feat(library/algebra/ordered_ring): add theorems used for rational upper bounds
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Rob Lewis
Leonardo de Moura
parent
cf7c85e5fd
commit
1f4765e30a
4 changed files with 36 additions and 160 deletions
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@ -422,6 +422,13 @@ section
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theorem zero_gt_neg_one : -1 < (0:A) :=
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neg_zero ▸ (neg_lt_neg zero_lt_one)
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theorem le_of_mul_le_of_ge_one {a b c : A} (H : a * c ≤ b) (Hb : b ≥ 0) (Hc : c ≥ 1) : a ≤ b :=
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have H' : a * c ≤ b * c, from calc
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a * c ≤ b : H
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... = b * 1 : mul_one
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... ≤ b * c : mul_le_mul_of_nonneg_left Hc Hb,
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le_of_mul_le_mul_right H' (lt_of_lt_of_le zero_lt_one Hc)
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end
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/- TODO: Isabelle's library has all kinds of cancelation rules for the simplifier.
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@ -14,6 +14,7 @@ Basic types:
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* [fin](fin.lean) : finite ordinals
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* [int](int/int.md) : the integers
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* [rat](rat/rat.md) : the rationals
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* [pnat](pnat.lean) : the positive natural numbers
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Constructors:
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@ -180,8 +180,8 @@ by rewrite [*of_nat_eq, propext !of_int_le_of_int]; apply int.of_nat_le_of_nat
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theorem of_nat_pos (a : ℕ) : (of_nat a > 0) ↔ (#nat a > nat.zero) :=
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!of_nat_lt_of_nat
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theorem of_nat_nonneg (a : ℕ) : (of_nat a ≥ 0) ↔ (#nat a ≥ nat.zero) :=
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!of_nat_le_of_nat
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theorem of_nat_nonneg (a : ℕ) : (of_nat a ≥ 0) :=
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iff.mp' !of_nat_le_of_nat !nat.zero_le
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theorem le.refl (a : ℚ) : a ≤ a :=
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by rewrite [↑rat.le, sub_self]; apply nonneg_zero
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@ -11,166 +11,13 @@ To do:
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o Rename things and possibly make theorems private
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-/
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import data.nat data.rat.order data.nat.pnat
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import data.nat data.rat.order data.pnat
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open nat eq eq.ops pnat
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open -[coercions] rat
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local notation 0 := rat.of_num 0
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local notation 1 := rat.of_num 1
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----------------------------------------------------------------------------------------------------
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-----------------------------------------------
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-- positive naturals
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/-inductive pnat : Type :=
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pos : Π n : nat, n > 0 → pnat
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notation `ℕ+` := pnat
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definition nat_of_pnat (p : pnat) : ℕ :=
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pnat.rec_on p (λ n H, n)
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local postfix `~` : std.prec.max_plus := nat_of_pnat
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theorem nat_of_pnat_pos (p : pnat) : nat_of_pnat p > 0 :=
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pnat.rec_on p (λ n H, H)
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definition add (p q : pnat) : pnat :=
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pnat.pos (p~ + q~) (nat.add_pos (nat_of_pnat_pos p) (nat_of_pnat_pos q))
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infix `+` := add
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definition mul (p q : pnat) : pnat :=
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pnat.pos (p~ * q~) (nat.mul_pos (nat_of_pnat_pos p) (nat_of_pnat_pos q))
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infix `*` := mul
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definition le (p q : pnat) := p~ ≤ q~
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infix `≤` := le
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notation p `≥` q := q ≤ p
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definition lt (p q : pnat) := p~ < q~
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infix `<` := lt
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definition pnat_le_decidable [instance] (p q : pnat) : decidable (p ≤ q) :=
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pnat.rec_on p (λ n H, pnat.rec_on q
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(λ m H2, if Hl : n ≤ m then decidable.inl Hl else decidable.inr Hl))
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definition pnat_lt_decidable [instance] {p q : pnat} : decidable (p < q) :=
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pnat.rec_on p (λ n H, pnat.rec_on q
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(λ m H2, if Hl : n < m then decidable.inl Hl else decidable.inr Hl))
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theorem ple.trans {p q r : pnat} (H1 : p ≤ q) (H2 : q ≤ r) : p ≤ r := nat.le.trans H1 H2
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definition max (p q : pnat) :=
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pnat.pos (nat.max (p~) (q~)) (nat.lt_of_lt_of_le (!nat_of_pnat_pos) (!le_max_right))
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theorem max_right (a b : ℕ+) : max a b ≥ b := !le_max_right
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theorem max_left (a b : ℕ+) : max a b ≥ a := !le_max_left
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theorem max_eq_right {a b : ℕ+} (H : a < b) : max a b = b := sorry -- nat.max_eq_right H
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theorem max_eq_left {a b : ℕ+} (H : ¬ a < b) : max a b = a := sorry
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theorem pnat.le_of_lt {a b : ℕ+} (H : a < b) : a ≤ b := nat.le_of_lt H
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theorem pnat.not_lt_of_le {a b : ℕ+} (H : a ≤ b) : ¬ (b < a) := nat.not_lt_of_ge H
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theorem pnat.le_of_not_lt {a b : ℕ+} (H : ¬ a < b) : b ≤ a := nat.le_of_not_gt H
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theorem pnat.eq_of_le_of_ge {a b : ℕ+} (H1 : a ≤ b) (H2 : b ≤ a) : a = b := sorry
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theorem pnat.le.refl (a : ℕ+) : a ≤ a := !nat.le.refl
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notation 2 := pnat.pos 2 dec_trivial
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notation 3 := pnat.pos 3 dec_trivial
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definition pone : pnat := pnat.pos 1 dec_trivial
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definition pnat.to_rat [reducible] (n : ℕ+) : ℚ :=
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pnat.rec_on n (λ n H, of_nat n)
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-- these will come in rat
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theorem rat_of_nat_nonneg (n : ℕ) : 0 ≤ of_nat n := sorry
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theorem rat_of_nat_is_pos (n : ℕ) (Hn : n > 0) : of_nat n > 0 := sorry
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theorem rat_of_nat_ge_one (n : ℕ) : n ≥ 1 → of_nat n ≥ 1 := sorry
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theorem ge_one_of_pos {n : ℕ} (Hn : n > 0) : n ≥ 1 := succ_le_of_lt Hn
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theorem rat_of_pnat_ge_one (n : ℕ+) : pnat.to_rat n ≥ 1 :=
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pnat.rec_on n (λ m h, rat_of_nat_ge_one m (ge_one_of_pos h))
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theorem rat_of_pnat_is_pos (n : ℕ+) : pnat.to_rat n > 0 :=
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pnat.rec_on n (λ m h, rat_of_nat_is_pos (m) h)
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-- not used, except maybe in following thm
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theorem nat_le_to_rat_le {m n : ℕ} (H : m ≤ n) : of_nat m ≤ of_nat n := sorry
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theorem pnat_le_to_rat_le {m n : ℕ+} (H : m ≤ n) : pnat.to_rat m ≤ pnat.to_rat n := sorry
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definition inv (n : ℕ+) : ℚ := (1 : ℚ) / pnat.to_rat n
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postfix `⁻¹` := inv
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theorem inv_pos (n : ℕ+) : n⁻¹ > 0 := div_pos_of_pos !rat_of_pnat_is_pos
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theorem inv_le_one (n : ℕ+) : n⁻¹ ≤ (1 : ℚ) := sorry
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theorem inv_lt_one_of_gt {n : ℕ+} (H : n~ > 1) : n⁻¹ < (1 : ℚ) := sorry
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theorem pone_inv : pone⁻¹ = 1 := rfl -- ? Why is this rfl?
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theorem add_invs_nonneg (m n : ℕ+) : 0 ≤ m⁻¹ + n⁻¹ :=
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begin
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apply rat.le_of_lt,
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apply rat.add_pos,
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repeat apply inv_pos,
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end
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theorem half_shrink_strong (n : ℕ+) : (2 * n)⁻¹ < n⁻¹ := sorry
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theorem half_shrink (n : ℕ+) : (2 * n)⁻¹ ≤ n⁻¹ := le_of_lt !half_shrink_strong
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theorem inv_ge_of_le {p q : ℕ+} (H : p ≤ q) : q⁻¹ ≤ p⁻¹ := sorry
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theorem ge_of_inv_le {p q : ℕ+} (H : p⁻¹ ≤ q⁻¹) : q < p := sorry
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theorem padd_halves (p : ℕ+) : (2 * p)⁻¹ + (2 * p)⁻¹ = p⁻¹ := sorry
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theorem p_add_fractions (n : ℕ+) : (2 * n)⁻¹ + (2 * 3 * n)⁻¹ + (3 * n)⁻¹ = n⁻¹ := sorry
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theorem add_halves_double (m n : ℕ+) :
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m⁻¹ + n⁻¹ = ((2 * m)⁻¹ + (2 * n)⁻¹) + ((2 * m)⁻¹ + (2 * n)⁻¹) :=
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have simp [visible] : ∀ a b : ℚ, (a + a) + (b + b) = (a + b) + (a + b), from sorry,
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by rewrite [-padd_halves m, -padd_halves n, simp]
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theorem pnat_div_helper {p q : ℕ+} : (p * q)⁻¹ = p⁻¹ * q⁻¹ := sorry
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theorem inv_mul_le_inv (p q : ℕ+) : (p * q)⁻¹ ≤ q⁻¹ :=
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begin
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rewrite [pnat_div_helper, -{q⁻¹}rat.one_mul at {2}],
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apply rat.mul_le_mul,
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apply inv_le_one,
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apply rat.le.refl,
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apply rat.le_of_lt,
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apply inv_pos,
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apply rat.le_of_lt rat.zero_lt_one
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end
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theorem pnat_mul_le_mul_left' (a b c : ℕ+) (H : a ≤ b) : c * a ≤ c * b := sorry
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theorem pnat_mul_assoc (a b c : ℕ+) : a * b * c = a * (b * c) := sorry
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theorem pnat_mul_comm (a b : ℕ+) : a * b = b * a := sorry
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theorem pnat_add_assoc (a b c : ℕ+) : a + b + c = a + (b + c) := sorry
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theorem s_mul_assoc_lemma_3 (a b n : ℕ+) (p : ℚ) :
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p * ((a * n)⁻¹ + (b * n)⁻¹) = p * (a⁻¹ + b⁻¹) * n⁻¹ := sorry
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theorem pnat.mul_le_mul_left (p q : ℕ+) : q ≤ p * q := sorry
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theorem pnat.mul_le_mul_right (p q : ℕ+) : p ≤ p * q := sorry
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theorem one_lt_two : pone < 2 := sorry
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theorem pnat.lt_of_not_le {p q : ℕ+} (H : ¬ p ≤ q) : q < p := sorry
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theorem pnat.inv_cancel (p : ℕ+) : pnat.to_rat p * p⁻¹ = (1 : ℚ) := sorry
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theorem pnat.inv_cancel_right (p : ℕ+) : p⁻¹ * pnat.to_rat p = (1 : ℚ) := sorry-/
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-------------------------------------
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-- theorems to add to (ordered) field and/or rat
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@ -188,9 +35,30 @@ theorem find_midpoint {a b : ℚ} (H : a > b) : ∃ c : ℚ, a > b + c :=
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from div_lt_div_of_lt_of_pos H2 two_pos,
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by rewrite [div_two at H3, -div_add_div_same at H3, div_two at H3]; exact H3)
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constant ceil : ℚ → ℕ
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definition ceil : ℚ → ℕ := λ a, int.nat_abs (num a) + 1
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theorem ceil_ge (a : ℚ) : of_nat (ceil a) ≥ a := sorry
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theorem rat_of_nat_abs (z : ℤ) : abs (of_int z) = of_nat (int.nat_abs z) :=
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have simp [visible] : ∀ n : ℕ, of_int (int.neg_succ_of_nat n) = - of_nat (n + 1), from λ n, rfl,
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int.induction_on z
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(take a, abs_of_nonneg (!of_nat_nonneg))
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(take a, by rewrite [simp, abs_neg, abs_of_nonneg (!of_nat_nonneg)])
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theorem ceil_ge (a : ℚ) : of_nat (ceil a) ≥ a :=
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have H : abs a * abs (of_int (denom a)) = abs (of_int (num a)), from !abs_mul ▸ !mul_denom ▸ rfl,
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have H'' : 1 ≤ abs (of_int (denom a)), begin
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have J : of_int (denom a) > 0, from (iff.mp' !of_int_pos) !denom_pos,
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rewrite (abs_of_pos J),
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apply iff.mp' !of_int_le_of_int,
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apply denom_pos
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end,
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have H' : abs a ≤ abs (of_int (num a)), from
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le_of_mul_le_of_ge_one (H ▸ !le.refl) !abs_nonneg H'',
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calc
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a ≤ abs a : le_abs_self
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... ≤ abs (of_int (num a)) : H'
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... ≤ abs (of_int (num a)) + 1 : rat.le_add_of_nonneg_right trivial
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... = of_nat (int.nat_abs (num a)) + 1 : rat_of_nat_abs
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... = of_nat (int.nat_abs (num a) + 1) : of_nat_add
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theorem add_sub_comm (a b c d : ℚ) : a + b - (c + d) = (a - c) + (b - d) := sorry
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@ -200,7 +68,7 @@ theorem distrib_three_right (a b c d : ℚ) : (a + b + c) * d = a * d + b * d +
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theorem mul_le_mul_of_mul_div_le (a b c d : ℚ) : a * (b / c) ≤ d → b * a ≤ d * c := sorry
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definition pceil (a : ℚ) : ℕ+ := pnat.pos (ceil a + 1) (sorry)
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definition pceil (a : ℚ) : ℕ+ := pnat.pos (ceil a) (add_pos_right dec_trivial _)
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theorem pceil_helper {a : ℚ} {n : ℕ+} (H : pceil a ≤ n) : n⁻¹ ≤ 1 / a := sorry
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@ -216,7 +84,7 @@ theorem s_mul_assoc_lemma_4 {n : ℕ+} {ε q : ℚ} (Hε : ε > 0) (Hq : q > 0)
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apply mul_le_mul_of_mul_div_le,
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assumption
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end
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exit
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-------------------------------------
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-- small helper lemmas
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