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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|---|---|---|---|---|---|---|---|---|---|---|
right.one_le_inv_iff :
1 ≤ a⁻¹ ↔ a ≤ 1 | by { rw [← mul_le_mul_iff_right a], simp } | lemma | right.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_right"
] | Uses `right` co(ntra)variant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_le_iff_one_le_mul : a⁻¹ ≤ b ↔ 1 ≤ b * a | (mul_le_mul_iff_right a).symm.trans $ by rw inv_mul_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"
] | [
"inv_mul_self",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_inv_iff_mul_le_one_right : a ≤ b⁻¹ ↔ a * b ≤ 1 | (mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_self | lemma | le_inv_iff_mul_le_one_right | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_self",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_iff_le_mul : a * b⁻¹ ≤ c ↔ a ≤ c * b | (mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right | lemma | mul_inv_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"
] | [
"inv_mul_cancel_right",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_inv_iff_mul_le : c ≤ a * b⁻¹ ↔ c * b ≤ a | (mul_le_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right | lemma | le_mul_inv_iff_mul_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_cancel_right",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_one_iff_le : a * b⁻¹ ≤ 1 ↔ a ≤ b | mul_inv_le_iff_le_mul.trans $ by rw one_mul | lemma | mul_inv_le_one_iff_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_inv_iff_le : 1 ≤ a * b⁻¹ ↔ b ≤ a | by rw [← mul_le_mul_iff_right b, one_mul, inv_mul_cancel_right] | lemma | le_mul_inv_iff_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_cancel_right",
"mul_le_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_one_iff : b * a⁻¹ ≤ 1 ↔ b ≤ a | trans (mul_inv_le_iff_le_mul) $ by rw one_mul | lemma | mul_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_inv_le_iff_le_mul",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.inv_lt_one_iff :
a⁻¹ < 1 ↔ 1 < a | by rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul] | lemma | right.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"
] | [
"inv_mul_self",
"mul_lt_mul_iff_right",
"one_mul"
] | Uses `right` co(ntra)variant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.one_lt_inv_iff :
1 < a⁻¹ ↔ a < 1 | by rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul] | lemma | right.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"
] | [
"inv_mul_self",
"mul_lt_mul_iff_right",
"one_mul"
] | Uses `right` co(ntra)variant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_lt_iff_one_lt_mul : a⁻¹ < b ↔ 1 < b * a | (mul_lt_mul_iff_right a).symm.trans $ by rw inv_mul_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"
] | [
"inv_mul_self",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_inv_iff_mul_lt_one : a < b⁻¹ ↔ a * b < 1 | (mul_lt_mul_iff_right b).symm.trans $ by rw inv_mul_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"
] | [
"inv_mul_self",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_lt_iff_lt_mul : a * b⁻¹ < c ↔ a < c * b | by rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right] | lemma | mul_inv_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"
] | [
"inv_mul_cancel_right",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_inv_iff_mul_lt : c < a * b⁻¹ ↔ c * b < a | (mul_lt_mul_iff_right b).symm.trans $ by rw inv_mul_cancel_right | lemma | lt_mul_inv_iff_mul_lt | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_cancel_right",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_lt_one_iff_lt : a * b⁻¹ < 1 ↔ a < b | by rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right, one_mul] | lemma | inv_mul_lt_one_iff_lt | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_cancel_right",
"mul_lt_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_inv_iff_lt : 1 < a * b⁻¹ ↔ b < a | by rw [← mul_lt_mul_iff_right b, one_mul, inv_mul_cancel_right] | lemma | lt_mul_inv_iff_lt | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_cancel_right",
"mul_lt_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_lt_one_iff : b * a⁻¹ < 1 ↔ b < a | trans (mul_inv_lt_iff_lt_mul) $ by rw one_mul | lemma | mul_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_lt_iff_lt_mul",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a | by { rw [← mul_le_mul_iff_left a, ← mul_le_mul_iff_right b], simp } | lemma | inv_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",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_inv_mul_iff : a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b | by rw [← mul_le_mul_iff_left d, ← mul_le_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right] | lemma | mul_inv_le_inv_mul_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_cancel_right",
"mul_assoc",
"mul_inv_cancel_left",
"mul_le_mul_iff_left",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_self_iff (a : α) {b : α} : a / b ≤ a ↔ 1 ≤ b | by simp [div_eq_mul_inv] | lemma | div_le_self_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_self_iff (a : α) {b : α} : a ≤ a / b ↔ b ≤ 1 | by simp [div_eq_mul_inv] | lemma | le_div_self_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a | by { rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b], simp } | lemma | inv_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_lt_mul_iff_left",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt' : a⁻¹ < b ↔ b⁻¹ < a | by rw [← inv_lt_inv_iff, inv_inv] | lemma | inv_lt' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_inv",
"inv_lt_inv_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_inv' : a < b⁻¹ ↔ b < a⁻¹ | by rw [← inv_lt_inv_iff, inv_inv] | lemma | lt_inv' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_inv",
"inv_lt_inv_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_lt_inv_mul_iff : a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b | by rw [← mul_lt_mul_iff_left d, ← mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right] | lemma | mul_inv_lt_inv_mul_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_mul_cancel_right",
"mul_assoc",
"mul_inv_cancel_left",
"mul_lt_mul_iff_left",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_self_iff (a : α) {b : α} : a / b < a ↔ 1 < b | by simp [div_eq_mul_inv] | lemma | div_lt_self_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a | le_trans (left.inv_le_one_iff.mpr h) h | lemma | left.inv_le_self | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ | le_trans h (left.one_le_inv_iff.mpr h) | lemma | left.self_le_inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.inv_lt_self (h : 1 < a) : a⁻¹ < a | (left.inv_lt_one_iff.mpr h).trans h | lemma | left.inv_lt_self | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.self_lt_inv (h : a < 1) : a < a⁻¹ | lt_trans h (left.one_lt_inv_iff.mpr h) | lemma | left.self_lt_inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a | le_trans (right.inv_le_one_iff.mpr h) h | lemma | right.inv_le_self | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ | le_trans h (right.one_le_inv_iff.mpr h) | lemma | right.self_le_inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.inv_lt_self (h : 1 < a) : a⁻¹ < a | (right.inv_lt_one_iff.mpr h).trans h | lemma | right.inv_lt_self | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.self_lt_inv (h : a < 1) : a < a⁻¹ | lt_trans h (right.one_lt_inv_iff.mpr h) | lemma | right.self_lt_inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_le_iff_le_mul' : c⁻¹ * a ≤ b ↔ a ≤ b * c | by rw [inv_mul_le_iff_le_mul, mul_comm] | 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"
] | [
"inv_mul_le_iff_le_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_iff_le_mul' : a * b⁻¹ ≤ c ↔ a ≤ b * c | by rw [← inv_mul_le_iff_le_mul, mul_comm] | lemma | mul_inv_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"
] | [
"inv_mul_le_iff_le_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_le_mul_inv_iff' : a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b | by rw [mul_comm c, mul_inv_le_inv_mul_iff, mul_comm] | lemma | mul_inv_le_mul_inv_iff' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_comm",
"mul_inv_le_inv_mul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_lt_iff_lt_mul' : c⁻¹ * a < b ↔ a < b * c | by rw [inv_mul_lt_iff_lt_mul, mul_comm] | 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"
] | [
"inv_mul_lt_iff_lt_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_lt_iff_le_mul' : a * b⁻¹ < c ↔ a < b * c | by rw [← inv_mul_lt_iff_lt_mul, mul_comm] | lemma | mul_inv_lt_iff_le_mul' | 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_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_lt_mul_inv_iff' : a * b⁻¹ < c * d⁻¹ ↔ a * d < c * b | by rw [mul_comm c, mul_inv_lt_inv_mul_iff, mul_comm] | lemma | mul_inv_lt_mul_inv_iff' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_comm",
"mul_inv_lt_inv_mul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_iff_right (c : α) : a / c ≤ b / c ↔ a ≤ b | by simpa only [div_eq_mul_inv] using mul_le_mul_iff_right _ | lemma | div_le_div_iff_right | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_right' (h : a ≤ b) (c : α) : a / c ≤ b / c | (div_le_div_iff_right c).2 h | lemma | div_le_div_right' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_le_div_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_div' : 1 ≤ a / b ↔ b ≤ a | by rw [← mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right] | lemma | one_le_div' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right",
"mul_le_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_one' : a / b ≤ 1 ↔ a ≤ b | by rw [← mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right] | lemma | div_le_one' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right",
"mul_le_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff_mul_le : a ≤ c / b ↔ a * b ≤ c | by rw [← mul_le_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right] | lemma | le_div_iff_mul_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_iff_le_mul : a / c ≤ b ↔ a ≤ b * c | by rw [← mul_le_mul_iff_right c, div_eq_mul_inv, inv_mul_cancel_right] | lemma | div_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"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group.to_has_ordered_sub {α : Type*} [add_group α] [has_le α]
[covariant_class α α (swap (+)) (≤)] : has_ordered_sub α | ⟨λ a b c, sub_le_iff_le_add⟩ | instance | add_group.to_has_ordered_sub | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"add_group",
"covariant_class",
"has_ordered_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_iff_left (a : α) : a / b ≤ a / c ↔ c ≤ b | by rw [div_eq_mul_inv, div_eq_mul_inv, ← mul_le_mul_iff_left a⁻¹, inv_mul_cancel_left,
inv_mul_cancel_left, inv_le_inv_iff] | lemma | div_le_div_iff_left | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_le_inv_iff",
"inv_mul_cancel_left",
"mul_le_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_left' (h : a ≤ b) (c : α) : c / b ≤ c / a | (div_le_div_iff_left c).2 h | lemma | div_le_div_left' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_le_div_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_iff' : a / b ≤ c / d ↔ a * d ≤ c * b | by simpa only [div_eq_mul_inv] using mul_inv_le_mul_inv_iff' | lemma | div_le_div_iff' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"mul_inv_le_mul_inv_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff_mul_le' : b ≤ c / a ↔ a * b ≤ c | by rw [le_div_iff_mul_le, mul_comm] | lemma | le_div_iff_mul_le' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"le_div_iff_mul_le",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_iff_le_mul' : a / b ≤ c ↔ a ≤ b * c | by rw [div_le_iff_le_mul, mul_comm] | lemma | div_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"
] | [
"div_le_iff_le_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_div_iff_le_mul : b⁻¹ ≤ a / c ↔ c ≤ a * b | le_div_iff_mul_le.trans inv_mul_le_iff_le_mul' | lemma | inv_le_div_iff_le_mul | 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'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_div_iff_le_mul' : a⁻¹ ≤ b / c ↔ c ≤ a * b | by rw [inv_le_div_iff_le_mul, mul_comm] | lemma | inv_le_div_iff_le_mul' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"inv_le_div_iff_le_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_comm : a / b ≤ c ↔ a / c ≤ b | div_le_iff_le_mul'.trans div_le_iff_le_mul.symm | lemma | div_le_comm | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_comm : a ≤ b / c ↔ c ≤ b / a | le_div_iff_mul_le'.trans le_div_iff_mul_le.symm | lemma | le_div_comm | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div'' (hab : a ≤ b) (hcd : c ≤ d) :
a / d ≤ b / c | begin
rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_le_inv_mul_iff, mul_comm],
exact mul_le_mul' hab hcd
end | lemma | div_le_div'' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"mul_comm",
"mul_inv_le_inv_mul_iff",
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b | by simpa only [div_eq_mul_inv] using mul_lt_mul_iff_right _ | lemma | div_lt_div_iff_right | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_right' (h : a < b) (c : α) : a / c < b / c | (div_lt_div_iff_right c).2 h | lemma | div_lt_div_right' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_lt_div_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_div' : 1 < a / b ↔ b < a | by rw [← mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right] | lemma | one_lt_div' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right",
"mul_lt_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_one' : a / b < 1 ↔ a < b | by rw [← mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right] | lemma | div_lt_one' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right",
"mul_lt_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_div_iff_mul_lt : a < c / b ↔ a * b < c | by rw [← mul_lt_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right] | lemma | lt_div_iff_mul_lt | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_iff_lt_mul : a / c < b ↔ a < b * c | by rw [← mul_lt_mul_iff_right c, div_eq_mul_inv, inv_mul_cancel_right] | lemma | div_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"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right",
"mul_lt_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_iff_left (a : α) : a / b < a / c ↔ c < b | by rw [div_eq_mul_inv, div_eq_mul_inv, ← mul_lt_mul_iff_left a⁻¹, inv_mul_cancel_left,
inv_mul_cancel_left, inv_lt_inv_iff] | lemma | div_lt_div_iff_left | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_lt_inv_iff",
"inv_mul_cancel_left",
"mul_lt_mul_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_div_iff_lt_mul : a⁻¹ < b / c ↔ c < a * b | by rw [div_eq_mul_inv, lt_mul_inv_iff_mul_lt, inv_mul_lt_iff_lt_mul] | lemma | inv_lt_div_iff_lt_mul | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"inv_mul_lt_iff_lt_mul",
"lt_mul_inv_iff_mul_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_left' (h : a < b) (c : α) : c / b < c / a | (div_lt_div_iff_left c).2 h | lemma | div_lt_div_left' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_lt_div_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div_iff' : a / b < c / d ↔ a * d < c * b | by simpa only [div_eq_mul_inv] using mul_inv_lt_mul_inv_iff' | lemma | div_lt_div_iff' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"mul_inv_lt_mul_inv_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_div_iff_mul_lt' : b < c / a ↔ a * b < c | by rw [lt_div_iff_mul_lt, mul_comm] | lemma | lt_div_iff_mul_lt' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"lt_div_iff_mul_lt",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_iff_lt_mul' : a / b < c ↔ a < b * c | by rw [div_lt_iff_lt_mul, mul_comm] | lemma | div_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"
] | [
"div_lt_iff_lt_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_div_iff_lt_mul' : b⁻¹ < a / c ↔ c < a * b | lt_div_iff_mul_lt.trans inv_mul_lt_iff_lt_mul' | lemma | inv_lt_div_iff_lt_mul' | 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'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_comm : a / b < c ↔ a / c < b | div_lt_iff_lt_mul'.trans div_lt_iff_lt_mul.symm | lemma | div_lt_comm | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_div_comm : a < b / c ↔ c < b / a | lt_div_iff_mul_lt'.trans lt_div_iff_mul_lt.symm | lemma | lt_div_comm | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt_div'' (hab : a < b) (hcd : c < d) :
a / d < b / c | begin
rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_lt_inv_mul_iff, mul_comm],
exact mul_lt_mul_of_lt_of_lt hab hcd
end | lemma | div_lt_div'' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_eq_mul_inv",
"mul_comm",
"mul_inv_lt_inv_mul_iff",
"mul_lt_mul_of_lt_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cmp_div_one' [covariant_class α α (swap (*)) (≤)] (a b : α) : cmp (a / b) 1 = cmp a b | by rw [← cmp_mul_right' _ _ b, one_mul, div_mul_cancel'] | lemma | cmp_div_one' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"cmp_mul_right'",
"covariant_class",
"div_mul_cancel'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_one_lt_lt_mul (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b | le_of_not_lt (λ h₁, lt_irrefl a (by simpa using (h _ (lt_inv_mul_iff_lt.mpr h₁)))) | lemma | le_of_forall_one_lt_lt_mul | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_forall_one_lt_lt_mul : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε | ⟨λ h ε, lt_mul_of_le_of_one_lt h, le_of_forall_one_lt_lt_mul⟩ | lemma | le_iff_forall_one_lt_lt_mul | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"lt_mul_of_le_of_one_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_inv_mul_iff [covariant_class α α (swap (*)) (≤)] :
a / b ≤ a⁻¹ * b ↔ a ≤ b | begin
rw [div_eq_mul_inv, mul_inv_le_inv_mul_iff],
exact ⟨λ h, not_lt.mp (λ k, not_lt.mpr h (mul_lt_mul_of_lt_of_lt k k)), λ h, mul_le_mul' h h⟩,
end | lemma | div_le_inv_mul_iff | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"covariant_class",
"div_eq_mul_inv",
"mul_inv_le_inv_mul_iff",
"mul_le_mul'",
"mul_lt_mul_of_lt_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_flip {α : Type*} [comm_group α] [linear_order α] [covariant_class α α (*) (≤)]
{a b : α}:
a / b ≤ b / a ↔ a ≤ b | begin
rw [div_eq_mul_inv b, mul_comm],
exact div_le_inv_mul_iff,
end | lemma | div_le_div_flip | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"comm_group",
"covariant_class",
"div_eq_mul_inv",
"div_le_inv_mul_iff",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_add_comm_group (α : Type u) extends ordered_add_comm_group α, linear_order α | class | linear_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"
] | [
"ordered_add_comm_group"
] | A linearly ordered additive commutative group is an
additive commutative group with a linear order in which
addition is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_add_comm_group_with_top (α : Type*)
extends linear_ordered_add_comm_monoid_with_top α, sub_neg_monoid α, nontrivial α | (neg_top : - (⊤ : α) = ⊤)
(add_neg_cancel : ∀ a:α, a ≠ ⊤ → a + (- a) = 0) | class | linear_ordered_add_comm_group_with_top | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"linear_ordered_add_comm_monoid_with_top",
"nontrivial",
"sub_neg_monoid"
] | A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_comm_group (α : Type u) extends ordered_comm_group α, linear_order α | class | linear_ordered_comm_group | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"ordered_comm_group"
] | A linearly ordered commutative group is a
commutative group with a linear order in which
multiplication is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_group.mul_lt_mul_left'
(a b : α) (h : a < b) (c : α) : c * a < c * b | mul_lt_mul_left' h c | lemma | linear_ordered_comm_group.mul_lt_mul_left' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"mul_lt_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_inv_eq' (h : a⁻¹ = a) : a = 1 | match lt_trichotomy a 1 with
| or.inl h₁ :=
have 1 < a, from h ▸ one_lt_inv_of_inv h₁,
absurd h₁ this.asymm
| or.inr (or.inl h₁) := h₁
| or.inr (or.inr h₁) :=
have a < 1, from h ▸ inv_lt_one'.mpr h₁,
absurd h₁ this.asymm
end | lemma | eq_one_of_inv_eq' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_one_lt' [nontrivial α] : ∃ (a:α), 1 < a | begin
obtain ⟨y, hy⟩ := decidable.exists_ne (1 : α),
cases hy.lt_or_lt,
{ exact ⟨y⁻¹, one_lt_inv'.mpr h⟩ },
{ exact ⟨y, h⟩ }
end | lemma | exists_one_lt' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"decidable.exists_ne",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_group.to_no_max_order [nontrivial α] :
no_max_order α | ⟨ begin
obtain ⟨y, hy⟩ : ∃ (a:α), 1 < a := exists_one_lt',
exact λ a, ⟨a * y, lt_mul_of_one_lt_right' a hy⟩
end ⟩ | instance | linear_ordered_comm_group.to_no_max_order | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"exists_one_lt'",
"lt_mul_of_one_lt_right'",
"no_max_order",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_group.to_no_min_order [nontrivial α] : no_min_order α | ⟨ begin
obtain ⟨y, hy⟩ : ∃ (a:α), 1 < a := exists_one_lt',
exact λ a, ⟨a / y, (div_lt_self_iff a).mpr hy⟩
end ⟩ | instance | linear_ordered_comm_group.to_no_min_order | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"div_lt_self_iff",
"exists_one_lt'",
"no_min_order",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_group.to_linear_ordered_cancel_comm_monoid :
linear_ordered_cancel_comm_monoid α | { ..‹linear_ordered_comm_group α›, ..ordered_comm_group.to_ordered_cancel_comm_monoid } | instance | linear_ordered_comm_group.to_linear_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"
] | [
"linear_ordered_cancel_comm_monoid",
"ordered_comm_group.to_ordered_cancel_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positive_cone (α : Type*) [add_comm_group α] | (nonneg : α → Prop)
(pos : α → Prop := λ a, nonneg a ∧ ¬ nonneg (-a))
(pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬ nonneg (-a) . order_laws_tac)
(zero_nonneg : nonneg 0)
(add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b))
(nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0) | structure | add_comm_group.positive_cone | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"add_comm_group"
] | A collection of elements in an `add_comm_group` designated as "non-negative".
This is useful for constructing an `ordered_add_commm_group`
by choosing a positive cone in an exisiting `add_comm_group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
total_positive_cone (α : Type*) [add_comm_group α] extends positive_cone α | (nonneg_decidable : decidable_pred nonneg)
(nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a)) | structure | add_comm_group.total_positive_cone | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"add_comm_group"
] | A positive cone in an `add_comm_group` induces a linear order if
for every `a`, either `a` or `-a` is non-negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_positive_cone {α : Type*} [add_comm_group α] (C : positive_cone α) :
ordered_add_comm_group α | { le := λ a b, C.nonneg (b - a),
lt := λ a b, C.pos (b - a),
lt_iff_le_not_le := λ a b, by simp; rw [C.pos_iff]; simp,
le_refl := λ a, by simp [C.zero_nonneg],
le_trans := λ a b c nab nbc, by simp [-sub_eq_add_neg];
rw ← sub_add_sub_cancel; exact C.add_nonneg nbc... | def | ordered_add_comm_group.mk_of_positive_cone | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"add_comm_group",
"ordered_add_comm_group"
] | Construct an `ordered_add_comm_group` by
designating a positive cone in an existing `add_comm_group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_positive_cone {α : Type*} [add_comm_group α] (C : total_positive_cone α) :
linear_ordered_add_comm_group α | { le_total := λ a b, by { convert C.nonneg_total (b - a), change C.nonneg _ = _, congr, simp, },
decidable_le := λ a b, C.nonneg_decidable _,
..ordered_add_comm_group.mk_of_positive_cone C.to_positive_cone } | def | linear_ordered_add_comm_group.mk_of_positive_cone | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"add_comm_group",
"linear_ordered_add_comm_group",
"ordered_add_comm_group.mk_of_positive_cone"
] | Construct a `linear_ordered_add_comm_group` by
designating a positive cone in an existing `add_comm_group`
such that for every `a`, either `a` or `-a` is non-negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_le_inv' : a ≤ b → b⁻¹ ≤ a⁻¹ | inv_le_inv_iff.mpr | lemma | inv_le_inv' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_inv' : a < b → b⁻¹ < a⁻¹ | inv_lt_inv_iff.mpr | lemma | inv_lt_inv' | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_lt_one_of_one_lt : 1 < a → a⁻¹ < 1 | inv_lt_one_iff_one_lt.mpr | theorem | inv_lt_one_of_one_lt | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_one_of_one_le : 1 ≤ a → a⁻¹ ≤ 1 | inv_le_one'.mpr | lemma | inv_le_one_of_one_le | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_inv_of_le_one : a ≤ 1 → 1 ≤ a⁻¹ | one_le_inv'.mpr | lemma | one_le_inv_of_le_one | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.inv (hf : monotone f) : antitone (λ x, (f x)⁻¹) | λ x y hxy, inv_le_inv_iff.2 (hf hxy) | lemma | monotone.inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"antitone",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.inv (hf : antitone f) : monotone (λ x, (f x)⁻¹) | λ x y hxy, inv_le_inv_iff.2 (hf hxy) | lemma | antitone.inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"antitone",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.inv (hf : monotone_on f s) :
antitone_on (λ x, (f x)⁻¹) s | λ x hx y hy hxy, inv_le_inv_iff.2 (hf hx hy hxy) | lemma | monotone_on.inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"antitone_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.inv (hf : antitone_on f s) :
monotone_on (λ x, (f x)⁻¹) s | λ x hx y hy hxy, inv_le_inv_iff.2 (hf hx hy hxy) | lemma | antitone_on.inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"antitone_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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