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
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is_max_on_iff : is_max_on f s a ↔ ∀ x ∈ s, f x ≤ f a | iff.rfl | lemma | is_max_on_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on_univ_iff : is_min_on f univ a ↔ ∀ x, f a ≤ f x | univ_subset_iff.trans eq_univ_iff_forall | lemma | is_min_on_univ_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on_univ_iff : is_max_on f univ a ↔ ∀ x, f x ≤ f a | univ_subset_iff.trans eq_univ_iff_forall | lemma | is_max_on_univ_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.tendsto_principal_Ici (h : is_min_filter f l a) :
tendsto f l (𝓟 $ Ici (f a)) | tendsto_principal.2 h | lemma | is_min_filter.tendsto_principal_Ici | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.tendsto_principal_Iic (h : is_max_filter f l a) :
tendsto f l (𝓟 $ Iic (f a)) | tendsto_principal.2 h | lemma | is_max_filter.tendsto_principal_Iic | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.is_extr : is_min_filter f l a → is_extr_filter f l a | or.inl | lemma | is_min_filter.is_extr | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.is_extr : is_max_filter f l a → is_extr_filter f l a | or.inr | lemma | is_max_filter.is_extr | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter",
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.is_extr (h : is_min_on f s a) : is_extr_on f s a | h.is_extr | lemma | is_min_on.is_extr | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.is_extr (h : is_max_on f s a) : is_extr_on f s a | h.is_extr | lemma | is_max_on.is_extr | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on",
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter_const {b : β} : is_min_filter (λ _, b) l a | univ_mem' $ λ _, le_rfl | lemma | is_min_filter_const | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter_const {b : β} : is_max_filter (λ _, b) l a | univ_mem' $ λ _, le_rfl | lemma | is_max_filter_const | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter_const {b : β} : is_extr_filter (λ _, b) l a | is_min_filter_const.is_extr | lemma | is_extr_filter_const | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on_const {b : β} : is_min_on (λ _, b) s a | is_min_filter_const | lemma | is_min_on_const | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter_const",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on_const {b : β} : is_max_on (λ _, b) s a | is_max_filter_const | lemma | is_max_on_const | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter_const",
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on_const {b : β} : is_extr_on (λ _, b) s a | is_extr_filter_const | lemma | is_extr_on_const | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter_const",
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter_dual_iff : is_min_filter (to_dual ∘ f) l a ↔ is_max_filter f l a | iff.rfl | lemma | is_min_filter_dual_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter_dual_iff : is_max_filter (to_dual ∘ f) l a ↔ is_min_filter f l a | iff.rfl | lemma | is_max_filter_dual_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter_dual_iff : is_extr_filter (to_dual ∘ f) l a ↔ is_extr_filter f l a | or_comm _ _ | lemma | is_extr_filter_dual_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on_dual_iff : is_min_on (to_dual ∘ f) s a ↔ is_max_on f s a | iff.rfl | lemma | is_min_on_dual_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on_dual_iff : is_max_on (to_dual ∘ f) s a ↔ is_min_on f s a | iff.rfl | lemma | is_max_on_dual_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on_dual_iff : is_extr_on (to_dual ∘ f) s a ↔ is_extr_on f s a | or_comm _ _ | lemma | is_extr_on_dual_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.filter_mono (h : is_min_filter f l a) (hl : l' ≤ l) :
is_min_filter f l' a | hl h | lemma | is_min_filter.filter_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.filter_mono (h : is_max_filter f l a) (hl : l' ≤ l) :
is_max_filter f l' a | hl h | lemma | is_max_filter.filter_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter.filter_mono (h : is_extr_filter f l a) (hl : l' ≤ l) :
is_extr_filter f l' a | h.elim (λ h, (h.filter_mono hl).is_extr) (λ h, (h.filter_mono hl).is_extr) | lemma | is_extr_filter.filter_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.filter_inf (h : is_min_filter f l a) (l') : is_min_filter f (l ⊓ l') a | h.filter_mono inf_le_left | lemma | is_min_filter.filter_inf | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"inf_le_left",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.filter_inf (h : is_max_filter f l a) (l') : is_max_filter f (l ⊓ l') a | h.filter_mono inf_le_left | lemma | is_max_filter.filter_inf | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"inf_le_left",
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter.filter_inf (h : is_extr_filter f l a) (l') : is_extr_filter f (l ⊓ l') a | h.filter_mono inf_le_left | lemma | is_extr_filter.filter_inf | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"inf_le_left",
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.on_subset (hf : is_min_on f t a) (h : s ⊆ t) : is_min_on f s a | hf.filter_mono $ principal_mono.2 h | lemma | is_min_on.on_subset | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.on_subset (hf : is_max_on f t a) (h : s ⊆ t) : is_max_on f s a | hf.filter_mono $ principal_mono.2 h | lemma | is_max_on.on_subset | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.on_subset (hf : is_extr_on f t a) (h : s ⊆ t) : is_extr_on f s a | hf.filter_mono $ principal_mono.2 h | lemma | is_extr_on.on_subset | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.inter (hf : is_min_on f s a) (t) : is_min_on f (s ∩ t) a | hf.on_subset (inter_subset_left s t) | lemma | is_min_on.inter | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.inter (hf : is_max_on f s a) (t) : is_max_on f (s ∩ t) a | hf.on_subset (inter_subset_left s t) | lemma | is_max_on.inter | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.inter (hf : is_extr_on f s a) (t) : is_extr_on f (s ∩ t) a | hf.on_subset (inter_subset_left s t) | lemma | is_extr_on.inter | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.comp_mono (hf : is_min_filter f l a) {g : β → γ} (hg : monotone g) :
is_min_filter (g ∘ f) l a | mem_of_superset hf $ λ x hx, hg hx | lemma | is_min_filter.comp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.comp_mono (hf : is_max_filter f l a) {g : β → γ} (hg : monotone g) :
is_max_filter (g ∘ f) l a | mem_of_superset hf $ λ x hx, hg hx | lemma | is_max_filter.comp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter.comp_mono (hf : is_extr_filter f l a) {g : β → γ} (hg : monotone g) :
is_extr_filter (g ∘ f) l a | hf.elim (λ hf, (hf.comp_mono hg).is_extr) (λ hf, (hf.comp_mono hg).is_extr) | lemma | is_extr_filter.comp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.comp_antitone (hf : is_min_filter f l a) {g : β → γ}
(hg : antitone g) :
is_max_filter (g ∘ f) l a | hf.dual.comp_mono (λ x y h, hg h) | lemma | is_min_filter.comp_antitone | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"antitone",
"is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.comp_antitone (hf : is_max_filter f l a) {g : β → γ}
(hg : antitone g) :
is_min_filter (g ∘ f) l a | hf.dual.comp_mono (λ x y h, hg h) | lemma | is_max_filter.comp_antitone | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"antitone",
"is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter.comp_antitone (hf : is_extr_filter f l a) {g : β → γ}
(hg : antitone g) :
is_extr_filter (g ∘ f) l a | hf.dual.comp_mono (λ x y h, hg h) | lemma | is_extr_filter.comp_antitone | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"antitone",
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.comp_mono (hf : is_min_on f s a) {g : β → γ} (hg : monotone g) :
is_min_on (g ∘ f) s a | hf.comp_mono hg | lemma | is_min_on.comp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.comp_mono (hf : is_max_on f s a) {g : β → γ} (hg : monotone g) :
is_max_on (g ∘ f) s a | hf.comp_mono hg | lemma | is_max_on.comp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.comp_mono (hf : is_extr_on f s a) {g : β → γ} (hg : monotone g) :
is_extr_on (g ∘ f) s a | hf.comp_mono hg | lemma | is_extr_on.comp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.comp_antitone (hf : is_min_on f s a) {g : β → γ}
(hg : antitone g) :
is_max_on (g ∘ f) s a | hf.comp_antitone hg | lemma | is_min_on.comp_antitone | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"antitone",
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.comp_antitone (hf : is_max_on f s a) {g : β → γ}
(hg : antitone g) :
is_min_on (g ∘ f) s a | hf.comp_antitone hg | lemma | is_max_on.comp_antitone | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"antitone",
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.comp_antitone (hf : is_extr_on f s a) {g : β → γ}
(hg : antitone g) :
is_extr_on (g ∘ f) s a | hf.comp_antitone hg | lemma | is_extr_on.comp_antitone | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"antitone",
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.bicomp_mono [preorder δ] {op : β → γ → δ} (hop : ((≤) ⇒ (≤) ⇒ (≤)) op op)
(hf : is_min_filter f l a) {g : α → γ} (hg : is_min_filter g l a) :
is_min_filter (λ x, op (f x) (g x)) l a | mem_of_superset (inter_mem hf hg) $ λ x ⟨hfx, hgx⟩, hop hfx hgx | lemma | is_min_filter.bicomp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.bicomp_mono [preorder δ] {op : β → γ → δ} (hop : ((≤) ⇒ (≤) ⇒ (≤)) op op)
(hf : is_max_filter f l a) {g : α → γ} (hg : is_max_filter g l a) :
is_max_filter (λ x, op (f x) (g x)) l a | mem_of_superset (inter_mem hf hg) $ λ x ⟨hfx, hgx⟩, hop hfx hgx | lemma | is_max_filter.bicomp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.bicomp_mono [preorder δ] {op : β → γ → δ} (hop : ((≤) ⇒ (≤) ⇒ (≤)) op op)
(hf : is_min_on f s a) {g : α → γ} (hg : is_min_on g s a) :
is_min_on (λ x, op (f x) (g x)) s a | hf.bicomp_mono hop hg | lemma | is_min_on.bicomp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.bicomp_mono [preorder δ] {op : β → γ → δ} (hop : ((≤) ⇒ (≤) ⇒ (≤)) op op)
(hf : is_max_on f s a) {g : α → γ} (hg : is_max_on g s a) :
is_max_on (λ x, op (f x) (g x)) s a | hf.bicomp_mono hop hg | lemma | is_max_on.bicomp_mono | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.comp_tendsto {g : δ → α} {l' : filter δ} {b : δ} (hf : is_min_filter f l (g b))
(hg : tendsto g l' l) :
is_min_filter (f ∘ g) l' b | hg hf | lemma | is_min_filter.comp_tendsto | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.comp_tendsto {g : δ → α} {l' : filter δ} {b : δ} (hf : is_max_filter f l (g b))
(hg : tendsto g l' l) :
is_max_filter (f ∘ g) l' b | hg hf | lemma | is_max_filter.comp_tendsto | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter.comp_tendsto {g : δ → α} {l' : filter δ} {b : δ}
(hf : is_extr_filter f l (g b)) (hg : tendsto g l' l) :
is_extr_filter (f ∘ g) l' b | hf.elim (λ hf, (hf.comp_tendsto hg).is_extr) (λ hf, (hf.comp_tendsto hg).is_extr) | lemma | is_extr_filter.comp_tendsto | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.on_preimage (g : δ → α) {b : δ} (hf : is_min_on f s (g b)) :
is_min_on (f ∘ g) (g ⁻¹' s) b | hf.comp_tendsto (tendsto_principal_principal.mpr $ subset.refl _) | lemma | is_min_on.on_preimage | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.on_preimage (g : δ → α) {b : δ} (hf : is_max_on f s (g b)) :
is_max_on (f ∘ g) (g ⁻¹' s) b | hf.comp_tendsto (tendsto_principal_principal.mpr $ subset.refl _) | lemma | is_max_on.on_preimage | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.on_preimage (g : δ → α) {b : δ} (hf : is_extr_on f s (g b)) :
is_extr_on (f ∘ g) (g ⁻¹' s) b | hf.elim (λ hf, (hf.on_preimage g).is_extr) (λ hf, (hf.on_preimage g).is_extr) | lemma | is_extr_on.on_preimage | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.comp_maps_to {t : set δ} {g : δ → α} {b : δ} (hf : is_min_on f s a)
(hg : maps_to g t s) (ha : g b = a) :
is_min_on (f ∘ g) t b | λ y hy, by simpa only [mem_set_of_eq, ha, (∘)] using hf (hg hy) | lemma | is_min_on.comp_maps_to | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.comp_maps_to {t : set δ} {g : δ → α} {b : δ} (hf : is_max_on f s a)
(hg : maps_to g t s) (ha : g b = a) :
is_max_on (f ∘ g) t b | hf.dual.comp_maps_to hg ha | lemma | is_max_on.comp_maps_to | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.comp_maps_to {t : set δ} {g : δ → α} {b : δ} (hf : is_extr_on f s a)
(hg : maps_to g t s) (ha : g b = a) :
is_extr_on (f ∘ g) t b | hf.elim (λ h, or.inl $ h.comp_maps_to hg ha) (λ h, or.inr $ h.comp_maps_to hg ha) | lemma | is_extr_on.comp_maps_to | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.add (hf : is_min_filter f l a) (hg : is_min_filter g l a) :
is_min_filter (λ x, f x + g x) l a | show is_min_filter (λ x, f x + g x) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, add_le_add hx hy) hg | lemma | is_min_filter.add | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.add (hf : is_max_filter f l a) (hg : is_max_filter g l a) :
is_max_filter (λ x, f x + g x) l a | show is_max_filter (λ x, f x + g x) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, add_le_add hx hy) hg | lemma | is_max_filter.add | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.add (hf : is_min_on f s a) (hg : is_min_on g s a) :
is_min_on (λ x, f x + g x) s a | hf.add hg | lemma | is_min_on.add | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.add (hf : is_max_on f s a) (hg : is_max_on g s a) :
is_max_on (λ x, f x + g x) s a | hf.add hg | lemma | is_max_on.add | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.neg (hf : is_min_filter f l a) : is_max_filter (λ x, -f x) l a | hf.comp_antitone (λ x y hx, neg_le_neg hx) | lemma | is_min_filter.neg | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.neg (hf : is_max_filter f l a) : is_min_filter (λ x, -f x) l a | hf.comp_antitone (λ x y hx, neg_le_neg hx) | lemma | is_max_filter.neg | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter.neg (hf : is_extr_filter f l a) : is_extr_filter (λ x, -f x) l a | hf.elim (λ hf, hf.neg.is_extr) (λ hf, hf.neg.is_extr) | lemma | is_extr_filter.neg | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.neg (hf : is_min_on f s a) : is_max_on (λ x, -f x) s a | hf.comp_antitone (λ x y hx, neg_le_neg hx) | lemma | is_min_on.neg | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.neg (hf : is_max_on f s a) : is_min_on (λ x, -f x) s a | hf.comp_antitone (λ x y hx, neg_le_neg hx) | lemma | is_max_on.neg | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_on.neg (hf : is_extr_on f s a) : is_extr_on (λ x, -f x) s a | hf.elim (λ hf, hf.neg.is_extr) (λ hf, hf.neg.is_extr) | lemma | is_extr_on.neg | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_extr_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.sub (hf : is_min_filter f l a) (hg : is_max_filter g l a) :
is_min_filter (λ x, f x - g x) l a | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | is_min_filter.sub | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.sub (hf : is_max_filter f l a) (hg : is_min_filter g l a) :
is_max_filter (λ x, f x - g x) l a | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | is_max_filter.sub | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.sub (hf : is_min_on f s a) (hg : is_max_on g s a) :
is_min_on (λ x, f x - g x) s a | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | is_min_on.sub | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.sub (hf : is_max_on f s a) (hg : is_min_on g s a) :
is_max_on (λ x, f x - g x) s a | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | is_max_on.sub | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.sup (hf : is_min_filter f l a) (hg : is_min_filter g l a) :
is_min_filter (λ x, f x ⊔ g x) l a | show is_min_filter (λ x, f x ⊔ g x) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, sup_le_sup hx hy) hg | lemma | is_min_filter.sup | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.sup (hf : is_max_filter f l a) (hg : is_max_filter g l a) :
is_max_filter (λ x, f x ⊔ g x) l a | show is_max_filter (λ x, f x ⊔ g x) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, sup_le_sup hx hy) hg | lemma | is_max_filter.sup | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.sup (hf : is_min_on f s a) (hg : is_min_on g s a) :
is_min_on (λ x, f x ⊔ g x) s a | hf.sup hg | lemma | is_min_on.sup | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.sup (hf : is_max_on f s a) (hg : is_max_on g s a) :
is_max_on (λ x, f x ⊔ g x) s a | hf.sup hg | lemma | is_max_on.sup | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.inf (hf : is_min_filter f l a) (hg : is_min_filter g l a) :
is_min_filter (λ x, f x ⊓ g x) l a | show is_min_filter (λ x, f x ⊓ g x) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, inf_le_inf hx hy) hg | lemma | is_min_filter.inf | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"inf_le_inf",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.inf (hf : is_max_filter f l a) (hg : is_max_filter g l a) :
is_max_filter (λ x, f x ⊓ g x) l a | show is_max_filter (λ x, f x ⊓ g x) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, inf_le_inf hx hy) hg | lemma | is_max_filter.inf | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"inf_le_inf",
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.inf (hf : is_min_on f s a) (hg : is_min_on g s a) :
is_min_on (λ x, f x ⊓ g x) s a | hf.inf hg | lemma | is_min_on.inf | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.inf (hf : is_max_on f s a) (hg : is_max_on g s a) :
is_max_on (λ x, f x ⊓ g x) s a | hf.inf hg | lemma | is_max_on.inf | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.min (hf : is_min_filter f l a) (hg : is_min_filter g l a) :
is_min_filter (λ x, min (f x) (g x)) l a | show is_min_filter (λ x, min (f x) (g x)) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, min_le_min hx hy) hg | lemma | is_min_filter.min | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter",
"min_le_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.min (hf : is_max_filter f l a) (hg : is_max_filter g l a) :
is_max_filter (λ x, min (f x) (g x)) l a | show is_max_filter (λ x, min (f x) (g x)) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, min_le_min hx hy) hg | lemma | is_max_filter.min | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"min_le_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.min (hf : is_min_on f s a) (hg : is_min_on g s a) :
is_min_on (λ x, min (f x) (g x)) s a | hf.min hg | lemma | is_min_on.min | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.min (hf : is_max_on f s a) (hg : is_max_on g s a) :
is_max_on (λ x, min (f x) (g x)) s a | hf.min hg | lemma | is_max_on.min | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.max (hf : is_min_filter f l a) (hg : is_min_filter g l a) :
is_min_filter (λ x, max (f x) (g x)) l a | show is_min_filter (λ x, max (f x) (g x)) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, max_le_max hx hy) hg | lemma | is_min_filter.max | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_filter",
"max_le_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.max (hf : is_max_filter f l a) (hg : is_max_filter g l a) :
is_max_filter (λ x, max (f x) (g x)) l a | show is_max_filter (λ x, max (f x) (g x)) l a,
from hf.bicomp_mono (λ x x' hx y y' hy, max_le_max hx hy) hg | lemma | is_max_filter.max | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_filter",
"max_le_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.max (hf : is_min_on f s a) (hg : is_min_on g s a) :
is_min_on (λ x, max (f x) (g x)) s a | hf.max hg | lemma | is_min_on.max | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.max (hf : is_max_on f s a) (hg : is_max_on g s a) :
is_max_on (λ x, max (f x) (g x)) s a | hf.max hg | lemma | is_max_on.max | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_le.is_max_filter {α β : Type*} [preorder β] {f g : α → β} {a : α}
{l : filter α} (hle : g ≤ᶠ[l] f) (hfga : f a = g a) (h : is_max_filter f l a) :
is_max_filter g l a | begin
refine hle.mp (h.mono $ λ x hf hgf, _),
rw ← hfga,
exact le_trans hgf hf
end | lemma | filter.eventually_le.is_max_filter | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_filter.congr {α β : Type*} [preorder β] {f g : α → β} {a : α} {l : filter α}
(h : is_max_filter f l a) (heq : f =ᶠ[l] g) (hfga : f a = g a) :
is_max_filter g l a | heq.symm.le.is_max_filter hfga h | lemma | is_max_filter.congr | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_max_filter_iff {α β : Type*} [preorder β] {f g : α → β} {a : α}
{l : filter α} (heq : f =ᶠ[l] g) (hfga : f a = g a) :
is_max_filter f l a ↔ is_max_filter g l a | ⟨λ h, h.congr heq hfga, λ h, h.congr heq.symm hfga.symm⟩ | lemma | filter.eventually_eq.is_max_filter_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_max_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_le.is_min_filter {α β : Type*} [preorder β] {f g : α → β} {a : α}
{l : filter α} (hle : f ≤ᶠ[l] g) (hfga : f a = g a) (h : is_min_filter f l a) :
is_min_filter g l a | @filter.eventually_le.is_max_filter _ βᵒᵈ _ _ _ _ _ hle hfga h | lemma | filter.eventually_le.is_min_filter | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"filter.eventually_le.is_max_filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_filter.congr {α β : Type*} [preorder β] {f g : α → β} {a : α} {l : filter α}
(h : is_min_filter f l a) (heq : f =ᶠ[l] g) (hfga : f a = g a) :
is_min_filter g l a | heq.le.is_min_filter hfga h | lemma | is_min_filter.congr | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_min_filter_iff {α β : Type*} [preorder β] {f g : α → β} {a : α}
{l : filter α} (heq : f =ᶠ[l] g) (hfga : f a = g a) :
is_min_filter f l a ↔ is_min_filter g l a | ⟨λ h, h.congr heq hfga, λ h, h.congr heq.symm hfga.symm⟩ | lemma | filter.eventually_eq.is_min_filter_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_min_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_extr_filter.congr {α β : Type*} [preorder β] {f g : α → β} {a : α} {l : filter α}
(h : is_extr_filter f l a) (heq : f =ᶠ[l] g) (hfga : f a = g a) :
is_extr_filter g l a | begin
rw is_extr_filter at *,
rwa [← heq.is_max_filter_iff hfga, ← heq.is_min_filter_iff hfga],
end | lemma | is_extr_filter.congr | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_extr_filter_iff {α β : Type*} [preorder β] {f g : α → β} {a : α}
{l : filter α} (heq : f =ᶠ[l] g) (hfga : f a = g a) :
is_extr_filter f l a ↔ is_extr_filter g l a | ⟨λ h, h.congr heq hfga, λ h, h.congr heq.symm hfga.symm⟩ | lemma | filter.eventually_eq.is_extr_filter_iff | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"filter",
"is_extr_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_max_on.supr_eq (hx₀ : x₀ ∈ s) (h : is_max_on f s x₀) :
(⨆ x : s, f x) = f x₀ | begin
haveI : nonempty s := ⟨⟨x₀, hx₀⟩⟩,
exact csupr_eq_of_forall_le_of_forall_lt_exists_gt (λ x, h x.prop) (λ w hw, ⟨⟨x₀, hx₀⟩, hw⟩),
end | lemma | is_max_on.supr_eq | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"csupr_eq_of_forall_le_of_forall_lt_exists_gt",
"is_max_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_min_on.infi_eq (hx₀ : x₀ ∈ s) (h : is_min_on f s x₀) : (⨅ x : s, f x) = f x₀ | @is_max_on.supr_eq αᵒᵈ β _ _ _ _ hx₀ h | lemma | is_min_on.infi_eq | order.filter | src/order/filter/extr.lean | [
"order.filter.basic"
] | [
"is_max_on.supr_eq",
"is_min_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt [preorder β] {f g : α → β} : (f : β*) < g ↔ ∀* x, f x < g x | by simp only [lt_iff_le_not_le, eventually_and, coe_le, eventually_not, eventually_le] | lemma | filter.germ.coe_lt | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pos [preorder β] [has_zero β] {f : α → β} : 0 < (f : β*) ↔ ∀* x, 0 < f x | coe_lt | lemma | filter.germ.coe_pos | order.filter | src/order/filter/filter_product.lean | [
"order.filter.ultrafilter",
"order.filter.germ"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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