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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|---|---|---|---|---|---|---|---|---|---|---|
eventually_top {p : α → Prop} : (∀ᶠ x in ⊤, p x) ↔ (∀ x, p x) | iff.rfl | lemma | filter.eventually_top | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_sup {p : α → Prop} {f g : filter α} :
(∀ᶠ x in f ⊔ g, p x) ↔ (∀ᶠ x in f, p x) ∧ (∀ᶠ x in g, p x) | iff.rfl | lemma | filter.eventually_sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_Sup {p : α → Prop} {fs : set (filter α)} :
(∀ᶠ x in Sup fs, p x) ↔ (∀ f ∈ fs, ∀ᶠ x in f, p x) | iff.rfl | lemma | filter.eventually_Sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_supr {p : α → Prop} {fs : ι → filter α} :
(∀ᶠ x in (⨆ b, fs b), p x) ↔ (∀ b, ∀ᶠ x in fs b, p x) | mem_supr | lemma | filter.eventually_supr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_principal {a : set α} {p : α → Prop} :
(∀ᶠ x in 𝓟 a, p x) ↔ (∀ x ∈ a, p x) | iff.rfl | lemma | filter.eventually_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_inf {f g : filter α} {p : α → Prop} :
(∀ᶠ x in f ⊓ g, p x) ↔ ∃ (s ∈ f) (t ∈ g), ∀ x ∈ s ∩ t, p x | mem_inf_iff_superset | lemma | filter.eventually_inf | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_inf_principal {f : filter α} {p : α → Prop} {s : set α} :
(∀ᶠ x in f ⊓ 𝓟 s, p x) ↔ ∀ᶠ x in f, x ∈ s → p x | mem_inf_principal | theorem | filter.eventually_inf_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently (p : α → Prop) (f : filter α) : Prop | ¬∀ᶠ x in f, ¬p x | def | filter.frequently | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | `f.frequently p` or `∃ᶠ x in f, p x` mean that `{x | ¬p x} ∉ f`. E.g., `∃ᶠ x in at_top, p x`
means that there exist arbitrarily large `x` for which `p` holds true. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually.frequently {f : filter α} [ne_bot f] {p : α → Prop} (h : ∀ᶠ x in f, p x) :
∃ᶠ x in f, p x | compl_not_mem h | lemma | filter.eventually.frequently | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_of_forall {f : filter α} [ne_bot f] {p : α → Prop} (h : ∀ x, p x) :
∃ᶠ x in f, p x | eventually.frequently (eventually_of_forall h) | lemma | filter.frequently_of_forall | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently.mp {p q : α → Prop} {f : filter α} (h : ∃ᶠ x in f, p x)
(hpq : ∀ᶠ x in f, p x → q x) :
∃ᶠ x in f, q x | mt (λ hq, hq.mp $ hpq.mono $ λ x, mt) h | lemma | filter.frequently.mp | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently.filter_mono {p : α → Prop} {f g : filter α} (h : ∃ᶠ x in f, p x) (hle : f ≤ g) :
∃ᶠ x in g, p x | mt (λ h', h'.filter_mono hle) h | lemma | filter.frequently.filter_mono | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently.mono {p q : α → Prop} {f : filter α} (h : ∃ᶠ x in f, p x)
(hpq : ∀ x, p x → q x) :
∃ᶠ x in f, q x | h.mp (eventually_of_forall hpq) | lemma | filter.frequently.mono | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently.and_eventually {p q : α → Prop} {f : filter α}
(hp : ∃ᶠ x in f, p x) (hq : ∀ᶠ x in f, q x) :
∃ᶠ x in f, p x ∧ q x | begin
refine mt (λ h, hq.mp $ h.mono _) hp,
exact λ x hpq hq hp, hpq ⟨hp, hq⟩
end | lemma | filter.frequently.and_eventually | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.and_frequently {p q : α → Prop} {f : filter α}
(hp : ∀ᶠ x in f, p x) (hq : ∃ᶠ x in f, q x) :
∃ᶠ x in f, p x ∧ q x | by simpa only [and.comm] using hq.and_eventually hp | lemma | filter.eventually.and_frequently | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently.exists {p : α → Prop} {f : filter α} (hp : ∃ᶠ x in f, p x) : ∃ x, p x | begin
by_contradiction H,
replace H : ∀ᶠ x in f, ¬ p x, from eventually_of_forall (not_exists.1 H),
exact hp H
end | lemma | filter.frequently.exists | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"by_contradiction",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.exists {p : α → Prop} {f : filter α} [ne_bot f] (hp : ∀ᶠ x in f, p x) :
∃ x, p x | hp.frequently.exists | lemma | filter.eventually.exists | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_iff_forall_eventually_exists_and {p : α → Prop} {f : filter α} :
(∃ᶠ x in f, p x) ↔ ∀ {q : α → Prop}, (∀ᶠ x in f, q x) → ∃ x, p x ∧ q x | ⟨λ hp q hq, (hp.and_eventually hq).exists,
λ H hp, by simpa only [and_not_self, exists_false] using H hp⟩ | lemma | filter.frequently_iff_forall_eventually_exists_and | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"exists_false",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_iff {f : filter α} {P : α → Prop} :
(∃ᶠ x in f, P x) ↔ ∀ {U}, U ∈ f → ∃ x ∈ U, P x | begin
simp only [frequently_iff_forall_eventually_exists_and, exists_prop, and_comm (P _)],
refl
end | lemma | filter.frequently_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"exists_prop",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_eventually {p : α → Prop} {f : filter α} :
(¬ ∀ᶠ x in f, p x) ↔ (∃ᶠ x in f, ¬ p x) | by simp [filter.frequently] | lemma | filter.not_eventually | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.frequently"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_frequently {p : α → Prop} {f : filter α} :
(¬ ∃ᶠ x in f, p x) ↔ (∀ᶠ x in f, ¬ p x) | by simp only [filter.frequently, not_not] | lemma | filter.not_frequently | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.frequently",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_true_iff_ne_bot (f : filter α) : (∃ᶠ x in f, true) ↔ ne_bot f | by simp [filter.frequently, -not_eventually, eventually_false_iff_eq_bot, ne_bot_iff] | lemma | filter.frequently_true_iff_ne_bot | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.frequently"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_false (f : filter α) : ¬ ∃ᶠ x in f, false | by simp | lemma | filter.frequently_false | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_const {f : filter α} [ne_bot f] {p : Prop} :
(∃ᶠ x in f, p) ↔ p | classical.by_cases (λ h : p, by simpa [h]) (λ h, by simp [h]) | lemma | filter.frequently_const | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_or_distrib {f : filter α} {p q : α → Prop} :
(∃ᶠ x in f, p x ∨ q x) ↔ (∃ᶠ x in f, p x) ∨ (∃ᶠ x in f, q x) | by simp only [filter.frequently, ← not_and_distrib, not_or_distrib, eventually_and] | lemma | filter.frequently_or_distrib | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.frequently",
"not_and_distrib",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_or_distrib_left {f : filter α} [ne_bot f] {p : Prop} {q : α → Prop} :
(∃ᶠ x in f, p ∨ q x) ↔ (p ∨ ∃ᶠ x in f, q x) | by simp | lemma | filter.frequently_or_distrib_left | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_or_distrib_right {f : filter α} [ne_bot f] {p : α → Prop} {q : Prop} :
(∃ᶠ x in f, p x ∨ q) ↔ (∃ᶠ x in f, p x) ∨ q | by simp | lemma | filter.frequently_or_distrib_right | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_imp_distrib {f : filter α} {p q : α → Prop} :
(∃ᶠ x in f, p x → q x) ↔ ((∀ᶠ x in f, p x) → ∃ᶠ x in f, q x) | by simp [imp_iff_not_or, not_eventually, frequently_or_distrib] | lemma | filter.frequently_imp_distrib | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"imp_iff_not_or"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_imp_distrib_left {f : filter α} [ne_bot f] {p : Prop} {q : α → Prop} :
(∃ᶠ x in f, p → q x) ↔ (p → ∃ᶠ x in f, q x) | by simp | lemma | filter.frequently_imp_distrib_left | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_imp_distrib_right {f : filter α} [ne_bot f] {p : α → Prop} {q : Prop} :
(∃ᶠ x in f, p x → q) ↔ ((∀ᶠ x in f, p x) → q) | by simp | lemma | filter.frequently_imp_distrib_right | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_imp_distrib_right {f : filter α} {p : α → Prop} {q : Prop} :
(∀ᶠ x in f, p x → q) ↔ ((∃ᶠ x in f, p x) → q) | by simp only [imp_iff_not_or, eventually_or_distrib_right, not_frequently] | lemma | filter.eventually_imp_distrib_right | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"imp_iff_not_or"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_and_distrib_left {f : filter α} {p : Prop} {q : α → Prop} :
(∃ᶠ x in f, p ∧ q x) ↔ (p ∧ ∃ᶠ x in f, q x) | by simp only [filter.frequently, not_and, eventually_imp_distrib_left, not_imp] | lemma | filter.frequently_and_distrib_left | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.frequently",
"not_and",
"not_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_and_distrib_right {f : filter α} {p : α → Prop} {q : Prop} :
(∃ᶠ x in f, p x ∧ q) ↔ ((∃ᶠ x in f, p x) ∧ q) | by simp only [and_comm _ q, frequently_and_distrib_left] | lemma | filter.frequently_and_distrib_right | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_bot {p : α → Prop} : ¬ ∃ᶠ x in ⊥, p x | by simp | lemma | filter.frequently_bot | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_top {p : α → Prop} : (∃ᶠ x in ⊤, p x) ↔ (∃ x, p x) | by simp [filter.frequently] | lemma | filter.frequently_top | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.frequently"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_principal {a : set α} {p : α → Prop} :
(∃ᶠ x in 𝓟 a, p x) ↔ (∃ x ∈ a, p x) | by simp [filter.frequently, not_forall] | lemma | filter.frequently_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.frequently",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_sup {p : α → Prop} {f g : filter α} :
(∃ᶠ x in f ⊔ g, p x) ↔ (∃ᶠ x in f, p x) ∨ (∃ᶠ x in g, p x) | by simp only [filter.frequently, eventually_sup, not_and_distrib] | lemma | filter.frequently_sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.frequently",
"not_and_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_Sup {p : α → Prop} {fs : set (filter α)} :
(∃ᶠ x in Sup fs, p x) ↔ (∃ f ∈ fs, ∃ᶠ x in f, p x) | by simp [filter.frequently, -not_eventually, not_forall] | lemma | filter.frequently_Sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.frequently",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_supr {p : α → Prop} {fs : β → filter α} :
(∃ᶠ x in (⨆ b, fs b), p x) ↔ (∃ b, ∃ᶠ x in fs b, p x) | by simp [filter.frequently, -not_eventually, not_forall] | lemma | filter.frequently_supr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.frequently",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.choice {r : α → β → Prop} {l : filter α}
[l.ne_bot] (h : ∀ᶠ x in l, ∃ y, r x y) : ∃ f : α → β, ∀ᶠ x in l, r x (f x) | begin
classical,
use (λ x, if hx : ∃ y, r x y then classical.some hx
else classical.some (classical.some_spec h.exists)),
filter_upwards [h],
intros x hx,
rw dif_pos hx,
exact classical.some_spec hx
end | lemma | filter.eventually.choice | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq (l : filter α) (f g : α → β) : Prop | ∀ᶠ x in l, f x = g x | def | filter.eventually_eq | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | Two functions `f` and `g` are *eventually equal* along a filter `l` if the set of `x` such that
`f x = g x` belongs to `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_eq.eventually {l : filter α} {f g : α → β} (h : f =ᶠ[l] g) :
∀ᶠ x in l, f x = g x | h | lemma | filter.eventually_eq.eventually | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.rw {l : filter α} {f g : α → β} (h : f =ᶠ[l] g) (p : α → β → Prop)
(hf : ∀ᶠ x in l, p x (f x)) :
∀ᶠ x in l, p x (g x) | hf.congr $ h.mono $ λ x hx, hx ▸ iff.rfl | lemma | filter.eventually_eq.rw | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_set {s t : set α} {l : filter α} :
s =ᶠ[l] t ↔ ∀ᶠ x in l, x ∈ s ↔ x ∈ t | eventually_congr $ eventually_of_forall $ λ x, ⟨eq.to_iff, iff.to_eq⟩ | lemma | filter.eventually_eq_set | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_univ {s : set α} {l : filter α} : s =ᶠ[l] univ ↔ s ∈ l | by simp [eventually_eq_set] | lemma | filter.eventually_eq_univ | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.exists_mem {l : filter α} {f g : α → β} (h : f =ᶠ[l] g) :
∃ s ∈ l, eq_on f g s | h.exists_mem | lemma | filter.eventually_eq.exists_mem | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_of_mem {l : filter α} {f g : α → β} {s : set α}
(hs : s ∈ l) (h : eq_on f g s) : f =ᶠ[l] g | eventually_of_mem hs h | lemma | filter.eventually_eq_of_mem | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_iff_exists_mem {l : filter α} {f g : α → β} :
(f =ᶠ[l] g) ↔ ∃ s ∈ l, eq_on f g s | eventually_iff_exists_mem | lemma | filter.eventually_eq_iff_exists_mem | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.filter_mono {l l' : filter α} {f g : α → β} (h₁ : f =ᶠ[l] g) (h₂ : l' ≤ l) :
f =ᶠ[l'] g | h₂ h₁ | lemma | filter.eventually_eq.filter_mono | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.refl (l : filter α) (f : α → β) :
f =ᶠ[l] f | eventually_of_forall $ λ x, rfl | lemma | filter.eventually_eq.refl | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.rfl {l : filter α} {f : α → β} : f =ᶠ[l] f | eventually_eq.refl l f | lemma | filter.eventually_eq.rfl | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.symm {f g : α → β} {l : filter α} (H : f =ᶠ[l] g) :
g =ᶠ[l] f | H.mono $ λ _, eq.symm | lemma | filter.eventually_eq.symm | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.trans {l : filter α} {f g h : α → β}
(H₁ : f =ᶠ[l] g) (H₂ : g =ᶠ[l] h) : f =ᶠ[l] h | H₂.rw (λ x y, f x = y) H₁ | lemma | filter.eventually_eq.trans | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.prod_mk {l} {f f' : α → β} (hf : f =ᶠ[l] f') {g g' : α → γ} (hg : g =ᶠ[l] g') :
(λ x, (f x, g x)) =ᶠ[l] (λ x, (f' x, g' x)) | hf.mp $ hg.mono $ by { intros, simp only * } | lemma | filter.eventually_eq.prod_mk | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.fun_comp {f g : α → β} {l : filter α} (H : f =ᶠ[l] g) (h : β → γ) :
(h ∘ f) =ᶠ[l] (h ∘ g) | H.mono $ λ x hx, congr_arg h hx | lemma | filter.eventually_eq.fun_comp | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.comp₂ {δ} {f f' : α → β} {g g' : α → γ} {l} (Hf : f =ᶠ[l] f') (h : β → γ → δ)
(Hg : g =ᶠ[l] g') :
(λ x, h (f x) (g x)) =ᶠ[l] (λ x, h (f' x) (g' x)) | (Hf.prod_mk Hg).fun_comp (uncurry h) | lemma | filter.eventually_eq.comp₂ | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.mul [has_mul β] {f f' g g' : α → β} {l : filter α} (h : f =ᶠ[l] g)
(h' : f' =ᶠ[l] g') :
((λ x, f x * f' x) =ᶠ[l] (λ x, g x * g' x)) | h.comp₂ (*) h' | lemma | filter.eventually_eq.mul | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.inv [has_inv β] {f g : α → β} {l : filter α} (h : f =ᶠ[l] g) :
((λ x, (f x)⁻¹) =ᶠ[l] (λ x, (g x)⁻¹)) | h.fun_comp has_inv.inv | lemma | filter.eventually_eq.inv | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.div [has_div β] {f f' g g' : α → β} {l : filter α} (h : f =ᶠ[l] g)
(h' : f' =ᶠ[l] g') :
((λ x, f x / f' x) =ᶠ[l] (λ x, g x / g' x)) | h.comp₂ (/) h' | lemma | filter.eventually_eq.div | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.const_smul {𝕜} [has_smul 𝕜 β] {l : filter α} {f g : α → β}
(h : f =ᶠ[l] g) (c : 𝕜) :
(λ x, c • f x) =ᶠ[l] (λ x, c • g x) | h.fun_comp (λ x, c • x) | lemma | filter.eventually_eq.const_smul | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.smul {𝕜} [has_smul 𝕜 β] {l : filter α} {f f' : α → 𝕜}
{g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
(λ x, f x • g x) =ᶠ[l] λ x, f' x • g' x | hf.comp₂ (•) hg | lemma | filter.eventually_eq.smul | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.sup [has_sup β] {l : filter α} {f f' g g' : α → β}
(hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
(λ x, f x ⊔ g x) =ᶠ[l] λ x, f' x ⊔ g' x | hf.comp₂ (⊔) hg | lemma | filter.eventually_eq.sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"has_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.inf [has_inf β] {l : filter α} {f f' g g' : α → β}
(hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
(λ x, f x ⊓ g x) =ᶠ[l] λ x, f' x ⊓ g' x | hf.comp₂ (⊓) hg | lemma | filter.eventually_eq.inf | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"has_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.preimage {l : filter α} {f g : α → β}
(h : f =ᶠ[l] g) (s : set β) : f ⁻¹' s =ᶠ[l] g ⁻¹' s | h.fun_comp s | lemma | filter.eventually_eq.preimage | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.inter {s t s' t' : set α} {l : filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
(s ∩ s' : set α) =ᶠ[l] (t ∩ t' : set α) | h.comp₂ (∧) h' | lemma | filter.eventually_eq.inter | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.union {s t s' t' : set α} {l : filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
(s ∪ s' : set α) =ᶠ[l] (t ∪ t' : set α) | h.comp₂ (∨) h' | lemma | filter.eventually_eq.union | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.compl {s t : set α} {l : filter α} (h : s =ᶠ[l] t) :
(sᶜ : set α) =ᶠ[l] (tᶜ : set α) | h.fun_comp not | lemma | filter.eventually_eq.compl | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.diff {s t s' t' : set α} {l : filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
(s \ s' : set α) =ᶠ[l] (t \ t' : set α) | h.inter h'.compl | lemma | filter.eventually_eq.diff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_empty {s : set α} {l : filter α} :
s =ᶠ[l] (∅ : set α) ↔ ∀ᶠ x in l, x ∉ s | eventually_eq_set.trans $ by simp | lemma | filter.eventually_eq_empty | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_eventually_eq_left {s t : set α} {l : filter α} :
(s ∩ t : set α) =ᶠ[l] s ↔ ∀ᶠ x in l, x ∈ s → x ∈ t | by simp only [eventually_eq_set, mem_inter_iff, and_iff_left_iff_imp] | lemma | filter.inter_eventually_eq_left | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"and_iff_left_iff_imp",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_eventually_eq_right {s t : set α} {l : filter α} :
(s ∩ t : set α) =ᶠ[l] t ↔ ∀ᶠ x in l, x ∈ t → x ∈ s | by rw [inter_comm, inter_eventually_eq_left] | lemma | filter.inter_eventually_eq_right | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_principal {s : set α} {f g : α → β} :
f =ᶠ[𝓟 s] g ↔ eq_on f g s | iff.rfl | lemma | filter.eventually_eq_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_inf_principal_iff {F : filter α} {s : set α} {f g : α → β} :
(f =ᶠ[F ⊓ 𝓟 s] g) ↔ ∀ᶠ x in F, x ∈ s → f x = g x | eventually_inf_principal | lemma | filter.eventually_eq_inf_principal_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.sub_eq [add_group β] {f g : α → β} {l : filter α} (h : f =ᶠ[l] g) :
f - g =ᶠ[l] 0 | by simpa using (eventually_eq.sub (eventually_eq.refl l f) h).symm | lemma | filter.eventually_eq.sub_eq | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"add_group",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_iff_sub [add_group β] {f g : α → β} {l : filter α} :
f =ᶠ[l] g ↔ f - g =ᶠ[l] 0 | ⟨λ h, h.sub_eq, λ h, by simpa using h.add (eventually_eq.refl l g)⟩ | lemma | filter.eventually_eq_iff_sub | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"add_group",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le (l : filter α) (f g : α → β) : Prop | ∀ᶠ x in l, f x ≤ g x | def | filter.eventually_le | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | A function `f` is eventually less than or equal to a function `g` at a filter `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_le.congr {f f' g g' : α → β} (H : f ≤ᶠ[l] g) (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
f' ≤ᶠ[l] g' | H.mp $ hg.mp $ hf.mono $ λ x hf hg H, by rwa [hf, hg] at H | lemma | filter.eventually_le.congr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le_congr {f f' g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
f ≤ᶠ[l] g ↔ f' ≤ᶠ[l] g' | ⟨λ H, H.congr hf hg, λ H, H.congr hf.symm hg.symm⟩ | lemma | filter.eventually_le_congr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.le (h : f =ᶠ[l] g) : f ≤ᶠ[l] g | h.mono $ λ x, le_of_eq | lemma | filter.eventually_eq.le | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.refl (l : filter α) (f : α → β) :
f ≤ᶠ[l] f | eventually_eq.rfl.le | lemma | filter.eventually_le.refl | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.rfl : f ≤ᶠ[l] f | eventually_le.refl l f | lemma | filter.eventually_le.rfl | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.trans (H₁ : f ≤ᶠ[l] g) (H₂ : g ≤ᶠ[l] h) : f ≤ᶠ[l] h | H₂.mp $ H₁.mono $ λ x, le_trans | lemma | filter.eventually_le.trans | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.trans_le (H₁ : f =ᶠ[l] g) (H₂ : g ≤ᶠ[l] h) : f ≤ᶠ[l] h | H₁.le.trans H₂ | lemma | filter.eventually_eq.trans_le | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.trans_eq (H₁ : f ≤ᶠ[l] g) (H₂ : g =ᶠ[l] h) : f ≤ᶠ[l] h | H₁.trans H₂.le | lemma | filter.eventually_le.trans_eq | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.antisymm [partial_order β] {l : filter α} {f g : α → β}
(h₁ : f ≤ᶠ[l] g) (h₂ : g ≤ᶠ[l] f) :
f =ᶠ[l] g | h₂.mp $ h₁.mono $ λ x, le_antisymm | lemma | filter.eventually_le.antisymm | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le_antisymm_iff [partial_order β] {l : filter α} {f g : α → β} :
f =ᶠ[l] g ↔ f ≤ᶠ[l] g ∧ g ≤ᶠ[l] f | by simp only [eventually_eq, eventually_le, le_antisymm_iff, eventually_and] | lemma | filter.eventually_le_antisymm_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.le_iff_eq [partial_order β] {l : filter α} {f g : α → β} (h : f ≤ᶠ[l] g) :
g ≤ᶠ[l] f ↔ g =ᶠ[l] f | ⟨λ h', h'.antisymm h, eventually_eq.le⟩ | lemma | filter.eventually_le.le_iff_eq | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.ne_of_lt [preorder β] {l : filter α} {f g : α → β}
(h : ∀ᶠ x in l, f x < g x) : ∀ᶠ x in l, f x ≠ g x | h.mono (λ x hx, hx.ne) | lemma | filter.eventually.ne_of_lt | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.ne_top_of_lt [partial_order β] [order_top β] {l : filter α} {f g : α → β}
(h : ∀ᶠ x in l, f x < g x) : ∀ᶠ x in l, f x ≠ ⊤ | h.mono (λ x hx, hx.ne_top) | lemma | filter.eventually.ne_top_of_lt | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.lt_top_of_ne [partial_order β] [order_top β] {l : filter α} {f : α → β}
(h : ∀ᶠ x in l, f x ≠ ⊤) : ∀ᶠ x in l, f x < ⊤ | h.mono (λ x hx, hx.lt_top) | lemma | filter.eventually.lt_top_of_ne | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.lt_top_iff_ne_top [partial_order β] [order_top β] {l : filter α} {f : α → β} :
(∀ᶠ x in l, f x < ⊤) ↔ ∀ᶠ x in l, f x ≠ ⊤ | ⟨eventually.ne_of_lt, eventually.lt_top_of_ne⟩ | lemma | filter.eventually.lt_top_iff_ne_top | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.inter {s t s' t' : set α} {l : filter α} (h : s ≤ᶠ[l] t)
(h' : s' ≤ᶠ[l] t') :
(s ∩ s' : set α) ≤ᶠ[l] (t ∩ t' : set α) | h'.mp $ h.mono $ λ x, and.imp | lemma | filter.eventually_le.inter | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.union {s t s' t' : set α} {l : filter α} (h : s ≤ᶠ[l] t)
(h' : s' ≤ᶠ[l] t') :
(s ∪ s' : set α) ≤ᶠ[l] (t ∪ t' : set α) | h'.mp $ h.mono $ λ x, or.imp | lemma | filter.eventually_le.union | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.compl {s t : set α} {l : filter α} (h : s ≤ᶠ[l] t) :
(tᶜ : set α) ≤ᶠ[l] (sᶜ : set α) | h.mono $ λ x, mt | lemma | filter.eventually_le.compl | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.diff {s t s' t' : set α} {l : filter α} (h : s ≤ᶠ[l] t)
(h' : t' ≤ᶠ[l] s') :
(s \ s' : set α) ≤ᶠ[l] (t \ t' : set α) | h.inter h'.compl | lemma | filter.eventually_le.diff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_eventually_le_iff_mem_inf_principal {s t : set α} {l : filter α} :
s ≤ᶠ[l] t ↔ t ∈ l ⊓ 𝓟 s | mem_inf_principal.symm | lemma | filter.set_eventually_le_iff_mem_inf_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_eventually_le_iff_inf_principal_le {s t : set α} {l : filter α} :
s ≤ᶠ[l] t ↔ l ⊓ 𝓟 s ≤ l ⊓ 𝓟 t | set_eventually_le_iff_mem_inf_principal.trans $
by simp only [le_inf_iff, inf_le_left, true_and, le_principal_iff] | lemma | filter.set_eventually_le_iff_inf_principal_le | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"inf_le_left",
"le_inf_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_eventually_eq_iff_inf_principal {s t : set α} {l : filter α} :
s =ᶠ[l] t ↔ l ⊓ 𝓟 s = l ⊓ 𝓟 t | by simp only [eventually_le_antisymm_iff, le_antisymm_iff, set_eventually_le_iff_inf_principal_le] | lemma | filter.set_eventually_eq_iff_inf_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.mul_le_mul
[mul_zero_class β] [partial_order β] [pos_mul_mono β] [mul_pos_mono β]
{l : filter α} {f₁ f₂ g₁ g₂ : α → β}
(hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) (hg₀ : 0 ≤ᶠ[l] g₁) (hf₀ : 0 ≤ᶠ[l] f₂) :
f₁ * g₁ ≤ᶠ[l] f₂ * g₂ | by filter_upwards [hf, hg, hg₀, hf₀] with x using mul_le_mul | lemma | filter.eventually_le.mul_le_mul | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"mul_le_mul",
"mul_pos_mono",
"mul_zero_class",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.mul_le_mul' [has_mul β] [preorder β]
[covariant_class β β (*) (≤)] [covariant_class β β (swap (*)) (≤)]
{l : filter α} {f₁ f₂ g₁ g₂ : α → β} (hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) :
f₁ * g₁ ≤ᶠ[l] f₂ * g₂ | by filter_upwards [hf, hg] with x hfx hgx using mul_le_mul' hfx hgx | lemma | filter.eventually_le.mul_le_mul' | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"covariant_class",
"filter",
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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