feat(library/algebra/field): define discrete field

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Jeremy Avigad 2015-02-19 15:17:21 -05:00 committed by Leonardo de Moura
parent eef2e99a1c
commit a8cf58d97c

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@ -30,32 +30,26 @@ section division_ring
definition divide (a b : A) : A := a * b⁻¹ definition divide (a b : A) : A := a * b⁻¹
infix `/` := divide infix `/` := divide
-- only in this file
local attribute divide [reducible]
theorem mul_inv_cancel (H : a ≠ 0) : a * a⁻¹ = 1 := theorem mul_inv_cancel (H : a ≠ 0) : a * a⁻¹ = 1 :=
division_ring.mul_inv_cancel H division_ring.mul_inv_cancel H
theorem inv_mul_cancel (H : a ≠ 0) : a⁻¹ * a = 1 := theorem inv_mul_cancel (H : a ≠ 0) : a⁻¹ * a = 1 :=
division_ring.inv_mul_cancel H division_ring.inv_mul_cancel H
theorem inv_eq_one_div (H : a ≠ 0) : a⁻¹ = 1 / a := theorem inv_eq_one_div : a⁻¹ = 1 / a := !one_mul⁻¹
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 := theorem mul_one_div_cancel (H : a ≠ 0) : a * (1 / a) = 1 :=
calc by rewrite [-inv_eq_one_div, (mul_inv_cancel H)]
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 := theorem one_div_mul_cancel (H : a ≠ 0) : (1 / a) * a = 1 :=
calc calc
(1 / a) * a = a⁻¹ * a : inv_eq_one_div H (1 / a) * a = a⁻¹ * a : inv_eq_one_div
... = 1 : inv_mul_cancel H ... = 1 : inv_mul_cancel H
theorem div_self (H : a ≠ 0) : a / a = 1 := theorem div_self (H : a ≠ 0) : a / a = 1 := mul_inv_cancel H
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) := theorem mul_div_assoc (Hc : c ≠ 0) : (a * b) / c = a * (b / c) :=
eq.symm (calc eq.symm (calc
@ -79,8 +73,8 @@ section division_ring
theorem div_one : a / 1 = a := theorem div_one : a / 1 = a :=
calc calc
a / 1 = a * 1⁻¹ : rfl a / 1 = /- a * 1⁻¹ : rfl
... = a * 1 : inv_one_is_one ... = -/ a * 1 : inv_one_is_one
... = a : mul_one ... = a : mul_one
-- note: integral domain has a "mul_ne_zero". When we get to "field", show it is an -- note: integral domain has a "mul_ne_zero". When we get to "field", show it is an
@ -200,10 +194,10 @@ section division_ring
theorem mul_inv (Ha : a ≠ 0) (Hb : b ≠ 0) : (b * a)⁻¹ = a⁻¹ * b⁻¹ := 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, have H1 : b * a ≠ 0, from mul_ne_zero' Hb Ha,
eq.symm (calc eq.symm (calc
a⁻¹ * b⁻¹ = (1 / a) * b⁻¹ : inv_eq_one_div Ha a⁻¹ * b⁻¹ = (1 / a) * b⁻¹ : inv_eq_one_div
... = (1 / a) * (1 / b) : inv_eq_one_div Hb ... = (1 / a) * (1 / b) : inv_eq_one_div
... = (1 / (b * a)) : one_div_mul_one_div Ha Hb ... = (1 / (b * a)) : one_div_mul_one_div Ha Hb
... = (b * a)⁻¹ : inv_eq_one_div H1) ... = (b * a)⁻¹ : inv_eq_one_div)
theorem mul_div_cancel (Hb : b ≠ 0) : a * b / b = a := theorem mul_div_cancel (Hb : b ≠ 0) : a * b / b = a :=
calc calc
@ -257,7 +251,7 @@ section division_ring
end division_ring end division_ring
structure field [class] (A : Type) extends division_ring A, comm_ring A structure field [class] (A : Type) extends division_ring A, comm_ring A
section field section field
variables [s : field A] {a b c d: A} variables [s : field A] {a b c d: A}
@ -265,23 +259,23 @@ section field
-- I think of "div_cancel" has being something like a * b / b = a or a / b * b = a. The name -- 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 -- 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 := theorem div_mul_right (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), have Ha : a ≠ 0, from and.left (mul_ne_zero_imp_ne_zero H),
symm (calc symm (calc
1 / b = 1 * (1 / b) : one_mul 1 / b = 1 * (1 / b) : one_mul
... = (a * a⁻¹) * (1 / b) : mul_inv_cancel Ha ... = (a * a⁻¹) * (1 / b) : mul_inv_cancel Ha
... = a * (a⁻¹ * (1 / b)) : mul.assoc ... = a * (a⁻¹ * (1 / b)) : mul.assoc
... = a * ((1 / a) * (1 / b)) :inv_eq_one_div Ha ... = a * ((1 / a) * (1 / b)) :inv_eq_one_div
... = a * (1 / (b * a)) : one_div_mul_one_div Ha Hb ... = a * (1 / (b * a)) : one_div_mul_one_div Ha Hb
... = a * (1 / (a * b)) : mul.comm ... = a * (1 / (a * b)) : mul.comm
... = a * (a * b)⁻¹ : inv_eq_one_div H ... = a * (a * b)⁻¹ : inv_eq_one_div
... = a / (a * b) : rfl) ... = a / (a * b) : rfl)
theorem div_mul_self_right (Ha : a ≠ 0) (H : a * b ≠ 0) : b / (a * b) = 1 / a := theorem div_mul_left (Ha : a ≠ 0) (H : a * b ≠ 0) : b / (a * b) = 1 / a :=
have H1 : b * a ≠ 0, from mul_ne_zero_comm H, have H1 : b * a ≠ 0, from mul_ne_zero_comm H,
calc calc
(b / (a * b)) = (b / (b * a)) : mul.comm (b / (a * b)) = (b / (b * a)) : mul.comm
... = 1 / a : div_mul_self_left Ha H1 ... = 1 / a : div_mul_right Ha H1
theorem mul_div_cancel_left (Ha : a ≠ 0) : a * b / a = b := theorem mul_div_cancel_left (Ha : a ≠ 0) : a * b / a = b :=
calc calc
@ -296,12 +290,12 @@ section field
theorem one_div_add_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b) := 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), have H : a * b ≠ 0, from (mul_ne_zero' Ha Hb),
symm (calc symm (calc
(a + b) / (a * b) = (a + b) * (a * b)⁻¹ : rfl (a + b) / (a * b)/- = (a + b) * (a * b)⁻¹ : rfl
... = a * (a * b)⁻¹ + b * (a * b)⁻¹ : right_distrib ...-/ = a * (a * b)⁻¹ + b * (a * b)⁻¹ : right_distrib
... = a / (a * b) + b * (a * b)⁻¹ : rfl ... = 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)⁻¹ : div_mul_right Hb H
... = 1 / b + b / (a * b) : rfl ... = 1 / b + b / (a * b) : rfl
... = 1 / b + 1 / a : div_mul_self_right Ha H ... = 1 / b + 1 / a : div_mul_left Ha H
... = 1 / a + 1 / b : add.comm) ... = 1 / a + 1 / b : add.comm)
theorem div_mul_div (Hb : b ≠ 0) (Hd : d ≠ 0) : (a / b) * (c / d) = (a * c) / (b * d) := theorem div_mul_div (Hb : b ≠ 0) (Hd : d ≠ 0) : (a / b) * (c / d) = (a * c) / (b * d) :=
@ -369,4 +363,37 @@ section field
end field end field
structure discrete_field [class] (A : Type) extends field A :=
(decidable_equality : ∀x y : A, decidable (x = y))
section discrete_field
variable [s : discrete_field A]
include s
variables {a b c : A}
definition decidable_eq [instance] (a b : A) : decidable (a = b) :=
@discrete_field.decidable_equality A s a b
theorem discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero
(x y : A) (H : x * y = 0) : x = 0 y = 0 :=
decidable.by_cases
(assume H : x = 0, or.inl H)
(assume H1 : x ≠ 0,
or.inr (calc
y = 1 * y : one_mul
... = x⁻¹ * x * y : inv_mul_cancel H1
... = x⁻¹ * (x * y) : mul.assoc
... = x⁻¹ * 0 : H
... = 0 : mul_zero))
definition discrete_field.to_integral_domain [instance] [reducible] [coercion] :
integral_domain A :=
⦃ integral_domain, s,
eq_zero_or_eq_zero_of_mul_eq_zero := discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero⦄
example (H1 : a ≠ 0) (H2 : b ≠ 0) : a * b ≠ 0 :=
@mul_ne_zero A s a b H1 H2
end discrete_field
end algebra end algebra