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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|---|---|---|---|---|---|---|---|---|---|---|
div_pos_iff : 0 < a / b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 | by simp [division_def, mul_pos_iff] | lemma | div_pos_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"mul_pos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_neg_iff : a / b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b | by simp [division_def, mul_neg_iff] | lemma | div_neg_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"mul_neg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_nonneg_iff : 0 ≤ a / b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 | by simp [division_def, mul_nonneg_iff] | lemma | div_nonneg_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"mul_nonneg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_nonpos_iff : a / b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b | by simp [division_def, mul_nonpos_iff] | lemma | div_nonpos_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"mul_nonpos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_nonneg_of_nonpos (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a / b | div_nonneg_iff.2 $ or.inr ⟨ha, hb⟩ | lemma | div_nonneg_of_nonpos | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_pos_of_neg_of_neg (ha : a < 0) (hb : b < 0) : 0 < a / b | div_pos_iff.2 $ or.inr ⟨ha, hb⟩ | lemma | div_pos_of_neg_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_neg_of_neg_of_pos (ha : a < 0) (hb : 0 < b) : a / b < 0 | div_neg_iff.2 $ or.inr ⟨ha, hb⟩ | lemma | div_neg_of_neg_of_pos | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_neg_of_pos_of_neg (ha : 0 < a) (hb : b < 0) : a / b < 0 | div_neg_iff.2 $ or.inl ⟨ha, hb⟩ | lemma | div_neg_of_pos_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_iff_of_neg (hc : c < 0) : b / c ≤ a ↔ a * c ≤ b | ⟨λ h, div_mul_cancel b (ne_of_lt hc) ▸ mul_le_mul_of_nonpos_right h hc.le,
λ h, calc
a = a * c * (1 / c) : mul_mul_div a (ne_of_lt hc)
... ≥ b * (1 / c) : mul_le_mul_of_nonpos_right h (one_div_neg.2 hc).le
... = b / c : (div_eq_mul_one_div b c).symm⟩ | lemma | div_le_iff_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"div_mul_cancel",
"mul_le_mul_of_nonpos_right",
"mul_mul_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_iff_of_neg' (hc : c < 0) : b / c ≤ a ↔ c * a ≤ b | by rw [mul_comm, div_le_iff_of_neg hc] | lemma | div_le_iff_of_neg' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff_of_neg",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff_of_neg (hc : c < 0) : a ≤ b / c ↔ b ≤ a * c | by rw [← neg_neg c, mul_neg, div_neg, le_neg,
div_le_iff (neg_pos.2 hc), neg_mul] | lemma | le_div_iff_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff",
"div_neg",
"mul_neg",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff_of_neg' (hc : c < 0) : a ≤ b / c ↔ b ≤ c * a | by rw [mul_comm, le_div_iff_of_neg hc] | lemma | le_div_iff_of_neg' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_div_iff_of_neg",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_iff_of_neg (hc : c < 0) : b / c < a ↔ a * c < b | lt_iff_lt_of_le_iff_le $ le_div_iff_of_neg hc | lemma | div_lt_iff_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_div_iff_of_neg",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_iff_of_neg' (hc : c < 0) : b / c < a ↔ c * a < b | by rw [mul_comm, div_lt_iff_of_neg hc] | lemma | div_lt_iff_of_neg' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff_of_neg",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_div_iff_of_neg (hc : c < 0) : a < b / c ↔ b < a * c | lt_iff_lt_of_le_iff_le $ div_le_iff_of_neg hc | lemma | lt_div_iff_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff_of_neg",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_div_iff_of_neg' (hc : c < 0) : a < b / c ↔ b < c * a | by rw [mul_comm, lt_div_iff_of_neg hc] | lemma | lt_div_iff_of_neg' | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_div_iff_of_neg",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_one_of_ge (h : b ≤ a) (hb : b ≤ 0) : a / b ≤ 1 | by simpa only [neg_div_neg_eq] using div_le_one_of_le (neg_le_neg h) (neg_nonneg_of_nonpos hb) | lemma | div_le_one_of_ge | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_one_of_le",
"neg_div_neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_inv_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a | by rw [← one_div, div_le_iff_of_neg ha, ← div_eq_inv_mul, div_le_iff_of_neg hb, one_mul] | lemma | inv_le_inv_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_inv_mul",
"div_le_iff_of_neg",
"one_div",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a | by rw [← inv_le_inv_of_neg hb (inv_lt_zero.2 ha), inv_inv] | lemma | inv_le_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_inv",
"inv_le_inv_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_inv_of_neg (ha : a < 0) (hb : b < 0) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ | by rw [← inv_le_inv_of_neg (inv_lt_zero.2 hb) ha, inv_inv] | lemma | le_inv_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_inv",
"inv_le_inv_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_inv_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ < b⁻¹ ↔ b < a | lt_iff_lt_of_le_iff_le (inv_le_inv_of_neg hb ha) | lemma | inv_lt_inv_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le_inv_of_neg",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ < b ↔ b⁻¹ < a | lt_iff_lt_of_le_iff_le (le_inv_of_neg hb ha) | lemma | inv_lt_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_inv_of_neg",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_inv_of_neg (ha : a < 0) (hb : b < 0) : a < b⁻¹ ↔ b < a⁻¹ | lt_iff_lt_of_le_iff_le (inv_le_of_neg hb ha) | lemma | lt_inv_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le_of_neg",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_of_nonpos_of_le (hc : c ≤ 0) (h : b ≤ a) : a / c ≤ b / c | begin
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],
exact mul_le_mul_of_nonpos_right h (one_div_nonpos.2 hc)
end | lemma | div_le_div_of_nonpos_of_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"mul_le_mul_of_nonpos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_of_neg_of_lt (hc : c < 0) (h : b < a) : a / c < b / c | begin
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c],
exact mul_lt_mul_of_neg_right h (one_div_neg.2 hc)
end | lemma | div_lt_div_of_neg_of_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"mul_lt_mul_of_neg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_right_of_neg (hc : c < 0) : a / c ≤ b / c ↔ b ≤ a | ⟨le_imp_le_of_lt_imp_lt $ div_lt_div_of_neg_of_lt hc, div_le_div_of_nonpos_of_le $ hc.le⟩ | lemma | div_le_div_right_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_of_nonpos_of_le",
"div_lt_div_of_neg_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_right_of_neg (hc : c < 0) : a / c < b / c ↔ b < a | lt_iff_lt_of_le_iff_le $ div_le_div_right_of_neg hc | lemma | div_lt_div_right_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_div_right_of_neg",
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_div_of_neg (hb : b < 0) : 1 ≤ a / b ↔ a ≤ b | by rw [le_div_iff_of_neg hb, one_mul] | lemma | one_le_div_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_div_iff_of_neg",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_one_of_neg (hb : b < 0) : a / b ≤ 1 ↔ b ≤ a | by rw [div_le_iff_of_neg hb, one_mul] | lemma | div_le_one_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_iff_of_neg",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_div_of_neg (hb : b < 0) : 1 < a / b ↔ a < b | by rw [lt_div_iff_of_neg hb, one_mul] | lemma | one_lt_div_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_div_iff_of_neg",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_one_of_neg (hb : b < 0) : a / b < 1 ↔ b < a | by rw [div_lt_iff_of_neg hb, one_mul] | lemma | div_lt_one_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_iff_of_neg",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_le_of_neg (ha : a < 0) (hb : b < 0) : 1 / a ≤ b ↔ 1 / b ≤ a | by simpa using inv_le_of_neg ha hb | lemma | one_div_le_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_lt_of_neg (ha : a < 0) (hb : b < 0) : 1 / a < b ↔ 1 / b < a | by simpa using inv_lt_of_neg ha hb | lemma | one_div_lt_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_lt_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_div_of_neg (ha : a < 0) (hb : b < 0) : a ≤ 1 / b ↔ b ≤ 1 / a | by simpa using le_inv_of_neg ha hb | lemma | le_one_div_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"le_inv_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_one_div_of_neg (ha : a < 0) (hb : b < 0) : a < 1 / b ↔ b < 1 / a | by simpa using lt_inv_of_neg ha hb | lemma | lt_one_div_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_inv_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_div_iff : 1 < a / b ↔ 0 < b ∧ b < a ∨ b < 0 ∧ a < b | begin
rcases lt_trichotomy b 0 with (hb|rfl|hb),
{ simp [hb, hb.not_lt, one_lt_div_of_neg] },
{ simp [lt_irrefl, zero_le_one] },
{ simp [hb, hb.not_lt, one_lt_div] }
end | lemma | one_lt_div_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_lt_div",
"one_lt_div_of_neg",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_div_iff : 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b | begin
rcases lt_trichotomy b 0 with (hb|rfl|hb),
{ simp [hb, hb.not_lt, one_le_div_of_neg] },
{ simp [lt_irrefl, zero_lt_one.not_le, zero_lt_one] },
{ simp [hb, hb.not_lt, one_le_div] }
end | lemma | one_le_div_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_le_div",
"one_le_div_of_neg",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_one_iff : a / b < 1 ↔ 0 < b ∧ a < b ∨ b = 0 ∨ b < 0 ∧ b < a | begin
rcases lt_trichotomy b 0 with (hb|rfl|hb),
{ simp [hb, hb.not_lt, hb.ne, div_lt_one_of_neg] },
{ simp [zero_lt_one], },
{ simp [hb, hb.not_lt, div_lt_one, hb.ne.symm] }
end | lemma | div_lt_one_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_one",
"div_lt_one_of_neg",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_one_iff : a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a | begin
rcases lt_trichotomy b 0 with (hb|rfl|hb),
{ simp [hb, hb.not_lt, hb.ne, div_le_one_of_neg] },
{ simp [zero_le_one], },
{ simp [hb, hb.not_lt, div_le_one, hb.ne.symm] }
end | lemma | div_le_one_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_le_one",
"div_le_one_of_neg",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_le_one_div_of_neg_of_le (hb : b < 0) (h : a ≤ b) : 1 / b ≤ 1 / a | by rwa [div_le_iff_of_neg' hb, ← div_eq_mul_one_div, div_le_one_of_neg (h.trans_lt hb)] | lemma | one_div_le_one_div_of_neg_of_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"div_le_iff_of_neg'",
"div_le_one_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_lt_one_div_of_neg_of_lt (hb : b < 0) (h : a < b) : 1 / b < 1 / a | by rwa [div_lt_iff_of_neg' hb, ← div_eq_mul_one_div, div_lt_one_of_neg (h.trans hb)] | lemma | one_div_lt_one_div_of_neg_of_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_eq_mul_one_div",
"div_lt_iff_of_neg'",
"div_lt_one_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_neg_of_one_div_le_one_div (hb : b < 0) (h : 1 / a ≤ 1 / b) : b ≤ a | le_imp_le_of_lt_imp_lt (one_div_lt_one_div_of_neg_of_lt hb) h | lemma | le_of_neg_of_one_div_le_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_lt_one_div_of_neg_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_neg_of_one_div_lt_one_div (hb : b < 0) (h : 1 / a < 1 / b) : b < a | lt_imp_lt_of_le_imp_le (one_div_le_one_div_of_neg_of_le hb) h | lemma | lt_of_neg_of_one_div_lt_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_imp_lt_of_le_imp_le",
"one_div_le_one_div_of_neg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_le_one_div_of_neg (ha : a < 0) (hb : b < 0) : 1 / a ≤ 1 / b ↔ b ≤ a | by simpa [one_div] using inv_le_inv_of_neg ha hb | lemma | one_div_le_one_div_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_le_inv_of_neg",
"one_div"
] | For the single implications with fewer assumptions, see `one_div_lt_one_div_of_neg_of_lt` and
`lt_of_one_div_lt_one_div` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_div_lt_one_div_of_neg (ha : a < 0) (hb : b < 0) : 1 / a < 1 / b ↔ b < a | lt_iff_lt_of_le_iff_le (one_div_le_one_div_of_neg hb ha) | lemma | one_div_lt_one_div_of_neg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"lt_iff_lt_of_le_iff_le",
"one_div_le_one_div_of_neg"
] | For the single implications with fewer assumptions, see `one_div_lt_one_div_of_lt` and
`lt_of_one_div_lt_one_div` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_div_lt_neg_one (h1 : a < 0) (h2 : -1 < a) : 1 / a < -1 | suffices 1 / a < 1 / -1, by rwa one_div_neg_one_eq_neg_one at this,
one_div_lt_one_div_of_neg_of_lt h1 h2 | lemma | one_div_lt_neg_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_lt_one_div_of_neg_of_lt",
"one_div_neg_one_eq_neg_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_le_neg_one (h1 : a < 0) (h2 : -1 ≤ a) : 1 / a ≤ -1 | suffices 1 / a ≤ 1 / -1, by rwa one_div_neg_one_eq_neg_one at this,
one_div_le_one_div_of_neg_of_le h1 h2 | lemma | one_div_le_neg_one | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"one_div_le_one_div_of_neg_of_le",
"one_div_neg_one_eq_neg_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self_div_two (a : α) : a - a / 2 = a / 2 | suffices a / 2 + a / 2 - a / 2 = a / 2, by rwa add_halves at this,
by rw [add_sub_cancel] | lemma | sub_self_div_two | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"add_halves"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_two_sub_self (a : α) : a / 2 - a = - (a / 2) | suffices a / 2 - (a / 2 + a / 2) = - (a / 2), by rwa add_halves at this,
by rw [sub_add_eq_sub_sub, sub_self, zero_sub] | lemma | div_two_sub_self | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"add_halves"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_sub_div_two_lt (h : a < b) : a + (b - a) / 2 < b | begin
rwa [← div_sub_div_same, sub_eq_add_neg, add_comm (b/2), ← add_assoc, ← sub_eq_add_neg,
← lt_sub_iff_add_lt, sub_self_div_two, sub_self_div_two, div_lt_div_right (zero_lt_two' α)]
end | lemma | add_sub_div_two_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_lt_div_right",
"div_sub_div_same",
"sub_self_div_two",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_one_div_inv_le_two (a2 : 2 ≤ a) : (1 - 1 / a)⁻¹ ≤ 2 | begin
-- Take inverses on both sides to obtain `2⁻¹ ≤ 1 - 1 / a`
refine (inv_le_inv_of_le (inv_pos.2 $ zero_lt_two' α) _).trans_eq (inv_inv (2 : α)),
-- move `1 / a` to the left and `1 - 1 / 2 = 1 / 2` to the right to obtain `1 / a ≤ ⅟ 2`
refine (le_sub_iff_add_le.2 (_ : _ + 2⁻¹ = _ ).le).trans ((sub_le_sub_iff... | lemma | sub_one_div_inv_le_two | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"inv_inv",
"inv_le_inv_of_le",
"mul_inv_cancel",
"one_div",
"two_mul",
"two_ne_zero",
"zero_lt_two",
"zero_lt_two'"
] | An inequality involving `2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_lub.mul_left {s : set α} (ha : 0 ≤ a) (hs : is_lub s b) :
is_lub ((λ b, a * b) '' s) (a * b) | begin
rcases lt_or_eq_of_le ha with ha | rfl,
{ exact (order_iso.mul_left₀ _ ha).is_lub_image'.2 hs, },
{ simp_rw zero_mul,
rw hs.nonempty.image_const,
exact is_lub_singleton },
end | lemma | is_lub.mul_left | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"is_lub",
"is_lub_singleton",
"order_iso.mul_left₀",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.mul_right {s : set α} (ha : 0 ≤ a) (hs : is_lub s b) :
is_lub ((λ b, b * a) '' s) (b * a) | by simpa [mul_comm] using hs.mul_left ha | lemma | is_lub.mul_right | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"is_lub",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub_mul_div_mul_neg_iff (hc : c ≠ 0) (hd : d ≠ 0) :
(a * d - b * c) / (c * d) < 0 ↔ a / c < b / d | by rw [mul_comm b c, ← div_sub_div _ _ hc hd, sub_lt_zero] | lemma | mul_sub_mul_div_mul_neg_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_sub_div",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub_mul_div_mul_nonpos_iff (hc : c ≠ 0) (hd : d ≠ 0) :
(a * d - b * c) / (c * d) ≤ 0 ↔ a / c ≤ b / d | by rw [mul_comm b c, ← div_sub_div _ _ hc hd, sub_nonpos] | lemma | mul_sub_mul_div_mul_nonpos_iff | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"div_sub_div",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_add_lt_and_pos_of_lt (h : b < a) : ∃ c, b + c < a ∧ 0 < c | ⟨(a - b) / 2, add_sub_div_two_lt h, div_pos (sub_pos_of_lt h) zero_lt_two⟩ | lemma | exists_add_lt_and_pos_of_lt | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"add_sub_div_two_lt",
"div_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_sub_le (h : ∀ ε > 0, b - ε ≤ a) : b ≤ a | begin
contrapose! h,
simpa only [and_comm ((0 : α) < _), lt_sub_iff_add_lt, gt_iff_lt]
using exists_add_lt_and_pos_of_lt h,
end | lemma | le_of_forall_sub_le | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"exists_add_lt_and_pos_of_lt",
"gt_iff_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_inj_of_nonneg (a0 : 0 ≤ a) (b0 : 0 ≤ b) : a * a = b * b ↔ a = b | mul_self_eq_mul_self_iff.trans $ or_iff_left_of_imp $
λ h, by { subst a, have : b = 0 := le_antisymm (neg_nonneg.1 a0) b0, rw [this, neg_zero] } | lemma | mul_self_inj_of_nonneg | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_div_div_right_of_nonpos (hc : c ≤ 0) (a b : α) : min (a / c) (b / c) = (max a b) / c | eq.symm $ antitone.map_max $ λ x y, div_le_div_of_nonpos_of_le hc | lemma | min_div_div_right_of_nonpos | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"antitone.map_max",
"div_le_div_of_nonpos_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_div_div_right_of_nonpos (hc : c ≤ 0) (a b : α) : max (a / c) (b / c) = (min a b) / c | eq.symm $ antitone.map_min $ λ x y, div_le_div_of_nonpos_of_le hc | lemma | max_div_div_right_of_nonpos | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"antitone.map_min",
"div_le_div_of_nonpos_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_inv (a : α) : |a⁻¹| = (|a|)⁻¹ | map_inv₀ (abs_hom : α →*₀ α) a | lemma | abs_inv | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"abs_hom",
"map_inv₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_div (a b : α) : |a / b| = |a| / |b| | map_div₀ (abs_hom : α →*₀ α) a b | lemma | abs_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"abs_hom",
"map_div₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_one_div (a : α) : |1 / a| = 1 / |a| | by rw [abs_div, abs_one] | lemma | abs_one_div | algebra.order.field | src/algebra/order/field/basic.lean | [
"order.bounds.order_iso",
"algebra.field.basic",
"algebra.order.field.defs",
"algebra.group_power.order"
] | [
"abs_div",
"abs_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_semifield (α : Type*) extends linear_ordered_comm_semiring α, semifield α | class | linear_ordered_semifield | algebra.order.field | src/algebra/order/field/defs.lean | [
"algebra.field.defs",
"algebra.order.ring.defs"
] | [
"linear_ordered_comm_semiring",
"semifield"
] | A linear ordered semifield is a field with a linear order respecting the operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_field (α : Type*) extends linear_ordered_comm_ring α, field α | class | linear_ordered_field | algebra.order.field | src/algebra/order/field/defs.lean | [
"algebra.field.defs",
"algebra.order.ring.defs"
] | [
"field",
"linear_ordered_comm_ring"
] | A linear ordered field is a field with a linear order respecting the operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_field.to_linear_ordered_semifield [linear_ordered_field α] :
linear_ordered_semifield α | { ..linear_ordered_ring.to_linear_ordered_semiring, ..‹linear_ordered_field α› } | instance | linear_ordered_field.to_linear_ordered_semifield | algebra.order.field | src/algebra/order/field/defs.lean | [
"algebra.field.defs",
"algebra.order.ring.defs"
] | [
"linear_ordered_field",
"linear_ordered_ring.to_linear_ordered_semiring",
"linear_ordered_semifield"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective.linear_ordered_semifield [linear_ordered_semifield α] [has_zero β] [has_one β]
[has_add β] [has_mul β] [has_pow β ℕ] [has_smul ℕ β] [has_nat_cast β] [has_inv β] [has_div β]
[has_pow β ℤ] [has_sup β] [has_inf β] (f : β → α) (hf : injective f) (zero : f 0 = 0)
(one : f 1 = 1) (add : ∀ x y, f (x + y) = f x... | { ..hf.linear_ordered_semiring f zero one add mul nsmul npow nat_cast hsup hinf,
..hf.semifield f zero one add mul inv div nsmul npow zpow nat_cast } | def | function.injective.linear_ordered_semifield | algebra.order.field | src/algebra/order/field/inj_surj.lean | [
"algebra.order.field.defs",
"algebra.field.basic",
"algebra.order.ring.inj_surj"
] | [
"has_inf",
"has_nat_cast",
"has_smul",
"has_sup",
"linear_ordered_semifield"
] | Pullback a `linear_ordered_semifield` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective.linear_ordered_field [linear_ordered_field α] [has_zero β] [has_one β] [has_add β]
[has_mul β] [has_neg β] [has_sub β] [has_pow β ℕ] [has_smul ℕ β] [has_smul ℤ β] [has_smul ℚ β]
[has_nat_cast β] [has_int_cast β] [has_rat_cast β] [has_inv β] [has_div β] [has_pow β ℤ]
[has_sup β] [has_inf β]
(f : β → α)... | { .. hf.linear_ordered_ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast hsup hinf,
.. hf.field f zero one add mul neg sub inv div nsmul zsmul qsmul npow zpow nat_cast int_cast
rat_cast } | def | function.injective.linear_ordered_field | algebra.order.field | src/algebra/order/field/inj_surj.lean | [
"algebra.order.field.defs",
"algebra.field.basic",
"algebra.order.ring.inj_surj"
] | [
"has_inf",
"has_int_cast",
"has_nat_cast",
"has_rat_cast",
"has_smul",
"has_sup",
"linear_ordered_field"
] | Pullback a `linear_ordered_field` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.exists_forall_pos_add_lt [has_exists_add_of_le α] [finite ι] {x y : ι → α}
(h : ∀ i, x i < y i) : ∃ ε, 0 < ε ∧ ∀ i, x i + ε < y i | begin
casesI nonempty_fintype ι,
casesI is_empty_or_nonempty ι,
{ exact ⟨1, zero_lt_one, is_empty_elim⟩ },
choose ε hε hxε using λ i, exists_pos_add_of_lt' (h i),
obtain rfl : x + ε = y := funext hxε,
have hε : 0 < finset.univ.inf' finset.univ_nonempty ε := (finset.lt_inf'_iff _).2 (λ i _, hε _),
exact ⟨_... | lemma | pi.exists_forall_pos_add_lt | algebra.order.field | src/algebra/order/field/pi.lean | [
"algebra.order.field.basic",
"data.fintype.lattice"
] | [
"finite",
"finset.inf'_le",
"finset.lt_inf'_iff",
"finset.mem_univ",
"finset.univ_nonempty",
"half_pos",
"has_exists_add_of_le",
"is_empty_or_nonempty",
"nonempty_fintype",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_of_le (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n | begin
have ha₀ : 0 < a, from one_pos.trans_le ha,
lift n - m to ℕ using sub_nonneg.2 h with k hk,
calc a ^ m = a ^ m * 1 : (mul_one _).symm
... ≤ a ^ m * a ^ k : mul_le_mul_of_nonneg_left (one_le_pow_of_one_le ha _) (zpow_nonneg ha₀.le _)
... = a ^ n : by rw [← zpow_coe_nat, ← zpow_add₀ ha₀.ne', hk, add_sub_c... | lemma | zpow_le_of_le | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"lift",
"mul_le_mul_of_nonneg_left",
"mul_one",
"one_le_pow_of_one_le",
"zpow_add₀",
"zpow_coe_nat",
"zpow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_one_of_nonpos (ha : 1 ≤ a) (hn : n ≤ 0) : a ^ n ≤ 1 | (zpow_le_of_le ha hn).trans_eq $ zpow_zero _ | lemma | zpow_le_one_of_nonpos | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_le_of_le",
"zpow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_zpow_of_nonneg (ha : 1 ≤ a) (hn : 0 ≤ n) : 1 ≤ a ^ n | (zpow_zero _).symm.trans_le $ zpow_le_of_le ha hn | lemma | one_le_zpow_of_nonneg | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_le_of_le",
"zpow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.zpow_pos_of_pos {a : ℕ} (h : 0 < a) (n : ℤ) : 0 < (a : α)^n | by { apply zpow_pos_of_pos, exact_mod_cast h } | lemma | nat.zpow_pos_of_pos | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.zpow_ne_zero_of_pos {a : ℕ} (h : 0 < a) (n : ℤ) : (a : α)^n ≠ 0 | (nat.zpow_pos_of_pos h n).ne' | lemma | nat.zpow_ne_zero_of_pos | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"nat.zpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_zpow (ha : 1 < a) : ∀ n : ℤ, 0 < n → 1 < a ^ n | | (n : ℕ) h := (zpow_coe_nat _ _).symm.subst (one_lt_pow ha $ int.coe_nat_ne_zero.mp h.ne')
| -[1+ n] h := ((int.neg_succ_not_pos _).mp h).elim | lemma | one_lt_zpow | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"int.neg_succ_not_pos",
"one_lt_pow",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_strict_mono (hx : 1 < a) : strict_mono ((^) a : ℤ → α) | strict_mono_int_of_lt_succ $ λ n,
have xpos : 0 < a, from zero_lt_one.trans hx,
calc a ^ n < a ^ n * a : lt_mul_of_one_lt_right (zpow_pos_of_pos xpos _) hx
... = a ^ (n + 1) : (zpow_add_one₀ xpos.ne' _).symm | lemma | zpow_strict_mono | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"lt_mul_of_one_lt_right",
"strict_mono",
"strict_mono_int_of_lt_succ",
"zpow_add_one₀",
"zpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_strict_anti (h₀ : 0 < a) (h₁ : a < 1) : strict_anti ((^) a : ℤ → α) | strict_anti_int_of_succ_lt $ λ n,
calc a ^ (n + 1) = a ^ n * a : zpow_add_one₀ h₀.ne' _
... < a ^ n * 1 : (mul_lt_mul_left $ zpow_pos_of_pos h₀ _).2 h₁
... = a ^ n : mul_one _ | lemma | zpow_strict_anti | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"mul_lt_mul_left",
"mul_one",
"strict_anti",
"strict_anti_int_of_succ_lt",
"zpow_add_one₀",
"zpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_lt_iff_lt (hx : 1 < a) : a ^ m < a ^ n ↔ m < n | (zpow_strict_mono hx).lt_iff_lt | lemma | zpow_lt_iff_lt | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_iff_le (hx : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n | (zpow_strict_mono hx).le_iff_le | lemma | zpow_le_iff_le | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_pow_le (ha : 0 ≤ a) (hb : 1 ≤ b) (k : ℕ) : a/b^k ≤ a | div_le_self ha $ one_le_pow_of_one_le hb _ | lemma | div_pow_le | 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_self",
"one_le_pow_of_one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_injective (h₀ : 0 < a) (h₁ : a ≠ 1) : injective ((^) a : ℤ → α) | begin
rcases h₁.lt_or_lt with H|H,
{ exact (zpow_strict_anti h₀ H).injective },
{ exact (zpow_strict_mono H).injective }
end | lemma | zpow_injective | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_strict_anti",
"zpow_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_inj (h₀ : 0 < a) (h₁ : a ≠ 1) : a ^ m = a ^ n ↔ m = n | (zpow_injective h₀ h₁).eq_iff | lemma | zpow_inj | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_max_of_min_le {x : α} (hx : 1 ≤ x) {a b c : ℤ} (h : min a b ≤ c) :
x ^ -c ≤ max (x ^ -a) (x ^ -b) | begin
have : antitone (λ n : ℤ, x ^ -n) := λ m n h, zpow_le_of_le hx (neg_le_neg h),
exact (this h).trans_eq this.map_min,
end | lemma | zpow_le_max_of_min_le | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"antitone",
"zpow_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_max_iff_min_le {x : α} (hx : 1 < x) {a b c : ℤ} :
x ^ -c ≤ max (x ^ -a) (x ^ -b) ↔ min a b ≤ c | by simp_rw [le_max_iff, min_le_iff, zpow_le_iff_le hx, neg_le_neg_iff] | lemma | zpow_le_max_iff_min_le | 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_max_iff",
"min_le_iff",
"zpow_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit0_nonneg (a : α) (n : ℤ) : 0 ≤ a ^ bit0 n | (mul_self_nonneg _).trans_eq $ (zpow_bit0 _ _).symm | lemma | zpow_bit0_nonneg | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"mul_self_nonneg",
"zpow_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_two_nonneg (a : α) : 0 ≤ a ^ (2 : ℤ) | zpow_bit0_nonneg _ _ | lemma | zpow_two_nonneg | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_bit0_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_neg_two_nonneg (a : α) : 0 ≤ a ^ (-2 : ℤ) | zpow_bit0_nonneg _ (-1) | lemma | zpow_neg_two_nonneg | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_bit0_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit0_pos (h : a ≠ 0) (n : ℤ) : 0 < a ^ bit0 n | (zpow_bit0_nonneg a n).lt_of_ne (zpow_ne_zero _ h).symm | lemma | zpow_bit0_pos | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_bit0_nonneg",
"zpow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_two_pos_of_ne_zero (h : a ≠ 0) : 0 < a ^ (2 : ℤ) | zpow_bit0_pos h _ | lemma | zpow_two_pos_of_ne_zero | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_bit0_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit0_pos_iff (hn : n ≠ 0) : 0 < a ^ bit0 n ↔ a ≠ 0 | ⟨by { rintro h rfl, refine (zero_zpow _ _).not_gt h, rwa bit0_ne_zero }, λ h, zpow_bit0_pos h _⟩ | lemma | zpow_bit0_pos_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"bit0_ne_zero",
"zero_zpow",
"zpow_bit0_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit1_neg_iff : a ^ bit1 n < 0 ↔ a < 0 | ⟨λ h, not_le.1 $ λ h', not_le.2 h $ zpow_nonneg h' _,
λ h, by rw [bit1, zpow_add_one₀ h.ne]; exact mul_neg_of_pos_of_neg (zpow_bit0_pos h.ne _) h⟩ | lemma | zpow_bit1_neg_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"mul_neg_of_pos_of_neg",
"zpow_add_one₀",
"zpow_bit0_pos",
"zpow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit1_nonneg_iff : 0 ≤ a ^ bit1 n ↔ 0 ≤ a | le_iff_le_iff_lt_iff_lt.2 zpow_bit1_neg_iff | lemma | zpow_bit1_nonneg_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_bit1_neg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit1_nonpos_iff : a ^ bit1 n ≤ 0 ↔ a ≤ 0 | by rw [le_iff_lt_or_eq, le_iff_lt_or_eq, zpow_bit1_neg_iff, zpow_eq_zero_iff (int.bit1_ne_zero n)] | lemma | zpow_bit1_nonpos_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"int.bit1_ne_zero",
"zpow_bit1_neg_iff",
"zpow_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit1_pos_iff : 0 < a ^ bit1 n ↔ 0 < a | lt_iff_lt_of_le_iff_le zpow_bit1_nonpos_iff | lemma | zpow_bit1_pos_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"lt_iff_lt_of_le_iff_le",
"zpow_bit1_nonpos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.zpow_nonneg (hn : even n) (a : α) : 0 ≤ a ^ n | by obtain ⟨k, rfl⟩ := hn; exact zpow_bit0_nonneg _ _ | lemma | even.zpow_nonneg | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_bit0_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.zpow_pos_iff (hn : even n) (h : n ≠ 0) : 0 < a ^ n ↔ a ≠ 0 | by obtain ⟨k, rfl⟩ := hn; exact zpow_bit0_pos_iff (by rintro rfl; simpa using h) | lemma | even.zpow_pos_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"zpow_bit0_pos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.zpow_neg_iff (hn : odd n) : a ^ n < 0 ↔ a < 0 | by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit1_neg_iff | lemma | odd.zpow_neg_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"odd",
"two_mul",
"zpow_bit1_neg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.zpow_nonneg_iff (hn : odd n) : 0 ≤ a ^ n ↔ 0 ≤ a | by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit1_nonneg_iff | lemma | odd.zpow_nonneg_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"odd",
"two_mul",
"zpow_bit1_nonneg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.zpow_nonpos_iff (hn : odd n) : a ^ n ≤ 0 ↔ a ≤ 0 | by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit1_nonpos_iff | lemma | odd.zpow_nonpos_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"odd",
"two_mul",
"zpow_bit1_nonpos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.zpow_pos_iff (hn : odd n) : 0 < a ^ n ↔ 0 < a | by cases hn with k hk; simpa only [hk, two_mul] using zpow_bit1_pos_iff | lemma | odd.zpow_pos_iff | algebra.order.field | src/algebra/order/field/power.lean | [
"algebra.parity",
"algebra.char_zero.lemmas",
"algebra.group_with_zero.power",
"algebra.order.field.basic"
] | [
"odd",
"two_mul",
"zpow_bit1_pos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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