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
value |
|---|---|---|---|---|---|---|---|---|---|---|
is_compact.exists_local_min_on_mem_subset {f : β → α} {s t : set β} {z : β}
(ht : is_compact t) (hf : continuous_on f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') :
∃ x ∈ s, is_local_min_on f t x | begin
obtain ⟨x, hx, hfx⟩ : ∃ x ∈ t, ∀ y ∈ t, f x ≤ f y := ht.exists_forall_le ⟨z, hz⟩ hf,
have key : ∀ ⦃y⦄, y ∈ t → (∀ z' ∈ t \ s, f y < f z') → y ∈ s := λ y hy hfy,
by { by_contra; simpa using ((hfy y ((mem_diff y).mpr ⟨hy,h⟩))) },
have h1 : ∀ z' ∈ t \ s, f x < f z' := λ z' hz', (hfx z hz).trans_lt (hfz z' ... | lemma | is_compact.exists_local_min_on_mem_subset | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"by_contra",
"continuous_on",
"eventually_nhds_within_of_forall",
"is_compact",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.exists_local_min_mem_open {f : β → α} {s t : set β} {z : β} (ht : is_compact t)
(hst : s ⊆ t) (hf : continuous_on f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
(hs : is_open s) :
∃ x ∈ s, is_local_min f x | begin
obtain ⟨x, hx, hfx⟩ := ht.exists_local_min_on_mem_subset hf hz hfz,
exact ⟨x, hx, hfx.is_local_min (filter.mem_of_superset (hs.mem_nhds hx) hst)⟩
end | lemma | is_compact.exists_local_min_mem_open | topology.algebra.order | src/topology/algebra/order/compact.lean | [
"topology.algebra.order.intermediate_value",
"topology.local_extr"
] | [
"continuous_on",
"filter.mem_of_superset",
"is_compact",
"is_local_min",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_Icc_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [regular_space β] {f : α → β} {a b : α}
{la lb : β} (hab : a ≠ b) (hf : continuous_on f (Ioo a b))
(ha : tendsto f (𝓝[>] a) (𝓝 la)) (hb : tendsto f (𝓝[<] b) (𝓝 lb)) :
continu... | begin
apply continuous_on_extend_from,
{ rw closure_Ioo hab },
{ intros x x_in,
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with rfl | rfl | h,
{ exact ⟨la, ha.mono_left $ nhds_within_mono _ Ioo_subset_Ioi_self⟩ },
{ exact ⟨lb, hb.mono_left $ nhds_within_mono _ Ioo_subset_Iio_self⟩ },
{ use [f ... | lemma | continuous_on_Icc_extend_from_Ioo | topology.algebra.order | src/topology/algebra/order/extend_from.lean | [
"topology.order.basic",
"topology.extend_from"
] | [
"closure_Ioo",
"continuous_on",
"continuous_on_extend_from",
"densely_ordered",
"extend_from",
"nhds_within_mono",
"order_topology",
"regular_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_lim_at_left_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [t2_space β] {f : α → β} {a b : α}
{la : β} (hab : a < b) (ha : tendsto f (𝓝[>] a) (𝓝 la)) :
extend_from (Ioo a b) f a = la | begin
apply extend_from_eq,
{ rw closure_Ioo hab.ne,
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc] },
{ simpa [hab] }
end | lemma | eq_lim_at_left_extend_from_Ioo | topology.algebra.order | src/topology/algebra/order/extend_from.lean | [
"topology.order.basic",
"topology.extend_from"
] | [
"closure_Ioo",
"densely_ordered",
"extend_from",
"extend_from_eq",
"order_topology",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_lim_at_right_extend_from_Ioo [topological_space α] [linear_order α] [densely_ordered α]
[order_topology α] [topological_space β] [t2_space β] {f : α → β} {a b : α}
{lb : β} (hab : a < b) (hb : tendsto f (𝓝[<] b) (𝓝 lb)) :
extend_from (Ioo a b) f b = lb | begin
apply extend_from_eq,
{ rw closure_Ioo hab.ne,
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc] },
{ simpa [hab] }
end | lemma | eq_lim_at_right_extend_from_Ioo | topology.algebra.order | src/topology/algebra/order/extend_from.lean | [
"topology.order.basic",
"topology.extend_from"
] | [
"closure_Ioo",
"densely_ordered",
"extend_from",
"extend_from_eq",
"order_topology",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_Ico_extend_from_Ioo [topological_space α]
[linear_order α] [densely_ordered α] [order_topology α] [topological_space β]
[regular_space β] {f : α → β} {a b : α} {la : β} (hab : a < b) (hf : continuous_on f (Ioo a b))
(ha : tendsto f (𝓝[>] a) (𝓝 la)) :
continuous_on (extend_from (Ioo a b) f) (Ico ... | begin
apply continuous_on_extend_from,
{ rw [closure_Ioo hab.ne], exact Ico_subset_Icc_self, },
{ intros x x_in,
rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with rfl | h,
{ use la,
simpa [hab] },
{ use [f x, hf x h] } }
end | lemma | continuous_on_Ico_extend_from_Ioo | topology.algebra.order | src/topology/algebra/order/extend_from.lean | [
"topology.order.basic",
"topology.extend_from"
] | [
"closure_Ioo",
"continuous_on",
"continuous_on_extend_from",
"densely_ordered",
"extend_from",
"order_topology",
"regular_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_Ioc_extend_from_Ioo [topological_space α]
[linear_order α] [densely_ordered α] [order_topology α] [topological_space β]
[regular_space β] {f : α → β} {a b : α} {lb : β} (hab : a < b) (hf : continuous_on f (Ioo a b))
(hb : tendsto f (𝓝[<] b) (𝓝 lb)) :
continuous_on (extend_from (Ioo a b) f) (Ioc ... | begin
have := @continuous_on_Ico_extend_from_Ioo αᵒᵈ _ _ _ _ _ _ _ f _ _ _ hab,
erw [dual_Ico, dual_Ioi, dual_Ioo] at this,
exact this hf hb
end | lemma | continuous_on_Ioc_extend_from_Ioo | topology.algebra.order | src/topology/algebra/order/extend_from.lean | [
"topology.order.basic",
"topology.extend_from"
] | [
"continuous_on",
"continuous_on_Ico_extend_from_Ioo",
"densely_ordered",
"extend_from",
"order_topology",
"regular_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.closure (h : is_max_on f s a) (hc : continuous_on f (closure s)) :
is_max_on f (closure s) a | λ x hx, continuous_within_at.closure_le hx ((hc x hx).mono subset_closure)
continuous_within_at_const h | lemma | is_max_on.closure | topology.algebra.order | src/topology/algebra/order/extr_closure.lean | [
"topology.local_extr",
"topology.order.basic"
] | [
"closure",
"continuous_on",
"continuous_within_at.closure_le",
"continuous_within_at_const",
"is_max_on",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.closure (h : is_min_on f s a) (hc : continuous_on f (closure s)) :
is_min_on f (closure s) a | h.dual.closure hc | lemma | is_min_on.closure | topology.algebra.order | src/topology/algebra/order/extr_closure.lean | [
"topology.local_extr",
"topology.order.basic"
] | [
"closure",
"continuous_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.closure (h : is_extr_on f s a) (hc : continuous_on f (closure s)) :
is_extr_on f (closure s) a | h.elim (λ h, or.inl $ h.closure hc) (λ h, or.inr $ h.closure hc) | lemma | is_extr_on.closure | topology.algebra.order | src/topology/algebra/order/extr_closure.lean | [
"topology.local_extr",
"topology.order.basic"
] | [
"closure",
"continuous_on",
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_max_on.closure (h : is_local_max_on f s a)
(hc : continuous_on f (closure s)) :
is_local_max_on f (closure s) a | begin
rcases mem_nhds_within.1 h with ⟨U, Uo, aU, hU⟩,
refine mem_nhds_within.2 ⟨U, Uo, aU, _⟩,
rintro x ⟨hxU, hxs⟩,
refine continuous_within_at.closure_le _ _ continuous_within_at_const hU,
{ rwa [mem_closure_iff_nhds_within_ne_bot, nhds_within_inter_of_mem,
← mem_closure_iff_nhds_within_ne_bot],
e... | lemma | is_local_max_on.closure | topology.algebra.order | src/topology/algebra/order/extr_closure.lean | [
"topology.local_extr",
"topology.order.basic"
] | [
"closure",
"continuous_on",
"continuous_within_at.closure_le",
"continuous_within_at_const",
"is_local_max_on",
"mem_closure_iff_nhds_within_ne_bot",
"nhds_within_inter_of_mem",
"nhds_within_le_nhds",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_min_on.closure (h : is_local_min_on f s a)
(hc : continuous_on f (closure s)) :
is_local_min_on f (closure s) a | is_local_max_on.closure h.dual hc | lemma | is_local_min_on.closure | topology.algebra.order | src/topology/algebra/order/extr_closure.lean | [
"topology.local_extr",
"topology.order.basic"
] | [
"closure",
"continuous_on",
"is_local_max_on.closure",
"is_local_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_extr_on.closure (h : is_local_extr_on f s a)
(hc : continuous_on f (closure s)) :
is_local_extr_on f (closure s) a | h.elim (λ h, or.inl $ h.closure hc) (λ h, or.inr $ h.closure hc) | lemma | is_local_extr_on.closure | topology.algebra.order | src/topology/algebra/order/extr_closure.lean | [
"topology.local_extr",
"topology.order.basic"
] | [
"closure",
"continuous_on",
"is_local_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_tendsto_nhds_zero_right (x : α) :
tendsto (uncurry ((*) : α → α → α)) (𝓝 0 ×ᶠ 𝓝 x) $ 𝓝 0 | begin
have hx : 0 < 2 * (1 + |x|) := by positivity,
rw ((nhds_basis_zero_abs_sub_lt α).prod $ nhds_basis_abs_sub_lt x).tendsto_iff
(nhds_basis_zero_abs_sub_lt α),
refine λ ε ε_pos, ⟨(ε/(2 * (1 + |x|)), 1), ⟨div_pos ε_pos hx, zero_lt_one⟩, _⟩,
suffices : ∀ (a b : α), |a| < ε / (2 * (1 + |x|)) → |b - x| < 1 ... | lemma | mul_tendsto_nhds_zero_right | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"abs_add",
"abs_mul",
"abs_nonneg",
"and_imp",
"lt_div_iff",
"mul_le_mul_of_nonneg_left",
"nhds_basis_abs_sub_lt",
"nhds_basis_zero_abs_sub_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_tendsto_nhds_zero_left (x : α) :
tendsto (uncurry ((*) : α → α → α)) (𝓝 x ×ᶠ 𝓝 0) $ 𝓝 0 | begin
intros s hs,
have := mul_tendsto_nhds_zero_right x hs,
rw [filter.mem_map, mem_prod_iff] at this ⊢,
obtain ⟨U, hU, V, hV, h⟩ := this,
exact ⟨V, hV, U, hU, λ y hy, ((mul_comm y.2 y.1) ▸
h (⟨hy.2, hy.1⟩ : (prod.mk y.2 y.1) ∈ U ×ˢ V) : y.1 * y.2 ∈ s)⟩,
end | lemma | mul_tendsto_nhds_zero_left | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"filter.mem_map",
"mul_comm",
"mul_tendsto_nhds_zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_map_mul_left_nhds_one {x₀ : α} (hx₀ : x₀ ≠ 0) :
𝓝 x₀ = map (λ x, x₀*x) (𝓝 1) | begin
have hx₀' : 0 < |x₀| := abs_pos.2 hx₀,
refine filter.ext (λ t, _),
simp only [exists_prop, set_of_subset_set_of, (nhds_basis_abs_sub_lt x₀).mem_iff,
(nhds_basis_abs_sub_lt (1 : α)).mem_iff, filter.mem_map'],
refine ⟨λ h, _, λ h, _⟩,
{ obtain ⟨i, hi, hit⟩ := h,
refine ⟨i / (|x₀|), div_pos hi (abs... | lemma | nhds_eq_map_mul_left_nhds_one | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"abs_div",
"abs_mul",
"div_lt_div_of_lt",
"div_pos",
"div_self",
"exists_prop",
"filter.ext",
"filter.mem_map'",
"le_rfl",
"mul_div_assoc'",
"mul_div_cancel_left",
"mul_lt_mul'",
"mul_one",
"nhds_basis_abs_sub_lt",
"ring",
"sub_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_map_mul_right_nhds_one {x₀ : α} (hx₀ : x₀ ≠ 0) :
𝓝 x₀ = map (λ x, x*x₀) (𝓝 1) | by simp_rw [mul_comm _ x₀, nhds_eq_map_mul_left_nhds_one hx₀] | lemma | nhds_eq_map_mul_right_nhds_one | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"mul_comm",
"nhds_eq_map_mul_left_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_tendsto_nhds_one_nhds_one :
tendsto (uncurry ((*) : α → α → α)) (𝓝 1 ×ᶠ 𝓝 1) $ 𝓝 1 | begin
rw ((nhds_basis_Ioo_pos (1 : α)).prod $ nhds_basis_Ioo_pos (1 : α)).tendsto_iff
(nhds_basis_Ioo_pos_of_pos (zero_lt_one : (0 : α) < 1)),
intros ε hε,
have hε' : 0 ≤ 1 - ε / 4 := by linarith,
have ε_pos : 0 < ε / 4 := by linarith,
have ε_pos' : 0 < ε / 2 := by linarith,
simp only [and_imp, prod.fo... | lemma | mul_tendsto_nhds_one_nhds_one | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"and_imp",
"div_le_div",
"mul_le_mul",
"mul_le_mul_left",
"mul_lt_mul''",
"mul_one",
"nhds_basis_Ioo_pos",
"nhds_basis_Ioo_pos_of_pos",
"zero_lt_one",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_field.has_continuous_mul : has_continuous_mul α | ⟨begin
rw continuous_iff_continuous_at,
rintro ⟨x₀, y₀⟩,
by_cases hx₀ : x₀ = 0,
{ rw [hx₀, continuous_at, zero_mul, nhds_prod_eq],
exact mul_tendsto_nhds_zero_right y₀ },
by_cases hy₀ : y₀ = 0,
{ rw [hy₀, continuous_at, mul_zero, nhds_prod_eq],
exact mul_tendsto_nhds_zero_left x₀ },
have hxy : x₀ ... | instance | linear_ordered_field.has_continuous_mul | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"continuous_at",
"continuous_iff_continuous_at",
"filter.map_map",
"has_continuous_mul",
"mul_assoc",
"mul_ne_zero",
"mul_tendsto_nhds_one_nhds_one",
"mul_tendsto_nhds_zero_left",
"mul_tendsto_nhds_zero_right",
"mul_zero",
"nhds_eq_map_mul_left_nhds_one",
"nhds_eq_map_mul_right_nhds_one",
"n... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.at_top_mul {C : α} (hC : 0 < C) (hf : tendsto f l at_top)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_top | begin
refine tendsto_at_top_mono' _ _ (hf.at_top_mul_const (half_pos hC)),
filter_upwards [hg.eventually (lt_mem_nhds (half_lt_self hC)),
hf.eventually (eventually_ge_at_top 0)] with x hg hf using mul_le_mul_of_nonneg_left hg.le hf,
end | lemma | filter.tendsto.at_top_mul | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"half_pos",
"lt_mem_nhds",
"mul_le_mul_of_nonneg_left"
] | In a linearly ordered field with the order topology, if `f` tends to `at_top` and `g` tends to
a positive constant `C` then `f * g` tends to `at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.mul_at_top {C : α} (hC : 0 < C) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_top) :
tendsto (λ x, (f x * g x)) l at_top | by simpa only [mul_comm] using hg.at_top_mul hC hf | lemma | filter.tendsto.mul_at_top | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"mul_comm"
] | In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and
`g` tends to `at_top` then `f * g` tends to `at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.at_top_mul_neg {C : α} (hC : C < 0) (hf : tendsto f l at_top)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_bot | by simpa only [(∘), neg_mul_eq_mul_neg, neg_neg]
using tendsto_neg_at_top_at_bot.comp (hf.at_top_mul (neg_pos.2 hC) hg.neg) | lemma | filter.tendsto.at_top_mul_neg | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"neg_mul_eq_mul_neg"
] | In a linearly ordered field with the order topology, if `f` tends to `at_top` and `g` tends to
a negative constant `C` then `f * g` tends to `at_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.neg_mul_at_top {C : α} (hC : C < 0) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_top) :
tendsto (λ x, (f x * g x)) l at_bot | by simpa only [mul_comm] using hg.at_top_mul_neg hC hf | lemma | filter.tendsto.neg_mul_at_top | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"mul_comm"
] | In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and
`g` tends to `at_top` then `f * g` tends to `at_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.at_bot_mul {C : α} (hC : 0 < C) (hf : tendsto f l at_bot)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_bot | by simpa [(∘)]
using tendsto_neg_at_top_at_bot.comp ((tendsto_neg_at_bot_at_top.comp hf).at_top_mul hC hg) | lemma | filter.tendsto.at_bot_mul | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [] | In a linearly ordered field with the order topology, if `f` tends to `at_bot` and `g` tends to
a positive constant `C` then `f * g` tends to `at_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.at_bot_mul_neg {C : α} (hC : C < 0) (hf : tendsto f l at_bot)
(hg : tendsto g l (𝓝 C)) :
tendsto (λ x, (f x * g x)) l at_top | by simpa [(∘)]
using tendsto_neg_at_bot_at_top.comp ((tendsto_neg_at_bot_at_top.comp hf).at_top_mul_neg hC hg) | lemma | filter.tendsto.at_bot_mul_neg | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [] | In a linearly ordered field with the order topology, if `f` tends to `at_bot` and `g` tends to
a negative constant `C` then `f * g` tends to `at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.mul_at_bot {C : α} (hC : 0 < C) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_bot) :
tendsto (λ x, (f x * g x)) l at_bot | by simpa only [mul_comm] using hg.at_bot_mul hC hf | lemma | filter.tendsto.mul_at_bot | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"mul_comm"
] | In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and
`g` tends to `at_bot` then `f * g` tends to `at_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.neg_mul_at_bot {C : α} (hC : C < 0) (hf : tendsto f l (𝓝 C))
(hg : tendsto g l at_bot) :
tendsto (λ x, (f x * g x)) l at_top | by simpa only [mul_comm] using hg.at_bot_mul_neg hC hf | lemma | filter.tendsto.neg_mul_at_bot | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"mul_comm"
] | In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and
`g` tends to `at_bot` then `f * g` tends to `at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_inv_zero_at_top : tendsto (λx:α, x⁻¹) (𝓝[>] (0:α)) at_top | begin
refine (at_top_basis' 1).tendsto_right_iff.2 (λ b hb, _),
have hb' : 0 < b := by positivity,
filter_upwards [Ioc_mem_nhds_within_Ioi ⟨le_rfl, inv_pos.2 hb'⟩]
with x hx using (le_inv hx.1 hb').1 hx.2,
end | lemma | tendsto_inv_zero_at_top | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"Ioc_mem_nhds_within_Ioi",
"le_inv"
] | The function `x ↦ x⁻¹` tends to `+∞` on the right of `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_inv_at_top_zero' : tendsto (λr:α, r⁻¹) at_top (𝓝[>] (0:α)) | begin
refine (has_basis.tendsto_iff at_top_basis ⟨λ s, mem_nhds_within_Ioi_iff_exists_Ioc_subset⟩).2 _,
refine λ b hb, ⟨b⁻¹, trivial, λ x hx, _⟩,
have : 0 < x := lt_of_lt_of_le (inv_pos.2 hb) hx,
exact ⟨inv_pos.2 this, (inv_le this hb).2 hx⟩
end | lemma | tendsto_inv_at_top_zero' | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"inv_le"
] | The function `r ↦ r⁻¹` tends to `0` on the right as `r → +∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_inv_at_top_zero : tendsto (λr:α, r⁻¹) at_top (𝓝 0) | tendsto_inv_at_top_zero'.mono_right inf_le_left | lemma | tendsto_inv_at_top_zero | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.div_at_top [has_continuous_mul α] {f g : β → α} {l : filter β} {a : α}
(h : tendsto f l (𝓝 a)) (hg : tendsto g l at_top) : tendsto (λ x, f x / g x) l (𝓝 0) | by { simp only [div_eq_mul_inv], exact mul_zero a ▸ h.mul (tendsto_inv_at_top_zero.comp hg) } | lemma | filter.tendsto.div_at_top | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"div_eq_mul_inv",
"filter",
"has_continuous_mul",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.inv_tendsto_at_top (h : tendsto f l at_top) : tendsto (f⁻¹) l (𝓝 0) | tendsto_inv_at_top_zero.comp h | lemma | filter.tendsto.inv_tendsto_at_top | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.inv_tendsto_zero (h : tendsto f l (𝓝[>] 0)) :
tendsto (f⁻¹) l at_top | tendsto_inv_zero_at_top.comp h | lemma | filter.tendsto.inv_tendsto_zero | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pow_neg_at_top {n : ℕ} (hn : n ≠ 0) : tendsto (λ x : α, x ^ (-(n:ℤ))) at_top (𝓝 0) | by simpa only [zpow_neg, zpow_coe_nat] using (@tendsto_pow_at_top α _ _ hn).inv_tendsto_at_top | lemma | tendsto_pow_neg_at_top | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"zpow_coe_nat",
"zpow_neg"
] | The function `x^(-n)` tends to `0` at `+∞` for any positive natural `n`.
A version for positive real powers exists as `tendsto_rpow_neg_at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_zpow_at_top_zero {n : ℤ} (hn : n < 0) :
tendsto (λ x : α, x^n) at_top (𝓝 0) | begin
lift -n to ℕ using le_of_lt (neg_pos.mpr hn) with N,
rw [← neg_pos, ← h, nat.cast_pos] at hn,
simpa only [h, neg_neg] using tendsto_pow_neg_at_top hn.ne'
end | lemma | tendsto_zpow_at_top_zero | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"lift",
"nat.cast_pos",
"tendsto_pow_neg_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_mul_zpow_at_top_zero {n : ℤ} {c : α} (hn : n < 0) :
tendsto (λ x, c * x ^ n) at_top (𝓝 0) | (mul_zero c) ▸ (filter.tendsto.const_mul c (tendsto_zpow_at_top_zero hn)) | lemma | tendsto_const_mul_zpow_at_top_zero | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"filter.tendsto.const_mul",
"mul_zero",
"tendsto_zpow_at_top_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_mul_pow_nhds_iff' {n : ℕ} {c d : α} :
tendsto (λ x : α, c * x ^ n) at_top (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d | begin
rcases eq_or_ne n 0 with (rfl|hn),
{ simp [tendsto_const_nhds_iff] },
rcases lt_trichotomy c 0 with hc|rfl|hc,
{ have := tendsto_const_mul_pow_at_bot_iff.2 ⟨hn, hc⟩,
simp [not_tendsto_nhds_of_tendsto_at_bot this, hc.ne, hn] },
{ simp [tendsto_const_nhds_iff] },
{ have := tendsto_const_mul_pow_at_t... | lemma | tendsto_const_mul_pow_nhds_iff' | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"eq_or_ne",
"not_tendsto_nhds_of_tendsto_at_bot",
"not_tendsto_nhds_of_tendsto_at_top",
"tendsto_const_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_mul_pow_nhds_iff {n : ℕ} {c d : α} (hc : c ≠ 0) :
tendsto (λ x : α, c * x ^ n) at_top (𝓝 d) ↔ n = 0 ∧ c = d | by simp [tendsto_const_mul_pow_nhds_iff', hc] | lemma | tendsto_const_mul_pow_nhds_iff | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"tendsto_const_mul_pow_nhds_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_mul_zpow_at_top_nhds_iff {n : ℤ} {c d : α} (hc : c ≠ 0) :
tendsto (λ x : α, c * x ^ n) at_top (𝓝 d) ↔ (n = 0 ∧ c = d) ∨ (n < 0 ∧ d = 0) | begin
refine ⟨λ h, _, λ h, _⟩,
{ by_cases hn : 0 ≤ n,
{ lift n to ℕ using hn,
simp only [zpow_coe_nat] at h,
rw [tendsto_const_mul_pow_nhds_iff hc, ← int.coe_nat_eq_zero] at h,
exact or.inl h },
{ rw not_le at hn,
refine or.inr ⟨hn, tendsto_nhds_unique h (tendsto_const_mul_zpow_at_to... | lemma | tendsto_const_mul_zpow_at_top_nhds_iff | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"int.coe_nat_eq_zero",
"lift",
"mul_one",
"tendsto_const_mul_pow_nhds_iff",
"tendsto_const_mul_zpow_at_top_zero",
"tendsto_const_nhds",
"tendsto_nhds_unique",
"zpow_coe_nat",
"zpow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_field.to_topological_division_ring : topological_division_ring α | { continuous_at_inv₀ :=
begin
suffices : ∀ {x : α}, 0 < x → continuous_at has_inv.inv x,
{ intros x hx,
cases hx.symm.lt_or_lt,
{ exact this h },
convert (this $ neg_pos.mpr h).neg.comp continuous_neg.continuous_at,
ext,
simp [neg_inv] },
intros t ht,
rw [continuous_at,
... | instance | linear_ordered_field.to_topological_division_ring | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"abs_div",
"abs_of_pos",
"aux",
"continuous_at",
"div_eq_mul_inv",
"div_mul_cancel",
"half_pos",
"inv_div",
"inv_lt_of_inv_lt",
"inv_sub_inv",
"lt_min_iff",
"min_le_of_left_le",
"mul_comm",
"mul_div_cancel'",
"mul_lt_mul''",
"mul_lt_mul_of_pos_left",
"mul_min_of_nonneg",
"neg_inv",... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_pos_comap_mul_left {x : α} (hx : 0 < x) :
comap (λ ε, x * ε) (𝓝[>] 0) = 𝓝[>] 0 | begin
suffices : ∀ {x : α} (hx : 0 < x), 𝓝[>] 0 ≤ comap (λ ε, x * ε) (𝓝[>] 0),
{ refine le_antisymm _ (this hx),
have hr : 𝓝[>] (0 : α) = ((𝓝[>] (0 : α)).comap (λ ε, x⁻¹ * ε)).comap (λ ε, x * ε),
{ simp [comap_comap, inv_mul_cancel hx.ne.symm, comap_id, one_mul_eq_id], },
conv_rhs { rw hr, },
rw... | lemma | nhds_within_pos_comap_mul_left | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"continuous_mul_left",
"continuous_within_at",
"inv_mul_cancel",
"mul_left_surjective₀",
"mul_zero",
"nhds_within_le_comap",
"one_mul_eq_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_within_pos_mul_left {x : α} (hx : 0 < x)
{p : α → Prop} (h : ∀ᶠ ε in 𝓝[>] 0, p ε) : ∀ᶠ ε in 𝓝[>] 0, p (x * ε) | begin
convert h.comap (λ ε, x * ε),
exact (nhds_within_pos_comap_mul_left hx).symm,
end | lemma | eventually_nhds_within_pos_mul_left | topology.algebra.order | src/topology/algebra/order/field.lean | [
"tactic.positivity",
"tactic.linarith",
"topology.algebra.order.group",
"topology.algebra.field"
] | [
"nhds_within_pos_comap_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_at_top [no_max_order X] : tendsto 𝓝 (at_top : filter X) (𝓝 at_top) | filter.tendsto_nhds_at_top_iff.2 $ λ x, (eventually_gt_at_top x).mono $ λ y, le_mem_nhds | lemma | filter.tendsto_nhds_at_top | topology.algebra.order | src/topology/algebra/order/filter.lean | [
"topology.order.basic",
"topology.filter"
] | [
"filter",
"le_mem_nhds",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_at_bot [no_min_order X] : tendsto 𝓝 (at_bot : filter X) (𝓝 at_bot) | @filter.tendsto_nhds_at_top Xᵒᵈ _ _ _ _ | lemma | filter.tendsto_nhds_at_bot | topology.algebra.order | src/topology/algebra/order/filter.lean | [
"topology.order.basic",
"topology.filter"
] | [
"filter",
"filter.tendsto_nhds_at_top",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.nhds_at_top [no_max_order X] {f : α → X} {l : filter α} (h : tendsto f l at_top) :
tendsto (𝓝 ∘ f) l (𝓝 at_top) | filter.tendsto_nhds_at_top.comp h | lemma | filter.tendsto.nhds_at_top | topology.algebra.order | src/topology/algebra/order/filter.lean | [
"topology.order.basic",
"topology.filter"
] | [
"filter",
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.nhds_at_bot [no_min_order X] {f : α → X} {l : filter α} (h : tendsto f l at_bot) :
tendsto (𝓝 ∘ f) l (𝓝 at_bot) | @tendsto.nhds_at_top α Xᵒᵈ _ _ _ _ _ _ h | lemma | filter.tendsto.nhds_at_bot | topology.algebra.order | src/topology/algebra/order/filter.lean | [
"topology.order.basic",
"topology.filter"
] | [
"filter",
"no_min_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_floor_at_top : tendsto (floor : α → ℤ) at_top at_top | floor_mono.tendsto_at_top_at_top $ λ b, ⟨(b + 1 : ℤ),
by { rw floor_int_cast, exact (lt_add_one _).le }⟩ | lemma | tendsto_floor_at_top | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_floor_at_bot : tendsto (floor : α → ℤ) at_bot at_bot | floor_mono.tendsto_at_bot_at_bot $ λ b, ⟨b, (floor_int_cast _).le⟩ | lemma | tendsto_floor_at_bot | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_ceil_at_top : tendsto (ceil : α → ℤ) at_top at_top | ceil_mono.tendsto_at_top_at_top $ λ b, ⟨b, (ceil_int_cast _).ge⟩ | lemma | tendsto_ceil_at_top | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_ceil_at_bot : tendsto (ceil : α → ℤ) at_bot at_bot | ceil_mono.tendsto_at_bot_at_bot $ λ b, ⟨(b - 1 : ℤ),
by { rw ceil_int_cast, exact (sub_one_lt _).le }⟩ | lemma | tendsto_ceil_at_bot | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"sub_one_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_floor (n : ℤ) : continuous_on (λ x, floor x : α → α) (Ico n (n+1) : set α) | (continuous_on_congr $ floor_eq_on_Ico' n).mpr continuous_on_const | lemma | continuous_on_floor | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"continuous_on",
"continuous_on_congr",
"continuous_on_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_ceil (n : ℤ) : continuous_on (λ x, ceil x : α → α) (Ioc (n-1) n : set α) | (continuous_on_congr $ ceil_eq_on_Ioc' n).mpr continuous_on_const | lemma | continuous_on_ceil | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"continuous_on",
"continuous_on_congr",
"continuous_on_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_floor_right' [order_closed_topology α] (n : ℤ) :
tendsto (λ x, floor x : α → α) (𝓝[≥] n) (𝓝 n) | begin
rw ← nhds_within_Ico_eq_nhds_within_Ici (lt_add_one (n : α)),
simpa only [floor_int_cast] using
(continuous_on_floor n _ (left_mem_Ico.mpr $ lt_add_one (_ : α))).tendsto
end | lemma | tendsto_floor_right' | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"continuous_on_floor",
"lt_add_one",
"nhds_within_Ico_eq_nhds_within_Ici",
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_ceil_left' [order_closed_topology α] (n : ℤ) :
tendsto (λ x, ceil x : α → α) (𝓝[≤] n) (𝓝 n) | begin
rw ← nhds_within_Ioc_eq_nhds_within_Iic (sub_one_lt (n : α)),
simpa only [ceil_int_cast] using
(continuous_on_ceil _ _ (right_mem_Ioc.mpr $ sub_one_lt (_ : α))).tendsto
end | lemma | tendsto_ceil_left' | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"continuous_on_ceil",
"nhds_within_Ioc_eq_nhds_within_Iic",
"order_closed_topology",
"sub_one_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_floor_right [order_closed_topology α] (n : ℤ) :
tendsto (λ x, floor x : α → α) (𝓝[≥] n) (𝓝[≥] n) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _ (tendsto_floor_right' _)
begin
refine (eventually_nhds_within_of_forall $ λ x (hx : (n : α) ≤ x), _),
change _ ≤ _,
norm_cast,
convert ← floor_mono hx,
rw floor_eq_iff,
exact ⟨le_rfl, lt_add_one _⟩
end | lemma | tendsto_floor_right | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"eventually_nhds_within_of_forall",
"lt_add_one",
"order_closed_topology",
"tendsto_floor_right'",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_ceil_left [order_closed_topology α] (n : ℤ) :
tendsto (λ x, ceil x : α → α) (𝓝[≤] n) (𝓝[≤] n) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _ (tendsto_ceil_left' _)
begin
refine (eventually_nhds_within_of_forall $ λ x (hx : x ≤ (n : α)), _),
change _ ≤ _,
norm_cast,
convert ← ceil_mono hx,
rw ceil_eq_iff,
exact ⟨sub_one_lt _, le_rfl⟩
end | lemma | tendsto_ceil_left | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"eventually_nhds_within_of_forall",
"order_closed_topology",
"tendsto_ceil_left'",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_floor_left [order_closed_topology α] (n : ℤ) :
tendsto (λ x, floor x : α → α) (𝓝[<] n) (𝓝[≤] (n-1)) | begin
rw ← nhds_within_Ico_eq_nhds_within_Iio (sub_one_lt (n : α)),
convert (tendsto_nhds_within_congr $ (λ x hx, (floor_eq_on_Ico' (n-1) x hx).symm))
(tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _ tendsto_const_nhds
(eventually_of_forall (λ _, mem_Iic.mpr $ le_rfl)));
norm_cast <|> apply_i... | lemma | tendsto_floor_left | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"le_rfl",
"nhds_within_Ico_eq_nhds_within_Iio",
"order_closed_topology",
"ring",
"sub_one_lt",
"tendsto_const_nhds",
"tendsto_nhds_within_congr",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_ceil_right [order_closed_topology α] (n : ℤ) :
tendsto (λ x, ceil x : α → α) (𝓝[>] n) (𝓝[≥] (n+1)) | begin
rw ← nhds_within_Ioc_eq_nhds_within_Ioi (lt_add_one (n : α)),
convert (tendsto_nhds_within_congr $ (λ x hx, (ceil_eq_on_Ioc' (n+1) x hx).symm))
(tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _ tendsto_const_nhds
(eventually_of_forall (λ _, mem_Ici.mpr $ le_rfl)));
norm_cast <|> apply_in... | lemma | tendsto_ceil_right | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"le_rfl",
"lt_add_one",
"nhds_within_Ioc_eq_nhds_within_Ioi",
"order_closed_topology",
"ring",
"tendsto_const_nhds",
"tendsto_nhds_within_congr",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_floor_left' [order_closed_topology α] (n : ℤ) :
tendsto (λ x, floor x : α → α) (𝓝[<] n) (𝓝 (n-1)) | begin
rw ← nhds_within_univ,
exact tendsto_nhds_within_mono_right (subset_univ _) (tendsto_floor_left n),
end | lemma | tendsto_floor_left' | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"nhds_within_univ",
"order_closed_topology",
"tendsto_floor_left",
"tendsto_nhds_within_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_ceil_right' [order_closed_topology α] (n : ℤ) :
tendsto (λ x, ceil x : α → α) (𝓝[>] n) (𝓝 (n+1)) | begin
rw ← nhds_within_univ,
exact tendsto_nhds_within_mono_right (subset_univ _) (tendsto_ceil_right n),
end | lemma | tendsto_ceil_right' | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"nhds_within_univ",
"order_closed_topology",
"tendsto_ceil_right",
"tendsto_nhds_within_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_fract [topological_add_group α] (n : ℤ) :
continuous_on (fract : α → α) (Ico n (n+1) : set α) | continuous_on_id.sub (continuous_on_floor n) | lemma | continuous_on_fract | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"continuous_on",
"continuous_on_floor",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_fract_left' [order_closed_topology α] [topological_add_group α]
(n : ℤ) : tendsto (fract : α → α) (𝓝[<] n) (𝓝 1) | begin
convert (tendsto_nhds_within_of_tendsto_nhds tendsto_id).sub (tendsto_floor_left' n);
[{norm_cast, ring}, apply_instance, apply_instance]
end | lemma | tendsto_fract_left' | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"order_closed_topology",
"ring",
"tendsto_floor_left'",
"tendsto_nhds_within_of_tendsto_nhds",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_fract_left [order_closed_topology α] [topological_add_group α]
(n : ℤ) : tendsto (fract : α → α) (𝓝[<] n) (𝓝[<] 1) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _
(tendsto_fract_left' _) (eventually_of_forall fract_lt_one) | lemma | tendsto_fract_left | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"order_closed_topology",
"tendsto_fract_left'",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_fract_right' [order_closed_topology α] [topological_add_group α]
(n : ℤ) : tendsto (fract : α → α) (𝓝[≥] n) (𝓝 0) | begin
convert (tendsto_nhds_within_of_tendsto_nhds tendsto_id).sub (tendsto_floor_right' n);
[exact (sub_self _).symm, apply_instance, apply_instance]
end | lemma | tendsto_fract_right' | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"order_closed_topology",
"tendsto_floor_right'",
"tendsto_nhds_within_of_tendsto_nhds",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_fract_right [order_closed_topology α] [topological_add_group α]
(n : ℤ) : tendsto (fract : α → α) (𝓝[≥] n) (𝓝[≥] 0) | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _
(tendsto_fract_right' _) (eventually_of_forall fract_nonneg) | lemma | tendsto_fract_right | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"order_closed_topology",
"tendsto_fract_right'",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.comp_fract' {f : β → α → γ}
(h : continuous_on (uncurry f) $ univ ×ˢ I) (hf : ∀ s, f s 0 = f s 1) :
continuous (λ st : β × α, f st.1 $ fract st.2) | begin
change continuous ((uncurry f) ∘ (prod.map id (fract))),
rw continuous_iff_continuous_at,
rintro ⟨s, t⟩,
by_cases ht : t = floor t,
{ rw ht,
rw ← continuous_within_at_univ,
have : (univ : set (β × α)) ⊆ (univ ×ˢ Iio ↑⌊t⌋) ∪ (univ ×ˢ Ici ↑⌊t⌋),
{ rintros p -,
rw ← prod_union,
exac... | lemma | continuous_on.comp_fract' | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"Ioo_mem_nhds",
"continuous",
"continuous_at",
"continuous_iff_continuous_at",
"continuous_on",
"continuous_on_fract",
"continuous_within_at",
"continuous_within_at.mono",
"continuous_within_at.union",
"continuous_within_at_univ",
"nhds_prod_eq",
"nhds_within_Icc_eq_nhds_within_Ici",
"nhds_w... | Do not use this, use `continuous_on.comp_fract` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.comp_fract
{s : β → α}
{f : β → α → γ}
(h : continuous_on (uncurry f) $ univ ×ˢ Icc 0 1)
(hs : continuous s)
(hf : ∀ s, f s 0 = f s 1) :
continuous (λ x : β, f x $ int.fract (s x)) | (h.comp_fract' hf).comp (continuous_id.prod_mk hs) | lemma | continuous_on.comp_fract | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"continuous",
"continuous_on",
"int.fract"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.comp_fract'' {f : α → β} (h : continuous_on f I) (hf : f 0 = f 1) :
continuous (f ∘ fract) | continuous_on.comp_fract (h.comp continuous_on_snd $ λ x hx, (mem_prod.mp hx).2)
continuous_id (λ _, hf) | lemma | continuous_on.comp_fract'' | topology.algebra.order | src/topology/algebra/order/floor.lean | [
"algebra.order.floor",
"topology.algebra.order.group"
] | [
"continuous",
"continuous_id",
"continuous_on",
"continuous_on.comp_fract",
"continuous_on_snd"
] | A special case of `continuous_on.comp_fract`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_add_comm_group.topological_add_group : topological_add_group G | { continuous_add :=
begin
refine continuous_iff_continuous_at.2 _,
rintro ⟨a, b⟩,
refine linear_ordered_add_comm_group.tendsto_nhds.2 (λ ε ε0, _),
rcases dense_or_discrete 0 ε with (⟨δ, δ0, δε⟩|⟨h₁, h₂⟩),
{ -- If there exists `δ ∈ (0, ε)`, then we choose `δ`-nhd of `a` and `(ε-δ)`-nhd ... | instance | linear_ordered_add_comm_group.topological_add_group | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [
"abs_add",
"abs_sub_comm",
"dense_or_discrete",
"eventually_abs_sub_lt",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_abs : continuous (abs : G → G) | continuous_id.max continuous_neg | lemma | continuous_abs | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.abs {a : G} (h : tendsto f l (𝓝 a)) :
tendsto (λ x, |f x|) l (𝓝 (|a|)) | (continuous_abs.tendsto _).comp h | lemma | filter.tendsto.abs | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_zero_iff_abs_tendsto_zero (f : α → G) :
tendsto f l (𝓝 0) ↔ tendsto (abs ∘ f) l (𝓝 0) | begin
refine ⟨λ h, (abs_zero : |(0 : G)| = 0) ▸ h.abs, λ h, _⟩,
have : tendsto (λ a, -|f a|) l (𝓝 0) := (neg_zero : -(0 : G) = 0) ▸ h.neg,
exact tendsto_of_tendsto_of_tendsto_of_le_of_le this h
(λ x, neg_abs_le_self $ f x) (λ x, le_abs_self $ f x),
end | lemma | tendsto_zero_iff_abs_tendsto_zero | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [
"abs_zero",
"le_abs_self",
"neg_abs_le_self",
"tendsto_of_tendsto_of_tendsto_of_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.abs (h : continuous f) : continuous (λ x, |f x|) | continuous_abs.comp h | lemma | continuous.abs | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.abs (h : continuous_at f a) : continuous_at (λ x, |f x|) a | h.abs | lemma | continuous_at.abs | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.abs (h : continuous_within_at f s a) :
continuous_within_at (λ x, |f x|) s a | h.abs | lemma | continuous_within_at.abs | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.abs (h : continuous_on f s) : continuous_on (λ x, |f x|) s | λ x hx, (h x hx).abs | lemma | continuous_on.abs | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abs_nhds_within_zero : tendsto (abs : G → G) (𝓝[≠] 0) (𝓝[>] 0) | (continuous_abs.tendsto' (0 : G) 0 abs_zero).inf $ tendsto_principal_principal.2 $ λ x, abs_pos.2 | lemma | tendsto_abs_nhds_within_zero | topology.algebra.order | src/topology/algebra/order/group.lean | [
"topology.order.basic",
"topology.algebra.group.basic"
] | [
"abs_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intermediate_value_univ₂ [preconnected_space X] {a b : X} {f g : X → α} (hf : continuous f)
(hg : continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) :
∃ x, f x = g x | begin
obtain ⟨x, h, hfg, hgf⟩ : (univ ∩ {x | f x ≤ g x ∧ g x ≤ f x}).nonempty,
from is_preconnected_closed_iff.1 preconnected_space.is_preconnected_univ _ _
(is_closed_le hf hg) (is_closed_le hg hf) (λ x hx, le_total _ _) ⟨a, trivial, ha⟩
⟨b, trivial, hb⟩,
exact ⟨x, le_antisymm hfg hgf⟩
end | lemma | intermediate_value_univ₂ | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous",
"is_closed_le",
"preconnected_space"
] | Intermediate value theorem for two functions: if `f` and `g` are two continuous functions
on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intermediate_value_univ₂_eventually₁ [preconnected_space X] {a : X} {l : filter X} [ne_bot l]
{f g : X → α} (hf : continuous f) (hg : continuous g) (ha : f a ≤ g a) (he : g ≤ᶠ[l] f) :
∃ x, f x = g x | let ⟨c, hc⟩ := he.frequently.exists in intermediate_value_univ₂ hf hg ha hc | lemma | intermediate_value_univ₂_eventually₁ | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous",
"filter",
"intermediate_value_univ₂",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intermediate_value_univ₂_eventually₂ [preconnected_space X] {l₁ l₂ : filter X}
[ne_bot l₁] [ne_bot l₂] {f g : X → α} (hf : continuous f) (hg : continuous g)
(he₁ : f ≤ᶠ[l₁] g ) (he₂ : g ≤ᶠ[l₂] f) :
∃ x, f x = g x | let ⟨c₁, hc₁⟩ := he₁.frequently.exists, ⟨c₂, hc₂⟩ := he₂.frequently.exists in
intermediate_value_univ₂ hf hg hc₁ hc₂ | lemma | intermediate_value_univ₂_eventually₂ | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous",
"filter",
"intermediate_value_univ₂",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value₂ {s : set X} (hs : is_preconnected s)
{a b : X} (ha : a ∈ s) (hb : b ∈ s) {f g : X → α}
(hf : continuous_on f s) (hg : continuous_on g s) (ha' : f a ≤ g a) (hb' : g b ≤ f b) :
∃ x ∈ s, f x = g x | let ⟨x, hx⟩ := @intermediate_value_univ₂ s α _ _ _ _ (subtype.preconnected_space hs) ⟨a, ha⟩ ⟨b, hb⟩
_ _ (continuous_on_iff_continuous_restrict.1 hf) (continuous_on_iff_continuous_restrict.1 hg)
ha' hb'
in ⟨x, x.2, hx⟩ | lemma | is_preconnected.intermediate_value₂ | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"intermediate_value_univ₂",
"is_preconnected",
"subtype.preconnected_space"
] | Intermediate value theorem for two functions: if `f` and `g` are two functions continuous
on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`,
then for some `x ∈ s` we have `f x = g x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.intermediate_value₂_eventually₁ {s : set X} (hs : is_preconnected s)
{a : X} {l : filter X} (ha : a ∈ s) [ne_bot l] (hl : l ≤ 𝓟 s) {f g : X → α}
(hf : continuous_on f s) (hg : continuous_on g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) :
∃ x ∈ s, f x = g x | begin
rw continuous_on_iff_continuous_restrict at hf hg,
obtain ⟨b, h⟩ := @intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (subtype.preconnected_space hs)
⟨a, ha⟩ _ (comap_coe_ne_bot_of_le_principal hl) _ _ hf hg ha' (he.comap _),
exact ⟨b, b.prop, h⟩,
end | lemma | is_preconnected.intermediate_value₂_eventually₁ | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_iff_continuous_restrict",
"filter",
"intermediate_value_univ₂_eventually₁",
"is_preconnected",
"subtype.preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value₂_eventually₂ {s : set X} (hs : is_preconnected s)
{l₁ l₂ : filter X} [ne_bot l₁] [ne_bot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α}
(hf : continuous_on f s) (hg : continuous_on g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) :
∃ x ∈ s, f x = g x | begin
rw continuous_on_iff_continuous_restrict at hf hg,
obtain ⟨b, h⟩ := @intermediate_value_univ₂_eventually₂ _ _ _ _ _ _ (subtype.preconnected_space hs)
_ _ (comap_coe_ne_bot_of_le_principal hl₁) (comap_coe_ne_bot_of_le_principal hl₂)
_ _ hf hg (he₁.comap _) (he₂.comap _),
exact ⟨b, b.prop, h⟩,
end | lemma | is_preconnected.intermediate_value₂_eventually₂ | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_iff_continuous_restrict",
"filter",
"intermediate_value_univ₂_eventually₂",
"is_preconnected",
"subtype.preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value {s : set X} (hs : is_preconnected s)
{a b : X} (ha : a ∈ s) (hb : b ∈ s) {f : X → α} (hf : continuous_on f s) :
Icc (f a) (f b) ⊆ f '' s | λ x hx, mem_image_iff_bex.2 $ hs.intermediate_value₂ ha hb hf continuous_on_const hx.1 hx.2 | lemma | is_preconnected.intermediate_value | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_const",
"is_preconnected"
] | **Intermediate Value Theorem** for continuous functions on connected sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.intermediate_value_Ico {s : set X} (hs : is_preconnected s)
{a : X} {l : filter X} (ha : a ∈ s) [ne_bot l] (hl : l ≤ 𝓟 s) {f : X → α}
(hf : continuous_on f s) {v : α} (ht : tendsto f l (𝓝 v)) :
Ico (f a) v ⊆ f '' s | λ y h, bex_def.1 $ hs.intermediate_value₂_eventually₁ ha hl
hf continuous_on_const h.1 (eventually_ge_of_tendsto_gt h.2 ht) | lemma | is_preconnected.intermediate_value_Ico | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_const",
"eventually_ge_of_tendsto_gt",
"filter",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value_Ioc {s : set X} (hs : is_preconnected s)
{a : X} {l : filter X} (ha : a ∈ s) [ne_bot l] (hl : l ≤ 𝓟 s) {f : X → α}
(hf : continuous_on f s) {v : α} (ht : tendsto f l (𝓝 v)) :
Ioc v (f a) ⊆ f '' s | λ y h, bex_def.1 $ bex.imp_right (λ x _, eq.symm) $ hs.intermediate_value₂_eventually₁ ha hl
continuous_on_const hf h.2 (eventually_le_of_tendsto_lt h.1 ht) | lemma | is_preconnected.intermediate_value_Ioc | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bex.imp_right",
"continuous_on",
"continuous_on_const",
"eventually_le_of_tendsto_lt",
"filter",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value_Ioo {s : set X} (hs : is_preconnected s)
{l₁ l₂ : filter X} [ne_bot l₁] [ne_bot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α}
(hf : continuous_on f s) {v₁ v₂ : α} (ht₁ : tendsto f l₁ (𝓝 v₁)) (ht₂ : tendsto f l₂ (𝓝 v₂)) :
Ioo v₁ v₂ ⊆ f '' s | λ y h, bex_def.1 $ hs.intermediate_value₂_eventually₂ hl₁ hl₂
hf continuous_on_const (eventually_le_of_tendsto_lt h.1 ht₁) (eventually_ge_of_tendsto_gt h.2 ht₂) | lemma | is_preconnected.intermediate_value_Ioo | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_const",
"eventually_ge_of_tendsto_gt",
"eventually_le_of_tendsto_lt",
"filter",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value_Ici {s : set X} (hs : is_preconnected s)
{a : X} {l : filter X} (ha : a ∈ s) [ne_bot l] (hl : l ≤ 𝓟 s) {f : X → α}
(hf : continuous_on f s) (ht : tendsto f l at_top) :
Ici (f a) ⊆ f '' s | λ y h, bex_def.1 $ hs.intermediate_value₂_eventually₁ ha hl
hf continuous_on_const h (tendsto_at_top.1 ht y) | lemma | is_preconnected.intermediate_value_Ici | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_const",
"filter",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value_Iic {s : set X} (hs : is_preconnected s)
{a : X} {l : filter X} (ha : a ∈ s) [ne_bot l] (hl : l ≤ 𝓟 s) {f : X → α}
(hf : continuous_on f s) (ht : tendsto f l at_bot) :
Iic (f a) ⊆ f '' s | λ y h, bex_def.1 $ bex.imp_right (λ x _, eq.symm) $ hs.intermediate_value₂_eventually₁ ha hl
continuous_on_const hf h (tendsto_at_bot.1 ht y) | lemma | is_preconnected.intermediate_value_Iic | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bex.imp_right",
"continuous_on",
"continuous_on_const",
"filter",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value_Ioi {s : set X} (hs : is_preconnected s)
{l₁ l₂ : filter X} [ne_bot l₁] [ne_bot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α}
(hf : continuous_on f s) {v : α} (ht₁ : tendsto f l₁ (𝓝 v)) (ht₂ : tendsto f l₂ at_top) :
Ioi v ⊆ f '' s | λ y h, bex_def.1 $ hs.intermediate_value₂_eventually₂ hl₁ hl₂
hf continuous_on_const (eventually_le_of_tendsto_lt h ht₁) (tendsto_at_top.1 ht₂ y) | lemma | is_preconnected.intermediate_value_Ioi | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_const",
"eventually_le_of_tendsto_lt",
"filter",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value_Iio {s : set X} (hs : is_preconnected s)
{l₁ l₂ : filter X} [ne_bot l₁] [ne_bot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α}
(hf : continuous_on f s) {v : α} (ht₁ : tendsto f l₁ at_bot) (ht₂ : tendsto f l₂ (𝓝 v)) :
Iio v ⊆ f '' s | λ y h, bex_def.1 $ hs.intermediate_value₂_eventually₂ hl₁ hl₂
hf continuous_on_const (tendsto_at_bot.1 ht₁ y) (eventually_ge_of_tendsto_gt h ht₂) | lemma | is_preconnected.intermediate_value_Iio | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_const",
"eventually_ge_of_tendsto_gt",
"filter",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.intermediate_value_Iii {s : set X} (hs : is_preconnected s)
{l₁ l₂ : filter X} [ne_bot l₁] [ne_bot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α}
(hf : continuous_on f s) (ht₁ : tendsto f l₁ at_bot) (ht₂ : tendsto f l₂ at_top) :
univ ⊆ f '' s | λ y h, bex_def.1 $ hs.intermediate_value₂_eventually₂ hl₁ hl₂
hf continuous_on_const (tendsto_at_bot.1 ht₁ y) (tendsto_at_top.1 ht₂ y) | lemma | is_preconnected.intermediate_value_Iii | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on",
"continuous_on_const",
"filter",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intermediate_value_univ [preconnected_space X] (a b : X) {f : X → α} (hf : continuous f) :
Icc (f a) (f b) ⊆ range f | λ x hx, intermediate_value_univ₂ hf continuous_const hx.1 hx.2 | lemma | intermediate_value_univ | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous",
"continuous_const",
"intermediate_value_univ₂",
"preconnected_space"
] | **Intermediate Value Theorem** for continuous functions on connected spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_range_of_exists_le_of_exists_ge [preconnected_space X] {c : α} {f : X → α}
(hf : continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) :
c ∈ range f | let ⟨a, ha⟩ := h₁, ⟨b, hb⟩ := h₂ in intermediate_value_univ a b hf ⟨ha, hb⟩ | lemma | mem_range_of_exists_le_of_exists_ge | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous",
"intermediate_value_univ",
"preconnected_space"
] | **Intermediate Value Theorem** for continuous functions on connected spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.Icc_subset {s : set α} (hs : is_preconnected s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) :
Icc a b ⊆ s | by simpa only [image_id] using hs.intermediate_value ha hb continuous_on_id | lemma | is_preconnected.Icc_subset | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"continuous_on_id",
"is_preconnected"
] | If a preconnected set contains endpoints of an interval, then it includes the whole interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.ord_connected {s : set α} (h : is_preconnected s) :
ord_connected s | ⟨λ x hx y hy, h.Icc_subset hx hy⟩ | lemma | is_preconnected.ord_connected | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected.Icc_subset {s : set α} (hs : is_connected s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) :
Icc a b ⊆ s | hs.2.Icc_subset ha hb | lemma | is_connected.Icc_subset | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"is_connected"
] | If a preconnected set contains endpoints of an interval, then it includes the whole interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.eq_univ_of_unbounded {s : set α} (hs : is_preconnected s) (hb : ¬bdd_below s)
(ha : ¬bdd_above s) :
s = univ | begin
refine eq_univ_of_forall (λ x, _),
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := not_bdd_below_iff.1 hb x,
obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bdd_above_iff.1 ha x,
exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩
end | lemma | is_preconnected.eq_univ_of_unbounded | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bdd_above",
"bdd_below",
"is_preconnected"
] | If preconnected set in a linear order space is unbounded below and above, then it is the whole
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.Ioo_cInf_cSup_subset {s : set α} (hs : is_connected s) (hb : bdd_below s)
(ha : bdd_above s) :
Ioo (Inf s) (Sup s) ⊆ s | λ x hx, let ⟨y, ys, hy⟩ := (is_glb_lt_iff (is_glb_cInf hs.nonempty hb)).1 hx.1 in
let ⟨z, zs, hz⟩ := (lt_is_lub_iff (is_lub_cSup hs.nonempty ha)).1 hx.2 in
hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩ | lemma | is_connected.Ioo_cInf_cSup_subset | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bdd_above",
"bdd_below",
"is_connected",
"is_glb_cInf",
"is_glb_lt_iff",
"is_lub_cSup",
"lt_is_lub_iff"
] | A bounded connected subset of a conditionally complete linear order includes the open interval
`(Inf s, Sup s)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_Icc_cInf_cSup_of_connected_bdd_closed {s : set α} (hc : is_connected s) (hb : bdd_below s)
(ha : bdd_above s) (hcl : is_closed s) :
s = Icc (Inf s) (Sup s) | subset.antisymm (subset_Icc_cInf_cSup hb ha) $
hc.Icc_subset (hcl.cInf_mem hc.nonempty hb) (hcl.cSup_mem hc.nonempty ha) | lemma | eq_Icc_cInf_cSup_of_connected_bdd_closed | topology.algebra.order | src/topology/algebra/order/intermediate_value.lean | [
"order.complete_lattice_intervals",
"topology.order.basic"
] | [
"bdd_above",
"bdd_below",
"is_closed",
"is_connected",
"subset_Icc_cInf_cSup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.