diff --git a/library/algebra/field.lean b/library/algebra/field.lean new file mode 100644 index 000000000..41559a8df --- /dev/null +++ b/library/algebra/field.lean @@ -0,0 +1,343 @@ +/- +Copyright (c) 2014 Robert Lewis. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. + +Module: algebra.field +Authors: Robert Lewis + +Structures with multiplicative and additive components, including semirings, rings, and fields. +The development is modeled after Isabelle's library. +-/ + +import logic.eq logic.connectives data.unit data.sigma data.prod +import algebra.function algebra.binary algebra.group algebra.ring +open eq eq.ops + +namespace algebra + +variable {A : Type} + +structure division_ring [class] (A : Type) extends ring A, has_inv A := + (mul_inv_cancel : ∀{a}, a ≠ zero → mul a (inv a) = one) + (inv_mul_cancel : ∀{a}, a ≠ zero → mul (inv a) a = one) + +-- theorem div_is_mul [s : division_ring A] {a b : A} : a / b = a * b⁻¹ := rfl + +section division_ring + variables [s : division_ring A] {a b c : A} + include s + + definition divide (a b : A) : A := a * b⁻¹ + infix `/` := divide + + theorem mul_inv_cancel (H : a ≠ 0) : a * a⁻¹ = 1 := + division_ring.mul_inv_cancel H + + theorem inv_mul_cancel (H : a ≠ 0) : a⁻¹ * a = 1 := + division_ring.inv_mul_cancel H + + theorem inv_eq_one_div (H : a ≠ 0) : a⁻¹ = 1 / a := + symm (calc + 1 / a = 1 * a⁻¹ : rfl --div_is_mul _ _ H + ... = a⁻¹ : one_mul + ) + + theorem mul_one_div_cancel (H : a ≠ 0) : a * (1 / a) = 1 := + calc + a * (1 / a) = a * a⁻¹ : inv_eq_one_div H + ... = 1 : mul_inv_cancel H + + theorem one_div_mul_cancel (H : a ≠ 0) : (1 / a) * a = 1 := + calc + (1 / a) * a = a⁻¹ * a : inv_eq_one_div H + ... = 1 : inv_mul_cancel H + + theorem div_self (H : a ≠ 0) : a / a = 1 := + calc + a / a = a * a⁻¹ : rfl + ... = 1 : mul_inv_cancel H + + theorem mul_div_assoc (Hc : c ≠ 0) : (a * b) / c = a * (b / c) := + eq.symm (calc + a * (b / c) = a * (b * c⁻¹) : rfl + ... = (a * b) * c⁻¹ : mul.assoc + ... = (a * b) / c : rfl) + + theorem one_div_ne_zero (H : a ≠ 0) : 1 / a ≠ 0 := + assume H2 : 1 / a = 0, + have C1 : 0 = 1, from symm (calc + 1 = a * (1 / a) : mul_one_div_cancel H + ... = a * 0 : H2 + ... = 0 : mul_zero), + absurd C1 zero_ne_one + + -- the analogue in group is called inv_one + theorem inv_one_is_one : 1⁻¹ = 1 := + calc + 1⁻¹ = 1⁻¹ * 1 : mul_one + ... = 1 : inv_mul_cancel (ne.symm zero_ne_one) + + theorem div_one : a / 1 = a := + calc + a / 1 = a * 1⁻¹ : rfl + ... = a * 1 : inv_one_is_one + ... = a : mul_one + + -- note: integral domain has a "mul_ne_zero". When we get to "field", show it is an + -- instance of an integral domain, so we can use that theorem. + -- check @mul_ne_zero + theorem mul_ne_zero' (Ha : a ≠ 0) (Hb : b ≠ 0) : a * b ≠ 0 := + assume H : a * b = 0, + have C1 : a = 0, from (calc + a = a * 1 : mul_one + ... = a * (b * (1 / b)) : mul_one_div_cancel Hb + ... = (a * b) * (1 / b) : mul.assoc + ... = 0 * (1 / b) : H + ... = 0 : zero_mul), + absurd C1 Ha + + theorem mul_ne_zero_imp_ne_zero (H : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 := + have Ha : a ≠ 0, from + (assume Ha1 : a = 0, + have H1 : a * b = 0, from (calc + a * b = 0 * b : Ha1 + ... = 0 : zero_mul), + absurd H1 H), + have Hb : b ≠ 0, from + (assume Hb1 : b = 0, + have H1 : a * b = 0, from (calc + a * b = a * 0 : Hb1 + ... = 0 : mul_zero), + absurd H1 H), + and.intro Ha Hb + + theorem mul_ne_zero_comm (H : a * b ≠ 0) : b * a ≠ 0 := + have H2 : a ≠ 0 ∧ b ≠ 0, from mul_ne_zero_imp_ne_zero H, + mul_ne_zero' (and.right H2) (and.left H2) + +-- theorem inv_zero_imp_zero (H : a⁻¹ = 0) : a = 0 := +-- classical? + + -- make "left" and "right" versions? + theorem eq_one_div_of_mul_eq_one (H : a * b = 1) : b = 1 / a := + have H2 : a ≠ 0, from + (assume A : a = 0, + have B : 0 = 1, from symm (calc + 1 = a * b : symm H + ... = 0 * b : A + ... = 0 : zero_mul), + absurd B zero_ne_one), + show b = 1 / a, from symm (calc + 1 / a = (1 / a) * 1 : mul_one + ... = (1 / a) * (a * b) : H + ... = (1 / a) * a * b : mul.assoc + ... = 1 * b : one_div_mul_cancel H2 + ... = b : one_mul) + + -- which one is left and which is right? + theorem eq_one_div_of_mul_eq_one_left (H : b * a = 1) : b = 1 / a := + have H2 : a ≠ 0, from + (assume A : a = 0, + have B : 0 = 1, from symm (calc + 1 = b * a : symm H + ... = b * 0 : A + ... = 0 : mul_zero), + absurd B zero_ne_one), + show b = 1 / a, from symm (calc + 1 / a = 1 * (1 / a) : one_mul + ... = b * a * (1 / a) : H + ... = b * (a * (1 / a)) : mul.assoc + ... = b * 1 : mul_one_div_cancel H2 + ... = b : mul_one) + + theorem one_div_mul_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : (1 / a) * (1 / b) = 1 / (b * a) := + have H : (b * a) * ((1 / a) * (1 / b)) = 1, from (calc + (b * a) * ((1 / a) * (1 / b)) = b * (a * ((1 / a) * (1 / b))) : mul.assoc + ... = b * ((a * (1 / a)) * (1 / b)) : mul.assoc + ... = b * (1 * (1 / b)) : mul_one_div_cancel Ha + ... = b * (1 / b) : one_mul + ... = 1 : mul_one_div_cancel Hb), + eq_one_div_of_mul_eq_one H + + theorem one_div_neg_one_eq_neg_one : 1 / (-1) = -1 := + have H : (-1) * (-1) = 1, from calc + (-1) * (-1) = - (-1) : neg_eq_neg_one_mul + ... = 1 : neg_neg, + symm (eq_one_div_of_mul_eq_one H) + + -- this should be in ring + theorem mul_neg_one_eq_neg : a * (-1) = -a := + have H : a + a * -1 = 0, from calc + a + a * -1 = a * 1 + a * -1 : mul_one + ... = a * (1 + -1) : left_distrib + ... = a * 0 : add.right_inv + ... = 0 : mul_zero, + symm (neg_eq_of_add_eq_zero H) + + theorem one_div_neg_eq_neg_one_div (H : a ≠ 0) : 1 / (- a) = - (1 / a) := + have H1 : -1 ≠ 0, from + (assume H2 : -1 = 0, absurd (symm (calc + 1 = -(-1) : neg_neg + ... = -0 : H2 + ... = 0 : neg_zero)) zero_ne_one), + calc + 1 / (- a) = 1 / ((-1) * a) : neg_eq_neg_one_mul + ... = (1 / a) * (1 / (- 1)) : one_div_mul_one_div H H1 + ... = (1 / a) * (-1) : one_div_neg_one_eq_neg_one + ... = - (1 / a) : mul_neg_one_eq_neg + + theorem div_div (H : a ≠ 0) : 1 / (1 / a) = a := + symm (eq_one_div_of_mul_eq_one_left (mul_one_div_cancel H)) + + theorem eq_of_invs_eq (Ha : a ≠ 0) (Hb : b ≠ 0) (H : 1 / a = 1 / b) : a = b := + calc + a = 1 / (1 / a) : div_div Ha + ... = 1 / (1 / b) : H + ... = b : div_div Hb + + -- oops, the analogous theorem in group is called inv_mul, but it *should* be called + -- mul_inv, in which case, we will have to rename this one + theorem mul_inv (Ha : a ≠ 0) (Hb : b ≠ 0) : (b * a)⁻¹ = a⁻¹ * b⁻¹ := + have H1 : b * a ≠ 0, from mul_ne_zero' Hb Ha, + eq.symm (calc + a⁻¹ * b⁻¹ = (1 / a) * b⁻¹ : inv_eq_one_div Ha + ... = (1 / a) * (1 / b) : inv_eq_one_div Hb + ... = (1 / (b * a)) : one_div_mul_one_div Ha Hb + ... = (b * a)⁻¹ : inv_eq_one_div H1) + + theorem mul_div_cancel (Hb : b ≠ 0) : a * b / b = a := + calc + (a * b) / b = a * b * b⁻¹ : rfl + ... = a * (b * b⁻¹) : mul.assoc + ... = a * 1 : mul_inv_cancel Hb + ... = a : mul_one + + theorem div_mul_cancel (Hb : b ≠ 0) : a / b * b = a := + calc + (a / b) * b = (a * b⁻¹) * b : rfl + ... = a * (b⁻¹ * b) : mul.assoc + ... = a * 1 : inv_mul_cancel Hb + ... = a : mul_one + + theorem div_add_div_same (Hc : c ≠ 0) : a / c + b / c = (a + b) / c := + calc + (a / c) + (b / c) = (a * c⁻¹) + (b / c) : rfl + ... = a * c⁻¹ + b * c⁻¹ : rfl + ... = (a + b) * c⁻¹ : right_distrib + ... = (a + b) / c : rfl + +end division_ring + +structure field [class] (A : Type) extends division_ring A, comm_ring A + +section field + variables [s : field A] {a b c d: A} + include s + + -- I think of "div_cancel" has being something like a * b / b = a or a / b * b = a. The name + -- I chose is clunky, but it has the right prefix + theorem div_mul_self_left (Hb : b ≠ 0) (H : a * b ≠ 0) : a / (a * b) = 1 / b := + have Ha : a ≠ 0, from and.left (mul_ne_zero_imp_ne_zero H), + symm (calc + 1 / b = 1 * (1 / b) : one_mul + ... = (a * a⁻¹) * (1 / b) : mul_inv_cancel Ha + ... = a * (a⁻¹ * (1 / b)) : mul.assoc + ... = a * ((1 / a) * (1 / b)) :inv_eq_one_div Ha + ... = a * (1 / (b * a)) : one_div_mul_one_div Ha Hb + ... = a * (1 / (a * b)) : mul.comm + ... = a * (a * b)⁻¹ : inv_eq_one_div H + ... = a / (a * b) : rfl) + + theorem div_mul_self_right (Ha : a ≠ 0) (H : a * b ≠ 0) : b / (a * b) = 1 / a := + have H1 : b * a ≠ 0, from mul_ne_zero_comm H, + calc + (b / (a * b)) = (b / (b * a)) : mul.comm + ... = 1 / a : div_mul_self_left Ha H1 + + theorem mul_div_cancel_left (Ha : a ≠ 0) : a * b / a = b := + calc + (a * b) / a = (b * a) / a : mul.comm + ... = b : mul_div_cancel Ha + + theorem mul_div_cancel' (Hb : b ≠ 0) : b * (a / b) = a := + calc + b * (a / b) = a / b * b : mul.comm + ... = a : div_mul_cancel Hb + + theorem one_div_add_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b) := + have H : a * b ≠ 0, from (mul_ne_zero' Ha Hb), + symm (calc + (a + b) / (a * b) = (a + b) * (a * b)⁻¹ : rfl + ... = a * (a * b)⁻¹ + b * (a * b)⁻¹ : right_distrib + ... = a / (a * b) + b * (a * b)⁻¹ : rfl + ... = 1 / b + b * (a * b)⁻¹ : div_mul_self_left Hb H + ... = 1 / b + b / (a * b) : rfl + ... = 1 / b + 1 / a : div_mul_self_right Ha H + ... = 1 / a + 1 / b : add.comm) + + theorem div_mul_div (Hb : b ≠ 0) (Hd : d ≠ 0) : (a / b) * (c / d) = (a * c) / (b * d) := + calc + (a / b) * (c / d) = (a * b⁻¹) * (c / d) : rfl + ... = (a * b⁻¹) * (c * d⁻¹) : rfl + ... = (a * c) * (d⁻¹ * b⁻¹) : by rewrite [2 mul.assoc, (mul.comm b⁻¹), mul.assoc] + ... = (a * c) * (b * d)⁻¹ : mul_inv Hd Hb + ... = (a * c) / (b * d) : rfl + + theorem mul_div_mul_left (Hb : b ≠ 0) (Hc : c ≠ 0) : (c * a) / (c * b) = a / b := + have H : c * b ≠ 0, from mul_ne_zero' Hc Hb, + calc + (c * a) / (c * b) = (c / c) * (a / b) : div_mul_div Hc Hb + ... = 1 * (a / b) : div_self Hc + ... = a / b : one_mul + + theorem mul_div_mul_right (Hb : b ≠ 0) (Hc : c ≠ 0) : (a * c) / (b * c) = a / b := + calc + (a * c) / (b * c) = (c * a) / (b * c) : mul.comm + ... = (c * a) / (c * b) : mul.comm + ... = a / b : mul_div_mul_left Hb Hc + + theorem div_mul_eq_mul_div (Hc : c ≠ 0) : (b / c) * a = (b * a) / c := + calc + (b / c) * a = (b * c⁻¹) * a : rfl + ... = (b * a) * c⁻¹ : by rewrite [mul.assoc, (mul.comm c⁻¹), -mul.assoc ] + ... = (b * a) / c : rfl + + -- this one is odd -- I am not sure what to call it, but again, the prefix is right + theorem div_mul_eq_mul_div_comm (Hc : c ≠ 0) : (b / c) * a = b * (a / c) := + calc + (b / c) * a = (b * a) / c : div_mul_eq_mul_div Hc + ... = (b * a) / (1 * c) : one_mul + ... = (b / 1) * (a / c) : div_mul_div (ne.symm zero_ne_one) Hc + ... = b * (a / c) : div_one + + theorem div_add_div (Hb : b ≠ 0) (Hd : d ≠ 0) : + (a / b) + (c / d) = ((a * d) + (b * c)) / (b * d) := + have H : b * d ≠ 0, from mul_ne_zero' Hb Hd, + calc + a / b + c / d = (a * d) / (b * d) + c / d : mul_div_mul_right Hb Hd + ... = (a * d) / (b * d) + (b * c) / (b * d) : mul_div_mul_left Hd Hb + ... = ((a * d) + (b * c)) / (b * d) : div_add_div_same H + + + theorem div_sub_div (Hb : b ≠ 0) (Hd : d ≠ 0) : + (a / b) - (c / d) = ((a * d) - (b * c)) / (b * d) := + calc + (a / b) - (c / d) = (a / b) + -1 * (c / d) : neg_eq_neg_one_mul + ... = (a / b) + ((-1 * c) / d) : mul_div_assoc Hd + ... = ((a * d) + (b * (-1 * c))) / (b * d) : div_add_div Hb Hd + ... = ((a * d) + -1 * (b * c)) / (b * d) : by rewrite [-mul.assoc, (mul.comm b), mul.assoc] + ... = ((a * d) + -(b * c)) / (b * d) : neg_eq_neg_one_mul + + + theorem mul_eq_mul_of_div_eq_div (Hb : b ≠ 0) (Hd : d ≠ 0) (H : a / b = c / d) : a * d = c * b := + calc + a * d = a * 1 * d : by rewrite [-mul_one, mul.assoc, (mul.comm d), -mul.assoc] + ... = (a * (b / b)) * d : div_self Hb + ... = ((a / b) * b) * d : div_mul_eq_mul_div_comm Hb + ... = ((c / d) * b) * d : H + ... = ((c * b) / d) * d : div_mul_eq_mul_div Hd + ... = c * b : div_mul_cancel Hd + +end field + +end algebra