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refactor(library/algebra/ordered_field): rename theorems to maintain consistency with other parts of the library

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This commit is contained in:
Jeremy Avigad 2015-08-27 10:24:48 -04:00 committed by Leonardo de Moura
parent 92af727daf
commit 93c2d1e9d0
4 changed files with 85 additions and 90 deletions
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141
library/algebra/ordered_field.lean
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@ -33,10 +33,10 @@ section linear_ordered_field
... = a * a⁻¹ : mul_inv_cancel (ne_of_lt H)
... = a * (1 / a) : inv_eq_one_div
theorem div_pos_of_pos (H : 0 < a) : 0 < 1 / a :=
theorem one_div_pos_of_pos (H : 0 < a) : 0 < 1 / a :=
lt_of_mul_lt_mul_left (mul_zero_lt_mul_inv_of_pos H) (le_of_lt H)
theorem div_neg_of_neg (H : a < 0) : 1 / a < 0 :=
theorem one_div_neg_of_neg (H : a < 0) : 1 / a < 0 :=
gt_of_mul_lt_mul_neg_left (mul_zero_lt_mul_inv_of_neg H) (le_of_lt H)
@ -55,7 +55,7 @@ section linear_ordered_field
... ≤ b * (a / b) : le_mul_of_ge_one_right (le_of_lt Hb) H
... = a : mul_div_cancel' Hb')
(assume H : b ≤ a,
have Hbinv : 1 / b > 0, from div_pos_of_pos Hb, calc
have Hbinv : 1 / b > 0, from one_div_pos_of_pos Hb, calc
1 = b * (1 / b) : mul_one_div_cancel Hb'
... ≤ a * (1 / b) : mul_le_mul_of_nonneg_right H (le_of_lt Hbinv)
... = a / b : div_eq_mul_one_div)
@ -74,7 +74,7 @@ section linear_ordered_field
b < b * (a / b) : lt_mul_of_gt_one_right Hb H
... = a : mul_div_cancel' Hb')
(assume H : b < a,
have Hbinv : 1 / b > 0, from div_pos_of_pos Hb, calc
have Hbinv : 1 / b > 0, from one_div_pos_of_pos Hb, calc
1 = b * (1 / b) : mul_one_div_cancel Hb'
... < a * (1 / b) : mul_lt_mul_of_pos_right H Hbinv
... = a / b : div_eq_mul_one_div)
@ -101,7 +101,7 @@ section linear_ordered_field
theorem le_div_of_mul_le (Hc : 0 < c) (H : a * c ≤ b) : a ≤ b / c :=
calc
a = a * c * (1 / c) : mul_mul_div (ne.symm (ne_of_lt Hc))
... ≤ b * (1 / c) : mul_le_mul_of_nonneg_right H (le_of_lt (div_pos_of_pos Hc))
... ≤ b * (1 / c) : mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hc))
... = b / c : div_eq_mul_one_div
theorem mul_lt_of_lt_div (Hc : 0 < c) (H : a < b / c) : a * c < b :=
@ -110,25 +110,25 @@ section linear_ordered_field
theorem lt_div_of_mul_lt (Hc : 0 < c) (H : a * c < b) : a < b / c :=
calc
a = a * c * (1 / c) : mul_mul_div (ne.symm (ne_of_lt Hc))
... < b * (1 / c) : mul_lt_mul_of_pos_right H (div_pos_of_pos Hc)
... < b * (1 / c) : mul_lt_mul_of_pos_right H (one_div_pos_of_pos Hc)
... = b / c : div_eq_mul_one_div
theorem mul_le_of_ge_div_neg (Hc : c < 0) (H : a ≥ b / c) : a * c ≤ b :=
theorem mul_le_of_div_le_of_neg (Hc : c < 0) (H : b / c ≤ a) : a * c ≤ b :=
div_mul_cancel (ne_of_lt Hc) ▸ mul_le_mul_of_nonpos_right H (le_of_lt Hc)
theorem ge_div_of_mul_le_neg (Hc : c < 0) (H : a * c ≤ b) : a ≥ b / c :=
theorem div_le_of_mul_le_of_neg (Hc : c < 0) (H : a * c ≤ b) : b / c ≤ a :=
calc
a = a * c * (1 / c) : mul_mul_div (ne_of_lt Hc)
... ≥ b * (1 / c) : mul_le_mul_of_nonpos_right H (le_of_lt (div_neg_of_neg Hc))
... ≥ b * (1 / c) : mul_le_mul_of_nonpos_right H (le_of_lt (one_div_neg_of_neg Hc))
... = b / c : div_eq_mul_one_div
theorem mul_lt_of_gt_div_neg (Hc : c < 0) (H : a > b / c) : a * c < b :=
theorem mul_lt_of_gt_div_of_neg (Hc : c < 0) (H : a > b / c) : a * c < b :=
div_mul_cancel (ne_of_lt Hc) ▸ mul_lt_mul_of_neg_right H Hc
theorem gt_div_of_mul_gt_neg (Hc : c < 0) (H : a * c < b) : a > b / c :=
theorem gt_div_of_mul_gt_of_neg (Hc : c < 0) (H : a * c < b) : a > b / c :=
calc
a = a * c * (1 / c) : mul_mul_div (ne_of_lt Hc)
... > b * (1 / c) : mul_lt_mul_of_neg_right H (div_neg_of_neg Hc)
... > b * (1 / c) : mul_lt_mul_of_neg_right H (one_div_neg_of_neg Hc)
... = b / c : div_eq_mul_one_div
@ -137,7 +137,7 @@ section linear_ordered_field
theorem div_le_of_le_mul (Hb : b > 0) (H : a ≤ b * c) : a / b ≤ c :=
calc
a / b = a * (1 / b) : div_eq_mul_one_div
... ≤ (b * c) * (1 / b) : mul_le_mul_of_nonneg_right H (le_of_lt (div_pos_of_pos Hb))
... ≤ (b * c) * (1 / b) : mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hb))
... = (b * c) / b : div_eq_mul_one_div
... = c : mul_div_cancel_left (ne.symm (ne_of_lt Hb))
@ -183,106 +183,106 @@ section linear_ordered_field
assert H3 : a / c - b / d + b / d ≤ 0 + b / d, from add_le_add_right H2 _,
begin rewrite [zero_add at H3, neg_add_cancel_right at H3], exact H3 end
theorem pos_div_of_pos_of_pos (Ha : 0 < a) (Hb : 0 < b) : 0 < a / b :=
theorem div_pos_of_pos_of_pos (Ha : 0 < a) (Hb : 0 < b) : 0 < a / b :=
begin
rewrite div_eq_mul_one_div,
apply mul_pos,
exact Ha,
apply div_pos_of_pos,
apply one_div_pos_of_pos,
exact Hb
end
theorem nonneg_div_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 ≤ a / b :=
theorem div_nonneg_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 ≤ a / b :=
begin
rewrite div_eq_mul_one_div,
apply mul_nonneg,
exact Ha,
apply le_of_lt,
apply div_pos_of_pos,
apply one_div_pos_of_pos,
exact Hb
end
theorem neg_div_of_neg_of_pos (Ha : a < 0) (Hb : 0 < b) : a / b < 0:=
theorem div_neg_of_neg_of_pos (Ha : a < 0) (Hb : 0 < b) : a / b < 0:=
begin
rewrite div_eq_mul_one_div,
apply mul_neg_of_neg_of_pos,
exact Ha,
apply div_pos_of_pos,
apply one_div_pos_of_pos,
exact Hb
end
theorem nonpos_div_of_nonpos_of_pos (Ha : a ≤ 0) (Hb : 0 < b) : a / b ≤ 0 :=
theorem div_nonpos_of_nonpos_of_pos (Ha : a ≤ 0) (Hb : 0 < b) : a / b ≤ 0 :=
begin
rewrite div_eq_mul_one_div,
apply mul_nonpos_of_nonpos_of_nonneg,
exact Ha,
apply le_of_lt,
apply div_pos_of_pos,
apply one_div_pos_of_pos,
exact Hb
end
theorem neg_div_of_pos_of_neg (Ha : 0 < a) (Hb : b < 0) : a / b < 0 :=
theorem div_neg_of_pos_of_neg (Ha : 0 < a) (Hb : b < 0) : a / b < 0 :=
begin
rewrite div_eq_mul_one_div,
apply mul_neg_of_pos_of_neg,
exact Ha,
apply div_neg_of_neg,
apply one_div_neg_of_neg,
exact Hb
end
theorem nonpos_div_of_nonneg_of_neg (Ha : 0 ≤ a) (Hb : b < 0) : a / b ≤ 0 :=
theorem div_nonpos_of_nonneg_of_neg (Ha : 0 ≤ a) (Hb : b < 0) : a / b ≤ 0 :=
begin
rewrite div_eq_mul_one_div,
apply mul_nonpos_of_nonneg_of_nonpos,
exact Ha,
apply le_of_lt,
apply div_neg_of_neg,
apply one_div_neg_of_neg,
exact Hb
end
theorem pos_div_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a / b :=
theorem div_pos_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a / b :=
begin
rewrite div_eq_mul_one_div,
apply mul_pos_of_neg_of_neg,
exact Ha,
apply div_neg_of_neg,
apply one_div_neg_of_neg,
exact Hb
end
theorem nonneg_div_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : 0 ≤ a / b :=
theorem div_nonneg_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : 0 ≤ a / b :=
begin
rewrite div_eq_mul_one_div,
apply mul_nonneg_of_nonpos_of_nonpos,
exact Ha,
apply le_of_lt,
apply div_neg_of_neg,
apply one_div_neg_of_neg,
exact Hb
end
theorem div_lt_div_of_lt_of_pos (H : a < b) (Hc : 0 < c) : a / c < b / c :=
begin
rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
exact mul_lt_mul_of_pos_right H (div_pos_of_pos Hc)
exact mul_lt_mul_of_pos_right H (one_div_pos_of_pos Hc)
end
theorem div_le_div_of_le_of_pos (H : a ≤ b) (Hc : 0 < c) : a / c ≤ b / c :=
begin
rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
exact mul_le_mul_of_nonneg_right H (le_of_lt (div_pos_of_pos Hc))
exact mul_le_mul_of_nonneg_right H (le_of_lt (one_div_pos_of_pos Hc))
end
theorem div_lt_div_of_lt_of_neg (H : b < a) (Hc : c < 0) : a / c < b / c :=
begin
rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
exact mul_lt_mul_of_neg_right H (div_neg_of_neg Hc)
exact mul_lt_mul_of_neg_right H (one_div_neg_of_neg Hc)
end
theorem div_le_div_of_le_of_neg (H : b ≤ a) (Hc : c < 0) : a / c ≤ b / c :=
begin
rewrite [{a/c}div_eq_mul_one_div, {b/c}div_eq_mul_one_div],
exact mul_le_mul_of_nonpos_right H (le_of_lt (div_neg_of_neg Hc))
exact mul_le_mul_of_nonpos_right H (le_of_lt (one_div_neg_of_neg Hc))
end
theorem two_pos : (1 : A) + 1 > 0 :=
@ -317,7 +317,7 @@ theorem nonneg_le_nonneg_of_squares_le (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a
apply (not_le_of_gt H') H
end
theorem div_two : (a + a) / 2 = a :=
theorem add_self_div_two : (a + a) / 2 = a :=
symm (iff.mpr (eq_div_iff_mul_eq (ne_of_gt (add_pos zero_lt_one zero_lt_one)))
(by rewrite [left_distrib, *mul_one]))
@ -331,7 +331,7 @@ theorem nonneg_le_nonneg_of_squares_le (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a
end
theorem div_two_lt_of_pos (H : a > 0) : a / (1 + 1) < a :=
have Ha : a / (1 + 1) > 0, from pos_div_of_pos_of_pos H (add_pos zero_lt_one zero_lt_one),
have Ha : a / (1 + 1) > 0, from div_pos_of_pos_of_pos H (add_pos zero_lt_one zero_lt_one),
calc
a / (1 + 1) < a / (1 + 1) + a / (1 + 1) : lt_add_of_pos_left Ha
... = a : add_halves
@ -343,10 +343,10 @@ theorem nonneg_le_nonneg_of_squares_le (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a
rewrite [div_mul_eq_div_mul_one_div Hb (ne_of_gt He), div_mul_eq_div_mul_one_div Hd (ne_of_gt He)],
apply mul_le_mul_of_nonneg_right H,
apply le_of_lt,
apply div_pos_of_pos He
apply one_div_pos_of_pos He
end
theorem find_midpoint (H : a > b) : ∃ c : A, a > b + c ∧ c > 0 :=
theorem exists_add_lt_and_pos_of_lt (H : b < a) : ∃ c : A, b + c < a ∧ c > 0 :=
exists.intro ((a - b) / (1 + 1))
(and.intro (assert H2 : a + a > (b + b) + (a - b), from calc
a + a > b + a : add_lt_add_right H
@ -355,8 +355,8 @@ theorem nonneg_le_nonneg_of_squares_le (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a
... = (b + b) + (a - b) : add_sub,
assert H3 : (a + a) / (1 + 1) > ((b + b) + (a - b)) / (1 + 1),
from div_lt_div_of_lt_of_pos H2 two_pos,
by rewrite [div_two at H3, -div_add_div_same at H3, div_two at H3]; exact H3)
(pos_div_of_pos_of_pos (iff.mpr !sub_pos_iff_lt H) two_pos))
by rewrite [add_self_div_two at H3, -div_add_div_same at H3, add_self_div_two at H3]; exact H3)
(div_pos_of_pos_of_pos (iff.mpr !sub_pos_iff_lt H) two_pos))
end linear_ordered_field
@ -384,24 +384,24 @@ section discrete_linear_ordered_field
: discrete_field A :=
⦃ discrete_field, s, has_decidable_eq := dec_eq_of_dec_lt⦄
theorem pos_of_div_pos (H : 0 < 1 / a) : 0 < a :=
have H1 : 0 < 1 / (1 / a), from div_pos_of_pos H,
theorem pos_of_one_div_pos (H : 0 < 1 / a) : 0 < a :=
have H1 : 0 < 1 / (1 / a), from one_div_pos_of_pos H,
have H2 : 1 / a ≠ 0, from
(assume H3 : 1 / a = 0,
have H4 : 1 / (1 / a) = 0, from H3⁻¹ ▸ div_zero,
absurd H4 (ne.symm (ne_of_lt H1))),
(div_div (ne_zero_of_one_div_ne_zero H2)) ▸ H1
theorem neg_of_div_neg (H : 1 / a < 0) : a < 0 :=
theorem neg_of_one_div_neg (H : 1 / a < 0) : a < 0 :=
have H1 : 0 < - (1 / a), from neg_pos_of_neg H,
have Ha : a ≠ 0, from ne_zero_of_one_div_ne_zero (ne_of_lt H),
have H2 : 0 < 1 / (-a), from (one_div_neg_eq_neg_one_div Ha)⁻¹ ▸ H1,
have H3 : 0 < -a, from pos_of_div_pos H2,
have H3 : 0 < -a, from pos_of_one_div_pos H2,
neg_of_neg_pos H3
theorem le_of_div_le (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
have Hb : 0 < b, from pos_of_div_pos (calc
0 < 1 / a : div_pos_of_pos H
theorem le_of_one_div_le_one_div (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
have Hb : 0 < b, from pos_of_one_div_pos (calc
0 < 1 / a : one_div_pos_of_pos H
... ≤ 1 / b : Hl),
have H' : 1 ≤ a / b, from (calc
1 = a / a : div_self (ne.symm (ne_of_lt H))
@ -410,21 +410,21 @@ section discrete_linear_ordered_field
... = a / b : div_eq_mul_one_div
), le_of_one_le_div Hb H'
theorem le_of_div_le_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
assert Ha : a ≠ 0, from ne_of_lt (neg_of_div_neg (calc
theorem le_of_one_div_le_one_div_of_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a :=
assert Ha : a ≠ 0, from ne_of_lt (neg_of_one_div_neg (calc
1 / a ≤ 1 / b : Hl
... < 0 : div_neg_of_neg H)),
... < 0 : one_div_neg_of_neg H)),
have H' : -b > 0, from neg_pos_of_neg H,
have Hl' : - (1 / b) ≤ - (1 / a), from neg_le_neg Hl,
have Hl'' : 1 / - b ≤ 1 / - a, from calc
1 / -b = - (1 / b) : by rewrite [one_div_neg_eq_neg_one_div (ne_of_lt H)]
... ≤ - (1 / a) : Hl'
... = 1 / -a : by rewrite [one_div_neg_eq_neg_one_div Ha],
le_of_neg_le_neg (le_of_div_le H' Hl'')
le_of_neg_le_neg (le_of_one_div_le_one_div H' Hl'')
theorem lt_of_div_lt (H : 0 < a) (Hl : 1 / a < 1 / b) : b < a :=
have Hb : 0 < b, from pos_of_div_pos (calc
0 < 1 / a : div_pos_of_pos H
theorem lt_of_one_div_lt_one_div (H : 0 < a) (Hl : 1 / a < 1 / b) : b < a :=
have Hb : 0 < b, from pos_of_one_div_pos (calc
0 < 1 / a : one_div_pos_of_pos H
... < 1 / b : Hl),
have H : 1 < a / b, from (calc
1 = a / a : div_self (ne.symm (ne_of_lt H))
@ -433,47 +433,44 @@ section discrete_linear_ordered_field
... = a / b : div_eq_mul_one_div),
lt_of_one_lt_div Hb H
theorem lt_of_div_lt_neg (H : b < 0) (Hl : 1 / a < 1 / b) : b < a :=
have H1 : b ≤ a, from le_of_div_le_neg H (le_of_lt Hl),
theorem lt_of_one_div_lt_one_div_of_neg (H : b < 0) (Hl : 1 / a < 1 / b) : b < a :=
have H1 : b ≤ a, from le_of_one_div_le_one_div_of_neg H (le_of_lt Hl),
have Hn : b ≠ a, from
(assume Hn' : b = a,
have Hl' : 1 / a = 1 / b, from Hn' ▸ refl _,
absurd Hl' (ne_of_lt Hl)),
lt_of_le_of_ne H1 Hn
theorem div_lt_div_of_lt (Ha : 0 < a) (H : a < b) : 1 / b < 1 / a :=
lt_of_not_ge
(assume H',
absurd H (not_lt_of_ge (le_of_div_le Ha H')))
absurd H (not_lt_of_ge (le_of_one_div_le_one_div Ha H')))
theorem div_le_div_of_le (Ha : 0 < a) (H : a ≤ b) : 1 / b ≤ 1 / a :=
le_of_not_gt
(assume H',
absurd H (not_le_of_gt (lt_of_div_lt Ha H')))
absurd H (not_le_of_gt (lt_of_one_div_lt_one_div Ha H')))
theorem div_lt_div_of_lt_neg (Hb : b < 0) (H : a < b) : 1 / b < 1 / a :=
lt_of_not_ge
(assume H',
absurd H (not_lt_of_ge (le_of_div_le_neg Hb H')))
absurd H (not_lt_of_ge (le_of_one_div_le_one_div_of_neg Hb H')))
theorem div_le_div_of_le_neg (Hb : b < 0) (H : a ≤ b) : 1 / b ≤ 1 / a :=
le_of_not_gt
(assume H',
absurd H (not_le_of_gt (lt_of_div_lt_neg Hb H')))
absurd H (not_le_of_gt (lt_of_one_div_lt_one_div_of_neg Hb H')))
theorem one_lt_div (H1 : 0 < a) (H2 : a < 1) : 1 < 1 / a :=
theorem one_lt_one_div (H1 : 0 < a) (H2 : a < 1) : 1 < 1 / a :=
one_div_one ▸ div_lt_div_of_lt H1 H2
theorem one_le_div (H1 : 0 < a) (H2 : a ≤ 1) : 1 ≤ 1 / a :=
theorem one_le_one_div (H1 : 0 < a) (H2 : a ≤ 1) : 1 ≤ 1 / a :=
one_div_one ▸ div_le_div_of_le H1 H2
theorem neg_one_lt_div_neg (H1 : a < 0) (H2 : -1 < a) : 1 / a < -1 :=
theorem one_div_lt_neg_one (H1 : a < 0) (H2 : -1 < a) : 1 / a < -1 :=
one_div_neg_one_eq_neg_one ▸ div_lt_div_of_lt_neg H1 H2
theorem neg_one_le_div_neg (H1 : a < 0) (H2 : -1 ≤ a) : 1 / a ≤ -1 :=
theorem one_div_le_neg_one (H1 : a < 0) (H2 : -1 ≤ a) : 1 / a ≤ -1 :=
one_div_neg_one_eq_neg_one ▸ div_le_div_of_le_neg H1 H2
theorem div_lt_div_of_pos_of_lt_of_pos (Hb : 0 < b) (H : b < a) (Hc : 0 < c) : c / a < c / b :=
@ -493,21 +490,21 @@ section discrete_linear_ordered_field
rewrite [2 div_mul_eq_div_mul_one_div'],
apply mul_le_mul_of_nonneg_right H,
apply le_of_lt,
apply div_pos_of_pos He
apply one_div_pos_of_pos He
end
theorem abs_one_div : abs (1 / a) = 1 / abs a :=
if H : a > 0 then
by rewrite [abs_of_pos H, abs_of_pos (div_pos_of_pos H)]
by rewrite [abs_of_pos H, abs_of_pos (one_div_pos_of_pos H)]
else
(if H' : a < 0 then
by rewrite [abs_of_neg H', abs_of_neg (div_neg_of_neg H'),
by rewrite [abs_of_neg H', abs_of_neg (one_div_neg_of_neg H'),
-(one_div_neg_eq_neg_one_div (ne_of_lt H'))]
else
assert Heq : a = 0, from eq_of_le_of_ge (le_of_not_gt H) (le_of_not_gt H'),
by rewrite [Heq, div_zero, *abs_zero, div_zero])
theorem ge_sub_of_abs_sub_le_left (H : abs (a - b) ≤ c) : a ≥ b - c :=
theorem sub_le_of_abs_sub_le_left (H : abs (a - b) ≤ c) : b - c ≤ a :=
if Hz : 0 ≤ a - b then
(calc
a ≥ b : (iff.mp !sub_nonneg_iff_le) Hz
@ -517,8 +514,8 @@ section discrete_linear_ordered_field
have Habs' : b ≤ c + a, from (iff.mpr !le_add_iff_sub_right_le) Habs,
(iff.mp !le_add_iff_sub_left_le) Habs')
theorem ge_sub_of_abs_sub_le_right (H : abs (a - b) ≤ c) : b ≥ a - c :=
ge_sub_of_abs_sub_le_left (!abs_sub ▸ H)
theorem sub_le_of_abs_sub_le_right (H : abs (a - b) ≤ c) : a - c ≤ b :=
sub_le_of_abs_sub_le_left (!abs_sub ▸ H)
theorem abs_sub_square : abs (a - b) * abs (a - b) = a * a + b * b - 2 * a * b :=
by rewrite [abs_mul_abs_self, *mul_sub_left_distrib, *mul_sub_right_distrib,

10
library/data/pnat.lean
View file

@ -7,9 +7,7 @@ Basic facts about the positive natural numbers.
Developed primarily for use in the construction of ℝ. For the most part, the only theorems here
are those needed for that construction.
-/
import data.rat.order data.nat
open nat rat subtype eq.ops
@ -113,7 +111,7 @@ theorem pnat_le_of_rat_of_pnat_le {m n : ℕ+} (H : rat_of_pnat m ≤ rat_of_pna
definition inv (n : ℕ+) : ℚ := (1 : ℚ) / rat_of_pnat n
postfix `⁻¹` := inv
theorem inv_pos (n : ℕ+) : n⁻¹ > 0 := div_pos_of_pos !rat_of_pnat_is_pos
theorem inv_pos (n : ℕ+) : n⁻¹ > 0 := one_div_pos_of_pos !rat_of_pnat_is_pos
theorem inv_le_one (n : ℕ+) : n⁻¹ ≤ (1 : ℚ) :=
begin
@ -180,7 +178,7 @@ theorem inv_gt_of_lt {p q : ℕ+} (H : p < q) : q⁻¹ < p⁻¹ :=
div_lt_div_of_lt !rat_of_pnat_is_pos (rat_of_pnat_lt_of_pnat_lt H)
theorem ge_of_inv_le {p q : ℕ+} (H : p⁻¹ ≤ q⁻¹) : q ≤ p :=
pnat_le_of_rat_of_pnat_le (le_of_div_le !rat_of_pnat_is_pos H)
pnat_le_of_rat_of_pnat_le (le_of_one_div_le_one_div !rat_of_pnat_is_pos H)
theorem two_mul (p : ℕ+) : rat_of_pnat (2 * p) = (1 + 1) * rat_of_pnat p :=
by rewrite pnat_to_rat_mul
@ -280,13 +278,13 @@ theorem pceil_helper {a : ℚ} {n : ℕ+} (H : pceil a ≤ n) (Ha : a > 0) : n
theorem inv_pceil_div (a b : ℚ) (Ha : a > 0) (Hb : b > 0) : (pceil (a / b))⁻¹ ≤ b / a :=
div_div' ▸ div_le_div_of_le
(div_pos_of_pos (pos_div_of_pos_of_pos Hb Ha))
(one_div_pos_of_pos (div_pos_of_pos_of_pos Hb Ha))
((div_div_eq_mul_div (ne_of_gt Hb) (ne_of_gt Ha))⁻¹ ▸
!rat.one_mul⁻¹ ▸ !ubound_ge)
theorem sep_by_inv {a b : ℚ} (H : a > b) : ∃ N : ℕ+, a > (b + N⁻¹ + N⁻¹) :=
begin
apply exists.elim (find_midpoint H),
apply exists.elim (exists_add_lt_and_pos_of_lt H),
intro c Hc,
existsi (pceil ((1 + 1 + 1) / c)),
apply rat.lt.trans,

12
library/data/real/basic.lean
View file

@ -24,8 +24,8 @@ theorem s_mul_assoc_lemma_3 (a b n : ℕ+) (p : ℚ) :
theorem s_mul_assoc_lemma_4 {n : ℕ+} {ε q : ℚ} (Hε : ε > 0) (Hq : q > 0) (H : n ≥ pceil (q / ε)) :
q * n⁻¹ ≤ ε :=
begin
let H2 := pceil_helper H (pos_div_of_pos_of_pos Hq Hε),
let H3 := mul_le_of_le_div (pos_div_of_pos_of_pos Hq Hε) H2,
let H2 := pceil_helper H (div_pos_of_pos_of_pos Hq Hε),
let H3 := mul_le_of_le_div (div_pos_of_pos_of_pos Hq Hε) H2,
rewrite -(one_mul ε),
apply mul_le_mul_of_mul_div_le,
repeat assumption
@ -49,7 +49,7 @@ theorem squeeze {a b : ℚ} (H : ∀ j : ℕ+, a ≤ b + j⁻¹ + j⁻¹ + j⁻
begin
apply rat.le_of_not_gt,
intro Hb,
cases find_midpoint Hb with [c, Hc],
cases exists_add_lt_and_pos_of_lt Hb with [c, Hc],
cases find_thirds b c (and.right Hc) with [j, Hbj],
have Ha : a > b + j⁻¹ + j⁻¹ + j⁻¹, from lt.trans Hbj (and.left Hc),
apply (not_le_of_gt Ha) !H
@ -59,7 +59,7 @@ theorem squeeze_2 {a b : ℚ} (H : ∀ ε : ℚ, ε > 0 → a ≥ b - ε) : a
begin
apply rat.le_of_not_gt,
intro Hb,
cases find_midpoint Hb with [c, Hc],
cases exists_add_lt_and_pos_of_lt Hb with [c, Hc],
let Hc' := H c (and.right Hc),
apply (rat.not_le_of_gt (and.left Hc)) (iff.mpr !le_add_iff_sub_right_le Hc')
end
@ -206,7 +206,7 @@ theorem pnat_bound {ε : ℚ} (Hε : ε > 0) : ∃ p : ℕ+, p⁻¹ ≤ ε :=
rewrite -(rat.div_div (rat.ne_of_gt Hε)) at {2},
apply pceil_helper,
apply le.refl,
apply div_pos_of_pos Hε
apply one_div_pos_of_pos Hε
end
theorem bdd_of_eq_var {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) :
@ -700,7 +700,7 @@ theorem mul_zero_equiv_zero {s t : seq} (Hs : regular s) (Ht : regular t) (Htz :
apply zero_is_reg,
intro ε Hε,
let Bd := bdd_of_eq_var Ht zero_is_reg Htz (ε / (Kq s))
(pos_div_of_pos_of_pos Hε (Kq_bound_pos Hs)),
(div_pos_of_pos_of_pos Hε (Kq_bound_pos Hs)),
cases Bd with [N, HN],
existsi N,
intro n Hn,

12
library/data/real/order.lean
View file

@ -82,7 +82,7 @@ theorem nonneg_of_bdd_within {s : seq} (Hs : regular s)
cases H (pceil ((1 + 1) / ε)) with [N, HN],
apply le.trans,
rotate 1,
apply ge_sub_of_abs_sub_le_left,
apply sub_le_of_abs_sub_le_left,
apply Hs,
apply (max (pceil ((1+1)/ε)) N),
rewrite [↑rat.sub, neg_add, {_ + (-k⁻¹ + _)}add.comm, *add.assoc],
@ -115,7 +115,7 @@ theorem pos_of_pos_equiv {s t : seq} (Hs : regular s) (Heq : s ≡ t) (Hp : pos
existsi 2 * 2 * N,
apply lt_of_lt_of_le,
rotate 1,
apply ge_sub_of_abs_sub_le_right,
apply sub_le_of_abs_sub_le_right,
apply Heq,
have Hs4 : N⁻¹ ≤ s (2 * 2 * N), from HN _ (!mul_le_mul_left),
apply lt_of_lt_of_le,
@ -138,7 +138,7 @@ theorem nonneg_of_nonneg_equiv {s t : seq} (Hs : regular s) (Ht : regular t) (He
intro m Hm,
apply le.trans,
rotate 1,
apply ge_sub_of_abs_sub_le_right,
apply sub_le_of_abs_sub_le_right,
apply Heq,
apply le.trans,
rotate 1,
@ -384,7 +384,7 @@ theorem le_and_sep_of_lt {s t : seq} (Hs : regular s) (Ht : regular t) (Lst : s_
cases Lst with [N, HN],
let Rns := reg_neg_reg Hs,
let Rtns := reg_add_reg Ht Rns,
let Habs := ge_sub_of_abs_sub_le_right (Rtns N n),
let Habs := sub_le_of_abs_sub_le_right (Rtns N n),
rewrite [sub_add_eq_sub_sub at Habs],
exact (calc
sadd t (sneg s) n ≥ sadd t (sneg s) N - N⁻¹ - n⁻¹ : Habs
@ -885,8 +885,8 @@ theorem nat_inv_lt_rat {a : ℚ} (H : a > 0) : ∃ n : ℕ+, n⁻¹ < a :=
apply pceil_helper,
rewrite div_div',
apply pnat.le.refl,
apply div_pos_of_pos,
apply pos_div_of_pos_of_pos H dec_trivial
apply one_div_pos_of_pos,
apply div_pos_of_pos_of_pos H dec_trivial
end