statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
even.zpow_abs {p : ℤ} (hp : even p) (a : α) : |a| ^ p = a ^ p | by cases abs_choice a with h h; simp only [h, hp.neg_zpow _] | lemma | even.zpow_abs | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"abs_choice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit0_abs (a : α) (p : ℤ) : |a| ^ bit0 p = a ^ bit0 p | (even_bit0 _).zpow_abs _ | lemma | zpow_bit0_abs | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"even_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.cast_le_pow_sub_div_sub (H : 1 < a) (n : ℕ) : (n : α) ≤ (a ^ n - 1) / (a - 1) | (le_div_iff (sub_pos.2 H)).2 $ le_sub_left_of_add_le $
one_add_mul_sub_le_pow ((neg_le_self zero_le_one).trans H.le) _ | lemma | nat.cast_le_pow_sub_div_sub | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"le_div_iff",
"one_add_mul_sub_le_pow",
"zero_le_one"
] | Bernoulli's inequality reformulated to estimate `(n : α)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.cast_le_pow_div_sub (H : 1 < a) (n : ℕ) : (n : α) ≤ a ^ n / (a - 1) | (n.cast_le_pow_sub_div_sub H).trans $ div_le_div_of_le (sub_nonneg.2 H.le)
(sub_le_self _ zero_le_one) | theorem | nat.cast_le_pow_div_sub | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"div_le_div_of_le",
"zero_le_one"
] | For any `a > 1` and a natural `n` we have `n ≤ a ^ n / (a - 1)`. See also
`nat.cast_le_pow_sub_div_sub` for a stronger inequality with `a ^ n - 1` in the numerator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_div (a b c : α) : (a - b) / c = a / c - b / c | by simp_rw [div_eq_mul_inv, tsub_mul] | lemma | tsub_div | algebra.order.field.canonical | src/algebra/order/field/canonical/basic.lean | [
"algebra.order.field.canonical.defs"
] | [
"div_eq_mul_inv",
"tsub_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_linear_ordered_semifield (α : Type*)
extends canonically_ordered_comm_semiring α, linear_ordered_semifield α | class | canonically_linear_ordered_semifield | algebra.order.field.canonical | src/algebra/order/field/canonical/defs.lean | [
"algebra.order.field.defs",
"algebra.order.ring.canonical",
"algebra.order.with_zero"
] | [
"canonically_ordered_comm_semiring",
"linear_ordered_semifield"
] | A canonically linear ordered field is a linear ordered field in which `a ≤ b` iff there exists
`c` with `b = a + c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_linear_ordered_semifield.to_linear_ordered_comm_group_with_zero
[canonically_linear_ordered_semifield α] : linear_ordered_comm_group_with_zero α | { mul_le_mul_left := λ a b h c, mul_le_mul_of_nonneg_left h $ zero_le _,
..‹canonically_linear_ordered_semifield α› } | instance | canonically_linear_ordered_semifield.to_linear_ordered_comm_group_with_zero | algebra.order.field.canonical | src/algebra/order/field/canonical/defs.lean | [
"algebra.order.field.defs",
"algebra.order.ring.canonical",
"algebra.order.with_zero"
] | [
"canonically_linear_ordered_semifield",
"linear_ordered_comm_group_with_zero",
"mul_le_mul_left",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_linear_ordered_semifield.to_canonically_linear_ordered_add_monoid
[canonically_linear_ordered_semifield α] : canonically_linear_ordered_add_monoid α | { ..‹canonically_linear_ordered_semifield α› } | instance | canonically_linear_ordered_semifield.to_canonically_linear_ordered_add_monoid | algebra.order.field.canonical | src/algebra/order/field/canonical/defs.lean | [
"algebra.order.field.defs",
"algebra.order.ring.canonical",
"algebra.order.with_zero"
] | [
"canonically_linear_ordered_add_monoid",
"canonically_linear_ordered_semifield"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_inv.to_has_abs [has_inv α] [has_sup α] : has_abs α | ⟨λ a, a ⊔ a⁻¹⟩ | instance | has_inv.to_has_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"has_abs",
"has_sup"
] | `abs a` is the absolute value of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_eq_sup_inv [has_inv α] [has_sup α] (a : α) : |a| = a ⊔ a⁻¹ | rfl | lemma | abs_eq_sup_inv | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"has_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq_max_neg : abs a = max a (-a) | rfl | lemma | abs_eq_max_neg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_choice (x : α) : |x| = x ∨ |x| = -x | max_choice _ _ | lemma | abs_choice | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"max_choice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le' : |a| ≤ b ↔ a ≤ b ∧ -a ≤ b | max_le_iff | lemma | abs_le' | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"max_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_abs : a ≤ |b| ↔ a ≤ b ∨ a ≤ -b | le_max_iff | lemma | le_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"le_max_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_abs_self (a : α) : a ≤ |a| | le_max_left _ _ | lemma | le_abs_self | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_le_abs_self (a : α) : -a ≤ |a| | le_max_right _ _ | lemma | neg_le_abs_self | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_abs : a < |b| ↔ a < b ∨ a < -b | lt_max_iff | lemma | lt_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"lt_max_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_abs (h₀ : a ≤ b) (h₁ : -a ≤ b) : |a| ≤ |b| | (abs_le'.2 ⟨h₀, h₁⟩).trans (le_abs_self b) | theorem | abs_le_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_by_cases (P : α → Prop) {a : α} (h1 : P a) (h2 : P (-a)) : P (|a|) | sup_ind _ _ h1 h2 | lemma | abs_by_cases | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"sup_ind"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_neg (a : α) : | -a| = |a| | begin
rw [abs_eq_max_neg, max_comm, neg_neg, abs_eq_max_neg]
end | lemma | abs_neg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_eq_max_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_or_eq_neg_of_abs_eq {a b : α} (h : |a| = b) : a = b ∨ a = -b | by simpa only [← h, eq_comm, neg_eq_iff_eq_neg] using abs_choice a | lemma | eq_or_eq_neg_of_abs_eq | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_choice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq_abs {a b : α} : |a| = |b| ↔ a = b ∨ a = -b | begin
refine ⟨λ h, _, λ h, _⟩,
{ obtain rfl | rfl := eq_or_eq_neg_of_abs_eq h;
simpa only [neg_eq_iff_eq_neg, neg_inj, or.comm] using abs_choice b },
{ cases h; simp only [h, abs_neg] },
end | lemma | abs_eq_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_choice",
"abs_neg",
"eq_or_eq_neg_of_abs_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_comm (a b : α) : |a - b| = |b - a| | calc |a - b| = | - (b - a)| : congr_arg _ (neg_sub b a).symm
... = |b - a| : abs_neg (b - a) | lemma | abs_sub_comm | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_of_nonneg (h : 0 ≤ a) : |a| = a | max_eq_left $ (neg_nonpos.2 h).trans h | lemma | abs_of_nonneg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_of_pos (h : 0 < a) : |a| = a | abs_of_nonneg h.le | lemma | abs_of_pos | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_of_nonpos (h : a ≤ 0) : |a| = -a | max_eq_right $ h.trans (neg_nonneg.2 h) | lemma | abs_of_nonpos | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_of_neg (h : a < 0) : |a| = -a | abs_of_nonpos h.le | lemma | abs_of_neg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_abs_of_nonneg (ha : 0 ≤ a) (hab : a ≤ b) : |a| ≤ |b| | by rwa [abs_of_nonneg ha, abs_of_nonneg (ha.trans hab)] | lemma | abs_le_abs_of_nonneg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_zero : |0| = (0:α) | abs_of_nonneg le_rfl | lemma | abs_zero | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_nonneg",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_pos : 0 < |a| ↔ a ≠ 0 | begin
rcases lt_trichotomy a 0 with (ha|rfl|ha),
{ simp [abs_of_neg ha, neg_pos, ha.ne, ha] },
{ simp },
{ simp [abs_of_pos ha, ha, ha.ne.symm] }
end | lemma | abs_pos | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_neg",
"abs_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_pos_of_pos (h : 0 < a) : 0 < |a| | abs_pos.2 h.ne.symm | lemma | abs_pos_of_pos | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_pos_of_neg (h : a < 0) : 0 < |a| | abs_pos.2 h.ne | lemma | abs_pos_of_neg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_abs_le_self (a : α) : -|a| ≤ a | begin
cases le_total 0 a with h h,
{ calc -|a| = - a : congr_arg (has_neg.neg) (abs_of_nonneg h)
... ≤ 0 : neg_nonpos.mpr h
... ≤ a : h },
{ calc -|a| = - - a : congr_arg (has_neg.neg) (abs_of_nonpos h)
... ≤ a : (neg_neg a).le }
end | lemma | neg_abs_le_self | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_nonneg",
"abs_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_abs_nonneg (a : α) : 0 ≤ a + |a| | begin
rw ←add_right_neg a,
apply add_le_add_left,
exact (neg_le_abs_self a),
end | lemma | add_abs_nonneg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"neg_le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_abs_le_neg (a : α) : -|a| ≤ -a | by simpa using neg_abs_le_self (-a) | lemma | neg_abs_le_neg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"neg_abs_le_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_nonneg (a : α) : 0 ≤ |a| | (le_total 0 a).elim (λ h, h.trans (le_abs_self a)) (λ h, (neg_nonneg.2 h).trans $ neg_le_abs_self a) | lemma | abs_nonneg | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"le_abs_self",
"neg_le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_abs (a : α) : | |a| | = |a| | abs_of_nonneg $ abs_nonneg a | lemma | abs_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_nonneg",
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq_zero : |a| = 0 ↔ a = 0 | decidable.not_iff_not.1 $ ne_comm.trans $ (abs_nonneg a).lt_iff_ne.symm.trans abs_pos | lemma | abs_eq_zero | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_nonneg",
"abs_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_nonpos_iff {a : α} : |a| ≤ 0 ↔ a = 0 | (abs_nonneg a).le_iff_eq.trans abs_eq_zero | lemma | abs_nonpos_iff | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_eq_zero",
"abs_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_abs_of_nonpos (ha : a ≤ 0) (hab : b ≤ a) : |a| ≤ |b| | by { rw [abs_of_nonpos ha, abs_of_nonpos (hab.trans ha)], exact neg_le_neg_iff.mpr hab } | lemma | abs_le_abs_of_nonpos | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_lt : |a| < b ↔ - b < a ∧ a < b | max_lt_iff.trans $ and.comm.trans $ by rw [neg_lt] | lemma | abs_lt | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_lt_of_abs_lt (h : |a| < b) : -b < a | (abs_lt.mp h).1 | lemma | neg_lt_of_abs_lt | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_abs_lt (h : |a| < b) : a < b | (abs_lt.mp h).2 | lemma | lt_of_abs_lt | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_sub_min_eq_abs' (a b : α) : max a b - min a b = |a - b| | begin
cases le_total a b with ab ba,
{ rw [max_eq_right ab, min_eq_left ab, abs_of_nonpos, neg_sub], rwa sub_nonpos },
{ rw [max_eq_left ba, min_eq_right ba, abs_of_nonneg], rwa sub_nonneg }
end | lemma | max_sub_min_eq_abs' | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_nonneg",
"abs_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_sub_min_eq_abs (a b : α) : max a b - min a b = |b - a| | by { rw abs_sub_comm, exact max_sub_min_eq_abs' _ _ } | lemma | max_sub_min_eq_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_sub_comm",
"max_sub_min_eq_abs'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le : |a| ≤ b ↔ - b ≤ a ∧ a ≤ b | by rw [abs_le', and.comm, neg_le] | lemma | abs_le | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_abs' : a ≤ |b| ↔ b ≤ -a ∨ a ≤ b | by rw [le_abs, or.comm, le_neg] | lemma | le_abs' | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"le_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_le_of_abs_le (h : |a| ≤ b) : -b ≤ a | (abs_le.mp h).1 | lemma | neg_le_of_abs_le | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_abs_le (h : |a| ≤ b) : a ≤ b | (abs_le.mp h).2 | lemma | le_of_abs_le | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_abs_le_mul_of_one_le' {β : Type*} [mul_one_class β] [preorder β]
[covariant_class β β (*) (≤)] [covariant_class β β (swap (*)) (≤)] {f : α → β} {a : α}
(h₁ : 1 ≤ f a) (h₂ : 1 ≤ f (-a)) :
f (|a|) ≤ f a * f (-a) | (le_total a 0).by_cases (λ ha, (abs_of_nonpos ha).symm ▸ le_mul_of_one_le_left' h₁)
(λ ha, (abs_of_nonneg ha).symm ▸ le_mul_of_one_le_right' h₂) | lemma | apply_abs_le_mul_of_one_le' | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_of_nonneg",
"abs_of_nonpos",
"covariant_class",
"le_mul_of_one_le_left'",
"le_mul_of_one_le_right'",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_abs_le_mul_of_one_le {β : Type*} [mul_one_class β] [preorder β]
[covariant_class β β (*) (≤)] [covariant_class β β (swap (*)) (≤)] {f : α → β}
(h : ∀ x, 1 ≤ f x) (a : α) :
f (|a|) ≤ f a * f (-a) | apply_abs_le_mul_of_one_le' (h _) (h _) | lemma | apply_abs_le_mul_of_one_le | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"apply_abs_le_mul_of_one_le'",
"covariant_class",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_add (a b : α) : |a + b| ≤ |a| + |b| | abs_le.2 ⟨(neg_add (|a|) (|b|)).symm ▸
add_le_add (neg_le.2 $ neg_le_abs_self _) (neg_le.2 $ neg_le_abs_self _),
add_le_add (le_abs_self _) (le_abs_self _)⟩ | lemma | abs_add | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"le_abs_self",
"neg_le_abs_self"
] | The **triangle inequality** in `linear_ordered_add_comm_group`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_add' (a b : α) : |a| ≤ |b| + |b + a| | by simpa using abs_add (-b) (b + a) | lemma | abs_add' | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub (a b : α) : | |a - b| ≤ |a| + |b| :=
by { rw [sub_eq_add_neg, ←abs_neg b], exact abs_add a _ } | theorem | abs_sub | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_le_iff : |a - b| ≤ c ↔ a - b ≤ c ∧ b - a ≤ c | by rw [abs_le, neg_le_sub_iff_le_add, sub_le_iff_le_add', and_comm, sub_le_iff_le_add'] | lemma | abs_sub_le_iff | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_lt_iff : |a - b| < c ↔ a - b < c ∧ b - a < c | by rw [abs_lt, neg_lt_sub_iff_lt_add', sub_lt_iff_lt_add', and_comm, sub_lt_iff_lt_add'] | lemma | abs_sub_lt_iff | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_le_of_abs_sub_le_left (h : |a - b| ≤ c) : b - c ≤ a | sub_le_comm.1 $ (abs_sub_le_iff.1 h).2 | lemma | sub_le_of_abs_sub_le_left | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_le_of_abs_sub_le_right (h : |a - b| ≤ c) : a - c ≤ b | sub_le_of_abs_sub_le_left (abs_sub_comm a b ▸ h) | lemma | sub_le_of_abs_sub_le_right | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_sub_comm",
"sub_le_of_abs_sub_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_lt_of_abs_sub_lt_left (h : |a - b| < c) : b - c < a | sub_lt_comm.1 $ (abs_sub_lt_iff.1 h).2 | lemma | sub_lt_of_abs_sub_lt_left | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_lt_of_abs_sub_lt_right (h : |a - b| < c) : a - c < b | sub_lt_of_abs_sub_lt_left (abs_sub_comm a b ▸ h) | lemma | sub_lt_of_abs_sub_lt_right | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_sub_comm",
"sub_lt_of_abs_sub_lt_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_abs_le_abs_sub (a b : α) : |a| - |b| ≤ |a - b| | sub_le_iff_le_add.2 $
calc |a| = |a - b + b| : by rw [sub_add_cancel]
... ≤ |a - b| + |b| : abs_add _ _ | lemma | abs_sub_abs_le_abs_sub | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_abs_sub_abs_le_abs_sub (a b : α) : | |a| - |b| | ≤ |a - b| | abs_sub_le_iff.2 ⟨abs_sub_abs_le_abs_sub _ _, by rw abs_sub_comm; apply abs_sub_abs_le_abs_sub⟩ | lemma | abs_abs_sub_abs_le_abs_sub | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_sub_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq (hb : 0 ≤ b) : |a| = b ↔ a = b ∨ a = -b | begin
refine ⟨eq_or_eq_neg_of_abs_eq, _⟩,
rintro (rfl|rfl); simp only [abs_neg, abs_of_nonneg hb]
end | lemma | abs_eq | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_neg",
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_max_abs_abs (hab : a ≤ b) (hbc : b ≤ c) : |b| ≤ max (|a|) (|c|) | abs_le'.2
⟨by simp [hbc.trans (le_abs_self c)],
by simp [(neg_le_neg_iff.mpr hab).trans (neg_le_abs_self a)]⟩ | lemma | abs_le_max_abs_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"le_abs_self",
"neg_le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_abs_abs_le_abs_max : min (|a|) (|b|) ≤ |max a b| | (le_total a b).elim
(λ h, (min_le_right _ _).trans_eq $ congr_arg _ (max_eq_right h).symm)
(λ h, (min_le_left _ _).trans_eq $ congr_arg _ (max_eq_left h).symm) | lemma | min_abs_abs_le_abs_max | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_abs_abs_le_abs_min : min (|a|) (|b|) ≤ |min a b| | (le_total a b).elim
(λ h, (min_le_left _ _).trans_eq $ congr_arg _ (min_eq_left h).symm)
(λ h, (min_le_right _ _).trans_eq $ congr_arg _ (min_eq_right h).symm) | lemma | min_abs_abs_le_abs_min | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_max_le_max_abs_abs : |max a b| ≤ max (|a|) (|b|) | (le_total a b).elim
(λ h, (congr_arg _ $ max_eq_right h).trans_le $ le_max_right _ _)
(λ h, (congr_arg _ $ max_eq_left h).trans_le $ le_max_left _ _) | lemma | abs_max_le_max_abs_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_min_le_max_abs_abs : |min a b| ≤ max (|a|) (|b|) | (le_total a b).elim
(λ h, (congr_arg _ $ min_eq_left h).trans_le $ le_max_left _ _)
(λ h, (congr_arg _ $ min_eq_right h).trans_le $ le_max_right _ _) | lemma | abs_min_le_max_abs_abs | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_abs_sub_eq_zero {a b : α} (h : |a - b| = 0) : a = b | sub_eq_zero.1 $ abs_eq_zero.1 h | lemma | eq_of_abs_sub_eq_zero | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_le (a b c : α) : |a - c| ≤ |a - b| + |b - c| | calc
|a - c| = |a - b + (b - c)| : by rw [sub_add_sub_cancel]
... ≤ |a - b| + |b - c| : abs_add _ _ | lemma | abs_sub_le | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_add_three (a b c : α) : |a + b + c| ≤ |a| + |b| + |c| | (abs_add _ _).trans (add_le_add_right (abs_add _ _) _) | lemma | abs_add_three | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_bdd_within_interval {a b lb ub : α} (hal : lb ≤ a) (hau : a ≤ ub)
(hbl : lb ≤ b) (hbu : b ≤ ub) : |a - b| ≤ ub - lb | abs_sub_le_iff.2 ⟨sub_le_sub hau hbl, sub_le_sub hbu hal⟩ | lemma | dist_bdd_within_interval | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_abs_sub_nonpos (h : |a - b| ≤ 0) : a = b | eq_of_abs_sub_eq_zero (le_antisymm h (abs_nonneg (a - b))) | lemma | eq_of_abs_sub_nonpos | algebra.order.group | src/algebra/order/group/abs.lean | [
"algebra.abs",
"algebra.order.group.order_iso",
"order.min_max"
] | [
"abs_nonneg",
"eq_of_abs_sub_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.exists_between_self_add (h : is_glb s a) (hε : 0 < ε) :
∃ b ∈ s, a ≤ b ∧ b < a + ε | h.exists_between $ lt_add_of_pos_right _ hε | lemma | is_glb.exists_between_self_add | algebra.order.group | src/algebra/order/group/bounds.lean | [
"order.bounds.basic",
"algebra.order.group.defs"
] | [
"is_glb"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.exists_between_self_add' (h : is_glb s a) (h₂ : a ∉ s) (hε : 0 < ε) :
∃ b ∈ s, a < b ∧ b < a + ε | h.exists_between' h₂ $ lt_add_of_pos_right _ hε | lemma | is_glb.exists_between_self_add' | algebra.order.group | src/algebra/order/group/bounds.lean | [
"order.bounds.basic",
"algebra.order.group.defs"
] | [
"is_glb"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.exists_between_sub_self (h : is_lub s a) (hε : 0 < ε) : ∃ b ∈ s, a - ε < b ∧ b ≤ a | h.exists_between $ sub_lt_self _ hε | lemma | is_lub.exists_between_sub_self | algebra.order.group | src/algebra/order/group/bounds.lean | [
"order.bounds.basic",
"algebra.order.group.defs"
] | [
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.exists_between_sub_self' (h : is_lub s a) (h₂ : a ∉ s) (hε : 0 < ε) :
∃ b ∈ s, a - ε < b ∧ b < a | h.exists_between' h₂ $ sub_lt_self _ hε | lemma | is_lub.exists_between_sub_self' | algebra.order.group | src/algebra/order/group/bounds.lean | [
"order.bounds.basic",
"algebra.order.group.defs"
] | [
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_add_comm_group (α : Type u) extends add_comm_group α, partial_order α | (add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b) | class | ordered_add_comm_group | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"add_comm_group"
] | An ordered additive commutative group is an additive commutative group
with a partial order in which addition is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_group (α : Type u) extends comm_group α, partial_order α | (mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b) | class | ordered_comm_group | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"comm_group",
"mul_le_mul_left"
] | An ordered commutative group is an commutative group
with a partial order in which multiplication is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_group.to_covariant_class_left_le (α : Type u) [ordered_comm_group α] :
covariant_class α α (*) (≤) | { elim := λ a b c bc, ordered_comm_group.mul_le_mul_left b c bc a } | instance | ordered_comm_group.to_covariant_class_left_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"covariant_class",
"ordered_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_comm_group.to_ordered_cancel_comm_monoid [ordered_comm_group α] :
ordered_cancel_comm_monoid α | { le_of_mul_le_mul_left := λ a b c, le_of_mul_le_mul_left',
..‹ordered_comm_group α› } | instance | ordered_comm_group.to_ordered_cancel_comm_monoid | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"le_of_mul_le_mul_left",
"le_of_mul_le_mul_left'",
"ordered_cancel_comm_monoid",
"ordered_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_comm_group.to_contravariant_class_left_le (α : Type u) [ordered_comm_group α] :
contravariant_class α α (*) (≤) | { elim := λ a b c bc, by simpa using mul_le_mul_left' bc a⁻¹, } | theorem | ordered_comm_group.to_contravariant_class_left_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"contravariant_class",
"mul_le_mul_left'",
"ordered_comm_group"
] | A choice-free shortcut instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_group.to_contravariant_class_right_le (α : Type u) [ordered_comm_group α] :
contravariant_class α α (swap (*)) (≤) | { elim := λ a b c bc, by simpa using mul_le_mul_right' bc a⁻¹, } | theorem | ordered_comm_group.to_contravariant_class_right_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"contravariant_class",
"mul_le_mul_right'",
"ordered_comm_group"
] | A choice-free shortcut instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.inv_le_one_iff :
a⁻¹ ≤ 1 ↔ 1 ≤ a | by { rw [← mul_le_mul_iff_left a], simp } | lemma | left.inv_le_one_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_le_mul_iff_left"
] | Uses `left` co(ntra)variant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.one_le_inv_iff :
1 ≤ a⁻¹ ↔ a ≤ 1 | by { rw [← mul_le_mul_iff_left a], simp } | lemma | left.one_le_inv_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_le_mul_iff_left"
] | Uses `left` co(ntra)variant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c | by { rw ← mul_le_mul_iff_left a, simp } | lemma | le_inv_mul_iff_mul_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_le_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c | by rw [← mul_le_mul_iff_left b, mul_inv_cancel_left] | lemma | inv_mul_le_iff_le_mul | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_cancel_left",
"mul_le_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_iff_one_le_mul' : a⁻¹ ≤ b ↔ 1 ≤ a * b | (mul_le_mul_iff_left a).symm.trans $ by rw mul_inv_self | lemma | inv_le_iff_one_le_mul' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_self",
"mul_le_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_inv_iff_mul_le_one_left : a ≤ b⁻¹ ↔ b * a ≤ 1 | (mul_le_mul_iff_left b).symm.trans $ by rw mul_inv_self | lemma | le_inv_iff_mul_le_one_left | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_self",
"mul_le_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_inv_mul_iff_le : 1 ≤ b⁻¹ * a ↔ b ≤ a | by rw [← mul_le_mul_iff_left b, mul_one, mul_inv_cancel_left] | lemma | le_inv_mul_iff_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_cancel_left",
"mul_le_mul_iff_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_le_one_iff : a⁻¹ * b ≤ 1 ↔ b ≤ a | trans (inv_mul_le_iff_le_mul) $ by rw mul_one | lemma | inv_mul_le_one_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_le_iff_le_mul",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.one_lt_inv_iff :
1 < a⁻¹ ↔ a < 1 | by rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one] | lemma | left.one_lt_inv_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_self",
"mul_lt_mul_iff_left",
"mul_one"
] | Uses `left` co(ntra)variant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.inv_lt_one_iff :
a⁻¹ < 1 ↔ 1 < a | by rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one] | lemma | left.inv_lt_one_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_self",
"mul_lt_mul_iff_left",
"mul_one"
] | Uses `left` co(ntra)variant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c | by { rw [← mul_lt_mul_iff_left a], simp } | lemma | lt_inv_mul_iff_mul_lt | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_lt_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c | by rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left] | lemma | inv_mul_lt_iff_lt_mul | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_cancel_left",
"mul_lt_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_iff_one_lt_mul' : a⁻¹ < b ↔ 1 < a * b | (mul_lt_mul_iff_left a).symm.trans $ by rw mul_inv_self | lemma | inv_lt_iff_one_lt_mul' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_self",
"mul_lt_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_inv_iff_mul_lt_one' : a < b⁻¹ ↔ b * a < 1 | (mul_lt_mul_iff_left b).symm.trans $ by rw mul_inv_self | lemma | lt_inv_iff_mul_lt_one' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_self",
"mul_lt_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < a | by rw [← mul_lt_mul_iff_left b, mul_one, mul_inv_cancel_left] | lemma | lt_inv_mul_iff_lt | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_inv_cancel_left",
"mul_lt_mul_iff_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_lt_one_iff : a⁻¹ * b < 1 ↔ b < a | trans (inv_mul_lt_iff_lt_mul) $ by rw mul_one | lemma | inv_mul_lt_one_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_lt_iff_lt_mul",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.inv_le_one_iff :
a⁻¹ ≤ 1 ↔ 1 ≤ a | by { rw [← mul_le_mul_iff_right a], simp } | lemma | right.inv_le_one_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_le_mul_iff_right"
] | Uses `right` co(ntra)variant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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