (eq_add_neg_of_add_eq H)
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theorem eq_add_of_neg_add_eq {a b c : A} (H : -a + b = c) : b = a + c :=
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theorem eq_add_of_neg_add_eq {a b c : A} (H : -b + a = c) : a = b + c :=
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!neg_neg ▸ (eq_neg_add_of_add_eq H)
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theorem add_eq_of_eq_neg_add {a b c : A} (H : a = -b + c) : b + a = c :=
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theorem add_eq_of_eq_neg_add {a b c : A} (H : b = -a + c) : a + b = c :=
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!neg_neg ▸ (neg_add_eq_of_eq_add H)
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theorem add_eq_of_eq_add_neg {a b c : A} (H : a = b + -c) : a + c = b :=
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theorem add_eq_of_eq_add_neg {a b c : A} (H : a = c + -b) : a + b = c :=
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!neg_neg ▸ (add_neg_eq_of_eq_add H)
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theorem add_eq_iff_eq_neg_add (a b c : A) : a + b = c ↔ b = -a + c :=
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@ -420,14 +420,24 @@ section add_group
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... ↔ c - d = 0 : H ▸ !iff.refl
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... ↔ c = d : iff.symm (eq_iff_sub_eq_zero c d)
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theorem eq_sub_of_add_eq {a b c : A} (H : a + c = b) : a = b - c :=
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!eq_add_neg_of_add_eq H
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theorem sub_eq_of_eq_add {a b c : A} (H : a = c + b) : a - b = c :=
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!add_neg_eq_of_eq_add H
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theorem eq_add_of_sub_eq {a b c : A} (H : a - c = b) : a = b + c :=
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eq_add_of_add_neg_eq H
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theorem add_eq_of_eq_sub {a b c : A} (H : a = c - b) : a + b = c :=
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add_eq_of_eq_add_neg H
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end add_group
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structure add_comm_group [class] (A : Type) extends add_group A, add_comm_monoid A
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section add_comm_group
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variable [s : add_comm_group A]
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include s
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variable [s : add_comm_group A]
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include s
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theorem sub_add_eq_sub_sub (a b c : A) : a - (b + c) = a - b - c :=
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!add.comm ▸ !sub_add_eq_sub_sub_swap
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@ -444,6 +454,17 @@ include s
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theorem add_sub_add_left_eq_sub (a b c : A) : (c + a) - (c + b) = a - b :=
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by rewrite ⟨sub_add_eq_sub_sub, (add.comm c a), add_sub_cancel⟩
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theorem eq_sub_of_add_eq' {a b c : A} (H : c + a = b) : a = b - c :=
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!eq_sub_of_add_eq (!add.comm ▸ H)
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theorem sub_eq_of_eq_add' {a b c : A} (H : a = b + c) : a - b = c :=
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!sub_eq_of_eq_add (!add.comm ▸ H)
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theorem eq_add_of_sub_eq' {a b c : A} (H : a - b = c) : a = b + c :=
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!add.comm ▸ eq_add_of_sub_eq H
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theorem add_eq_of_eq_sub' {a b c : A} (H : b = c - a) : a + b = c :=
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!add.comm ▸ add_eq_of_eq_sub H
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end add_comm_group
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/- bundled structures -/
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@ -656,21 +656,21 @@ section port_algebra
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theorem add_neg_cancel_left : ∀a b : ℤ, a + (-a + b) = b := algebra.add_neg_cancel_left
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theorem add_neg_cancel_right : ∀a b : ℤ, a + b + -b = a := algebra.add_neg_cancel_right
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theorem neg_add_rev : ∀a b : ℤ, -(a + b) = -b + -a := algebra.neg_add_rev
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theorem eq_add_neg_of_add_eq : ∀{a b c : ℤ}, a + b = c → a = c + -b :=
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theorem eq_add_neg_of_add_eq : ∀{a b c : ℤ}, a + c = b → a = b + -c :=
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@algebra.eq_add_neg_of_add_eq _ _
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theorem eq_neg_add_of_add_eq : ∀{a b c : ℤ}, a + b = c → b = -a + c :=
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theorem eq_neg_add_of_add_eq : ∀{a b c : ℤ}, b + a = c → a = -b + c :=
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@algebra.eq_neg_add_of_add_eq _ _
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theorem neg_add_eq_of_eq_add : ∀{a b c : ℤ}, a = b + c → -b + a = c :=
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theorem neg_add_eq_of_eq_add : ∀{a b c : ℤ}, b = a + c → -a + b = c :=
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@algebra.neg_add_eq_of_eq_add _ _
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theorem add_neg_eq_of_eq_add : ∀{a b c : ℤ}, a = b + c → a + -c = b :=
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theorem add_neg_eq_of_eq_add : ∀{a b c : ℤ}, a = c + b → a + -b = c :=
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@algebra.add_neg_eq_of_eq_add _ _
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theorem eq_add_of_add_neg_eq : ∀{a b c : ℤ}, a + -b = c → a = c + b :=
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theorem eq_add_of_add_neg_eq : ∀{a b c : ℤ}, a + -c = b → a = b + c :=
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@algebra.eq_add_of_add_neg_eq _ _
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theorem eq_add_of_neg_add_eq : ∀{a b c : ℤ}, -a + b = c → b = a + c :=
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theorem eq_add_of_neg_add_eq : ∀{a b c : ℤ}, -b + a = c → a = b + c :=
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@algebra.eq_add_of_neg_add_eq _ _
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theorem add_eq_of_eq_neg_add : ∀{a b c : ℤ}, a = -b + c → b + a = c :=
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theorem add_eq_of_eq_neg_add : ∀{a b c : ℤ}, b = -a + c → a + b = c :=
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@algebra.add_eq_of_eq_neg_add _ _
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theorem add_eq_of_eq_add_neg : ∀{a b c : ℤ}, a = b + -c → a + c = b :=
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theorem add_eq_of_eq_add_neg : ∀{a b c : ℤ}, a = c + -b → a + b = c :=
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@algebra.add_eq_of_eq_add_neg _ _
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theorem add_eq_iff_eq_neg_add : ∀a b c : ℤ, a + b = c ↔ b = -a + c :=
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@algebra.add_eq_iff_eq_neg_add _ _
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@ -694,6 +694,10 @@ section port_algebra
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theorem eq_sub_iff_add_eq : ∀a b c : ℤ, a = b - c ↔ a + c = b := algebra.eq_sub_iff_add_eq
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theorem eq_iff_eq_of_sub_eq_sub : ∀{a b c d : ℤ}, a - b = c - d → a = b ↔ c = d :=
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@algebra.eq_iff_eq_of_sub_eq_sub _ _
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theorem eq_sub_of_add_eq : ∀{a b c : ℤ}, a + c = b → a = b - c := @algebra.eq_sub_of_add_eq _ _
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theorem sub_eq_of_eq_add : ∀{a b c : ℤ}, a = c + b → a - b = c := @algebra.sub_eq_of_eq_add _ _
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theorem eq_add_of_sub_eq : ∀{a b c : ℤ}, a - c = b → a = b + c := @algebra.eq_add_of_sub_eq _ _
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theorem add_eq_of_eq_sub : ∀{a b c : ℤ}, a = c - b → a + b = c := @algebra.add_eq_of_eq_sub _ _
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theorem sub_add_eq_sub_sub : ∀a b c : ℤ, a - (b + c) = a - b - c := algebra.sub_add_eq_sub_sub
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theorem neg_add_eq_sub : ∀a b : ℤ, -a + b = b - a := algebra.neg_add_eq_sub
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theorem neg_add : ∀a b : ℤ, -(a + b) = -a + -b := algebra.neg_add
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@ -701,6 +705,10 @@ section port_algebra
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theorem sub_sub_ : ∀a b c : ℤ, a - b - c = a - (b + c) := algebra.sub_sub
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theorem add_sub_add_left_eq_sub : ∀a b c : ℤ, (c + a) - (c + b) = a - b :=
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algebra.add_sub_add_left_eq_sub
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theorem eq_sub_of_add_eq' : ∀{a b c : ℤ}, c + a = b → a = b - c := @algebra.eq_sub_of_add_eq' _ _
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theorem sub_eq_of_eq_add' : ∀{a b c : ℤ}, a = b + c → a - b = c := @algebra.sub_eq_of_eq_add' _ _
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theorem eq_add_of_sub_eq' : ∀{a b c : ℤ}, a - b = c → a = b + c := @algebra.eq_add_of_sub_eq' _ _
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theorem add_eq_of_eq_sub' : ∀{a b c : ℤ}, b = c - a → a + b = c := @algebra.add_eq_of_eq_sub' _ _
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theorem ne_zero_of_mul_ne_zero_right : ∀{a b : ℤ}, a * b ≠ 0 → a ≠ 0 :=
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@algebra.ne_zero_of_mul_ne_zero_right _ _
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theorem ne_zero_of_mul_ne_zero_left : ∀{a b : ℤ}, a * b ≠ 0 → b ≠ 0 :=
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