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feat/refactor(data/real/complete): add another archimedean property, rename theorems

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This commit is contained in:
Jeremy Avigad 2015-09-12 20:35:43 -04:00
parent d9e166f77f
commit 8db9afbf1c
2 changed files with 63 additions and 47 deletions
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103
library/data/real/complete.lean
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@ -212,14 +212,15 @@ theorem approx_spec' (x : ℝ) (n : ℕ+) : abs ((of_rat (approx x n)) - x) ≤
notation `r_seq` := ℕ+ → ℝ
noncomputable definition converges_to (X : r_seq) (a : ℝ) (N : ℕ+ → ℕ+) :=
noncomputable definition converges_to_with_rate (X : r_seq) (a : ℝ) (N : ℕ+ → ℕ+) :=
∀ k : ℕ+, ∀ n : ℕ+, n ≥ N k → abs (X n - a) ≤ of_rat k⁻¹
noncomputable definition cauchy (X : r_seq) (M : ℕ+ → ℕ+) :=
noncomputable definition cauchy_with_rate (X : r_seq) (M : ℕ+ → ℕ+) :=
∀ k : ℕ+, ∀ m n : ℕ+, m ≥ M k → n ≥ M k → abs (X m - X n) ≤ of_rat k⁻¹
theorem cauchy_of_converges_to {X : r_seq} {a : ℝ} {N : ℕ+ → ℕ+} (Hc : converges_to X a N) :
cauchy X (λ k, N (2 * k)) :=
theorem cauchy_with_rate_of_converges_to_with_rate {X : r_seq} {a : ℝ} {N : ℕ+ → ℕ+}
(Hc : converges_to_with_rate X a N) :
cauchy_with_rate X (λ k, N (2 * k)) :=
begin
intro k m n Hm Hn,
rewrite (rewrite_helper9 a),
@ -247,7 +248,7 @@ private theorem Nb_spec_left (M : ℕ+ → ℕ+) (k : ℕ+) : 3 * k ≤ Nb M k :
section lim_seq
parameter {X : r_seq}
parameter {M : ℕ+ → ℕ+}
hypothesis Hc : cauchy X M
hypothesis Hc : cauchy_with_rate X M
include Hc
noncomputable definition lim_seq : ℕ+ → ℚ :=
@ -294,14 +295,16 @@ theorem lim_seq_reg : rat_seq.regular lim_seq :=
end
theorem lim_seq_spec (k : ℕ+) :
rat_seq.s_le (rat_seq.s_abs (rat_seq.sadd lim_seq (rat_seq.sneg (rat_seq.const (lim_seq k))))) (rat_seq.const k⁻¹) :=
rat_seq.s_le (rat_seq.s_abs (rat_seq.sadd lim_seq
(rat_seq.sneg (rat_seq.const (lim_seq k))))) (rat_seq.const k⁻¹) :=
by apply rat_seq.const_bound; apply lim_seq_reg
private noncomputable definition r_lim_seq : rat_seq.reg_seq :=
rat_seq.reg_seq.mk lim_seq lim_seq_reg
private theorem r_lim_seq_spec (k : ℕ+) : rat_seq.r_le
(rat_seq.r_abs ((rat_seq.radd r_lim_seq (rat_seq.rneg (rat_seq.r_const ((rat_seq.reg_seq.sq r_lim_seq) k))))))
(rat_seq.r_abs ((rat_seq.radd r_lim_seq (rat_seq.rneg
(rat_seq.r_const ((rat_seq.reg_seq.sq r_lim_seq) k))))))
(rat_seq.r_const k⁻¹) :=
lim_seq_spec k
@ -318,7 +321,7 @@ theorem lim_spec (k : ℕ+) :
abs ((of_rat (lim_seq k)) - lim) ≤ of_rat k⁻¹ :=
by rewrite abs_sub; apply lim_spec'
theorem converges_of_cauchy : converges_to X lim (Nb M) :=
theorem converges_to_with_rate_of_cauchy_with_rate : converges_to_with_rate X lim (Nb M) :=
begin
intro k n Hn,
rewrite (rewrite_helper10 (X (Nb M n)) (of_rat (lim_seq n))),
@ -419,7 +422,7 @@ theorem archimedean_lower_strict (x : ℝ) : ∃ z : ℤ, x > of_int z :=
apply iff.mp !neg_lt_iff_neg_lt Hz
end
definition ex_floor (x : ℝ) :=
private definition ex_floor (x : ℝ) :=
(@ex_largest_of_bdd (λ z, x ≥ of_int z) _
(begin
existsi some (archimedean_upper_strict x),
@ -441,58 +444,77 @@ noncomputable definition floor (x : ℝ) : ℤ :=
noncomputable definition ceil (x : ℝ) : ℤ := - floor (-x)
theorem floor_spec (x : ℝ) : of_int (floor x) ≤ x :=
theorem floor_le (x : ℝ) : floor x ≤ x :=
and.left (some_spec (ex_floor x))
theorem floor_largest {x : ℝ} {z : ℤ} (Hz : z > floor x) : x < of_int z :=
theorem lt_of_floor_lt {x : ℝ} {z : ℤ} (Hz : floor x < z) : x < z :=
begin
apply lt_of_not_ge,
cases some_spec (ex_floor x),
apply a_1 _ Hz
end
theorem ceil_spec (x : ℝ) : of_int (ceil x) ≥ x :=
theorem le_ceil (x : ℝ) : x ≤ ceil x :=
begin
rewrite [↑ceil, of_int_neg],
apply iff.mp !le_neg_iff_le_neg,
apply floor_spec
apply floor_le
end
theorem ceil_smallest {x : ℝ} {z : ℤ} (Hz : z < ceil x) : x > of_int z :=
theorem lt_of_lt_ceil {x : ℝ} {z : ℤ} (Hz : z < ceil x) : z < x :=
begin
rewrite ↑ceil at Hz,
let Hz' := floor_largest (iff.mp !int.lt_neg_iff_lt_neg Hz),
let Hz' := lt_of_floor_lt (iff.mp !int.lt_neg_iff_lt_neg Hz),
rewrite [of_int_neg at Hz'],
apply lt_of_neg_lt_neg Hz'
end
theorem floor_succ (x : ℝ) : (floor x) + 1 = floor (x + 1) :=
theorem floor_succ (x : ℝ) : floor (x + 1) = floor x + 1 :=
begin
apply by_contradiction,
intro H,
cases int.lt_or_gt_of_ne H with [Hlt, Hgt],
let Hl := floor_largest (iff.mp !int.add_lt_iff_lt_sub_right Hlt),
rewrite [of_int_sub at Hl],
apply not_le_of_gt (iff.mpr !add_lt_iff_lt_sub_right Hl) !floor_spec,
let Hl := floor_largest Hgt,
cases int.lt_or_gt_of_ne H with [Hgt, Hlt],
let Hl := lt_of_floor_lt Hgt,
rewrite [of_int_add at Hl],
apply not_le_of_gt (lt_of_add_lt_add_right Hl) !floor_spec
apply not_le_of_gt (lt_of_add_lt_add_right Hl) !floor_le,
let Hl := lt_of_floor_lt (iff.mp !int.add_lt_iff_lt_sub_right Hlt),
rewrite [of_int_sub at Hl],
apply not_le_of_gt (iff.mpr !add_lt_iff_lt_sub_right Hl) !floor_le
end
theorem floor_succ_lt (x : ℝ) : floor (x - 1) < floor x :=
theorem floor_sub_one_lt_floor (x : ℝ) : floor (x - 1) < floor x :=
begin
apply @int.lt_of_add_lt_add_right _ 1,
rewrite [floor_succ (x - 1), sub_add_cancel],
rewrite [-floor_succ (x - 1), sub_add_cancel],
apply int.lt_add_of_pos_right dec_trivial
end
theorem ceil_succ (x : ℝ) : ceil x < ceil (x + 1) :=
theorem ceil_lt_ceil_succ (x : ℝ) : ceil x < ceil (x + 1) :=
begin
rewrite [↑ceil, neg_add],
apply int.neg_lt_neg,
apply floor_succ_lt
apply floor_sub_one_lt_floor
end
section
open int
theorem archimedean_small {ε : ℝ} (H : ε > 0) : ∃ (n : ℕ), 1 / succ n < ε :=
let n := int.nat_abs (ceil (2 / ε)) in
have int.of_nat n ≥ ceil (2 / ε),
by rewrite int.of_nat_nat_abs; apply int.le_abs_self,
have int.of_nat (succ n) ≥ ceil (2 / ε),
from int.le.trans this (int.of_nat_le_of_nat_of_le !le_succ),
have H₁ : succ n ≥ ceil (2 / ε), from of_int_le_of_int_of_le this,
have H₂ : succ n ≥ 2 / ε, from !le.trans !le_ceil H₁,
have H₃ : 2 / ε > 0, from div_pos_of_pos_of_pos two_pos H,
have 1 / succ n < ε, from calc
1 / succ n ≤ 1 / (2 / ε) : div_le_div_of_le H₃ H₂
... = ε / 2 : one_div_div
... < ε : div_two_lt_of_pos H,
exists.intro n this
end
end ints
--------------------------------------------------
-- supremum property
@ -517,10 +539,10 @@ definition rpt {A : Type} (op : A → A) : ℕ → A → A
definition ub (x : ℝ) := ∀ y : ℝ, X y → y ≤ x
definition sup (x : ℝ) := ub x ∧ ∀ y : ℝ, ub y → x ≤ y
definition is_sup (x : ℝ) := ub x ∧ ∀ y : ℝ, ub y → x ≤ y
definition lb (x : ℝ) := ∀ y : ℝ, X y → x ≤ y
definition inf (x : ℝ) := lb x ∧ ∀ y : ℝ, lb y → y ≤ x
definition is_inf (x : ℝ) := lb x ∧ ∀ y : ℝ, lb y → y ≤ x
parameter elt : ℝ
hypothesis inh : X elt
@ -529,15 +551,6 @@ hypothesis bdd : ub bound
include inh bdd
-- this should exist somewhere, no? I can't find it
theorem not_forall_of_exists_not {A : Type} {P : A → Prop} (H : ∃ a : A, ¬ P a) :
¬ ∀ a : A, P a :=
begin
intro Hall,
cases H with [a, Ha],
apply Ha (Hall a)
end
definition avg (a b : ℚ) := a / 2 + b / 2
noncomputable definition bisect (ab : ℚ × ℚ) :=
@ -551,9 +564,9 @@ noncomputable definition under : ℚ := rat.of_int (floor (elt - 1))
theorem under_spec1 : of_rat under < elt :=
have H : of_rat under < of_int (floor elt), begin
apply of_int_lt_of_int_of_lt,
apply floor_succ_lt
apply floor_sub_one_lt_floor
end,
lt_of_lt_of_le H !floor_spec
lt_of_lt_of_le H !floor_le
theorem under_spec : ¬ ub under :=
begin
@ -571,9 +584,9 @@ noncomputable definition over : ℚ := rat.of_int (ceil (bound + 1)) -- b
theorem over_spec1 : bound < of_rat over :=
have H : of_int (ceil bound) < of_rat over, begin
apply of_int_lt_of_int_of_lt,
apply ceil_succ
apply ceil_lt_ceil_succ
end,
lt_of_le_of_lt !ceil_spec H
lt_of_le_of_lt !le_ceil H
theorem over_spec : ub over :=
begin
@ -922,15 +935,15 @@ theorem under_over_equiv : p_under_seq ≡ p_over_seq :=
theorem under_over_eq : sup_under = sup_over := quot.sound under_over_equiv
theorem ex_sup_of_inh_of_bdd : ∃ x : ℝ, sup x :=
theorem exists_is_sup_of_inh_of_bdd : ∃ x : ℝ, is_sup x :=
exists.intro sup_over (and.intro over_bound (under_over_eq ▸ under_lowest_bound))
end supremum
definition bounding_set (X : ℝ → Prop) (x : ℝ) : Prop := ∀ y : ℝ, X y → x ≤ y
theorem ex_inf_of_inh_of_bdd (X : ℝ → Prop) (elt : ℝ) (inh : X elt) (bound : ℝ)
(bdd : lb X bound) : ∃ x : ℝ, inf X x :=
theorem exists_is_inf_of_inh_of_bdd (X : ℝ → Prop) (elt : ℝ) (inh : X elt) (bound : ℝ)
(bdd : lb X bound) : ∃ x : ℝ, is_inf X x :=
begin
have Hinh : bounding_set X bound, begin
intros y Hy,
@ -942,7 +955,7 @@ theorem ex_inf_of_inh_of_bdd (X : ℝ → Prop) (elt : ℝ) (inh : X elt) (bound
apply Hy,
apply inh
end,
cases ex_sup_of_inh_of_bdd _ _ Hinh _ Hub with [supr, Hsupr],
cases exists_is_sup_of_inh_of_bdd _ _ Hinh _ Hub with [supr, Hsupr],
existsi supr,
cases Hsupr with [Hubs1, Hubs2],
apply and.intro,

7
library/logic/quantifiers.lean
View file

@ -14,16 +14,19 @@ iff.intro (λ e x H, e (exists.intro x H)) Exists.rec
theorem forall_iff_not_exists {A : Type} {P : A → Prop} : (¬ ∃ a : A, P a) ↔ ∀ a : A, ¬ P a :=
exists_imp_distrib
theorem not_forall_not_of_exists {A : Type} {p : A → Prop} (H : ∃x, p x) : ¬ ∀ x, ¬ p x :=
theorem not_forall_not_of_exists {A : Type} {p : A → Prop} (H : ∃ x, p x) : ¬ ∀ x, ¬ p x :=
assume H1 : ∀ x, ¬ p x,
obtain (w : A) (Hw : p w), from H,
absurd Hw (H1 w)
theorem not_exists_not_of_forall {A : Type} {p : A → Prop} (H2 : ∀x, p x) : ¬ ∃ x, ¬p x :=
theorem not_exists_not_of_forall {A : Type} {p : A → Prop} (H2 : ∀ x, p x) : ¬ ∃ x, ¬p x :=
assume H1 : ∃ x, ¬ p x,
obtain (w : A) (Hw : ¬ p w), from H1,
absurd (H2 w) Hw
theorem not_forall_of_exists_not {A : Type} {P : A → Prop} (H : ∃ a : A, ¬ P a) : ¬ ∀ a : A, P a :=
assume H', not_exists_not_of_forall H' H
theorem forall_congr {A : Type} {φ ψ : A → Prop} : (∀ x, φ x ↔ ψ x) → ((∀ x, φ x) ↔ (∀ x, ψ x)) :=
forall_iff_forall