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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strict_mono_on.compares (hf : strict_mono_on f s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) :
∀ {o : ordering}, o.compares (f a) (f b) ↔ o.compares a b | | ordering.lt := hf.lt_iff_lt ha hb
| ordering.eq := ⟨λ h, ((hf.le_iff_le ha hb).1 h.le).antisymm ((hf.le_iff_le hb ha).1 h.symm.le),
congr_arg _⟩
| ordering.gt := hf.lt_iff_lt hb ha | theorem | strict_mono_on.compares | 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.compares (hf : strict_anti_on f s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) {o : ordering} :
o.compares (f a) (f b) ↔ o.compares b a | to_dual_compares_to_dual.trans $ hf.dual_right.compares hb ha | theorem | strict_anti_on.compares | 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.compares (hf : strict_mono f) {a b : α} {o : ordering} :
o.compares (f a) (f b) ↔ o.compares a b | (hf.strict_mono_on set.univ).compares trivial trivial | theorem | strict_mono.compares | 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.compares (hf : strict_anti f) {a b : α} {o : ordering} :
o.compares (f a) (f b) ↔ o.compares b a | (hf.strict_anti_on set.univ).compares trivial trivial | theorem | strict_anti.compares | 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.injective (hf : strict_mono f) : injective f | λ x y h, show compares eq x y, from hf.compares.1 h | lemma | strict_mono.injective | 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.injective (hf : strict_anti f) : injective f | λ x y h, show compares eq x y, from hf.compares.1 h.symm | lemma | strict_anti.injective | 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.maximal_of_maximal_image (hf : strict_mono f) {a} (hmax : ∀ p, p ≤ f a) (x : α) :
x ≤ a | hf.le_iff_le.mp (hmax (f x)) | lemma | strict_mono.maximal_of_maximal_image | 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.minimal_of_minimal_image (hf : strict_mono f) {a} (hmin : ∀ p, f a ≤ p) (x : α) :
a ≤ x | hf.le_iff_le.mp (hmin (f x)) | lemma | strict_mono.minimal_of_minimal_image | 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.minimal_of_maximal_image (hf : strict_anti f) {a} (hmax : ∀ p, p ≤ f a) (x : α) :
a ≤ x | hf.le_iff_le.mp (hmax (f x)) | lemma | strict_anti.minimal_of_maximal_image | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.maximal_of_minimal_image (hf : strict_anti f) {a} (hmin : ∀ p, f a ≤ p) (x : α) :
x ≤ a | hf.le_iff_le.mp (hmin (f x)) | lemma | strict_anti.maximal_of_minimal_image | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.strict_mono_iff_injective (hf : monotone f) :
strict_mono f ↔ injective f | ⟨λ h, h.injective, hf.strict_mono_of_injective⟩ | lemma | monotone.strict_mono_iff_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_iff_injective (hf : antitone f) :
strict_anti f ↔ injective f | ⟨λ h, h.injective, hf.strict_anti_of_injective⟩ | lemma | antitone.strict_anti_iff_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 | |
not_monotone_not_antitone_iff_exists_le_le :
¬ monotone f ∧ ¬ antitone f ↔ ∃ a b c, a ≤ b ∧ b ≤ c ∧
(f a < f b ∧ f c < f b ∨ f b < f a ∧ f b < f c) | begin
simp_rw [monotone, antitone, not_forall, not_le],
refine iff.symm ⟨_, _⟩,
{ rintro ⟨a, b, c, hab, hbc, ⟨hfab, hfcb⟩ | ⟨hfba, hfbc⟩⟩,
exacts [⟨⟨_, _, hbc, hfcb⟩, _, _, hab, hfab⟩, ⟨⟨_, _, hab, hfba⟩, _, _, hbc, hfbc⟩] },
rintro ⟨⟨a, b, hab, hfba⟩, c, d, hcd, hfcd⟩,
obtain hda | had := le_total d a,
... | lemma | not_monotone_not_antitone_iff_exists_le_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"monotone",
"not_forall"
] | A function between linear orders which is neither monotone nor antitone makes a dent upright or
downright. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_monotone_not_antitone_iff_exists_lt_lt :
¬ monotone f ∧ ¬ antitone f ↔ ∃ a b c, a < b ∧ b < c ∧
(f a < f b ∧ f c < f b ∨ f b < f a ∧ f b < f c) | begin
simp_rw [not_monotone_not_antitone_iff_exists_le_le, ←and_assoc],
refine exists₃_congr (λ a b c, and_congr_left $ λ h, (ne.le_iff_lt _).and $ ne.le_iff_lt _);
rintro rfl; simpa using h,
end | lemma | not_monotone_not_antitone_iff_exists_lt_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"and_congr_left",
"antitone",
"exists₃_congr",
"monotone",
"ne.le_iff_lt",
"not_monotone_not_antitone_iff_exists_le_le"
] | A function between linear orders which is neither monotone nor antitone makes a dent upright or
downright. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.cmp_map_eq (hf : strict_mono_on f s) (hx : x ∈ s) (hy : y ∈ s) :
cmp (f x) (f y) = cmp x y | ((hf.compares hx hy).2 (cmp_compares x y)).cmp_eq | lemma | strict_mono_on.cmp_map_eq | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"cmp_compares",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.cmp_map_eq (hf : strict_mono f) (x y : α) : cmp (f x) (f y) = cmp x y | (hf.strict_mono_on set.univ).cmp_map_eq trivial trivial | lemma | strict_mono.cmp_map_eq | 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_on.cmp_map_eq (hf : strict_anti_on f s) (hx : x ∈ s) (hy : y ∈ s) :
cmp (f x) (f y) = cmp y x | hf.dual_right.cmp_map_eq hy hx | lemma | strict_anti_on.cmp_map_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_anti.cmp_map_eq (hf : strict_anti f) (x y : α) : cmp (f x) (f y) = cmp y x | (hf.strict_anti_on set.univ).cmp_map_eq trivial trivial | lemma | strict_anti.cmp_map_eq | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.rel_of_forall_rel_succ_of_le_of_lt (r : β → β → Prop) [is_trans β r]
{f : ℕ → β} {a : ℕ} (h : ∀ n, a ≤ n → r (f n) (f (n + 1))) ⦃b c : ℕ⦄
(hab : a ≤ b) (hbc : b < c) :
r (f b) (f c) | begin
induction hbc with k b_lt_k r_b_k,
exacts [h _ hab, trans r_b_k (h _ (hab.trans_lt b_lt_k).le)]
end | lemma | nat.rel_of_forall_rel_succ_of_le_of_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.rel_of_forall_rel_succ_of_le_of_le (r : β → β → Prop) [is_refl β r] [is_trans β r]
{f : ℕ → β} {a : ℕ} (h : ∀ n, a ≤ n → r (f n) (f (n + 1))) ⦃b c : ℕ⦄
(hab : a ≤ b) (hbc : b ≤ c) :
r (f b) (f c) | hbc.eq_or_lt.elim (λ h, h ▸ refl _) (nat.rel_of_forall_rel_succ_of_le_of_lt r h hab) | lemma | nat.rel_of_forall_rel_succ_of_le_of_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"nat.rel_of_forall_rel_succ_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.rel_of_forall_rel_succ_of_lt (r : β → β → Prop) [is_trans β r]
{f : ℕ → β} (h : ∀ n, r (f n) (f (n + 1))) ⦃a b : ℕ⦄ (hab : a < b) : r (f a) (f b) | nat.rel_of_forall_rel_succ_of_le_of_lt r (λ n _, h n) le_rfl hab | lemma | nat.rel_of_forall_rel_succ_of_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"le_rfl",
"nat.rel_of_forall_rel_succ_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.rel_of_forall_rel_succ_of_le (r : β → β → Prop) [is_refl β r] [is_trans β r]
{f : ℕ → β} (h : ∀ n, r (f n) (f (n + 1))) ⦃a b : ℕ⦄ (hab : a ≤ b) : r (f a) (f b) | nat.rel_of_forall_rel_succ_of_le_of_le r (λ n _, h n) le_rfl hab | lemma | nat.rel_of_forall_rel_succ_of_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"le_rfl",
"nat.rel_of_forall_rel_succ_of_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_nat_of_le_succ {f : ℕ → α} (hf : ∀ n, f n ≤ f (n + 1)) :
monotone f | nat.rel_of_forall_rel_succ_of_le (≤) hf | lemma | monotone_nat_of_le_succ | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone",
"nat.rel_of_forall_rel_succ_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_nat_of_succ_le {f : ℕ → α} (hf : ∀ n, f (n + 1) ≤ f n) : antitone f | @monotone_nat_of_le_succ αᵒᵈ _ _ hf | lemma | antitone_nat_of_succ_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"monotone_nat_of_le_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_nat_of_lt_succ {f : ℕ → α} (hf : ∀ n, f n < f (n + 1)) : strict_mono f | nat.rel_of_forall_rel_succ_of_lt (<) hf | lemma | strict_mono_nat_of_lt_succ | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"nat.rel_of_forall_rel_succ_of_lt",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_nat_of_succ_lt {f : ℕ → α} (hf : ∀ n, f (n + 1) < f n) : strict_anti f | @strict_mono_nat_of_lt_succ αᵒᵈ _ f hf | lemma | strict_anti_nat_of_succ_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_mono_nat_of_lt_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_strict_mono' [no_max_order α] (a : α) : ∃ f : ℕ → α, strict_mono f ∧ f 0 = a | begin
have := (λ x : α, exists_gt x),
choose g hg,
exact ⟨λ n, nat.rec_on n a (λ _, g), strict_mono_nat_of_lt_succ $ λ n, hg _, rfl⟩
end | lemma | nat.exists_strict_mono' | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"no_max_order",
"strict_mono",
"strict_mono_nat_of_lt_succ"
] | If `α` is a preorder with no maximal elements, then there exists a strictly monotone function
`ℕ → α` with any prescribed value of `f 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_strict_anti' [no_min_order α] (a : α) : ∃ f : ℕ → α, strict_anti f ∧ f 0 = a | exists_strict_mono' (order_dual.to_dual a) | lemma | nat.exists_strict_anti' | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"no_min_order",
"order_dual.to_dual",
"strict_anti"
] | If `α` is a preorder with no maximal elements, then there exists a strictly antitone function
`ℕ → α` with any prescribed value of `f 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_strict_mono [nonempty α] [no_max_order α] : ∃ f : ℕ → α, strict_mono f | let ⟨a⟩ := ‹nonempty α›, ⟨f, hf, hfa⟩ := exists_strict_mono' a in ⟨f, hf⟩ | lemma | nat.exists_strict_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"no_max_order",
"strict_mono"
] | If `α` is a nonempty preorder with no maximal elements, then there exists a strictly monotone
function `ℕ → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_strict_anti [nonempty α] [no_min_order α] : ∃ f : ℕ → α, strict_anti f | exists_strict_mono αᵒᵈ | lemma | nat.exists_strict_anti | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"no_min_order",
"strict_anti"
] | If `α` is a nonempty preorder with no minimal elements, then there exists a strictly antitone
function `ℕ → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int.rel_of_forall_rel_succ_of_lt (r : β → β → Prop) [is_trans β r]
{f : ℤ → β} (h : ∀ n, r (f n) (f (n + 1))) ⦃a b : ℤ⦄ (hab : a < b) : r (f a) (f b) | begin
rcases hab.dest with ⟨n, rfl⟩, clear hab,
induction n with n ihn,
{ rw int.coe_nat_one, apply h },
{ rw [int.coe_nat_succ, ← int.add_assoc],
exact trans ihn (h _) }
end | lemma | int.rel_of_forall_rel_succ_of_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.rel_of_forall_rel_succ_of_le (r : β → β → Prop) [is_refl β r] [is_trans β r]
{f : ℤ → β} (h : ∀ n, r (f n) (f (n + 1))) ⦃a b : ℤ⦄ (hab : a ≤ b) : r (f a) (f b) | hab.eq_or_lt.elim (λ h, h ▸ refl _) (λ h', int.rel_of_forall_rel_succ_of_lt r h h') | lemma | int.rel_of_forall_rel_succ_of_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"int.rel_of_forall_rel_succ_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_int_of_le_succ {f : ℤ → α} (hf : ∀ n, f n ≤ f (n + 1)) : monotone f | int.rel_of_forall_rel_succ_of_le (≤) hf | lemma | monotone_int_of_le_succ | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"int.rel_of_forall_rel_succ_of_le",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_int_of_succ_le {f : ℤ → α} (hf : ∀ n, f (n + 1) ≤ f n) : antitone f | int.rel_of_forall_rel_succ_of_le (≥) hf | lemma | antitone_int_of_succ_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"int.rel_of_forall_rel_succ_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_int_of_lt_succ {f : ℤ → α} (hf : ∀ n, f n < f (n + 1)) : strict_mono f | int.rel_of_forall_rel_succ_of_lt (<) hf | lemma | strict_mono_int_of_lt_succ | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"int.rel_of_forall_rel_succ_of_lt",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_int_of_succ_lt {f : ℤ → α} (hf : ∀ n, f (n + 1) < f n) : strict_anti f | int.rel_of_forall_rel_succ_of_lt (>) hf | lemma | strict_anti_int_of_succ_lt | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"int.rel_of_forall_rel_succ_of_lt",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_strict_mono : ∃ f : ℤ → α, strict_mono f | begin
inhabit α,
rcases nat.exists_strict_mono' (default : α) with ⟨f, hf, hf₀⟩,
rcases nat.exists_strict_anti' (default : α) with ⟨g, hg, hg₀⟩,
refine ⟨λ n, int.cases_on n f (λ n, g (n + 1)), strict_mono_int_of_lt_succ _⟩,
rintro (n|_|n),
{ exact hf n.lt_succ_self },
{ show g 1 < f 0,
rw [hf₀, ← hg₀]... | lemma | int.exists_strict_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"nat.exists_strict_anti'",
"nat.exists_strict_mono'",
"strict_mono",
"strict_mono_int_of_lt_succ"
] | If `α` is a nonempty preorder with no minimal or maximal elements, then there exists a strictly
monotone function `f : ℤ → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_strict_anti : ∃ f : ℤ → α, strict_anti f | exists_strict_mono αᵒᵈ | lemma | int.exists_strict_anti | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti"
] | If `α` is a nonempty preorder with no minimal or maximal elements, then there exists a strictly
antitone function `f : ℤ → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.ne_of_lt_of_lt_nat {f : ℕ → α} (hf : monotone f) (n : ℕ) {x : α}
(h1 : f n < x) (h2 : x < f (n + 1)) (a : ℕ) :
f a ≠ x | by { rintro rfl, exact (hf.reflect_lt h1).not_le (nat.le_of_lt_succ $ hf.reflect_lt h2) } | lemma | monotone.ne_of_lt_of_lt_nat | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | If `f` is a monotone function from `ℕ` to a preorder such that `x` lies between `f n` and
`f (n + 1)`, then `x` doesn't lie in the range of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.ne_of_lt_of_lt_nat {f : ℕ → α} (hf : antitone f)
(n : ℕ) {x : α} (h1 : f (n + 1) < x) (h2 : x < f n) (a : ℕ) : f a ≠ x | by { rintro rfl, exact (hf.reflect_lt h2).not_le (nat.le_of_lt_succ $ hf.reflect_lt h1) } | lemma | antitone.ne_of_lt_of_lt_nat | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | If `f` is an antitone function from `ℕ` to a preorder such that `x` lies between `f (n + 1)` and
`f n`, then `x` doesn't lie in the range of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.ne_of_lt_of_lt_int {f : ℤ → α} (hf : monotone f) (n : ℤ) {x : α}
(h1 : f n < x) (h2 : x < f (n + 1)) (a : ℤ) :
f a ≠ x | by { rintro rfl, exact (hf.reflect_lt h1).not_le (int.le_of_lt_add_one $ hf.reflect_lt h2) } | lemma | monotone.ne_of_lt_of_lt_int | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | If `f` is a monotone function from `ℤ` to a preorder and `x` lies between `f n` and
`f (n + 1)`, then `x` doesn't lie in the range of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.ne_of_lt_of_lt_int {f : ℤ → α} (hf : antitone f)
(n : ℤ) {x : α} (h1 : f (n + 1) < x) (h2 : x < f n) (a : ℤ) : f a ≠ x | by { rintro rfl, exact (hf.reflect_lt h2).not_le (int.le_of_lt_add_one $ hf.reflect_lt h1) } | lemma | antitone.ne_of_lt_of_lt_int | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | If `f` is an antitone function from `ℤ` to a preorder and `x` lies between `f (n + 1)` and
`f n`, then `x` doesn't lie in the range of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono.id_le {φ : ℕ → ℕ} (h : strict_mono φ) : ∀ n, n ≤ φ n | λ n, nat.rec_on n (nat.zero_le _)
(λ n hn, nat.succ_le_of_lt (hn.trans_lt $ h $ nat.lt_succ_self n)) | lemma | strict_mono.id_le | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.mono_coe [preorder α] (t : set α) : monotone (coe : (subtype t) → α) | λ x y, id | lemma | subtype.mono_coe | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.strict_mono_coe [preorder α] (t : set α) : strict_mono (coe : (subtype t) → α) | λ x y, id | lemma | subtype.strict_mono_coe | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_fst : monotone (@prod.fst α β) | λ a b, and.left | lemma | monotone_fst | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_snd : monotone (@prod.snd α β) | λ a b, and.right | lemma | monotone_snd | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.prod_map (hf : monotone f) (hg : monotone g) : monotone (prod.map f g) | λ a b h, ⟨hf h.1, hg h.2⟩ | lemma | monotone.prod_map | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.prod_map (hf : antitone f) (hg : antitone g) : antitone (prod.map f g) | λ a b h, ⟨hf h.1, hg h.2⟩ | lemma | antitone.prod_map | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.prod_map (hf : strict_mono f) (hg : strict_mono g) : strict_mono (prod.map f g) | λ a b, by { simp_rw prod.lt_iff,
exact or.imp (and.imp hf.imp hg.monotone.imp) (and.imp hf.monotone.imp hg.imp) } | lemma | strict_mono.prod_map | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"prod.lt_iff",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.prod_map (hf : strict_anti f) (hg : strict_anti g) : strict_anti (prod.map f g) | λ a b, by { simp_rw prod.lt_iff,
exact or.imp (and.imp hf.imp hg.antitone.imp) (and.imp hf.antitone.imp hg.imp) } | lemma | strict_anti.prod_map | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"prod.lt_iff",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
update_mono : monotone (f.update i) | λ a b, update_le_update_iff'.2 | lemma | function.update_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
update_strict_mono : strict_mono (f.update i) | λ a b, update_lt_update_iff.2 | lemma | function.update_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 | |
const_mono : monotone (const β : α → β → α) | λ a b h i, h | lemma | function.const_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_strict_mono [nonempty β] : strict_mono (const β : α → β → α) | λ a b, const_lt_const.2 | lemma | function.const_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_on.exists_monotone_extension (h : monotone_on f s) (hl : bdd_below (f '' s))
(hu : bdd_above (f '' s)) :
∃ g : α → β, monotone g ∧ eq_on f g s | begin
/- The extension is defined by `f x = f a` for `x ≤ a`, and `f x` is the supremum of the values
of `f` to the left of `x` for `x ≥ a`. -/
classical,
rcases hl with ⟨a, ha⟩,
have hu' : ∀ x, bdd_above (f '' (Iic x ∩ s)),
from λ x, hu.mono (image_subset _ (inter_subset_right _ _)),
set g : α → β := ... | lemma | monotone_on.exists_monotone_extension | order.monotone | src/order/monotone/extension.lean | [
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below",
"cSup_le_cSup",
"disjoint",
"is_greatest",
"le_cSup_of_le",
"monotone",
"monotone_on"
] | If a function is monotone and is bounded on a set `s`, then it admits a monotone extension to
the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_on.exists_antitone_extension (h : antitone_on f s) (hl : bdd_below (f '' s))
(hu : bdd_above (f '' s)) :
∃ g : α → β, antitone g ∧ eq_on f g s | h.dual_right.exists_monotone_extension hu hl | lemma | antitone_on.exists_antitone_extension | order.monotone | src/order/monotone/extension.lean | [
"order.conditionally_complete_lattice.basic"
] | [
"antitone",
"antitone_on",
"bdd_above",
"bdd_below"
] | If a function is antitone and is bounded on a set `s`, then it admits an antitone extension to
the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary (f : ι → α) (g : ι → β) : Prop | ∀ ⦃i j⦄, g i < g j → f i ≤ f j | def | monovary | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [] | `f` monovaries with `g` if `g i < g j` implies `f i ≤ f j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary (f : ι → α) (g : ι → β) : Prop | ∀ ⦃i j⦄, g i < g j → f j ≤ f i | def | antivary | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [] | `f` antivaries with `g` if `g i < g j` implies `f j ≤ f i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on (f : ι → α) (g : ι → β) (s : set ι) : Prop | ∀ ⦃i⦄ (hi : i ∈ s) ⦃j⦄ (hj : j ∈ s), g i < g j → f i ≤ f j | def | monovary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [] | `f` monovaries with `g` on `s` if `g i < g j` implies `f i ≤ f j` for all `i, j ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on (f : ι → α) (g : ι → β) (s : set ι) : Prop | ∀ ⦃i⦄ (hi : i ∈ s) ⦃j⦄ (hj : j ∈ s), g i < g j → f j ≤ f i | def | antivary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [] | `f` antivaries with `g` on `s` if `g i < g j` implies `f j ≤ f i` for all `i, j ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.monovary_on (h : monovary f g) (s : set ι) : monovary_on f g s | λ i _ j _ hij, h hij | lemma | monovary.monovary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary.antivary_on (h : antivary f g) (s : set ι) : antivary_on f g s | λ i _ j _ hij, h hij | lemma | antivary.antivary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.empty : monovary_on f g ∅ | λ i, false.elim | lemma | monovary_on.empty | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.empty : antivary_on f g ∅ | λ i, false.elim | lemma | antivary_on.empty | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on_univ : monovary_on f g univ ↔ monovary f g | ⟨λ h i j, h trivial trivial, λ h i _ j _ hij, h hij⟩ | lemma | monovary_on_univ | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on_univ : antivary_on f g univ ↔ antivary f g | ⟨λ h i j, h trivial trivial, λ h i _ j _ hij, h hij⟩ | lemma | antivary_on_univ | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.subset (hst : s ⊆ t) (h : monovary_on f g t) : monovary_on f g s | λ i hi j hj, h (hst hi) (hst hj) | lemma | monovary_on.subset | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.subset (hst : s ⊆ t) (h : antivary_on f g t) : antivary_on f g s | λ i hi j hj, h (hst hi) (hst hj) | lemma | antivary_on.subset | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_const_left (g : ι → β) (a : α) : monovary (const ι a) g | λ i j _, le_rfl | lemma | monovary_const_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"le_rfl",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_const_left (g : ι → β) (a : α) : antivary (const ι a) g | λ i j _, le_rfl | lemma | antivary_const_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_const_right (f : ι → α) (b : β) : monovary f (const ι b) | λ i j h, (h.ne rfl).elim | lemma | monovary_const_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_const_right (f : ι → α) (b : β) : antivary f (const ι b) | λ i j h, (h.ne rfl).elim | lemma | antivary_const_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_self (f : ι → α) : monovary f f | λ i j, le_of_lt | lemma | monovary_self | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on_self (f : ι → α) (s : set ι) : monovary_on f f s | λ i _ j _, le_of_lt | lemma | monovary_on_self | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.monovary [subsingleton ι] (f : ι → α) (g : ι → β) : monovary f g | λ i j h, (ne_of_apply_ne _ h.ne $ subsingleton.elim _ _).elim | lemma | subsingleton.monovary | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary",
"ne_of_apply_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.antivary [subsingleton ι] (f : ι → α) (g : ι → β) : antivary f g | λ i j h, (ne_of_apply_ne _ h.ne $ subsingleton.elim _ _).elim | lemma | subsingleton.antivary | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"ne_of_apply_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.monovary_on [subsingleton ι] (f : ι → α) (g : ι → β) (s : set ι) :
monovary_on f g s | λ i _ j _ h, (ne_of_apply_ne _ h.ne $ subsingleton.elim _ _).elim | lemma | subsingleton.monovary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on",
"ne_of_apply_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.antivary_on [subsingleton ι] (f : ι → α) (g : ι → β) (s : set ι) :
antivary_on f g s | λ i _ j _ h, (ne_of_apply_ne _ h.ne $ subsingleton.elim _ _).elim | lemma | subsingleton.antivary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"ne_of_apply_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on_const_left (g : ι → β) (a : α) (s : set ι) : monovary_on (const ι a) g s | λ i _ j _ _, le_rfl | lemma | monovary_on_const_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"le_rfl",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on_const_left (g : ι → β) (a : α) (s : set ι) : antivary_on (const ι a) g s | λ i _ j _ _, le_rfl | lemma | antivary_on_const_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on_const_right (f : ι → α) (b : β) (s : set ι) : monovary_on f (const ι b) s | λ i _ j _ h, (h.ne rfl).elim | lemma | monovary_on_const_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on_const_right (f : ι → α) (b : β) (s : set ι) : antivary_on f (const ι b) s | λ i _ j _ h, (h.ne rfl).elim | lemma | antivary_on_const_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary.comp_right (h : monovary f g) (k : ι' → ι) : monovary (f ∘ k) (g ∘ k) | λ i j hij, h hij | lemma | monovary.comp_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary.comp_right (h : antivary f g) (k : ι' → ι) : antivary (f ∘ k) (g ∘ k) | λ i j hij, h hij | lemma | antivary.comp_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.comp_right (h : monovary_on f g s) (k : ι' → ι) :
monovary_on (f ∘ k) (g ∘ k) (k ⁻¹' s) | λ i hi j hj, h hi hj | lemma | monovary_on.comp_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.comp_right (h : antivary_on f g s) (k : ι' → ι) :
antivary_on (f ∘ k) (g ∘ k) (k ⁻¹' s) | λ i hi j hj, h hi hj | lemma | antivary_on.comp_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary.comp_monotone_left (h : monovary f g) (hf : monotone f') : monovary (f' ∘ f) g | λ i j hij, hf $ h hij | lemma | monovary.comp_monotone_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monotone",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary.comp_antitone_left (h : monovary f g) (hf : antitone f') : antivary (f' ∘ f) g | λ i j hij, hf $ h hij | lemma | monovary.comp_antitone_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone",
"antivary",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary.comp_monotone_left (h : antivary f g) (hf : monotone f') : antivary (f' ∘ f) g | λ i j hij, hf $ h hij | lemma | antivary.comp_monotone_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary.comp_antitone_left (h : antivary f g) (hf : antitone f') : monovary (f' ∘ f) g | λ i j hij, hf $ h hij | lemma | antivary.comp_antitone_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone",
"antivary",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.comp_monotone_on_left (h : monovary_on f g s) (hf : monotone f') :
monovary_on (f' ∘ f) g s | λ i hi j hj hij, hf $ h hi hj hij | lemma | monovary_on.comp_monotone_on_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monotone",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.comp_antitone_on_left (h : monovary_on f g s) (hf : antitone f') :
antivary_on (f' ∘ f) g s | λ i hi j hj hij, hf $ h hi hj hij | lemma | monovary_on.comp_antitone_on_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone",
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.comp_monotone_on_left (h : antivary_on f g s) (hf : monotone f') :
antivary_on (f' ∘ f) g s | λ i hi j hj hij, hf $ h hi hj hij | lemma | antivary_on.comp_monotone_on_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.comp_antitone_on_left (h : antivary_on f g s) (hf : antitone f') :
monovary_on (f' ∘ f) g s | λ i hi j hj hij, hf $ h hi hj hij | lemma | antivary_on.comp_antitone_on_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone",
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary.dual : monovary f g → monovary (to_dual ∘ f) (to_dual ∘ g) | swap | lemma | monovary.dual | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary.dual : antivary f g → antivary (to_dual ∘ f) (to_dual ∘ g) | swap | lemma | antivary.dual | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary.dual_left : monovary f g → antivary (to_dual ∘ f) g | id | lemma | monovary.dual_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary.dual_left : antivary f g → monovary (to_dual ∘ f) g | id | lemma | antivary.dual_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary.dual_right : monovary f g → antivary f (to_dual ∘ g) | swap | lemma | monovary.dual_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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