436 lines
17 KiB
Text
436 lines
17 KiB
Text
/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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The rational numbers as a field generated by the integers, defined as the usual quotient.
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-/
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import data.int algebra.field
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open int quot eq.ops
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record prerat : Type :=
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(num : ℤ) (denom : ℤ) (denom_pos : denom > 0)
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/-
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prerat: the representations of the rationals as integers num, denom, with denom > 0.
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note: names are not protected, because it is not expected that users will open prerat.
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-/
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namespace prerat
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/- the equivalence relation -/
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definition equiv (a b : prerat) : Prop := num a * denom b = num b * denom a
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infix `≡` := equiv
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theorem equiv.refl [refl] (a : prerat) : a ≡ a := rfl
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theorem equiv.symm [symm] {a b : prerat} (H : a ≡ b) : b ≡ a := !eq.symm H
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theorem num_eq_zero_of_equiv {a b : prerat} (H : a ≡ b) (na_zero : num a = 0) : num b = 0 :=
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have H1 : num a * denom b = 0, from !zero_mul ▸ na_zero ▸ rfl,
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have H2 : num b * denom a = 0, from H ▸ H1,
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show num b = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero H2) (ne_of_gt (denom_pos a))
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theorem num_pos_of_equiv {a b : prerat} (H : a ≡ b) (na_pos : num a > 0) : num b > 0 :=
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have H1 : num a * denom b > 0, from mul_pos na_pos (denom_pos b),
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have H2 : num b * denom a > 0, from H ▸ H1,
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show num b > 0, from pos_of_mul_pos_right H2 (le_of_lt (denom_pos a))
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theorem num_neg_of_equiv {a b : prerat} (H : a ≡ b) (na_neg : num a < 0) : num b < 0 :=
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have H1 : num a * denom b < 0, from mul_neg_of_neg_of_pos na_neg (denom_pos b),
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have H2 : -(-num b * denom a) < 0, from !neg_mul_eq_neg_mul⁻¹ ▸ !neg_neg⁻¹ ▸ H ▸ H1,
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have H3 : -num b > 0, from pos_of_mul_pos_right (pos_of_neg_neg H2) (le_of_lt (denom_pos a)),
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neg_of_neg_pos H3
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theorem equiv_of_num_eq_zero {a b : prerat} (H1 : num a = 0) (H2 : num b = 0) : a ≡ b :=
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by rewrite [↑equiv, H1, H2, *zero_mul]
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theorem equiv.trans [trans] {a b c : prerat} (H1 : a ≡ b) (H2 : b ≡ c) : a ≡ c :=
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decidable.by_cases
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(assume b0 : num b = 0,
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have a0 : num a = 0, from num_eq_zero_of_equiv (equiv.symm H1) b0,
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have c0 : num c = 0, from num_eq_zero_of_equiv H2 b0,
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equiv_of_num_eq_zero a0 c0)
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(assume bn0 : num b ≠ 0,
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have H3 : num b * denom b ≠ 0, from mul_ne_zero bn0 (ne_of_gt (denom_pos b)),
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have H4 : (num b * denom b) * (num a * denom c) = (num b * denom b) * (num c * denom a),
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from calc
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(num b * denom b) * (num a * denom c) = (num a * denom b) * (num b * denom c) :
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by rewrite [*mul.assoc, *mul.left_comm (num a), *mul.left_comm (num b)]
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... = (num b * denom a) * (num b * denom c) : {H1}
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... = (num b * denom a) * (num c * denom b) : {H2}
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... = (num b * denom b) * (num c * denom a) :
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by rewrite [*mul.assoc, *mul.left_comm (denom a),
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*mul.left_comm (denom b), mul.comm (denom a)],
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mul.cancel_left H3 H4)
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theorem equiv.is_equivalence : equivalence equiv :=
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mk_equivalence equiv equiv.refl @equiv.symm @equiv.trans
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definition setoid : setoid prerat :=
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setoid.mk equiv equiv.is_equivalence
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/- field operations -/
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theorem of_nat_succ_pos (n : nat) : of_nat (nat.succ n) > 0 :=
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of_nat_pos !nat.succ_pos
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definition of_int (i : int) : prerat := prerat.mk i 1 !of_nat_succ_pos
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definition zero : prerat := of_int 0
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definition one : prerat := of_int 1
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private theorem mul_denom_pos (a b : prerat) : denom a * denom b > 0 :=
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mul_pos (denom_pos a) (denom_pos b)
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definition add (a b : prerat) : prerat :=
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prerat.mk (num a * denom b + num b * denom a) (denom a * denom b) (mul_denom_pos a b)
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definition mul (a b : prerat) : prerat :=
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prerat.mk (num a * num b) (denom a * denom b) (mul_denom_pos a b)
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definition neg (a : prerat) : prerat :=
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prerat.mk (- num a) (denom a) (denom_pos a)
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theorem of_int_add (a b : ℤ) : of_int (#int a + b) ≡ add (of_int a) (of_int b) :=
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by esimp [equiv, num, denom, one, add, of_int]; rewrite [*int.mul_one]
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theorem of_int_mul (a b : ℤ) : of_int (#int a * b) ≡ mul (of_int a) (of_int b) :=
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!equiv.refl
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theorem of_int_neg (a : ℤ) : of_int (#int -a) ≡ neg (of_int a) :=
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!equiv.refl
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definition inv : prerat → prerat
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| inv (prerat.mk nat.zero d dp) := zero
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| inv (prerat.mk (nat.succ n) d dp) := prerat.mk d (nat.succ n) !of_nat_succ_pos
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| inv (prerat.mk -[n +1] d dp) := prerat.mk (-d) (nat.succ n) !of_nat_succ_pos
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theorem equiv_zero_of_num_eq_zero {a : prerat} (H : num a = 0) : a ≡ zero :=
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by rewrite [↑equiv, H, ↑zero, ↑num, ↑of_int, *zero_mul]
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theorem num_eq_zero_of_equiv_zero {a : prerat} : a ≡ zero → num a = 0 :=
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by rewrite [↑equiv, ↑zero, ↑of_int, mul_one, zero_mul]; intro H; exact H
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theorem inv_zero {d : int} (dp : d > 0) : inv (mk nat.zero d dp) = zero :=
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begin rewrite [↑inv, ▸*] end
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theorem inv_zero' : inv zero = zero := inv_zero (of_nat_succ_pos nat.zero)
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theorem inv_of_pos {n d : int} (np : n > 0) (dp : d > 0) : inv (mk n d dp) ≡ mk d n np :=
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obtain (n' : nat) (Hn' : n = of_nat n'), from exists_eq_of_nat (le_of_lt np),
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have H1 : (#nat n' > nat.zero), from lt_of_of_nat_lt_of_nat (Hn' ▸ np),
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obtain (k : nat) (Hk : n' = nat.succ k), from nat.exists_eq_succ_of_lt H1,
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have H2 : d * n = d * nat.succ k, by rewrite [Hn', Hk],
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Hn'⁻¹ ▸ (Hk⁻¹ ▸ H2)
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theorem inv_neg {n d : int} (np : n > 0) (dp : d > 0) : inv (mk (-n) d dp) ≡ mk (-d) n np :=
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obtain (n' : nat) (Hn' : n = of_nat n'), from exists_eq_of_nat (le_of_lt np),
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have H1 : (#nat n' > nat.zero), from lt_of_of_nat_lt_of_nat (Hn' ▸ np),
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obtain (k : nat) (Hk : n' = nat.succ k), from nat.exists_eq_succ_of_lt H1,
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have H2 : -d * n = -d * nat.succ k, by rewrite [Hn', Hk],
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have H3 : inv (mk -[k +1] d dp) ≡ mk (-d) n np, from H2,
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have H4 : -[k +1] = -n, from calc
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-[k +1] = -(nat.succ k) : rfl
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... = -n : by rewrite [Hk⁻¹, Hn'],
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H4 ▸ H3
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theorem inv_of_neg {n d : int} (nn : n < 0) (dp : d > 0) :
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inv (mk n d dp) ≡ mk (-d) (-n) (neg_pos_of_neg nn) :=
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have H : inv (mk (-(-n)) d dp) ≡ mk (-d) (-n) (neg_pos_of_neg nn),
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from inv_neg (neg_pos_of_neg nn) dp,
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!neg_neg ▸ H
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/- operations respect equiv -/
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theorem add_equiv_add {a1 b1 a2 b2 : prerat} (eqv1 : a1 ≡ a2) (eqv2 : b1 ≡ b2) :
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add a1 b1 ≡ add a2 b2 :=
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calc
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(num a1 * denom b1 + num b1 * denom a1) * (denom a2 * denom b2)
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= num a1 * denom a2 * denom b1 * denom b2 + num b1 * denom b2 * denom a1 * denom a2 :
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by rewrite [mul.right_distrib, *mul.assoc, mul.left_comm (denom b1),
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mul.comm (denom b2), *mul.assoc]
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... = num a2 * denom a1 * denom b1 * denom b2 + num b2 * denom b1 * denom a1 * denom a2 :
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by rewrite [↑equiv at *, eqv1, eqv2]
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... = (num a2 * denom b2 + num b2 * denom a2) * (denom a1 * denom b1) :
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by rewrite [mul.right_distrib, *mul.assoc, *mul.left_comm (denom b2),
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*mul.comm (denom b1), *mul.assoc, mul.left_comm (denom a2)]
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theorem mul_equiv_mul {a1 b1 a2 b2 : prerat} (eqv1 : a1 ≡ a2) (eqv2 : b1 ≡ b2) :
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mul a1 b1 ≡ mul a2 b2 :=
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calc
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(num a1 * num b1) * (denom a2 * denom b2)
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= (num a1 * denom a2) * (num b1 * denom b2) : by rewrite [*mul.assoc, mul.left_comm (num b1)]
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... = (num a2 * denom a1) * (num b2 * denom b1) : by rewrite [↑equiv at *, eqv1, eqv2]
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... = (num a2 * num b2) * (denom a1 * denom b1) : by rewrite [*mul.assoc, mul.left_comm (num b2)]
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theorem neg_equiv_neg {a b : prerat} (eqv : a ≡ b) : neg a ≡ neg b :=
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calc
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-num a * denom b = -(num a * denom b) : neg_mul_eq_neg_mul
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... = -(num b * denom a) : {eqv}
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... = -num b * denom a : neg_mul_eq_neg_mul
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theorem inv_equiv_inv : ∀{a b : prerat}, a ≡ b → inv a ≡ inv b
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| (mk an ad adp) (mk bn bd bdp) :=
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assume H,
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lt.by_cases
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(assume an_neg : an < 0,
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have bn_neg : bn < 0, from num_neg_of_equiv H an_neg,
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calc
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inv (mk an ad adp) ≡ mk (-ad) (-an) (neg_pos_of_neg an_neg) : inv_of_neg an_neg adp
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... ≡ mk (-bd) (-bn) (neg_pos_of_neg bn_neg) :
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by rewrite [↑equiv at *, ▸*, *neg_mul_neg, mul.comm ad, mul.comm bd, H]
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... ≡ inv (mk bn bd bdp) : (inv_of_neg bn_neg bdp)⁻¹)
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(assume an_zero : an = 0,
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have bn_zero : bn = 0, from num_eq_zero_of_equiv H an_zero,
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eq.subst (calc
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inv (mk an ad adp) = inv (mk 0 ad adp) : {an_zero}
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... = zero : inv_zero
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... = inv (mk 0 bd bdp) : inv_zero
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... = inv (mk bn bd bdp) : bn_zero) !equiv.refl)
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(assume an_pos : an > 0,
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have bn_pos : bn > 0, from num_pos_of_equiv H an_pos,
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calc
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inv (mk an ad adp) ≡ mk ad an an_pos : inv_of_pos an_pos adp
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... ≡ mk bd bn bn_pos :
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by rewrite [↑equiv at *, ▸*, mul.comm ad, mul.comm bd, H]
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... ≡ inv (mk bn bd bdp) : (inv_of_pos bn_pos bdp)⁻¹)
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/- properties -/
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theorem add.comm (a b : prerat) : add a b ≡ add b a :=
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by rewrite [↑add, ↑equiv, ▸*, add.comm, mul.comm (denom a)]
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theorem add.assoc (a b c : prerat) : add (add a b) c ≡ add a (add b c) :=
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by rewrite [↑add, ↑equiv, ▸*, *(mul.comm (num c)), *(λy, mul.comm y (denom a)), *mul.left_distrib,
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*mul.right_distrib, *mul.assoc, *add.assoc]
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theorem add_zero (a : prerat) : add a zero ≡ a :=
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by rewrite [↑add, ↑equiv, ↑zero, ↑of_int, ▸*, *mul_one, zero_mul, add_zero]
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theorem add.left_inv (a : prerat) : add (neg a) a ≡ zero :=
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by rewrite [↑add, ↑equiv, ↑neg, ↑zero, ↑of_int, ▸*, -neg_mul_eq_neg_mul, add.left_inv, *zero_mul]
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theorem mul.comm (a b : prerat) : mul a b ≡ mul b a :=
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by rewrite [↑mul, ↑equiv, mul.comm (num a), mul.comm (denom a)]
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theorem mul.assoc (a b c : prerat) : mul (mul a b) c ≡ mul a (mul b c) :=
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by rewrite [↑mul, ↑equiv, *mul.assoc]
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theorem mul_one (a : prerat) : mul a one ≡ a :=
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by rewrite [↑mul, ↑one, ↑of_int, ↑equiv, ▸*, *mul_one]
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-- with the simplifier this will be easy
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theorem mul.left_distrib (a b c : prerat) : mul a (add b c) ≡ add (mul a b) (mul a c) :=
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begin
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rewrite [↑mul, ↑add, ↑equiv, ▸*, *mul.left_distrib, *mul.right_distrib, -*int.mul.assoc],
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apply sorry
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end
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theorem mul_inv_cancel : ∀{a : prerat}, ¬ a ≡ zero → mul a (inv a) ≡ one
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| (mk an ad adp) :=
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assume H,
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let a := mk an ad adp in
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lt.by_cases
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(assume an_neg : an < 0,
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let ia := mk (-ad) (-an) (neg_pos_of_neg an_neg) in
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calc
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mul a (inv a) ≡ mul a ia : mul_equiv_mul !equiv.refl (inv_of_neg an_neg adp)
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... ≡ one : begin
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esimp [equiv, num, denom, one, mul, of_int],
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rewrite [*int.mul_one, *int.one_mul, int.mul.comm,
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neg_mul_comm]
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end)
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(assume an_zero : an = 0, absurd (equiv_zero_of_num_eq_zero an_zero) H)
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(assume an_pos : an > 0,
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let ia := mk ad an an_pos in
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calc
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mul a (inv a) ≡ mul a ia : mul_equiv_mul !equiv.refl (inv_of_pos an_pos adp)
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... ≡ one : begin
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esimp [equiv, num, denom, one, mul, of_int],
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rewrite [*int.mul_one, *int.one_mul, int.mul.comm]
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end)
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theorem zero_not_equiv_one : ¬ zero ≡ one :=
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begin
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esimp [equiv, zero, one, of_int],
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rewrite [zero_mul, int.mul_one],
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exact zero_ne_one
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end
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end prerat
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/-
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the rationals
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-/
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definition rat : Type.{1} := quot prerat.setoid
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notation `ℚ` := rat
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local attribute prerat.setoid [instance]
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namespace rat
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/- operations -/
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-- these coercions do not work: rat is not an inductive type
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definition of_int [coercion] (i : ℤ) : ℚ := ⟦prerat.of_int i⟧
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definition of_nat [coercion] (n : ℕ) : ℚ := ⟦prerat.of_int n⟧
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definition of_num [coercion] [reducible] (n : num) : ℚ := of_int (int.of_num n)
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definition add : ℚ → ℚ → ℚ :=
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quot.lift₂
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(λa b : prerat, ⟦prerat.add a b⟧)
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(take a1 a2 b1 b2, assume H1 H2, quot.sound (prerat.add_equiv_add H1 H2))
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definition mul : ℚ → ℚ → ℚ :=
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quot.lift₂
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(λa b : prerat, ⟦prerat.mul a b⟧)
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(take a1 a2 b1 b2, assume H1 H2, quot.sound (prerat.mul_equiv_mul H1 H2))
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definition neg : ℚ → ℚ :=
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quot.lift
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(λa : prerat, ⟦prerat.neg a⟧)
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(take a1 a2, assume H, quot.sound (prerat.neg_equiv_neg H))
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definition inv : ℚ → ℚ :=
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quot.lift
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(λa : prerat, ⟦prerat.inv a⟧)
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(take a1 a2, assume H, quot.sound (prerat.inv_equiv_inv H))
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definition zero := ⟦prerat.zero⟧
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definition one := ⟦prerat.one⟧
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infix `+` := rat.add
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infix `*` := rat.mul
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prefix `-` := rat.neg
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postfix `⁻¹` := rat.inv
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definition sub [reducible] (a b : rat) : rat := a + (-b)
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infix `-` := rat.sub
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-- TODO: this is a workaround, since the coercions from numerals do not work
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notation 0 := zero
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notation 1 := one
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/- properties -/
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theorem of_int_add (a b : ℤ) : of_int (#int a + b) = of_int a + of_int b :=
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quot.sound (prerat.of_int_add a b)
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theorem of_int_mul (a b : ℤ) : of_int (#int a * b) = of_int a * of_int b :=
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quot.sound (prerat.of_int_mul a b)
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theorem of_int_neg (a : ℤ) : of_int (#int -a) = -(of_int a) :=
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quot.sound (prerat.of_int_neg a)
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theorem of_int_sub (a b : ℤ) : of_int (#int a - b) = of_int a - of_int b :=
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calc
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of_int (#int a - b) = of_int a + of_int (#int -b) : of_int_add
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... = of_int a - of_int b : {of_int_neg b}
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theorem of_nat_eq (a : ℕ) : of_nat a = of_int (int.of_nat a) := rfl
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theorem of_nat_add (a b : ℕ) : of_nat (#nat a + b) = of_nat a + of_nat b :=
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by rewrite [*of_nat_eq, int.of_nat_add, rat.of_int_add]
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theorem of_nat_mul (a b : ℕ) : of_nat (#nat a * b) = of_nat a * of_nat b :=
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by rewrite [*of_nat_eq, int.of_nat_mul, rat.of_int_mul]
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theorem of_nat_sub {a b : ℕ} (H : #nat a ≥ b) : of_nat (#nat a - b) = of_nat a - of_nat b :=
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by rewrite [*of_nat_eq, int.of_nat_sub H, rat.of_int_sub]
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theorem add.comm (a b : ℚ) : a + b = b + a :=
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quot.induction_on₂ a b (take u v, quot.sound !prerat.add.comm)
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theorem add.assoc (a b c : ℚ) : a + b + c = a + (b + c) :=
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quot.induction_on₃ a b c (take u v w, quot.sound !prerat.add.assoc)
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theorem add_zero (a : ℚ) : a + 0 = a :=
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quot.induction_on a (take u, quot.sound !prerat.add_zero)
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theorem zero_add (a : ℚ) : 0 + a = a := !add.comm ▸ !add_zero
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||
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theorem add.left_inv (a : ℚ) : -a + a = 0 :=
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quot.induction_on a (take u, quot.sound !prerat.add.left_inv)
|
||
|
||
theorem mul.comm (a b : ℚ) : a * b = b * a :=
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quot.induction_on₂ a b (take u v, quot.sound !prerat.mul.comm)
|
||
|
||
theorem mul.assoc (a b c : ℚ) : a * b * c = a * (b * c) :=
|
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quot.induction_on₃ a b c (take u v w, quot.sound !prerat.mul.assoc)
|
||
|
||
theorem mul_one (a : ℚ) : a * 1 = a :=
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quot.induction_on a (take u, quot.sound !prerat.mul_one)
|
||
|
||
theorem one_mul (a : ℚ) : 1 * a = a := !mul.comm ▸ !mul_one
|
||
|
||
theorem mul.left_distrib (a b c : ℚ) : a * (b + c) = a * b + a * c :=
|
||
quot.induction_on₃ a b c (take u v w, quot.sound !prerat.mul.left_distrib)
|
||
|
||
theorem mul.right_distrib (a b c : ℚ) : (a + b) * c = a * c + b * c :=
|
||
by rewrite [mul.comm, mul.left_distrib, *mul.comm c]
|
||
|
||
theorem mul_inv_cancel {a : ℚ} : a ≠ 0 → a * a⁻¹ = 1 :=
|
||
quot.induction_on a
|
||
(take u,
|
||
assume H,
|
||
quot.sound (!prerat.mul_inv_cancel (assume H1, H (quot.sound H1))))
|
||
|
||
theorem inv_mul_cancel {a : ℚ} (H : a ≠ 0) : a⁻¹ * a = 1 :=
|
||
!mul.comm ▸ mul_inv_cancel H
|
||
|
||
theorem zero_ne_one : (#rat 0 ≠ 1) :=
|
||
assume H, prerat.zero_not_equiv_one (quot.exact H)
|
||
|
||
definition has_decidable_eq [instance] : decidable_eq ℚ :=
|
||
take a b, quot.rec_on_subsingleton₂ a b
|
||
(take u v,
|
||
if H : prerat.num u * prerat.denom v = prerat.num v * prerat.denom u
|
||
then decidable.inl (quot.sound H)
|
||
else decidable.inr (assume H1, H (quot.exact H1)))
|
||
|
||
theorem inv_zero : inv 0 = 0 :=
|
||
quot.sound (prerat.inv_zero' ▸ !prerat.equiv.refl)
|
||
|
||
section migrate_algebra
|
||
open [classes] algebra
|
||
|
||
protected definition discrete_field [reducible] : algebra.discrete_field rat :=
|
||
⦃algebra.discrete_field,
|
||
add := add,
|
||
add_assoc := add.assoc,
|
||
zero := 0,
|
||
zero_add := zero_add,
|
||
add_zero := add_zero,
|
||
neg := neg,
|
||
add_left_inv := add.left_inv,
|
||
add_comm := add.comm,
|
||
mul := mul,
|
||
mul_assoc := mul.assoc,
|
||
one := (of_num 1),
|
||
one_mul := one_mul,
|
||
mul_one := mul_one,
|
||
left_distrib := mul.left_distrib,
|
||
right_distrib := mul.right_distrib,
|
||
mul_comm := mul.comm,
|
||
mul_inv_cancel := @mul_inv_cancel,
|
||
inv_mul_cancel := @inv_mul_cancel,
|
||
zero_ne_one := zero_ne_one,
|
||
inv_zero := inv_zero,
|
||
has_decidable_eq := has_decidable_eq⦄
|
||
|
||
local attribute rat.discrete_field [instance]
|
||
definition divide (a b : rat) := algebra.divide a b
|
||
infix `/` := divide
|
||
definition dvd (a b : rat) := algebra.dvd a b
|
||
|
||
migrate from algebra with rat
|
||
replacing sub → rat.sub, divide → divide, dvd → dvd
|
||
|
||
end migrate_algebra
|
||
end rat
|