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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|---|---|---|---|---|---|---|---|---|---|---|
int_floor_nonneg_of_pos [linear_ordered_ring α] [floor_ring α] {a : α} (ha : 0 < a) :
0 ≤ ⌊a⌋ | int_floor_nonneg ha.le | lemma | tactic.int_floor_nonneg_of_pos | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"floor_ring",
"linear_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positivity_floor : expr → tactic strictness | | `(⌊%%a⌋) := do
strictness_a ← core a,
match strictness_a with
| positive p := nonnegative <$> mk_app ``int_floor_nonneg_of_pos [p]
| nonnegative p := nonnegative <$> mk_app ``int_floor_nonneg [p]
| _ := failed
end
| e := pp e >>= fail ∘ format.bracket "The expression `" "` is not o... | def | tactic.positivity_floor | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | Extension for the `positivity` tactic: `int.floor` is nonnegative if its input is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_ceil_pos [linear_ordered_semiring α] [floor_semiring α] {a : α} :
0 < a → 0 < ⌈a⌉₊ | nat.ceil_pos.2 | lemma | tactic.nat_ceil_pos | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"floor_semiring",
"linear_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_ceil_pos [linear_ordered_ring α] [floor_ring α] {a : α} : 0 < a → 0 < ⌈a⌉ | int.ceil_pos.2 | lemma | tactic.int_ceil_pos | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"floor_ring",
"linear_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positivity_ceil : expr → tactic strictness | | `(⌈%%a⌉₊) := do
positive p ← core a, -- We already know `0 ≤ n` for all `n : ℕ`
positive <$> mk_app ``nat_ceil_pos [p]
| `(⌈%%a⌉) := do
strictness_a ← core a,
match strictness_a with
| positive p := positive <$> mk_app ``int_ceil_pos [p]
| nonnegative p := nonnegative <$> mk_app ``... | def | tactic.positivity_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.ceil_nonneg"
] | Extension for the `positivity` tactic: `ceil` and `int.ceil` are positive/nonnegative if
their input is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_prod_one : (1 : nonempty_interval α).to_prod = 1 | rfl | lemma | nonempty_interval.to_prod_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_one : (1 : nonempty_interval α).fst = 1 | rfl | lemma | nonempty_interval.fst_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_one : (1 : nonempty_interval α).snd = 1 | rfl | lemma | nonempty_interval.snd_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one_interval : ((1 : nonempty_interval α) : interval α) = 1 | rfl | lemma | nonempty_interval.coe_one_interval | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval",
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_one : pure (1 : α) = 1 | rfl | lemma | nonempty_interval.pure_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_ne_bot : (1 : interval α) ≠ ⊥ | pure_ne_bot | lemma | interval.one_ne_bot | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_ne_one : (⊥ : interval α) ≠ 1 | bot_ne_pure | lemma | interval.bot_ne_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : nonempty_interval α) : set α) = 1 | coe_pure _ | lemma | nonempty_interval.coe_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem_one : (1 : α) ∈ (1 : nonempty_interval α) | ⟨le_rfl, le_rfl⟩ | lemma | nonempty_interval.one_mem_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : interval α) : set α) = 1 | Icc_self _ | lemma | interval.coe_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem_one : (1 : α) ∈ (1 : interval α) | ⟨le_rfl, le_rfl⟩ | lemma | interval.one_mem_one | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prod_mul : (s * t).to_prod = s.to_prod * t.to_prod | rfl | lemma | nonempty_interval.to_prod_mul | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_mul : (s * t).fst = s.fst * t.fst | rfl | lemma | nonempty_interval.fst_mul | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_mul : (s * t).snd = s.snd * t.snd | rfl | lemma | nonempty_interval.snd_mul | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_interval : (↑(s * t) : interval α) = s * t | rfl | lemma | nonempty_interval.coe_mul_interval | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_mul_pure : pure a * pure b = pure (a * b) | rfl | lemma | nonempty_interval.pure_mul_pure | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_mul : ⊥ * t = ⊥ | rfl | lemma | interval.bot_mul | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bot : s * ⊥ = ⊥ | option.map₂_none_right _ _ | lemma | interval.mul_bot | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"option.map₂_none_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_interval.has_nsmul [add_monoid α] [preorder α] [covariant_class α α (+) (≤)]
[covariant_class α α (swap (+)) (≤)] : has_smul ℕ (nonempty_interval α) | ⟨λ n s, ⟨(n • s.fst, n • s.snd), nsmul_le_nsmul_of_le_right s.fst_le_snd _⟩⟩ | instance | nonempty_interval.has_nsmul | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"add_monoid",
"covariant_class",
"has_smul",
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_interval.has_pow : has_pow (nonempty_interval α) ℕ | ⟨λ s n, ⟨s.to_prod ^ n, pow_le_pow_of_le_left' s.fst_le_snd _⟩⟩ | instance | nonempty_interval.has_pow | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"nonempty_interval",
"pow_le_pow_of_le_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prod_pow : (s ^ n).to_prod = s.to_prod ^ n | rfl | lemma | nonempty_interval.to_prod_pow | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_pow : (s ^ n).fst = s.fst ^ n | rfl | lemma | nonempty_interval.fst_pow | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_pow : (s ^ n).snd = s.snd ^ n | rfl | lemma | nonempty_interval.snd_pow | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_pow : pure a ^ n = pure (a ^ n) | rfl | lemma | nonempty_interval.pure_pow | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow_interval [ordered_comm_monoid α] (s : nonempty_interval α)
(n : ℕ) :
(↑(s ^ n) : interval α) = s ^ n | map_pow (⟨coe, coe_one_interval, coe_mul_interval⟩ : nonempty_interval α →* interval α) _ _ | lemma | nonempty_interval.coe_pow_interval | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval",
"map_pow",
"nonempty_interval",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_pow : ∀ {n : ℕ} (hn : n ≠ 0), (⊥ : interval α) ^ n = ⊥ | | 0 h := (h rfl).elim
| (nat.succ n) _ := bot_mul (⊥ ^ n) | lemma | interval.bot_pow | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_sub : (s - t).fst = s.fst - t.snd | rfl | lemma | nonempty_interval.fst_sub | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_sub : (s - t).snd = s.snd - t.fst | rfl | lemma | nonempty_interval.snd_sub | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub_interval : (↑(s - t) : interval α) = s - t | rfl | lemma | nonempty_interval.coe_sub_interval | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem_sub (ha : a ∈ s) (hb : b ∈ t) : a - b ∈ s - t | ⟨tsub_le_tsub ha.1 hb.2, tsub_le_tsub ha.2 hb.1⟩ | lemma | nonempty_interval.sub_mem_sub | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"tsub_le_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_sub_pure (a b : α) : pure a - pure b = pure (a - b) | rfl | lemma | nonempty_interval.pure_sub_pure | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_sub : ⊥ - t = ⊥ | rfl | lemma | interval.bot_sub | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_bot : s - ⊥ = ⊥ | option.map₂_none_right _ _ | lemma | interval.sub_bot | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"option.map₂_none_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_div : (s / t).fst = s.fst / t.snd | rfl | lemma | nonempty_interval.fst_div | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_div : (s / t).snd = s.snd / t.fst | rfl | lemma | nonempty_interval.snd_div | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div_interval : (↑(s / t) : interval α) = s / t | rfl | lemma | nonempty_interval.coe_div_interval | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mem_div (ha : a ∈ s) (hb : b ∈ t) : a / b ∈ s / t | ⟨div_le_div'' ha.1 hb.2, div_le_div'' ha.2 hb.1⟩ | lemma | nonempty_interval.div_mem_div | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"div_le_div''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_div_pure : pure a / pure b = pure (a / b) | rfl | lemma | nonempty_interval.pure_div_pure | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_div : ⊥ / t = ⊥ | rfl | lemma | interval.bot_div | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_bot : s / ⊥ = ⊥ | option.map₂_none_right _ _ | lemma | interval.div_bot | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"option.map₂_none_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_inv : s⁻¹.fst = s.snd⁻¹ | rfl | lemma | nonempty_interval.fst_inv | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_inv : s⁻¹.snd = s.fst⁻¹ | rfl | lemma | nonempty_interval.snd_inv | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv_interval : (↑(s⁻¹) : interval α) = s⁻¹ | rfl | lemma | nonempty_interval.coe_inv_interval | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mem_inv (ha : a ∈ s) : a⁻¹ ∈ s⁻¹ | ⟨inv_le_inv' ha.2, inv_le_inv' ha.1⟩ | lemma | nonempty_interval.inv_mem_inv | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"inv_le_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pure : (pure a)⁻¹ = pure a⁻¹ | rfl | lemma | nonempty_interval.inv_pure | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval.inv_bot : (⊥ : interval α)⁻¹ = ⊥ | rfl | lemma | interval.inv_bot | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff :
s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 | begin
refine ⟨λ h, _, _⟩,
{ rw [ext_iff, prod.ext_iff] at h,
have := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le,
refine ⟨s.fst, t.fst, _, _, h.1⟩; ext; try { refl },
exacts [this.1.symm, this.2.symm] },
{ rintro ⟨b, c, rfl, rfl, h⟩,
rw [pure_mul_pure, h, pure_one] }... | lemma | nonempty_interval.mul_eq_one_iff | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"mul_eq_one_iff",
"mul_le_mul_iff_of_ge",
"prod.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff :
s * t = 1 ↔ ∃ a b, s = pure a ∧ t = pure b ∧ a * b = 1 | begin
cases s,
{ simp [with_bot.none_eq_bot] },
cases t,
{ simp [with_bot.none_eq_bot] },
{ simp [with_bot.some_eq_coe, ←nonempty_interval.coe_mul_interval,
nonempty_interval.mul_eq_one_iff] }
end | lemma | interval.mul_eq_one_iff | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"mul_eq_one_iff",
"nonempty_interval.mul_eq_one_iff",
"with_bot.none_eq_bot",
"with_bot.some_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length : α | s.snd - s.fst | def | nonempty_interval.length | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | The length of an interval is its first component minus its second component. This measures the
accuracy of the approximation by an interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
length_nonneg : 0 ≤ s.length | sub_nonneg_of_le s.fst_le_snd | lemma | nonempty_interval.length_nonneg | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_pure : (pure a).length = 0 | sub_self _ | lemma | nonempty_interval.length_pure | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_zero : (0 : nonempty_interval α).length = 0 | length_pure _ | lemma | nonempty_interval.length_zero | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_neg : (-s).length = s.length | neg_sub_neg _ _ | lemma | nonempty_interval.length_neg | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_add : (s + t).length = s.length + t.length | add_sub_add_comm _ _ _ _ | lemma | nonempty_interval.length_add | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_sub : (s - t).length = s.length + t.length | by simp [sub_eq_add_neg] | lemma | nonempty_interval.length_sub | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_sum (f : ι → nonempty_interval α) (s : finset ι) :
(∑ i in s, f i).length = ∑ i in s, (f i).length | map_sum (⟨length, length_zero, length_add⟩ : nonempty_interval α →+ α) _ _ | lemma | nonempty_interval.length_sum | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"finset",
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length : interval α → α | | ⊥ := 0
| (s : nonempty_interval α) := s.length | def | interval.length | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval",
"nonempty_interval"
] | The length of an interval is its first component minus its second component. This measures the
accuracy of the approximation by an interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
length_nonneg : ∀ s : interval α, 0 ≤ s.length | | ⊥ := le_rfl
| (s : nonempty_interval α) := s.length_nonneg | lemma | interval.length_nonneg | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval",
"le_rfl",
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_pure : (pure a).length = 0 | nonempty_interval.length_pure _ | lemma | interval.length_pure | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"nonempty_interval.length_pure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_zero : (0 : interval α).length = 0 | length_pure _ | lemma | interval.length_zero | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_neg : ∀ s : interval α, (-s).length = s.length | | ⊥ := rfl
| (s : nonempty_interval α) := s.length_neg | lemma | interval.length_neg | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval",
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_add_le : ∀ s t : interval α, (s + t).length ≤ s.length + t.length | | ⊥ _ := by simp
| _ ⊥ := by simp
| (s : nonempty_interval α) (t : nonempty_interval α) := (s.length_add t).le | lemma | interval.length_add_le | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval",
"nonempty_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_sub_le : (s - t).length ≤ s.length + t.length | by simpa [sub_eq_add_neg] using length_add_le s (-t) | lemma | interval.length_sub_le | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_sum_le (f : ι → interval α) (s : finset ι) :
(∑ i in s, f i).length ≤ ∑ i in s, (f i).length | finset.le_sum_of_subadditive _ length_zero length_add_le _ _ | lemma | interval.length_sum_le | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"finset",
"interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positivity_interval_length : expr → tactic strictness | | `(nonempty_interval.length %%s) := nonnegative <$> mk_app `nonempty_interval.length_nonneg [s]
| `(interval.length %%s) := nonnegative <$> mk_app `interval.length_nonneg [s]
| e := pp e >>= fail ∘ format.bracket "The expression `"
"` isn't of the form `nonempty_interval.length s` or `interval.length s`" | def | tactic.positivity_interval_length | algebra.order | src/algebra/order/interval.lean | [
"algebra.big_operators.order",
"algebra.group.prod",
"data.option.n_ary",
"data.set.pointwise.basic",
"order.interval",
"tactic.positivity"
] | [
"interval.length_nonneg",
"nonempty_interval.length_nonneg"
] | Extension for the `positivity` tactic: The length of an interval is always nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_of_pos [invertible a] : 0 < ⅟a ↔ 0 < a | begin
have : 0 < a * ⅟a, by simp only [mul_inv_of_self, zero_lt_one],
exact ⟨λ h, pos_of_mul_pos_left this h.le, λ h, pos_of_mul_pos_right this h.le⟩
end | lemma | inv_of_pos | algebra.order | src/algebra/order/invertible.lean | [
"algebra.order.ring.defs",
"algebra.invertible"
] | [
"invertible",
"mul_inv_of_self",
"pos_of_mul_pos_left",
"pos_of_mul_pos_right",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_nonpos [invertible a] : ⅟a ≤ 0 ↔ a ≤ 0 | by simp only [← not_lt, inv_of_pos] | lemma | inv_of_nonpos | algebra.order | src/algebra/order/invertible.lean | [
"algebra.order.ring.defs",
"algebra.invertible"
] | [
"inv_of_pos",
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_nonneg [invertible a] : 0 ≤ ⅟a ↔ 0 ≤ a | begin
have : 0 < a * ⅟a, by simp only [mul_inv_of_self, zero_lt_one],
exact ⟨λ h, (pos_of_mul_pos_left this h).le, λ h, (pos_of_mul_pos_right this h).le⟩
end | lemma | inv_of_nonneg | algebra.order | src/algebra/order/invertible.lean | [
"algebra.order.ring.defs",
"algebra.invertible"
] | [
"invertible",
"mul_inv_of_self",
"pos_of_mul_pos_left",
"pos_of_mul_pos_right",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_lt_zero [invertible a] : ⅟a < 0 ↔ a < 0 | by simp only [← not_le, inv_of_nonneg] | lemma | inv_of_lt_zero | algebra.order | src/algebra/order/invertible.lean | [
"algebra.order.ring.defs",
"algebra.invertible"
] | [
"inv_of_nonneg",
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_le_one [invertible a] (h : 1 ≤ a) : ⅟a ≤ 1 | by haveI := @linear_order.decidable_le α _; exact
mul_inv_of_self a ▸ le_mul_of_one_le_left (inv_of_nonneg.2 $ zero_le_one.trans h) h | lemma | inv_of_le_one | algebra.order | src/algebra/order/invertible.lean | [
"algebra.order.ring.defs",
"algebra.invertible"
] | [
"invertible",
"le_mul_of_one_le_left",
"mul_inv_of_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
idem_semiring (α : Type u) extends semiring α, semilattice_sup α | (sup := (+))
(add_eq_sup : ∀ a b : α, a + b = a ⊔ b . try_refl_tac)
(bot : α := 0)
(bot_le : ∀ a, bot ≤ a) | class | idem_semiring | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"add_eq_sup",
"bot_le",
"semilattice_sup",
"semiring",
"try_refl_tac"
] | An idempotent semiring is a semiring with the additional property that addition is idempotent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
idem_comm_semiring (α : Type u) extends comm_semiring α, idem_semiring α | class | idem_comm_semiring | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"comm_semiring",
"idem_semiring"
] | An idempotent commutative semiring is a commutative semiring with the additional property that
addition is idempotent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_kstar (α : Type*) | (kstar : α → α) | class | has_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | Notation typeclass for the Kleene star `∗`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kleene_algebra (α : Type*) extends idem_semiring α, has_kstar α | (one_le_kstar : ∀ a : α, 1 ≤ a∗)
(mul_kstar_le_kstar : ∀ a : α, a * a∗ ≤ a∗)
(kstar_mul_le_kstar : ∀ a : α, a∗ * a ≤ a∗)
(mul_kstar_le_self : ∀ a b : α, b * a ≤ b → b * a∗ ≤ b)
(kstar_mul_le_self : ∀ a b : α, a * b ≤ b → a∗ * b ≤ b) | class | kleene_algebra | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"has_kstar",
"idem_semiring",
"kstar_mul_le_kstar",
"kstar_mul_le_self",
"mul_kstar_le_kstar",
"mul_kstar_le_self",
"one_le_kstar"
] | A Kleene Algebra is an idempotent semiring with an additional unary operator `kstar` (for Kleene
star) that satisfies the following properties:
* `1 + a * a∗ ≤ a∗`
* `1 + a∗ * a ≤ a∗`
* If `a * c + b ≤ c`, then `a∗ * b ≤ c`
* If `c * a + b ≤ c`, then `b * a∗ ≤ c` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
idem_semiring.to_order_bot [idem_semiring α] : order_bot α | { ..‹idem_semiring α› } | instance | idem_semiring.to_order_bot | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"idem_semiring",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
idem_semiring.of_semiring [semiring α] (h : ∀ a : α, a + a = a) : idem_semiring α | { le := λ a b, a + b = b,
le_refl := h,
le_trans := λ a b c (hab : _ = _) (hbc : _ = _), by { change _ = _, rw [←hbc, ←add_assoc, hab] },
le_antisymm := λ a b (hab : _ = _) (hba : _ = _), by rwa [←hba, add_comm],
sup := (+),
le_sup_left := λ a b, by { change _ = _, rw [←add_assoc, h] },
le_sup_right := λ a ... | def | idem_semiring.of_semiring | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"bot_le",
"idem_semiring",
"le_sup_left",
"le_sup_right",
"semiring",
"sup_le"
] | Construct an idempotent semiring from an idempotent addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_eq_sup (a b : α) : a + b = a ⊔ b | idem_semiring.add_eq_sup _ _ | lemma | add_eq_sup | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_idem (a : α) : a + a = a | by simp | lemma | add_idem | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_eq_self : ∀ {n : ℕ} (hn : n ≠ 0) (a : α), n • a = a | | 0 h := (h rfl).elim
| 1 h := one_nsmul
| (n + 2) h := λ a, by rw [succ_nsmul, nsmul_eq_self n.succ_ne_zero, add_idem] | lemma | nsmul_eq_self | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"add_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_left_iff_le : a + b = a ↔ b ≤ a | by simp | lemma | add_eq_left_iff_le | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_right_iff_le : a + b = b ↔ a ≤ b | by simp | lemma | add_eq_right_iff_le | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_iff : a + b ≤ c ↔ a ≤ c ∧ b ≤ c | by simp | lemma | add_le_iff | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le (ha : a ≤ c) (hb : b ≤ c) : a + b ≤ c | add_le_iff.2 ⟨ha, hb⟩ | lemma | add_le | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
idem_semiring.to_canonically_ordered_add_monoid : canonically_ordered_add_monoid α | { add_le_add_left := λ a b hbc c, by { simp_rw add_eq_sup, exact sup_le_sup_left hbc _ },
exists_add_of_le := λ a b h, ⟨b, h.add_eq_right.symm⟩,
le_self_add := λ a b, add_eq_right_iff_le.1 $ by rw [←add_assoc, add_idem],
..‹idem_semiring α› } | instance | idem_semiring.to_canonically_ordered_add_monoid | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"add_eq_sup",
"add_idem",
"canonically_ordered_add_monoid",
"sup_le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
idem_semiring.to_covariant_class_mul_le : covariant_class α α (*) (≤) | ⟨λ a b c hbc, add_eq_left_iff_le.1 $ by rw [←mul_add, hbc.add_eq_left]⟩ | instance | idem_semiring.to_covariant_class_mul_le | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
idem_semiring.to_covariant_class_swap_mul_le : covariant_class α α (swap (*)) (≤) | ⟨λ a b c hbc, add_eq_left_iff_le.1 $ by rw [←add_mul, hbc.add_eq_left]⟩ | instance | idem_semiring.to_covariant_class_swap_mul_le | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_kstar : 1 ≤ a∗ | kleene_algebra.one_le_kstar _ | lemma | one_le_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_kstar_le_kstar : a * a∗ ≤ a∗ | kleene_algebra.mul_kstar_le_kstar _ | lemma | mul_kstar_le_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_mul_le_kstar : a∗ * a ≤ a∗ | kleene_algebra.kstar_mul_le_kstar _ | lemma | kstar_mul_le_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_kstar_le_self : b * a ≤ b → b * a∗ ≤ b | kleene_algebra.mul_kstar_le_self _ _ | lemma | mul_kstar_le_self | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_mul_le_self : a * b ≤ b → a∗ * b ≤ b | kleene_algebra.kstar_mul_le_self _ _ | lemma | kstar_mul_le_self | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_kstar_le (hb : b ≤ c) (ha : c * a ≤ c) : b * a∗ ≤ c | (mul_le_mul_right' hb _).trans $ mul_kstar_le_self ha | lemma | mul_kstar_le | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"mul_kstar_le_self",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_mul_le (hb : b ≤ c) (ha : a * c ≤ c) : a∗ * b ≤ c | (mul_le_mul_left' hb _).trans $ kstar_mul_le_self ha | lemma | kstar_mul_le | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"kstar_mul_le_self",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_le_of_mul_le_left (hb : 1 ≤ b) : b * a ≤ b → a∗ ≤ b | by simpa using mul_kstar_le hb | lemma | kstar_le_of_mul_le_left | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"mul_kstar_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_le_of_mul_le_right (hb : 1 ≤ b) : a * b ≤ b → a∗ ≤ b | by simpa using kstar_mul_le hb | lemma | kstar_le_of_mul_le_right | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"kstar_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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