statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
antivary.dual_right : antivary f g → monovary f (to_dual ∘ g) | swap | lemma | antivary.dual_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.dual : monovary_on f g s → monovary_on (to_dual ∘ f) (to_dual ∘ g) s | swap₂ | lemma | monovary_on.dual | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.dual : antivary_on f g s → antivary_on (to_dual ∘ f) (to_dual ∘ g) s | swap₂ | lemma | antivary_on.dual | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.dual_left : monovary_on f g s → antivary_on (to_dual ∘ f) g s | id | lemma | monovary_on.dual_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.dual_left : antivary_on f g s → monovary_on (to_dual ∘ f) g s | id | lemma | antivary_on.dual_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.dual_right : monovary_on f g s → antivary_on f (to_dual ∘ g) s | swap₂ | lemma | monovary_on.dual_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.dual_right : antivary_on f g s → monovary_on f (to_dual ∘ g) s | swap₂ | lemma | antivary_on.dual_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_to_dual_left : monovary (to_dual ∘ f) g ↔ antivary f g | iff.rfl | lemma | monovary_to_dual_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_to_dual_right : monovary f (to_dual ∘ g) ↔ antivary f g | forall_swap | lemma | monovary_to_dual_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"forall_swap",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_to_dual_left : antivary (to_dual ∘ f) g ↔ monovary f g | iff.rfl | lemma | antivary_to_dual_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_to_dual_right : antivary f (to_dual ∘ g) ↔ monovary f g | forall_swap | lemma | antivary_to_dual_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary",
"forall_swap",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on_to_dual_left : monovary_on (to_dual ∘ f) g s ↔ antivary_on f g s | iff.rfl | lemma | monovary_on_to_dual_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on_to_dual_right : monovary_on f (to_dual ∘ g) s ↔ antivary_on f g s | forall₂_swap | lemma | monovary_on_to_dual_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"forall₂_swap",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on_to_dual_left : antivary_on (to_dual ∘ f) g s ↔ monovary_on f g s | iff.rfl | lemma | antivary_on_to_dual_left | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on_to_dual_right : antivary_on f (to_dual ∘ g) s ↔ monovary_on f g s | forall₂_swap | lemma | antivary_on_to_dual_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"forall₂_swap",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_id_iff : monovary f id ↔ monotone f | monotone_iff_forall_lt.symm | lemma | monovary_id_iff | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monotone",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_id_iff : antivary f id ↔ antitone f | antitone_iff_forall_lt.symm | lemma | antivary_id_iff | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone",
"antivary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on_id_iff : monovary_on f id s ↔ monotone_on f s | monotone_on_iff_forall_lt.symm | lemma | monovary_on_id_iff | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monotone_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on_id_iff : antivary_on f id s ↔ antitone_on f s | antitone_on_iff_forall_lt.symm | lemma | antivary_on_id_iff | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone_on",
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.monovary (hf : monotone f) (hg : monotone g) : monovary f g | λ i j hij, hf (hg.reflect_lt hij).le | lemma | monotone.monovary | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monotone",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.antivary (hf : monotone f) (hg : antitone g) : antivary f g | (hf.monovary hg.dual_right).dual_right | lemma | monotone.antivary | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone",
"antivary",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.monovary (hf : antitone f) (hg : antitone g) : monovary f g | (hf.dual_right.antivary hg).dual_left | lemma | antitone.monovary | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone",
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.antivary (hf : antitone f) (hg : monotone g) : antivary f g | (hf.monovary hg.dual_right).dual_right | lemma | antitone.antivary | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone",
"antivary",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.monovary_on (hf : monotone_on f s) (hg : monotone_on g s) :
monovary_on f g s | λ i hi j hj hij, hf hi hj (hg.reflect_lt hi hj hij).le | lemma | monotone_on.monovary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monotone_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.antivary_on (hf : monotone_on f s) (hg : antitone_on g s) :
antivary_on f g s | (hf.monovary_on hg.dual_right).dual_right | lemma | monotone_on.antivary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone_on",
"antivary_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.monovary_on (hf : antitone_on f s) (hg : antitone_on g s) :
monovary_on f g s | (hf.dual_right.antivary_on hg).dual_left | lemma | antitone_on.monovary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.antivary_on (hf : antitone_on f s) (hg : monotone_on g s) :
antivary_on f g s | (hf.monovary_on hg.dual_right).dual_right | lemma | antitone_on.antivary_on | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone_on",
"antivary_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.comp_monotone_on_right (h : monovary_on f g s) (hg : monotone_on g' (g '' s)) :
monovary_on f (g' ∘ g) s | λ i hi j hj hij, h hi hj $ hg.reflect_lt (mem_image_of_mem _ hi) (mem_image_of_mem _ hj) hij | lemma | monovary_on.comp_monotone_on_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monotone_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.comp_antitone_on_right (h : monovary_on f g s) (hg : antitone_on g' (g '' s)) :
antivary_on f (g' ∘ g) s | λ i hi j hj hij, h hj hi $ hg.reflect_lt (mem_image_of_mem _ hi) (mem_image_of_mem _ hj) hij | lemma | monovary_on.comp_antitone_on_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone_on",
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.comp_monotone_on_right (h : antivary_on f g s) (hg : monotone_on g' (g '' s)) :
antivary_on f (g' ∘ g) s | λ i hi j hj hij, h hi hj $ hg.reflect_lt (mem_image_of_mem _ hi) (mem_image_of_mem _ hj) hij | lemma | antivary_on.comp_monotone_on_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.comp_antitone_on_right (h : antivary_on f g s) (hg : antitone_on g' (g '' s)) :
monovary_on f (g' ∘ g) s | λ i hi j hj hij, h hj hi $ hg.reflect_lt (mem_image_of_mem _ hi) (mem_image_of_mem _ hj) hij | lemma | antivary_on.comp_antitone_on_right | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antitone_on",
"antivary_on",
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary.symm (h : monovary f g) : monovary g f | λ i j hf, le_of_not_lt $ λ hg, hf.not_le $ h hg | lemma | monovary.symm | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary.symm (h : antivary f g) : antivary g f | λ i j hf, le_of_not_lt $ λ hg, hf.not_le $ h hg | lemma | antivary.symm | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on.symm (h : monovary_on f g s) : monovary_on g f s | λ i hi j hj hf, le_of_not_lt $ λ hg, hf.not_le $ h hj hi hg | lemma | monovary_on.symm | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on.symm (h : antivary_on f g s) : antivary_on g f s | λ i hi j hj hf, le_of_not_lt $ λ hg, hf.not_le $ h hi hj hg | lemma | antivary_on.symm | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_comm : monovary f g ↔ monovary g f | ⟨monovary.symm, monovary.symm⟩ | lemma | monovary_comm | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_comm : antivary f g ↔ antivary g f | ⟨antivary.symm, antivary.symm⟩ | lemma | antivary_comm | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monovary_on_comm : monovary_on f g s ↔ monovary_on g f s | ⟨monovary_on.symm, monovary_on.symm⟩ | lemma | monovary_on_comm | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"monovary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antivary_on_comm : antivary_on f g s ↔ antivary_on g f s | ⟨antivary_on.symm, antivary_on.symm⟩ | lemma | antivary_on_comm | order.monotone | src/order/monotone/monovary.lean | [
"data.set.image"
] | [
"antivary_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_of_odd_strict_mono_on_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : strict_mono_on f (Ici 0)) :
strict_mono f | begin
refine strict_mono_on.Iic_union_Ici (λ x hx y hy hxy, neg_lt_neg_iff.1 _) h₂,
rw [← h₁, ← h₁],
exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_lt_neg hxy)
end | lemma | strict_mono_of_odd_strict_mono_on_nonneg | order.monotone | src/order/monotone/odd.lean | [
"order.monotone.union",
"algebra.order.group.instances"
] | [
"strict_mono",
"strict_mono_on",
"strict_mono_on.Iic_union_Ici"
] | An odd function on a linear ordered additive commutative group is strictly monotone on the whole
group provided that it is strictly monotone on `set.Ici 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_of_odd_strict_anti_on_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : strict_anti_on f (Ici 0)) :
strict_anti f | @strict_mono_of_odd_strict_mono_on_nonneg G Hᵒᵈ _ _ _ h₁ h₂ | lemma | strict_anti_of_odd_strict_anti_on_nonneg | order.monotone | src/order/monotone/odd.lean | [
"order.monotone.union",
"algebra.order.group.instances"
] | [
"strict_anti",
"strict_anti_on",
"strict_mono_of_odd_strict_mono_on_nonneg"
] | An odd function on a linear ordered additive commutative group is strictly antitone on the whole
group provided that it is strictly antitone on `set.Ici 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_of_odd_of_monotone_on_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : monotone_on f (Ici 0)) : monotone f | begin
refine monotone_on.Iic_union_Ici (λ x hx y hy hxy, neg_le_neg_iff.1 _) h₂,
rw [← h₁, ← h₁],
exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_le_neg hxy)
end | lemma | monotone_of_odd_of_monotone_on_nonneg | order.monotone | src/order/monotone/odd.lean | [
"order.monotone.union",
"algebra.order.group.instances"
] | [
"monotone",
"monotone_on",
"monotone_on.Iic_union_Ici"
] | An odd function on a linear ordered additive commutative group is monotone on the whole group
provided that it is monotone on `set.Ici 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_of_odd_of_monotone_on_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : antitone_on f (Ici 0)) : antitone f | @monotone_of_odd_of_monotone_on_nonneg G Hᵒᵈ _ _ _ h₁ h₂ | lemma | antitone_of_odd_of_monotone_on_nonneg | order.monotone | src/order/monotone/odd.lean | [
"order.monotone.union",
"algebra.order.group.instances"
] | [
"antitone",
"antitone_on",
"monotone_of_odd_of_monotone_on_nonneg"
] | An odd function on a linear ordered additive commutative group is antitone on the whole group
provided that it is monotone on `set.Ici 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.union {s t : set α} {c : α} (h₁ : strict_mono_on f s)
(h₂ : strict_mono_on f t) (hs : is_greatest s c) (ht : is_least t c) :
strict_mono_on f (s ∪ t) | begin
have A : ∀ x, x ∈ s ∪ t → x ≤ c → x ∈ s,
{ assume x hx hxc,
cases hx, { exact hx },
rcases eq_or_lt_of_le hxc with rfl|h'x, { exact hs.1 },
exact (lt_irrefl _ (h'x.trans_le (ht.2 hx))).elim },
have B : ∀ x, x ∈ s ∪ t → c ≤ x → x ∈ t,
{ assume x hx hxc,
cases hx, swap, { exact hx },
rca... | lemma | strict_mono_on.union | order.monotone | src/order/monotone/union.lean | [
"order.bounds.basic"
] | [
"eq_or_lt_of_le",
"is_greatest",
"is_least",
"strict_mono_on"
] | If `f` is strictly monotone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is strictly monotone on `s ∪ t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.Iic_union_Ici (h₁ : strict_mono_on f (Iic a))
(h₂ : strict_mono_on f (Ici a)) : strict_mono f | begin
rw [← strict_mono_on_univ, ← @Iic_union_Ici _ _ a],
exact strict_mono_on.union h₁ h₂ is_greatest_Iic is_least_Ici,
end | lemma | strict_mono_on.Iic_union_Ici | order.monotone | src/order/monotone/union.lean | [
"order.bounds.basic"
] | [
"is_greatest_Iic",
"is_least_Ici",
"strict_mono",
"strict_mono_on",
"strict_mono_on.union",
"strict_mono_on_univ"
] | If `f` is strictly monotone both on `(-∞, a]` and `[a, ∞)`, then it is strictly monotone on the
whole line. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_on.union {s t : set α} {c : α} (h₁ : strict_anti_on f s)
(h₂ : strict_anti_on f t) (hs : is_greatest s c) (ht : is_least t c) :
strict_anti_on f (s ∪ t) | (h₁.dual_right.union h₂.dual_right hs ht).dual_right | lemma | strict_anti_on.union | order.monotone | src/order/monotone/union.lean | [
"order.bounds.basic"
] | [
"is_greatest",
"is_least",
"strict_anti_on"
] | If `f` is strictly antitone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is strictly antitone on `s ∪ t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_on.Iic_union_Ici (h₁ : strict_anti_on f (Iic a))
(h₂ : strict_anti_on f (Ici a)) : strict_anti f | (h₁.dual_right.Iic_union_Ici h₂.dual_right).dual_right | lemma | strict_anti_on.Iic_union_Ici | order.monotone | src/order/monotone/union.lean | [
"order.bounds.basic"
] | [
"strict_anti",
"strict_anti_on"
] | If `f` is strictly antitone both on `(-∞, a]` and `[a, ∞)`, then it is strictly antitone on the
whole line. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_on.union_right {s t : set α} {c : α} (h₁ : monotone_on f s)
(h₂ : monotone_on f t) (hs : is_greatest s c) (ht : is_least t c) :
monotone_on f (s ∪ t) | begin
have A : ∀ x, x ∈ s ∪ t → x ≤ c → x ∈ s,
{ assume x hx hxc,
cases hx, { exact hx },
rcases eq_or_lt_of_le hxc with rfl|h'x, { exact hs.1 },
exact (lt_irrefl _ (h'x.trans_le (ht.2 hx))).elim },
have B : ∀ x, x ∈ s ∪ t → c ≤ x → x ∈ t,
{ assume x hx hxc,
cases hx, swap, { exact hx },
rca... | lemma | monotone_on.union_right | order.monotone | src/order/monotone/union.lean | [
"order.bounds.basic"
] | [
"eq_or_lt_of_le",
"is_greatest",
"is_least",
"monotone_on"
] | If `f` is monotone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is monotone on `s ∪ t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_on.Iic_union_Ici (h₁ : monotone_on f (Iic a))
(h₂ : monotone_on f (Ici a)) : monotone f | begin
rw [← monotone_on_univ, ← @Iic_union_Ici _ _ a],
exact monotone_on.union_right h₁ h₂ is_greatest_Iic is_least_Ici
end | lemma | monotone_on.Iic_union_Ici | order.monotone | src/order/monotone/union.lean | [
"order.bounds.basic"
] | [
"is_greatest_Iic",
"is_least_Ici",
"monotone",
"monotone_on",
"monotone_on.union_right",
"monotone_on_univ"
] | If `f` is monotone both on `(-∞, a]` and `[a, ∞)`, then it is monotone on the whole line. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_on.union_right {s t : set α} {c : α} (h₁ : antitone_on f s)
(h₂ : antitone_on f t) (hs : is_greatest s c) (ht : is_least t c) :
antitone_on f (s ∪ t) | (h₁.dual_right.union_right h₂.dual_right hs ht).dual_right | lemma | antitone_on.union_right | order.monotone | src/order/monotone/union.lean | [
"order.bounds.basic"
] | [
"antitone_on",
"is_greatest",
"is_least"
] | If `f` is antitone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is antitone on `s ∪ t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_on.Iic_union_Ici (h₁ : antitone_on f (Iic a))
(h₂ : antitone_on f (Ici a)) : antitone f | (h₁.dual_right.Iic_union_Ici h₂.dual_right).dual_right | lemma | antitone_on.Iic_union_Ici | order.monotone | src/order/monotone/union.lean | [
"order.bounds.basic"
] | [
"antitone",
"antitone_on"
] | If `f` is antitone both on `(-∞, a]` and `[a, ∞)`, then it is antitone on the whole line. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equipartition : Prop | (P.parts : set (finset α)).equitable_on card | def | finpartition.is_equipartition | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [
"finset"
] | An equipartition is a partition whose parts are all the same size, up to a difference of `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equipartition_iff_card_parts_eq_average : P.is_equipartition ↔
∀ a : finset α, a ∈ P.parts → a.card = s.card/P.parts.card ∨ a.card = s.card/P.parts.card + 1 | by simp_rw [is_equipartition, finset.equitable_on_iff, P.sum_card_parts] | lemma | finpartition.is_equipartition_iff_card_parts_eq_average | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [
"finset",
"finset.equitable_on_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.set.subsingleton.is_equipartition (h : (P.parts : set (finset α)).subsingleton) :
P.is_equipartition | h.equitable_on _ | lemma | set.subsingleton.is_equipartition | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equipartition.card_parts_eq_average (hP : P.is_equipartition) (ht : t ∈ P.parts) :
t.card = s.card / P.parts.card ∨ t.card = s.card / P.parts.card + 1 | P.is_equipartition_iff_card_parts_eq_average.1 hP _ ht | lemma | finpartition.is_equipartition.card_parts_eq_average | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equipartition.average_le_card_part (hP : P.is_equipartition) (ht : t ∈ P.parts) :
s.card / P.parts.card ≤ t.card | by { rw ←P.sum_card_parts, exact equitable_on.le hP ht } | lemma | finpartition.is_equipartition.average_le_card_part | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equipartition.card_part_le_average_add_one (hP : P.is_equipartition) (ht : t ∈ P.parts) :
t.card ≤ s.card / P.parts.card + 1 | by { rw ←P.sum_card_parts, exact equitable_on.le_add_one hP ht } | lemma | finpartition.is_equipartition.card_part_le_average_add_one | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_is_equipartition : (⊥ : finpartition s).is_equipartition | set.equitable_on_iff_exists_eq_eq_add_one.2 ⟨1, by simp⟩ | lemma | finpartition.bot_is_equipartition | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [
"finpartition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_is_equipartition : (⊤ : finpartition s).is_equipartition | (parts_top_subsingleton _).is_equipartition | lemma | finpartition.top_is_equipartition | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [
"finpartition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indiscrete_is_equipartition {hs : s ≠ ∅} : (indiscrete hs).is_equipartition | by { rw [is_equipartition, indiscrete_parts, coe_singleton], exact set.equitable_on_singleton s _ } | lemma | finpartition.indiscrete_is_equipartition | order.partition | src/order/partition/equipartition.lean | [
"data.set.equitable",
"order.partition.finpartition"
] | [
"set.equitable_on_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finpartition [lattice α] [order_bot α] (a : α) | (parts : finset α)
(sup_indep : parts.sup_indep id)
(sup_parts : parts.sup id = a)
(not_bot_mem : ⊥ ∉ parts) | structure | finpartition | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finset",
"lattice",
"order_bot"
] | A finite partition of `a : α` is a pairwise disjoint finite set of elements whose supremum is
`a`. We forbid `⊥` as a part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_erase [decidable_eq α] {a : α} (parts : finset α)
(sup_indep : parts.sup_indep id) (sup_parts : parts.sup id = a) :
finpartition a | { parts := parts.erase ⊥,
sup_indep := sup_indep.subset (erase_subset _ _),
sup_parts := (sup_erase_bot _).trans sup_parts,
not_bot_mem := not_mem_erase _ _ } | def | finpartition.of_erase | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset"
] | A `finpartition` constructor which does not insist on `⊥` not being a part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_subset {a b : α} (P : finpartition a) {parts : finset α}
(subset : parts ⊆ P.parts) (sup_parts : parts.sup id = b) :
finpartition b | { parts := parts,
sup_indep := P.sup_indep.subset subset,
sup_parts := sup_parts,
not_bot_mem := λ h, P.not_bot_mem (subset h) } | def | finpartition.of_subset | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset"
] | A `finpartition` constructor from a bigger existing finpartition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
copy {a b : α} (P : finpartition a) (h : a = b) : finpartition b | { parts := P.parts,
sup_indep := P.sup_indep,
sup_parts := h ▸ P.sup_parts,
not_bot_mem := P.not_bot_mem } | def | finpartition.copy | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition"
] | Changes the type of a finpartition to an equal one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty : finpartition (⊥ : α) | { parts := ∅,
sup_indep := sup_indep_empty _,
sup_parts := finset.sup_empty,
not_bot_mem := not_mem_empty ⊥ } | def | finpartition.empty | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset.sup_empty"
] | The empty finpartition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
default_eq_empty : (default : finpartition (⊥ : α)) = finpartition.empty α | rfl | lemma | finpartition.default_eq_empty | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finpartition.empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indiscrete (ha : a ≠ ⊥) : finpartition a | { parts := {a},
sup_indep := sup_indep_singleton _ _,
sup_parts := finset.sup_singleton,
not_bot_mem := λ h, ha (mem_singleton.1 h).symm } | def | finpartition.indiscrete | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset.sup_singleton"
] | The finpartition in one part, aka indiscrete finpartition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le {b : α} (hb : b ∈ P.parts) : b ≤ a | (le_sup hb).trans P.sup_parts.le | lemma | finpartition.le | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot {b : α} (hb : b ∈ P.parts) : b ≠ ⊥ | λ h, P.not_bot_mem $ h.subst hb | lemma | finpartition.ne_bot | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint : (P.parts : set α).pairwise_disjoint id | P.sup_indep.pairwise_disjoint | lemma | finpartition.disjoint | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parts_eq_empty_iff : P.parts = ∅ ↔ a = ⊥ | begin
simp_rw ←P.sup_parts,
refine ⟨λ h, _, λ h, eq_empty_iff_forall_not_mem.2 (λ b hb, P.not_bot_mem _)⟩,
{ rw h,
exact finset.sup_empty },
{ rwa ←le_bot_iff.1 ((le_sup hb).trans h.le) }
end | lemma | finpartition.parts_eq_empty_iff | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finset.sup_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parts_nonempty_iff : P.parts.nonempty ↔ a ≠ ⊥ | by rw [nonempty_iff_ne_empty, not_iff_not, parts_eq_empty_iff] | lemma | finpartition.parts_nonempty_iff | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parts_nonempty (P : finpartition a) (ha : a ≠ ⊥) : P.parts.nonempty | parts_nonempty_iff.2 ha | lemma | finpartition.parts_nonempty | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_atom.unique_finpartition (ha : is_atom a) : unique (finpartition a) | { default := indiscrete ha.1,
uniq := λ P, begin
have h : ∀ b ∈ P.parts, b = a,
{ exact λ b hb, (ha.le_iff.mp $ P.le hb).resolve_left (P.ne_bot hb) },
ext b,
refine iff.trans ⟨h b, _⟩ mem_singleton.symm,
rintro rfl,
obtain ⟨c, hc⟩ := P.parts_nonempty ha.1,
simp_rw ←h c hc,
exact hc,
... | def | is_atom.unique_finpartition | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"is_atom",
"unique"
] | There's a unique partition of an atom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
parts_top_subset (a : α) [decidable (a = ⊥)] : (⊤ : finpartition a).parts ⊆ {a} | begin
intros b hb,
change b ∈ finpartition.parts (dite _ _ _) at hb,
split_ifs at hb,
{ simp only [copy_parts, empty_parts, not_mem_empty] at hb,
exact hb.elim },
{ exact hb }
end | lemma | finpartition.parts_top_subset | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parts_top_subsingleton (a : α) [decidable (a = ⊥)] :
((⊤ : finpartition a).parts : set α).subsingleton | set.subsingleton_of_subset_singleton $ λ b hb, mem_singleton.1 $ parts_top_subset _ hb | lemma | finpartition.parts_top_subsingleton | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"set.subsingleton_of_subset_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parts_inf (P Q : finpartition a) :
(P ⊓ Q).parts = ((P.parts ×ˢ Q.parts).image $ λ bc : α × α, bc.1 ⊓ bc.2).erase ⊥ | rfl | lemma | finpartition.parts_inf | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_le_of_le {a b : α} {P Q : finpartition a} (h : P ≤ Q) (hb : b ∈ Q.parts) :
∃ c ∈ P.parts, c ≤ b | begin
by_contra' H,
refine Q.ne_bot hb (disjoint_self.1 $ disjoint.mono_right (Q.le hb) _),
rw [←P.sup_parts, finset.disjoint_sup_right],
rintro c hc,
obtain ⟨d, hd, hcd⟩ := h hc,
refine (Q.disjoint hb hd _).mono_right hcd,
rintro rfl,
exact H _ hc hcd,
end | lemma | finpartition.exists_le_of_le | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"disjoint.mono_right",
"finpartition",
"finset.disjoint_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_mono {a : α} {P Q : finpartition a} (h : P ≤ Q) : Q.parts.card ≤ P.parts.card | begin
classical,
have : ∀ b ∈ Q.parts, ∃ c ∈ P.parts, c ≤ b := λ b, exists_le_of_le h,
choose f hP hf using this,
rw ←card_attach,
refine card_le_card_of_inj_on (λ b, f _ b.2) (λ b _, hP _ b.2) (λ b hb c hc h, _),
exact subtype.coe_injective (Q.disjoint.elim b.2 c.2 $ λ H, P.ne_bot (hP _ b.2) $
disjoint... | lemma | finpartition.card_mono | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind (P : finpartition a) (Q : Π i ∈ P.parts, finpartition i) : finpartition a | { parts := P.parts.attach.bUnion (λ i, (Q i.1 i.2).parts),
sup_indep := begin
rw sup_indep_iff_pairwise_disjoint,
rintro a ha b hb h,
rw [finset.mem_coe, finset.mem_bUnion] at ha hb,
obtain ⟨⟨A, hA⟩, -, ha⟩ := ha,
obtain ⟨⟨B, hB⟩, -, hb⟩ := hb,
obtain rfl | hAB := eq_or_ne A B,
{ exact (Q ... | def | finpartition.bind | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"disjoint",
"eq_or_ne",
"finpartition",
"finset.mem_bUnion",
"finset.mem_coe"
] | Given a finpartition `P` of `a` and finpartitions of each part of `P`, this yields the
finpartition of `a` obtained by juxtaposing all the subpartitions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_bind : b ∈ (P.bind Q).parts ↔ ∃ A hA, b ∈ (Q A hA).parts | begin
rw [bind, mem_bUnion],
split,
{ rintro ⟨⟨A, hA⟩, -, h⟩,
exact ⟨A, hA, h⟩ },
{ rintro ⟨A, hA, h⟩,
exact ⟨⟨A, hA⟩, mem_attach _ ⟨A, hA⟩, h⟩ }
end | lemma | finpartition.mem_bind | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"mem_bind"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_bind (Q : Π i ∈ P.parts, finpartition i) :
(P.bind Q).parts.card = ∑ A in P.parts.attach, (Q _ A.2).parts.card | begin
apply card_bUnion,
rintro ⟨b, hb⟩ - ⟨c, hc⟩ - hbc,
rw finset.disjoint_left,
rintro d hdb hdc,
rw [ne.def, subtype.mk_eq_mk] at hbc,
exact (Q b hb).ne_bot hdb (eq_bot_iff.2 $
(le_inf ((Q b hb).le hdb) $ (Q c hc).le hdc).trans $ (P.disjoint hb hc hbc).le_bot),
end | lemma | finpartition.card_bind | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset.disjoint_left",
"le_inf",
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend (P : finpartition a) (hb : b ≠ ⊥) (hab : disjoint a b) (hc : a ⊔ b = c) :
finpartition c | { parts := insert b P.parts,
sup_indep :=
begin
rw [sup_indep_iff_pairwise_disjoint, coe_insert],
exact P.disjoint.insert (λ d hd hbd, hab.symm.mono_right $ P.le hd),
end,
sup_parts := by rwa [sup_insert, P.sup_parts, id, _root_.sup_comm],
not_bot_mem := λ h, (mem_insert.1 h).elim hb.symm P.not_bot_me... | def | finpartition.extend | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"disjoint",
"extend",
"finpartition"
] | Adds `b` to a finpartition of `a` to make a finpartition of `a ⊔ b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_extend (P : finpartition a) (b c : α) {hb : b ≠ ⊥} {hab : disjoint a b}
{hc : a ⊔ b = c} :
(P.extend hb hab hc).parts.card = P.parts.card + 1 | card_insert_of_not_mem $ λ h, hb $ hab.symm.eq_bot_of_le $ P.le h | lemma | finpartition.card_extend | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"disjoint",
"finpartition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
avoid (b : α) : finpartition (a \ b) | of_erase
(P.parts.image (\ b))
(P.disjoint.image_finset_of_le $ λ a, sdiff_le).sup_indep
(by rw [sup_image, comp.left_id, finset.sup_sdiff_right, ←id_def, P.sup_parts]) | def | finpartition.avoid | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset.sup_sdiff_right",
"sdiff_le"
] | Restricts a finpartition to avoid a given element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_avoid : c ∈ (P.avoid b).parts ↔ ∃ d ∈ P.parts, ¬ d ≤ b ∧ d \ b = c | begin
simp only [avoid, of_erase_parts, mem_erase, ne.def, mem_image, exists_prop,
←exists_and_distrib_left, @and.left_comm (c ≠ ⊥)],
refine exists_congr (λ d, and_congr_right' $ and_congr_left _),
rintro rfl,
rw sdiff_eq_bot_iff,
end | lemma | finpartition.mem_avoid | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"and_congr_left",
"and_congr_right'",
"exists_prop",
"sdiff_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_of_mem_parts {a : finset α} (ha : a ∈ P.parts) : a.nonempty | nonempty_iff_ne_empty.2 $ P.ne_bot ha | lemma | finpartition.nonempty_of_mem_parts | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem {a : α} (ha : a ∈ s) : ∃ t ∈ P.parts, a ∈ t | by { simp_rw ←P.sup_parts at ha, exact mem_sup.1 ha } | lemma | finpartition.exists_mem | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_parts : P.parts.bUnion id = s | (sup_eq_bUnion _ _).symm.trans P.sup_parts | lemma | finpartition.bUnion_parts | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_card_parts : ∑ i in P.parts, i.card = s.card | begin
convert congr_arg finset.card P.bUnion_parts,
rw card_bUnion P.sup_indep.pairwise_disjoint,
refl,
end | lemma | finpartition.sum_card_parts | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finset.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parts_bot (s : finset α) :
(⊥ : finpartition s).parts = s.map ⟨singleton, singleton_injective⟩ | rfl | lemma | finpartition.parts_bot | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_bot (s : finset α) : (⊥ : finpartition s).parts.card = s.card | finset.card_map _ | lemma | finpartition.card_bot | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset",
"finset.card_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot_iff : t ∈ (⊥ : finpartition s).parts ↔ ∃ a ∈ s, {a} = t | mem_map | lemma | finpartition.mem_bot_iff | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_parts_le_card (P : finpartition s) : P.parts.card ≤ s.card | by { rw ←card_bot s, exact card_mono bot_le } | lemma | finpartition.card_parts_le_card | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"bot_le",
"finpartition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
atomise (s : finset α) (F : finset (finset α)) : finpartition s | of_erase
(F.powerset.image $ λ Q, s.filter (λ i, ∀ t ∈ F, t ∈ Q ↔ i ∈ t))
(set.pairwise_disjoint.sup_indep $ λ x hx y hy h, disjoint_left.mpr $ λ z hz1 hz2, h begin
rw [mem_coe, mem_image] at hx hy,
obtain ⟨Q, hQ, rfl⟩ := hx,
obtain ⟨R, hR, rfl⟩ := hy,
suffices h : Q = R,
{ subst h },
rw [id... | def | finpartition.atomise | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"finpartition",
"finset"
] | Cuts `s` along the finsets in `F`: Two elements of `s` will be in the same part if they are
in the same finsets of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_atomise :
t ∈ (atomise s F).parts ↔ t.nonempty ∧ ∃ (Q ⊆ F), s.filter (λ i, ∀ u ∈ F, u ∈ Q ↔ i ∈ u) = t | by simp only [atomise, of_erase, bot_eq_empty, mem_erase, mem_image, nonempty_iff_ne_empty,
mem_singleton, and_comm, mem_powerset, exists_prop] | lemma | finpartition.mem_atomise | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
atomise_empty (hs : s.nonempty) : (atomise s ∅).parts = {s} | begin
simp only [atomise, powerset_empty, image_singleton, not_mem_empty, is_empty.forall_iff,
implies_true_iff, filter_true],
exact erase_eq_of_not_mem (not_mem_singleton.2 hs.ne_empty.symm),
end | lemma | finpartition.atomise_empty | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"is_empty.forall_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_atomise_le : (atomise s F).parts.card ≤ 2^F.card | (card_le_of_subset $ erase_subset _ _).trans $ finset.card_image_le.trans (card_powerset _).le | lemma | finpartition.card_atomise_le | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_filter_atomise (ht : t ∈ F) (hts : t ⊆ s) :
((atomise s F).parts.filter $ λ u, u ⊆ t ∧ u.nonempty).bUnion id = t | begin
ext a,
refine mem_bUnion.trans ⟨λ ⟨u, hu, ha⟩, (mem_filter.1 hu).2.1 ha, λ ha, _⟩,
obtain ⟨u, hu, hau⟩ := (atomise s F).exists_mem (hts ha),
refine ⟨u, mem_filter.2 ⟨hu, λ b hb, _, _, hau⟩, hau⟩,
obtain ⟨Q, hQ, rfl⟩ := (mem_atomise.1 hu).2,
rw mem_filter at hau hb,
rwa [←hb.2 _ ht, hau.2 _ ht],
end | lemma | finpartition.bUnion_filter_atomise | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_filter_atomise_le_two_pow (ht : t ∈ F) :
((atomise s F).parts.filter $ λ u, u ⊆ t ∧ u.nonempty).card ≤ 2 ^ (F.card - 1) | begin
suffices h : (atomise s F).parts.filter (λ u, u ⊆ t ∧ u.nonempty)
⊆ (F.erase t).powerset.image (λ P, s.filter $ λ i, ∀ x ∈ F, x ∈ insert t P ↔ i ∈ x),
{ refine (card_le_of_subset h).trans (card_image_le.trans _),
rw [card_powerset, card_erase_of_mem ht] },
rw subset_iff,
simp only [mem_erase, mem_... | lemma | finpartition.card_filter_atomise_le_two_pow | order.partition | src/order/partition/finpartition.lean | [
"algebra.big_operators.basic",
"order.atoms.finite",
"order.sup_indep"
] | [
"and_imp",
"exists_imp_distrib",
"exists_prop",
"finset.nonempty",
"forall_apply_eq_imp_iff₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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