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 |
|---|---|---|---|---|---|---|---|---|---|---|
strict_mono_on_to_dual_comp_iff :
strict_mono_on (to_dual ∘ f) s ↔ strict_anti_on f s | iff.rfl | lemma | strict_mono_on_to_dual_comp_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on_to_dual_comp_iff :
strict_anti_on (to_dual ∘ f) s ↔ strict_mono_on f s | iff.rfl | lemma | strict_anti_on_to_dual_comp_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.dual (hf : monotone f) : monotone (to_dual ∘ f ∘ of_dual) | swap hf | lemma | monotone.dual | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.dual (hf : antitone f) : antitone (to_dual ∘ f ∘ of_dual) | swap hf | lemma | antitone.dual | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.dual (hf : monotone_on f s) : monotone_on (to_dual ∘ f ∘ of_dual) s | swap₂ hf | lemma | monotone_on.dual | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.dual (hf : antitone_on f s) : antitone_on (to_dual ∘ f ∘ of_dual) s | swap₂ hf | lemma | antitone_on.dual | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.dual (hf : strict_mono f) : strict_mono (to_dual ∘ f ∘ of_dual) | swap hf | lemma | strict_mono.dual | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.dual (hf : strict_anti f) : strict_anti (to_dual ∘ f ∘ of_dual) | swap hf | lemma | strict_anti.dual | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.dual (hf : strict_mono_on f s) :
strict_mono_on (to_dual ∘ f ∘ of_dual) s | swap₂ hf | lemma | strict_mono_on.dual | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on.dual (hf : strict_anti_on f s) :
strict_anti_on (to_dual ∘ f ∘ of_dual) s | swap₂ hf | lemma | strict_anti_on.dual | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.comp_le_comp_left [preorder β]
{f : β → α} {g h : γ → β} (hf : monotone f) (le_gh : g ≤ h) :
has_le.le.{max w u} (f ∘ g) (f ∘ h) | λ x, hf (le_gh x) | theorem | monotone.comp_le_comp_left | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_lam {f : α → β → γ} (hf : ∀ b, monotone (λ a, f a b)) : monotone f | λ a a' h b, hf b h | theorem | monotone_lam | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_app (f : β → α → γ) (b : β) (hf : monotone (λ a b, f b a)) : monotone (f b) | λ a a' h, hf h b | theorem | monotone_app | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_lam {f : α → β → γ} (hf : ∀ b, antitone (λ a, f a b)) : antitone f | λ a a' h b, hf b h | theorem | antitone_lam | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_app (f : β → α → γ) (b : β) (hf : antitone (λ a b, f b a)) : antitone (f b) | λ a a' h, hf h b | theorem | antitone_app | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.monotone_eval {ι : Type u} {α : ι → Type v} [∀ i, preorder (α i)] (i : ι) :
monotone (function.eval i : (Π i, α i) → α i) | λ f g H, H i | lemma | function.monotone_eval | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"function.eval",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.imp (hf : monotone f) (h : a ≤ b) : f a ≤ f b | hf h | lemma | monotone.imp | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.imp (hf : antitone f) (h : a ≤ b) : f b ≤ f a | hf h | lemma | antitone.imp | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.imp (hf : strict_mono f) (h : a < b) : f a < f b | hf h | lemma | strict_mono.imp | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.imp (hf : strict_anti f) (h : a < b) : f b < f a | hf h | lemma | strict_anti.imp | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.monotone_on (hf : monotone f) (s : set α) : monotone_on f s | λ a _ b _, hf.imp | lemma | monotone.monotone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.antitone_on (hf : antitone f) (s : set α) : antitone_on f s | λ a _ b _, hf.imp | lemma | antitone.antitone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"antitone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on_univ : monotone_on f set.univ ↔ monotone f | ⟨λ h a b, h trivial trivial, λ h, h.monotone_on _⟩ | lemma | monotone_on_univ | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on_univ : antitone_on f set.univ ↔ antitone f | ⟨λ h a b, h trivial trivial, λ h, h.antitone_on _⟩ | lemma | antitone_on_univ | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"antitone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.strict_mono_on (hf : strict_mono f) (s : set α) : strict_mono_on f s | λ a _ b _, hf.imp | lemma | strict_mono.strict_mono_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.strict_anti_on (hf : strict_anti f) (s : set α) : strict_anti_on f s | λ a _ b _, hf.imp | lemma | strict_anti.strict_anti_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_univ : strict_mono_on f set.univ ↔ strict_mono f | ⟨λ h a b, h trivial trivial, λ h, h.strict_mono_on _⟩ | lemma | strict_mono_on_univ | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on_univ : strict_anti_on f set.univ ↔ strict_anti f | ⟨λ h a b, h trivial trivial, λ h, h.strict_anti_on _⟩ | lemma | strict_anti_on_univ | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.strict_mono_of_injective (h₁ : monotone f) (h₂ : injective f) : strict_mono f | λ a b h, (h₁ h.le).lt_of_ne (λ H, h.ne $ h₂ H) | lemma | monotone.strict_mono_of_injective | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.strict_anti_of_injective (h₁ : antitone f) (h₂ : injective f) : strict_anti f | λ a b h, (h₁ h.le).lt_of_ne (λ H, h.ne $ h₂ H.symm) | lemma | antitone.strict_anti_of_injective | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_iff_forall_lt : monotone f ↔ ∀ ⦃a b⦄, a < b → f a ≤ f b | forall₂_congr $ λ a b, ⟨λ hf h, hf h.le, λ hf h, h.eq_or_lt.elim (λ H, (congr_arg _ H).le) hf⟩ | lemma | monotone_iff_forall_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"forall₂_congr",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_iff_forall_lt : antitone f ↔ ∀ ⦃a b⦄, a < b → f b ≤ f a | forall₂_congr $ λ a b, ⟨λ hf h, hf h.le, λ hf h, h.eq_or_lt.elim (λ H, (congr_arg _ H).ge) hf⟩ | lemma | antitone_iff_forall_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on_iff_forall_lt :
monotone_on f s ↔ ∀ ⦃a⦄ (ha : a ∈ s) ⦃b⦄ (hb : b ∈ s), a < b → f a ≤ f b | ⟨λ hf a ha b hb h, hf ha hb h.le,
λ hf a ha b hb h, h.eq_or_lt.elim (λ H, (congr_arg _ H).le) (hf ha hb)⟩ | lemma | monotone_on_iff_forall_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on_iff_forall_lt :
antitone_on f s ↔ ∀ ⦃a⦄ (ha : a ∈ s) ⦃b⦄ (hb : b ∈ s), a < b → f b ≤ f a | ⟨λ hf a ha b hb h, hf ha hb h.le,
λ hf a ha b hb h, h.eq_or_lt.elim (λ H, (congr_arg _ H).ge) (hf ha hb)⟩ | lemma | antitone_on_iff_forall_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.monotone_on (hf : strict_mono_on f s) : monotone_on f s | monotone_on_iff_forall_lt.2 $ λ a ha b hb h, (hf ha hb h).le | lemma | strict_mono_on.monotone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on.antitone_on (hf : strict_anti_on f s) : antitone_on f s | antitone_on_iff_forall_lt.2 $ λ a ha b hb h, (hf ha hb h).le | lemma | strict_anti_on.antitone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.monotone (hf : strict_mono f) : monotone f | monotone_iff_forall_lt.2 $ λ a b h, (hf h).le | lemma | strict_mono.monotone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.antitone (hf : strict_anti f) : antitone f | antitone_iff_forall_lt.2 $ λ a b h, (hf h).le | lemma | strict_anti.antitone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone [subsingleton α] (f : α → β) : monotone f | λ a b _, (congr_arg _ $ subsingleton.elim _ _).le | lemma | subsingleton.monotone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone [subsingleton α] (f : α → β) : antitone f | λ a b _, (congr_arg _ $ subsingleton.elim _ _).le | lemma | subsingleton.antitone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone' [subsingleton β] (f : α → β) : monotone f | λ a b _, (subsingleton.elim _ _).le | lemma | subsingleton.monotone' | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone' [subsingleton β] (f : α → β) : antitone f | λ a b _, (subsingleton.elim _ _).le | lemma | subsingleton.antitone' | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono [subsingleton α] (f : α → β) : strict_mono f | λ a b h, (h.ne $ subsingleton.elim _ _).elim | lemma | subsingleton.strict_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti [subsingleton α] (f : α → β) : strict_anti f | λ a b h, (h.ne $ subsingleton.elim _ _).elim | lemma | subsingleton.strict_anti | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_id [preorder α] : monotone (id : α → α) | λ a b, id | lemma | monotone_id | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on_id [preorder α] {s : set α} : monotone_on id s | λ a ha b hb, id | lemma | monotone_on_id | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_id [preorder α] : strict_mono (id : α → α) | λ a b, id | lemma | strict_mono_id | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_id [preorder α] {s : set α} : strict_mono_on id s | λ a ha b hb, id | lemma | strict_mono_on_id | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_const [preorder α] [preorder β] {c : β} : monotone (λ (a : α), c) | λ a b _, le_rfl | theorem | monotone_const | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"le_rfl",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on_const [preorder α] [preorder β] {c : β} {s : set α} :
monotone_on (λ (a : α), c) s | λ a _ b _ _, le_rfl | theorem | monotone_on_const | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"le_rfl",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_const [preorder α] [preorder β] {c : β} : antitone (λ (a : α), c) | λ a b _, le_refl c | theorem | antitone_const | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on_const [preorder α] [preorder β] {c : β} {s : set α} :
antitone_on (λ (a : α), c) s | λ a _ b _ _, le_rfl | theorem | antitone_on_const | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_of_le_iff_le [preorder α] [preorder β] {f : α → β}
(h : ∀ x y, x ≤ y ↔ f x ≤ f y) : strict_mono f | λ a b, (lt_iff_lt_of_le_iff_le' (h _ _) (h _ _)).1 | lemma | strict_mono_of_le_iff_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"lt_iff_lt_of_le_iff_le'",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_of_le_iff_le [preorder α] [preorder β] {f : α → β}
(h : ∀ x y, x ≤ y ↔ f y ≤ f x) : strict_anti f | λ a b, (lt_iff_lt_of_le_iff_le' (h _ _) (h _ _)).1 | lemma | strict_anti_of_le_iff_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"lt_iff_lt_of_le_iff_le'",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_of_lt_imp_ne [linear_order α] {f : α → β} (h : ∀ x y, x < y → f x ≠ f y) :
injective f | begin
intros x y hxy,
contrapose hxy,
cases ne.lt_or_lt hxy with hxy hxy,
exacts [h _ _ hxy, (h _ _ hxy).symm]
end | lemma | injective_of_lt_imp_ne | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"ne.lt_or_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_of_le_imp_le [partial_order α] [preorder β] (f : α → β)
(h : ∀ {x y}, f x ≤ f y → x ≤ y) : injective f | λ x y hxy, (h hxy.le).antisymm (h hxy.ge) | lemma | injective_of_le_imp_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.is_max_of_apply (hf : strict_mono f) (ha : is_max (f a)) : is_max a | of_not_not $ λ h, let ⟨b, hb⟩ := not_is_max_iff.1 h in (hf hb).not_is_max ha | lemma | strict_mono.is_max_of_apply | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"is_max",
"not_is_max",
"of_not_not",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.is_min_of_apply (hf : strict_mono f) (ha : is_min (f a)) : is_min a | of_not_not $ λ h, let ⟨b, hb⟩ := not_is_min_iff.1 h in (hf hb).not_is_min ha | lemma | strict_mono.is_min_of_apply | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"is_min",
"not_is_min",
"of_not_not",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.is_max_of_apply (hf : strict_anti f) (ha : is_min (f a)) : is_max a | of_not_not $ λ h, let ⟨b, hb⟩ := not_is_max_iff.1 h in (hf hb).not_is_min ha | lemma | strict_anti.is_max_of_apply | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"is_max",
"is_min",
"not_is_min",
"of_not_not",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.is_min_of_apply (hf : strict_anti f) (ha : is_max (f a)) : is_min a | of_not_not $ λ h, let ⟨b, hb⟩ := not_is_min_iff.1 h in (hf hb).not_is_max ha | lemma | strict_anti.is_min_of_apply | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"is_max",
"is_min",
"not_is_max",
"of_not_not",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.ite' (hf : strict_mono f) (hg : strict_mono g) {p : α → Prop}
[decidable_pred p] (hp : ∀ ⦃x y⦄, x < y → p y → p x)
(hfg : ∀ ⦃x y⦄, p x → ¬p y → x < y → f x < g y) :
strict_mono (λ x, if p x then f x else g x) | begin
intros x y h,
by_cases hy : p y,
{ have hx : p x := hp h hy,
simpa [hx, hy] using hf h },
by_cases hx : p x,
{ simpa [hx, hy] using hfg hx hy h },
{ simpa [hx, hy] using hg h}
end | lemma | strict_mono.ite' | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.ite (hf : strict_mono f) (hg : strict_mono g) {p : α → Prop}
[decidable_pred p] (hp : ∀ ⦃x y⦄, x < y → p y → p x) (hfg : ∀ x, f x ≤ g x) :
strict_mono (λ x, if p x then f x else g x) | hf.ite' hg hp $ λ x y hx hy h, (hf h).trans_le (hfg y) | lemma | strict_mono.ite | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.ite' (hf : strict_anti f) (hg : strict_anti g) {p : α → Prop}
[decidable_pred p] (hp : ∀ ⦃x y⦄, x < y → p y → p x)
(hfg : ∀ ⦃x y⦄, p x → ¬p y → x < y → g y < f x) :
strict_anti (λ x, if p x then f x else g x) | (strict_mono.ite' hf.dual_right hg.dual_right hp hfg).dual_right | lemma | strict_anti.ite' | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_mono.ite'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.ite (hf : strict_anti f) (hg : strict_anti g) {p : α → Prop}
[decidable_pred p] (hp : ∀ ⦃x y⦄, x < y → p y → p x) (hfg : ∀ x, g x ≤ f x) :
strict_anti (λ x, if p x then f x else g x) | hf.ite' hg hp $ λ x y hx hy h, (hfg y).trans_lt (hf h) | lemma | strict_anti.ite | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.comp (hg : monotone g) (hf : monotone f) :
monotone (g ∘ f) | λ a b h, hg (hf h) | lemma | monotone.comp | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.comp_antitone (hg : monotone g) (hf : antitone f) :
antitone (g ∘ f) | λ a b h, hg (hf h) | lemma | monotone.comp_antitone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.comp (hg : antitone g) (hf : antitone f) :
monotone (g ∘ f) | λ a b h, hg (hf h) | lemma | antitone.comp | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.comp_monotone (hg : antitone g) (hf : monotone f) :
antitone (g ∘ f) | λ a b h, hg (hf h) | lemma | antitone.comp_monotone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.iterate {f : α → α} (hf : monotone f) (n : ℕ) : monotone (f^[n]) | nat.rec_on n monotone_id (λ n h, h.comp hf) | lemma | monotone.iterate | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone",
"monotone_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.comp_monotone_on (hg : monotone g) (hf : monotone_on f s) :
monotone_on (g ∘ f) s | λ a ha b hb h, hg (hf ha hb h) | lemma | monotone.comp_monotone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.comp_antitone_on (hg : monotone g) (hf : antitone_on f s) :
antitone_on (g ∘ f) s | λ a ha b hb h, hg (hf ha hb h) | lemma | monotone.comp_antitone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.comp_antitone_on (hg : antitone g) (hf : antitone_on f s) :
monotone_on (g ∘ f) s | λ a ha b hb h, hg (hf ha hb h) | lemma | antitone.comp_antitone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"antitone_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.comp_monotone_on (hg : antitone g) (hf : monotone_on f s) :
antitone_on (g ∘ f) s | λ a ha b hb h, hg (hf ha hb h) | lemma | antitone.comp_monotone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"antitone_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.comp (hg : strict_mono g) (hf : strict_mono f) :
strict_mono (g ∘ f) | λ a b h, hg (hf h) | lemma | strict_mono.comp | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.comp_strict_anti (hg : strict_mono g) (hf : strict_anti f) :
strict_anti (g ∘ f) | λ a b h, hg (hf h) | lemma | strict_mono.comp_strict_anti | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.comp (hg : strict_anti g) (hf : strict_anti f) :
strict_mono (g ∘ f) | λ a b h, hg (hf h) | lemma | strict_anti.comp | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.comp_strict_mono (hg : strict_anti g) (hf : strict_mono f) :
strict_anti (g ∘ f) | λ a b h, hg (hf h) | lemma | strict_anti.comp_strict_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.iterate {f : α → α} (hf : strict_mono f) (n : ℕ) :
strict_mono (f^[n]) | nat.rec_on n strict_mono_id (λ n h, h.comp hf) | lemma | strict_mono.iterate | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono",
"strict_mono_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.comp_strict_mono_on (hg : strict_mono g) (hf : strict_mono_on f s) :
strict_mono_on (g ∘ f) s | λ a ha b hb h, hg (hf ha hb h) | lemma | strict_mono.comp_strict_mono_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.comp_strict_anti_on (hg : strict_mono g) (hf : strict_anti_on f s) :
strict_anti_on (g ∘ f) s | λ a ha b hb h, hg (hf ha hb h) | lemma | strict_mono.comp_strict_anti_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti_on",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.comp_strict_anti_on (hg : strict_anti g) (hf : strict_anti_on f s) :
strict_mono_on (g ∘ f) s | λ a ha b hb h, hg (hf ha hb h) | lemma | strict_anti.comp_strict_anti_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_anti_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.comp_strict_mono_on (hg : strict_anti g) (hf : strict_mono_on f s) :
strict_anti_on (g ∘ f) s | λ a ha b hb h, hg (hf ha hb h) | lemma | strict_anti.comp_strict_mono_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_anti_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
foldl_monotone [preorder α] {f : α → β → α} (H : ∀ b, monotone (λ a, f a b)) (l : list β) :
monotone (λ a, l.foldl f a) | list.rec_on l (λ _ _, id) (λ i l hl _ _ h, hl (H _ h)) | theorem | list.foldl_monotone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
foldr_monotone [preorder β] {f : α → β → β} (H : ∀ a, monotone (f a)) (l : list α) :
monotone (λ b, l.foldr f b) | λ _ _ h, list.rec_on l h (λ i l hl, H i hl) | theorem | list.foldr_monotone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
foldl_strict_mono [preorder α] {f : α → β → α} (H : ∀ b, strict_mono (λ a, f a b))
(l : list β) : strict_mono (λ a, l.foldl f a) | list.rec_on l (λ _ _, id) (λ i l hl _ _ h, hl (H _ h)) | theorem | list.foldl_strict_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
foldr_strict_mono [preorder β] {f : α → β → β} (H : ∀ a, strict_mono (f a)) (l : list α) :
strict_mono (λ b, l.foldr f b) | λ _ _ h, list.rec_on l h (λ i l hl, H i hl) | theorem | list.foldr_strict_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.reflect_lt (hf : monotone f) {a b : α} (h : f a < f b) : a < b | lt_of_not_ge (λ h', h.not_le (hf h')) | lemma | monotone.reflect_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.reflect_lt (hf : antitone f) {a b : α} (h : f a < f b) : b < a | lt_of_not_ge (λ h', h.not_le (hf h')) | lemma | antitone.reflect_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.reflect_lt (hf : monotone_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s)
(h : f a < f b) :
a < b | lt_of_not_ge $ λ h', h.not_le $ hf hb ha h' | lemma | monotone_on.reflect_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.reflect_lt (hf : antitone_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s)
(h : f a < f b) :
b < a | lt_of_not_ge $ λ h', h.not_le $ hf ha hb h' | lemma | antitone_on.reflect_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.le_iff_le (hf : strict_mono_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
f a ≤ f b ↔ a ≤ b | ⟨λ h, le_of_not_gt $ λ h', (hf hb ha h').not_le h,
λ h, h.lt_or_eq_dec.elim (λ h', (hf ha hb h').le) (λ h', h' ▸ le_rfl)⟩ | lemma | strict_mono_on.le_iff_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"le_rfl",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on.le_iff_le (hf : strict_anti_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
f a ≤ f b ↔ b ≤ a | hf.dual_right.le_iff_le hb ha | lemma | strict_anti_on.le_iff_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.eq_iff_eq (hf : strict_mono_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
f a = f b ↔ a = b | ⟨λ h, le_antisymm ((hf.le_iff_le ha hb).mp h.le) ((hf.le_iff_le hb ha).mp h.ge),
by { rintro rfl, refl, }⟩ | lemma | strict_mono_on.eq_iff_eq | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on.eq_iff_eq (hf : strict_anti_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
f a = f b ↔ b = a | (hf.dual_right.eq_iff_eq ha hb).trans eq_comm | lemma | strict_anti_on.eq_iff_eq | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.lt_iff_lt (hf : strict_mono_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
f a < f b ↔ a < b | by rw [lt_iff_le_not_le, lt_iff_le_not_le, hf.le_iff_le ha hb, hf.le_iff_le hb ha] | lemma | strict_mono_on.lt_iff_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on.lt_iff_lt (hf : strict_anti_on f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
f a < f b ↔ b < a | hf.dual_right.lt_iff_lt hb ha | lemma | strict_anti_on.lt_iff_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.le_iff_le (hf : strict_mono f) {a b : α} :
f a ≤ f b ↔ a ≤ b | (hf.strict_mono_on set.univ).le_iff_le trivial trivial | lemma | strict_mono.le_iff_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.le_iff_le (hf : strict_anti f) {a b : α} :
f a ≤ f b ↔ b ≤ a | (hf.strict_anti_on set.univ).le_iff_le trivial trivial | lemma | strict_anti.le_iff_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.lt_iff_lt (hf : strict_mono f) {a b : α} :
f a < f b ↔ a < b | (hf.strict_mono_on set.univ).lt_iff_lt trivial trivial | lemma | strict_mono.lt_iff_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.lt_iff_lt (hf : strict_anti f) {a b : α} :
f a < f b ↔ b < a | (hf.strict_anti_on set.univ).lt_iff_lt trivial trivial | lemma | strict_anti.lt_iff_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | 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.