fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
coheight_le_iff {a : α} {n : ℕ∞} :
coheight a ≤ n ↔ ∀ ⦃p : LTSeries α⦄, a ≤ p.head → p.length ≤ n := by
rw [coheight_eq, iSup₂_le_iff] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_le_iff | null |
height_le {a : α} {n : ℕ∞} (h : ∀ (p : LTSeries α), p.last = a → p.length ≤ n) :
height a ≤ n := by
apply height_le_iff.mpr
intro p hlast
wlog hlenpos : p.length ≠ 0
· simp_all
let p' := p.eraseLast.snoc a (lt_of_lt_of_le (p.eraseLast_last_rel_last (by simp_all)) hlast)
rw [show p.length = p'.length by ... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_le | null |
height_le_iff' {a : α} {n : ℕ∞} :
height a ≤ n ↔ ∀ ⦃p : LTSeries α⦄, p.last = a → p.length ≤ n := by
constructor
· rw [height_le_iff]
exact fun h p hlast => h (le_of_eq hlast)
· exact height_le | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_le_iff' | Variant of `height_le_iff` ranging only over those series that end exactly on `a`. |
height_eq_iSup_last_eq (a : α) :
height a = ⨆ (p : LTSeries α) (_ : p.last = a), ↑(p.length) := by
apply eq_of_forall_ge_iff
intro n
rw [height_le_iff', iSup₂_le_iff] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_iSup_last_eq | Alternative definition of height, with the supremum ranging only over those series that end at `a`. |
coheight_eq_iSup_head_eq (a : α) :
coheight a = ⨆ (p : LTSeries α) (_ : p.head = a), ↑(p.length) := by
change height (α := αᵒᵈ) a = ⨆ (p : LTSeries α) (_ : p.head = a), ↑(p.length)
rw [height_eq_iSup_last_eq]
apply Equiv.iSup_congr ⟨RelSeries.reverse, RelSeries.reverse, fun _ ↦ RelSeries.reverse_reverse _,
... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_iSup_head_eq | Alternative definition of coheight, with the supremum only ranging over those series
that begin at `a`. |
coheight_le_iff' {a : α} {n : ℕ∞} :
coheight a ≤ n ↔ ∀ ⦃p : LTSeries α⦄, p.head = a → p.length ≤ n := by
rw [coheight_eq_iSup_head_eq, iSup₂_le_iff] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_le_iff' | Variant of `coheight_le_iff` ranging only over those series that begin exactly on `a`. |
coheight_le {a : α} {n : ℕ∞} (h : ∀ (p : LTSeries α), p.head = a → p.length ≤ n) :
coheight a ≤ n :=
coheight_le_iff'.mpr h | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_le | null |
length_le_height {p : LTSeries α} {x : α} (hlast : p.last ≤ x) :
p.length ≤ height x := by
by_cases hlen0 : p.length ≠ 0
· let p' := p.eraseLast.snoc x (by
apply lt_of_lt_of_le
· apply p.step ⟨p.length - 1, by cutsat⟩
· convert hlast
simp only [Fin.succ_mk, RelSeries.last, Fin.last]
... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | length_le_height | null |
length_le_coheight {x : α} {p : LTSeries α} (hhead : x ≤ p.head) :
p.length ≤ coheight x :=
length_le_height (α := αᵒᵈ) (p := p.reverse) (by simpa) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | length_le_coheight | null |
length_le_height_last {p : LTSeries α} : p.length ≤ height p.last :=
length_le_height le_rfl | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | length_le_height_last | The height of the last element in a series is larger or equal to the length of the series. |
length_le_coheight_head {p : LTSeries α} : p.length ≤ coheight p.head :=
length_le_coheight le_rfl | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | length_le_coheight_head | The coheight of the first element in a series is larger or equal to the length of the series. |
index_le_height (p : LTSeries α) (i : Fin (p.length + 1)) : i ≤ height (p i) :=
length_le_height_last (p := p.take i) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | index_le_height | The height of an element in a series is larger or equal to its index in the series. |
rev_index_le_coheight (p : LTSeries α) (i : Fin (p.length + 1)) : i.rev ≤ coheight (p i) := by
simpa using index_le_height (α := αᵒᵈ) p.reverse i.rev | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | rev_index_le_coheight | The coheight of an element in a series is larger or equal to its reverse index in the series. |
height_eq_index_of_length_eq_height_last {p : LTSeries α} (h : p.length = height p.last)
(i : Fin (p.length + 1)) : height (p i) = i := by
refine le_antisymm (height_le ?_) (index_le_height p i)
intro p' hp'
have hp'' := length_le_height_last (p := p'.smash (p.drop i) (by simpa))
simp [← h] at hp''; clear h... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_index_of_length_eq_height_last | In a maximally long series, i.e one as long as the height of the last element, the height of each
element is its index in the series. |
coheight_eq_index_of_length_eq_head_coheight {p : LTSeries α} (h : p.length = coheight p.head)
(i : Fin (p.length + 1)) : coheight (p i) = i.rev := by
simpa using height_eq_index_of_length_eq_height_last (α := αᵒᵈ) (p := p.reverse) (by simpa) i.rev | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_index_of_length_eq_head_coheight | In a maximally long series, i.e one as long as the coheight of the first element, the coheight of
each element is its reverse index in the series. |
height_mono : Monotone (α := α) height :=
fun _ _ hab ↦ biSup_mono (fun _ hla => hla.trans hab)
@[gcongr] protected lemma _root_.GCongr.height_le_height (a b : α) (hab : a ≤ b) :
height a ≤ height b := height_mono hab | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_mono | null |
coheight_anti : Antitone (α := α) coheight :=
(height_mono (α := αᵒᵈ)).dual_left
@[gcongr] protected lemma _root_.GCongr.coheight_le_coheight (a b : α) (hba : b ≤ a) :
coheight a ≤ coheight b := coheight_anti hba | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_anti | null |
private height_add_const (a : α) (n : ℕ∞) :
height a + n = ⨆ (p : LTSeries α) (_ : p.last = a), p.length + n := by
have hne : Nonempty { p : LTSeries α // p.last = a } := ⟨RelSeries.singleton _ a, rfl⟩
rw [height_eq_iSup_last_eq, iSup_subtype', iSup_subtype', ENat.iSup_add]
/- For elements of finite height, `he... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_add_const | null |
height_add_one_le {a b : α} (hab : a < b) : height a + 1 ≤ height b := by
cases hfin : height a with
| top =>
have : ⊤ ≤ height b := by
rw [← hfin]
gcongr
simp [this]
| coe n =>
apply Order.add_one_le_of_lt
rw [← hfin]
gcongr
simp [hfin]
/- For elements of finite height, `cohei... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_add_one_le | null |
coheight_add_one_le {a b : α} (hab : b < a) : coheight a + 1 ≤ coheight b := by
cases hfin : coheight a with
| top =>
have : ⊤ ≤ coheight b := by
rw [← hfin]
gcongr
simp [this]
| coe n =>
apply Order.add_one_le_of_lt
rw [← hfin]
gcongr
simp [hfin] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_add_one_le | null |
height_le_height_apply_of_strictMono (f : α → β) (hf : StrictMono f) (x : α) :
height x ≤ height (f x) := by
simp only [height_eq_iSup_last_eq]
apply iSup₂_le
intro p hlast
apply le_iSup₂_of_le (p.map f hf) (by simp [hlast]) (by simp) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_le_height_apply_of_strictMono | null |
coheight_le_coheight_apply_of_strictMono (f : α → β) (hf : StrictMono f) (x : α) :
coheight x ≤ coheight (f x) := by
apply height_le_height_apply_of_strictMono (α := αᵒᵈ)
exact fun _ _ h ↦ hf h
@[simp] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_le_coheight_apply_of_strictMono | null |
height_orderIso (f : α ≃o β) (x : α) : height (f x) = height x := by
apply le_antisymm
· simpa using height_le_height_apply_of_strictMono _ f.symm.strictMono (f x)
· exact height_le_height_apply_of_strictMono _ f.strictMono x | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_orderIso | null |
coheight_orderIso (f : α ≃o β) (x : α) : coheight (f x) = coheight x :=
height_orderIso (α := αᵒᵈ) f.dual x | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_orderIso | null |
private exists_eq_iSup_of_iSup_eq_coe {α : Type*} [Nonempty α] {f : α → ℕ∞} {n : ℕ}
(h : (⨆ x, f x) = n) : ∃ x, f x = n := by
obtain ⟨x, hx⟩ := ENat.sSup_mem_of_nonempty_of_lt_top (h ▸ ENat.coe_lt_top _)
use x
simpa [hx] using h | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | exists_eq_iSup_of_iSup_eq_coe | null |
exists_series_of_le_height (a : α) {n : ℕ} (h : n ≤ height a) :
∃ p : LTSeries α, p.last = a ∧ p.length = n := by
have hne : Nonempty { p : LTSeries α // p.last = a } := ⟨RelSeries.singleton _ a, rfl⟩
cases ha : height a with
| top =>
clear h
rw [height_eq_iSup_last_eq, iSup_subtype', ENat.iSup_coe_eq... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | exists_series_of_le_height | There exist a series ending in a element for any length up to the element’s height. |
exists_series_of_le_coheight (a : α) {n : ℕ} (h : n ≤ coheight a) :
∃ p : LTSeries α, p.head = a ∧ p.length = n := by
obtain ⟨p, hp, hl⟩ := exists_series_of_le_height (α := αᵒᵈ) a h
exact ⟨p.reverse, by simpa, by simpa⟩ | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | exists_series_of_le_coheight | null |
exists_series_of_height_eq_coe (a : α) {n : ℕ} (h : height a = n) :
∃ p : LTSeries α, p.last = a ∧ p.length = n :=
exists_series_of_le_height a (le_of_eq h.symm) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | exists_series_of_height_eq_coe | For an element of finite height there exists a series ending in that element of that height. |
exists_series_of_coheight_eq_coe (a : α) {n : ℕ} (h : coheight a = n) :
∃ p : LTSeries α, p.head = a ∧ p.length = n :=
exists_series_of_le_coheight a (le_of_eq h.symm) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | exists_series_of_coheight_eq_coe | null |
height_eq_iSup_lt_height (x : α) : height x = ⨆ y < x, height y + 1 := by
apply le_antisymm
· apply height_le
intro p hp
cases hlen : p.length with
| zero => simp
| succ n =>
apply le_iSup_of_le p.eraseLast.last
apply le_iSup_of_le (by rw [← hp]; exact p.eraseLast_last_rel_last (by cutsa... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_iSup_lt_height | Another characterization of height, based on the supremum of the heights of elements below. |
coheight_eq_iSup_gt_coheight (x : α) : coheight x = ⨆ y > x, coheight y + 1 :=
height_eq_iSup_lt_height (α := αᵒᵈ) x | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_iSup_gt_coheight | Another characterization of coheight, based on the supremum of the coheights of elements above. |
height_le_coe_iff {x : α} {n : ℕ} : height x ≤ n ↔ ∀ y < x, height y < n := by
conv_lhs => rw [height_eq_iSup_lt_height, iSup₂_le_iff]
congr! 2 with y _
cases height y
· simp
· norm_cast | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_le_coe_iff | null |
coheight_le_coe_iff {x : α} {n : ℕ} : coheight x ≤ n ↔ ∀ y > x, coheight y < n :=
height_le_coe_iff (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_le_coe_iff | null |
height_eq_top_iff {x : α} :
height x = ⊤ ↔ ∀ n, ∃ p : LTSeries α, p.last = x ∧ p.length = n where
mp h n := by
apply exists_series_of_le_height x (n := n)
simp [h]
mpr h := by
rw [height_eq_iSup_last_eq, iSup_subtype', ENat.iSup_coe_eq_top, bddAbove_def]
push_neg
intro n
obtain ⟨p, hlast... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_top_iff | The height of an element is infinite iff there exist series of arbitrary length ending in that
element. |
coheight_eq_top_iff {x : α} :
coheight x = ⊤ ↔ ∀ n, ∃ p : LTSeries α, p.head = x ∧ p.length = n := by
convert height_eq_top_iff (α := αᵒᵈ) (x := x) using 2 with n
constructor <;> (intro ⟨p, hp, hl⟩; use p.reverse; constructor <;> simpa) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_top_iff | The coheight of an element is infinite iff there exist series of arbitrary length ending in that
element. |
@[simp] height_eq_zero {x : α} : height x = 0 ↔ IsMin x := by
simpa [isMin_iff_forall_not_lt] using height_le_coe_iff (x := x) (n := 0)
protected alias ⟨_, IsMin.height_eq_zero⟩ := height_eq_zero | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_zero | The elements of height zero are the minimal elements. |
@[simp] coheight_eq_zero {x : α} : coheight x = 0 ↔ IsMax x :=
height_eq_zero (α := αᵒᵈ)
protected alias ⟨_, IsMax.coheight_eq_zero⟩ := coheight_eq_zero | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_zero | The elements of coheight zero are the maximal elements. |
height_ne_zero {x : α} : height x ≠ 0 ↔ ¬ IsMin x := height_eq_zero.not
@[simp] lemma height_pos {x : α} : 0 < height x ↔ ¬ IsMin x := by
simp [pos_iff_ne_zero] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_ne_zero | null |
coheight_ne_zero {x : α} : coheight x ≠ 0 ↔ ¬ IsMax x := coheight_eq_zero.not
@[simp] lemma coheight_pos {x : α} : 0 < coheight x ↔ ¬ IsMax x := by
simp [pos_iff_ne_zero]
@[simp] lemma height_bot (α : Type*) [Preorder α] [OrderBot α] : height (⊥ : α) = 0 := by simp
@[simp] lemma coheight_top (α : Type*) [Preorder α] ... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_ne_zero | null |
coe_lt_height_iff {x : α} {n : ℕ} (hfin : height x < ⊤) :
n < height x ↔ ∃ y < x, height y = n where
mp h := by
obtain ⟨m, hx : height x = m⟩ := Option.ne_none_iff_exists'.mp hfin.ne_top
rw [hx] at h; norm_cast at h
obtain ⟨p, hp, hlen⟩ := exists_series_of_height_eq_coe x hx
use p ⟨n, by omega⟩
... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coe_lt_height_iff | null |
coe_lt_coheight_iff {x : α} {n : ℕ} (hfin : coheight x < ⊤) :
n < coheight x ↔ ∃ y > x, coheight y = n :=
coe_lt_height_iff (α := αᵒᵈ) hfin | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coe_lt_coheight_iff | null |
height_eq_coe_add_one_iff {x : α} {n : ℕ} :
height x = n + 1 ↔ height x < ⊤ ∧ (∃ y < x, height y = n) ∧ (∀ y < x, height y ≤ n) := by
wlog hfin : height x < ⊤
· simp_all
exact ne_of_beq_false rfl
simp only [hfin, true_and]
trans n < height x ∧ height x ≤ n + 1
· rw [le_antisymm_iff, and_comm]
simp... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_coe_add_one_iff | null |
coheight_eq_coe_add_one_iff {x : α} {n : ℕ} :
coheight x = n + 1 ↔
coheight x < ⊤ ∧ (∃ y > x, coheight y = n) ∧ (∀ y > x, coheight y ≤ n) :=
height_eq_coe_add_one_iff (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_coe_add_one_iff | null |
height_eq_coe_iff {x : α} {n : ℕ} :
height x = n ↔
height x < ⊤ ∧ (n = 0 ∨ ∃ y < x, height y = n - 1) ∧ (∀ y < x, height y < n) := by
wlog hfin : height x < ⊤
· simp_all
simp only [hfin, true_and]
cases n
case zero => simp [isMin_iff_forall_not_lt]
case succ n =>
simp only [Nat.cast_add, Nat.c... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_coe_iff | null |
coheight_eq_coe_iff {x : α} {n : ℕ} :
coheight x = n ↔
coheight x < ⊤ ∧ (n = 0 ∨ ∃ y > x, coheight y = n - 1) ∧ (∀ y > x, coheight y < n) :=
height_eq_coe_iff (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_coe_iff | null |
height_eq_coe_iff_minimal_le_height {a : α} {n : ℕ} :
height a = n ↔ Minimal (fun y => n ≤ height y) a := by
by_cases hfin : height a < ⊤
· cases hn : n with
| zero => simp
| succ => simp [minimal_iff_forall_lt, height_eq_coe_add_one_iff, ENat.add_one_le_iff,
coe_lt_height_iff, *]
· suffices ∃... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_coe_iff_minimal_le_height | The elements of finite height `n` are the minimal elements among those of height `≥ n`. |
coheight_eq_coe_iff_maximal_le_coheight {a : α} {n : ℕ} :
coheight a = n ↔ Maximal (fun y => n ≤ coheight y) a :=
height_eq_coe_iff_minimal_le_height (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_coe_iff_maximal_le_coheight | The elements of finite coheight `n` are the maximal elements among those of coheight `≥ n`. |
one_lt_height_iff {x : α} : 1 < Order.height x ↔ ∃ y z, z < y ∧ y < x := by
rw [← ENat.add_one_le_iff ENat.one_ne_top, one_add_one_eq_two]
refine ⟨fun h ↦ ?_, ?_⟩
· obtain ⟨p, hp, hlen⟩ := Order.exists_series_of_le_height x (n := 2) h
refine ⟨p 1, p 0, p.rel_of_lt ?_, hp ▸ p.rel_of_lt ?_⟩ <;> simp [Fin.lt_def... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | one_lt_height_iff | null |
LTSeries.length_le_krullDim (p : LTSeries α) : p.length ≤ krullDim α := le_sSup ⟨_, rfl⟩
@[simp] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | LTSeries.length_le_krullDim | null |
krullDim_eq_bot_iff : krullDim α = ⊥ ↔ IsEmpty α := by
rw [eq_bot_iff, krullDim, iSup_le_iff]
simp only [le_bot_iff, WithBot.natCast_ne_bot, isEmpty_iff]
exact ⟨fun H x ↦ H ⟨0, fun _ ↦ x, by simp⟩, (· <| · 1)⟩ | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_bot_iff | null |
krullDim_nonneg_iff : 0 ≤ krullDim α ↔ Nonempty α := by
rw [← not_iff_not, not_le, not_nonempty_iff, ← krullDim_eq_bot_iff, ← WithBot.lt_coe_bot,
bot_eq_zero, WithBot.coe_zero] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_nonneg_iff | null |
krullDim_eq_bot [IsEmpty α] : krullDim α = ⊥ := krullDim_eq_bot_iff.mpr ‹_› | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_bot | null |
krullDim_nonneg [Nonempty α] : 0 ≤ krullDim α := krullDim_nonneg_iff.mpr ‹_› | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_nonneg | null |
krullDim_ne_bot_iff : krullDim α ≠ ⊥ ↔ Nonempty α := by
rw [ne_eq, krullDim_eq_bot_iff, not_isEmpty_iff] | theorem | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_ne_bot_iff | null |
bot_lt_krullDim_iff : ⊥ < krullDim α ↔ Nonempty α := by
rw [bot_lt_iff_ne_bot, krullDim_ne_bot_iff] | theorem | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | bot_lt_krullDim_iff | null |
bot_lt_krullDim [Nonempty α] : ⊥ < krullDim α :=
bot_lt_krullDim_iff.mpr ‹_› | theorem | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | bot_lt_krullDim | null |
krullDim_nonpos_iff_forall_isMax : krullDim α ≤ 0 ↔ ∀ x : α, IsMax x := by
simp only [krullDim, iSup_le_iff, isMax_iff_forall_not_lt]
refine ⟨fun H x y h ↦ (H ⟨1, ![x, y],
fun i ↦ by obtain rfl := Subsingleton.elim i 0; simpa⟩).not_gt (by simp), ?_⟩
· rintro H ⟨_ | n, l, h⟩
· simp
· cases H (l 0) (l 1... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_nonpos_iff_forall_isMax | null |
krullDim_nonpos_iff_forall_isMin : krullDim α ≤ 0 ↔ ∀ x : α, IsMin x := by
simp only [krullDim_nonpos_iff_forall_isMax, IsMax, IsMin]
exact forall_swap | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_nonpos_iff_forall_isMin | null |
krullDim_le_one_iff : krullDim α ≤ 1 ↔ ∀ x : α, IsMin x ∨ IsMax x := by
rw [← not_iff_not]
simp_rw [isMax_iff_forall_not_lt, isMin_iff_forall_not_lt, krullDim, iSup_le_iff]
push_neg
constructor
· rintro ⟨⟨_ | _ | n, l, hl⟩, hl'⟩
iterate 2 · cases hl'.not_ge (by simp)
exact ⟨l 1, ⟨l 0, hl 0⟩, l 2, hl 1... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_le_one_iff | null |
krullDim_le_one_iff_forall_isMax {α : Type*} [PartialOrder α] [OrderBot α] :
krullDim α ≤ 1 ↔ ∀ x : α, x ≠ ⊥ → IsMax x := by
simp [krullDim_le_one_iff, ← or_iff_not_imp_left] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_le_one_iff_forall_isMax | null |
krullDim_le_one_iff_forall_isMin {α : Type*} [PartialOrder α] [OrderTop α] :
krullDim α ≤ 1 ↔ ∀ x : α, x ≠ ⊤ → IsMin x := by
simp [krullDim_le_one_iff, ← or_iff_not_imp_right] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_le_one_iff_forall_isMin | null |
krullDim_pos_iff : 0 < krullDim α ↔ ∃ x y : α, x < y := by
rw [← not_iff_not]
push_neg
simp_rw [← isMax_iff_forall_not_lt, ← krullDim_nonpos_iff_forall_isMax] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_pos_iff | null |
one_le_krullDim_iff : 1 ≤ krullDim α ↔ ∃ x y : α, x < y := by
rw [← krullDim_pos_iff, ← Nat.cast_zero, ← WithBot.add_one_le_iff, Nat.cast_zero, zero_add] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | one_le_krullDim_iff | null |
krullDim_nonpos_of_subsingleton [Subsingleton α] : krullDim α ≤ 0 := by
rw [krullDim_nonpos_iff_forall_isMax]
exact fun x y h ↦ (Subsingleton.elim x y).ge | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_nonpos_of_subsingleton | null |
krullDim_eq_zero [Nonempty α] [Subsingleton α] :
krullDim α = 0 :=
le_antisymm krullDim_nonpos_of_subsingleton krullDim_nonneg | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_zero | null |
krullDim_eq_zero_of_unique [Unique α] : krullDim α = 0 :=
le_antisymm krullDim_nonpos_of_subsingleton krullDim_nonneg | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_zero_of_unique | null |
krullDim_eq_zero_iff_of_orderBot [OrderBot α] :
krullDim α = 0 ↔ Subsingleton α :=
⟨fun H ↦ subsingleton_of_forall_eq ⊥ fun _ ↦ le_bot_iff.mp
(krullDim_nonpos_iff_forall_isMax.mp H.le ⊥ bot_le), fun _ ↦ Order.krullDim_eq_zero⟩ | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_zero_iff_of_orderBot | null |
krullDim_pos_iff_of_orderBot [OrderBot α] :
0 < krullDim α ↔ Nontrivial α := by
rw [← not_subsingleton_iff_nontrivial, ← Order.krullDim_eq_zero_iff_of_orderBot,
← ne_eq, ← lt_or_lt_iff_ne, or_iff_right]
simp [Order.krullDim_nonneg] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_pos_iff_of_orderBot | null |
krullDim_eq_zero_iff_of_orderTop [OrderTop α] :
krullDim α = 0 ↔ Subsingleton α :=
⟨fun H ↦ subsingleton_of_forall_eq ⊤ fun _ ↦ top_le_iff.mp
(krullDim_nonpos_iff_forall_isMin.mp H.le ⊤ le_top), fun _ ↦ Order.krullDim_eq_zero⟩ | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_zero_iff_of_orderTop | null |
krullDim_pos_iff_of_orderTop [OrderTop α] :
0 < krullDim α ↔ Nontrivial α := by
rw [← not_subsingleton_iff_nontrivial, ← Order.krullDim_eq_zero_iff_of_orderTop,
← ne_eq, ← lt_or_lt_iff_ne, or_iff_right]
simp [Order.krullDim_nonneg] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_pos_iff_of_orderTop | null |
krullDim_eq_length_of_finiteDimensionalOrder [FiniteDimensionalOrder α] :
krullDim α = (LTSeries.longestOf α).length :=
le_antisymm
(iSup_le <| fun _ ↦ WithBot.coe_le_coe.mpr <| WithTop.coe_le_coe.mpr <|
RelSeries.length_le_length_longestOf _ _) <|
le_iSup (fun (i : LTSeries _) ↦ (i.length : WithBot... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_length_of_finiteDimensionalOrder | null |
krullDim_eq_top [InfiniteDimensionalOrder α] :
krullDim α = ⊤ :=
le_antisymm le_top <| le_iSup_iff.mpr <| fun m hm ↦ match m, hm with
| ⊥, hm => False.elim <| by
haveI : Inhabited α := ⟨LTSeries.withLength _ 0 0⟩
exact not_le_of_gt (WithBot.bot_lt_coe _ : ⊥ < (0 : WithBot (WithTop ℕ))) <| hm default
|... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_top | null |
krullDim_eq_top_iff : krullDim α = ⊤ ↔ InfiniteDimensionalOrder α := by
refine ⟨fun h ↦ ?_, fun _ ↦ krullDim_eq_top⟩
cases isEmpty_or_nonempty α
· simp [krullDim_eq_bot] at h
cases finiteDimensionalOrder_or_infiniteDimensionalOrder α
· rw [krullDim_eq_length_of_finiteDimensionalOrder] at h
cases h
· inf... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_top_iff | null |
le_krullDim_iff {n : ℕ} : n ≤ krullDim α ↔ ∃ l : LTSeries α, l.length = n := by
cases isEmpty_or_nonempty α
· simp [krullDim_eq_bot]
cases finiteDimensionalOrder_or_infiniteDimensionalOrder α
· rw [krullDim_eq_length_of_finiteDimensionalOrder, Nat.cast_le]
constructor
· exact fun H ↦ ⟨(LTSeries.longestO... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | le_krullDim_iff | null |
krullDim_eq_iSup_length [Nonempty α] :
krullDim α = ⨆ (p : LTSeries α), (p.length : ℕ∞) := by
unfold krullDim
rw [WithBot.coe_iSup (OrderTop.bddAbove _)]
rfl | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_iSup_length | A definition of krullDim for nonempty `α` that avoids `WithBot` |
krullDim_lt_coe_iff {n : ℕ} : krullDim α < n ↔ ∀ l : LTSeries α, l.length < n := by
rw [krullDim, ← WithBot.coe_natCast]
rcases n with - | n
· rw [ENat.coe_zero, ← bot_eq_zero, WithBot.lt_coe_bot]
simp
· simp [WithBot.lt_add_one_iff, WithBot.coe_natCast, Nat.lt_succ] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_lt_coe_iff | null |
krullDim_le_of_strictMono (f : α → β) (hf : StrictMono f) : krullDim α ≤ krullDim β :=
iSup_le fun p ↦ le_sSup ⟨p.map f hf, rfl⟩ | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_le_of_strictMono | null |
krullDim_le_of_strictComono_and_surj
(f : α → β) (hf : ∀ ⦃a b⦄, f a < f b → a < b) (hf' : Function.Surjective f) :
krullDim β ≤ krullDim α :=
iSup_le fun p ↦ le_sSup ⟨p.comap _ hf hf', rfl⟩ | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_le_of_strictComono_and_surj | null |
krullDim_eq_of_orderIso (f : α ≃o β) : krullDim α = krullDim β :=
le_antisymm (krullDim_le_of_strictMono _ f.strictMono) <|
krullDim_le_of_strictMono _ f.symm.strictMono
@[simp] lemma krullDim_orderDual : krullDim αᵒᵈ = krullDim α :=
le_antisymm (iSup_le fun i ↦ le_sSup ⟨i.reverse, rfl⟩) <|
iSup_le fun i ↦ ... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_of_orderIso | null |
height_le_krullDim (a : α) : height a ≤ krullDim α := by
have : Nonempty α := ⟨a⟩
rw [krullDim_eq_iSup_length]
simp only [WithBot.coe_le_coe]
exact height_le fun p _ ↦ le_iSup_of_le p le_rfl | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_le_krullDim | null |
coheight_le_krullDim (a : α) : coheight a ≤ krullDim α := by
simpa using height_le_krullDim (α := αᵒᵈ) a
@[simp] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_le_krullDim | null |
_root_.LTSeries.height_last_longestOf [FiniteDimensionalOrder α] :
height (LTSeries.longestOf α).last = krullDim α := by
refine le_antisymm (height_le_krullDim _) ?_
rw [krullDim_eq_length_of_finiteDimensionalOrder, height]
norm_cast
exact le_iSup_iff.mpr <| fun _ h ↦ iSup_le_iff.mp (h _) le_rfl | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | _root_.LTSeries.height_last_longestOf | null |
krullDim_eq_iSup_height_of_nonempty [Nonempty α] : krullDim α = ↑(⨆ (a : α), height a) := by
apply le_antisymm
· apply iSup_le
intro p
suffices p.length ≤ ⨆ (a : α), height a from (WithBot.unbotD_le_iff fun _ => this).mp this
apply le_iSup_of_le p.last (length_le_height_last (p := p))
· rw [WithBot.co... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_iSup_height_of_nonempty | The Krull dimension is the supremum of the elements' heights.
This version of the lemma assumes that `α` is nonempty. In this case, the coercion from `ℕ∞` to
`WithBot ℕ∞` is on the outside of the right-hand side, which is usually more convenient.
If `α` were empty, then `krullDim α = ⊥`. See `krullDim_eq_iSup_height`... |
krullDim_eq_iSup_coheight_of_nonempty [Nonempty α] :
krullDim α = ↑(⨆ (a : α), coheight a) := by
simpa using krullDim_eq_iSup_height_of_nonempty (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_iSup_coheight_of_nonempty | The Krull dimension is the supremum of the elements' coheights.
This version of the lemma assumes that `α` is nonempty. In this case, the coercion from `ℕ∞` to
`WithBot ℕ∞` is on the outside of the right-hand side, which is usually more convenient.
If `α` were empty, then `krullDim α = ⊥`. See `krullDim_eq_iSup_cohei... |
krullDim_eq_iSup_height_add_coheight_of_nonempty [Nonempty α] :
krullDim α = ↑(⨆ (a : α), height a + coheight a) := by
apply le_antisymm
· rw [krullDim_eq_iSup_height_of_nonempty, WithBot.coe_le_coe]
apply ciSup_mono (by bddDefault) (by simp)
· wlog hnottop : krullDim α < ⊤
· simp_all
rw [krullDim... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_iSup_height_add_coheight_of_nonempty | The Krull dimension is the supremum of the elements' height plus coheight. |
krullDim_eq_iSup_height : krullDim α = ⨆ (a : α), ↑(height a) := by
cases isEmpty_or_nonempty α with
| inl h => rw [krullDim_eq_bot, ciSup_of_empty]
| inr h => rw [krullDim_eq_iSup_height_of_nonempty, WithBot.coe_iSup (OrderTop.bddAbove _)] | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_iSup_height | The Krull dimension is the supremum of the elements' heights.
If `α` is `Nonempty`, then `krullDim_eq_iSup_height_of_nonempty`, with the coercion from
`ℕ∞` to `WithBot ℕ∞` outside the supremum, can be more convenient. |
krullDim_eq_iSup_coheight : krullDim α = ⨆ (a : α), ↑(coheight a) := by
cases isEmpty_or_nonempty α with
| inl h => rw [krullDim_eq_bot, ciSup_of_empty]
| inr h => rw [krullDim_eq_iSup_coheight_of_nonempty, WithBot.coe_iSup (OrderTop.bddAbove _)]
@[simp] -- not as useful as a simp lemma as it looks, due to the co... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_iSup_coheight | The Krull dimension is the supremum of the elements' coheights.
If `α` is `Nonempty`, then `krullDim_eq_iSup_coheight_of_nonempty`, with the coercion from
`ℕ∞` to `WithBot ℕ∞` outside the supremum, can be more convenient. |
height_top_eq_krullDim [OrderTop α] : height (⊤ : α) = krullDim α := by
rw [krullDim_eq_iSup_length]
simp only [WithBot.coe_inj]
apply le_antisymm
· exact height_le fun p _ ↦ le_iSup_of_le p le_rfl
· exact iSup_le fun _ => length_le_height le_top
@[simp] -- not as useful as a simp lemma as it looks, due to th... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_top_eq_krullDim | null |
coheight_bot_eq_krullDim [OrderBot α] : coheight (⊥ : α) = krullDim α := by
rw [← krullDim_orderDual]
exact height_top_eq_krullDim (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_bot_eq_krullDim | null |
height_eq_krullDim_Iic (x : α) : (height x : ℕ∞) = krullDim (Set.Iic x) := by
rw [← height_top_eq_krullDim, height, height, WithBot.coe_inj]
apply le_antisymm
· apply iSup_le; intro p; apply iSup_le; intro hp
let q := LTSeries.mk p.length (fun i ↦ (⟨p.toFun i, le_trans (p.monotone (Fin.le_last _)) hp⟩
: ... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_eq_krullDim_Iic | null |
coheight_eq_krullDim_Ici {α : Type*} [Preorder α] (x : α) :
(coheight x : ℕ∞) = krullDim (Set.Ici x) := by
rw [coheight, ← krullDim_orderDual, Order.krullDim_eq_of_orderIso (OrderIso.refl _)]
exact height_eq_krullDim_Iic _ | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_eq_krullDim_Ici | null |
finiteDimensionalOrder_iff_krullDim_ne_bot_and_top :
FiniteDimensionalOrder α ↔ krullDim α ≠ ⊥ ∧ krullDim α ≠ ⊤ := by
by_cases h : Nonempty α
· simp [← not_infiniteDimensionalOrder_iff, ← krullDim_eq_top_iff]
· constructor
· exact (fun h1 ↦ False.elim (h (LTSeries.nonempty_of_finiteDimensionalOrder α)))
... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | finiteDimensionalOrder_iff_krullDim_ne_bot_and_top | null |
krullDim_ne_bot_of_finiteDimensionalOrder [FiniteDimensionalOrder α] : krullDim α ≠ ⊥ :=
(finiteDimensionalOrder_iff_krullDim_ne_bot_and_top.mp ‹_›).1 | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_ne_bot_of_finiteDimensionalOrder | null |
krullDim_ne_top_of_finiteDimensionalOrder [FiniteDimensionalOrder α] : krullDim α ≠ ⊤ :=
(finiteDimensionalOrder_iff_krullDim_ne_bot_and_top.mp ‹_›).2 | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_ne_top_of_finiteDimensionalOrder | null |
@[mk_iff]
KrullDimLE (n : ℕ) (α : Type*) [Preorder α] : Prop where
krullDim_le : krullDim α ≤ n | class | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | KrullDimLE | Typeclass for orders with krull dimension at most `n`. |
KrullDimLE.mono {n m : ℕ} (e : n ≤ m) (α : Type*) [Preorder α] [KrullDimLE n α] :
KrullDimLE m α :=
⟨KrullDimLE.krullDim_le (n := n).trans (Nat.cast_le.mpr e)⟩ | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | KrullDimLE.mono | null |
krullDim_eq_one_iff_of_boundedOrder {α : Type*} [PartialOrder α] [BoundedOrder α] :
krullDim α = 1 ↔ IsSimpleOrder α := by
rw [le_antisymm_iff, krullDim_le_one_iff, WithBot.one_le_iff_pos,
Order.krullDim_pos_iff_of_orderBot, isSimpleOrder_iff]
simp only [isMin_iff_eq_bot, isMax_iff_eq_top, and_comm]
@[simp]... | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_eq_one_iff_of_boundedOrder | null |
coheight_nat (n : ℕ) : coheight n = ⊤ := coheight_of_noMaxOrder .. | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | coheight_nat | null |
krullDim_nat : krullDim ℕ = ⊤ := krullDim_of_noMaxOrder .. | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | krullDim_nat | null |
height_int (n : ℤ) : height n = ⊤ := height_of_noMinOrder .. | lemma | Order | [
"Mathlib.Algebra.Order.Group.Int",
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.RelSeries",
"Mathlib.Tactic.FinCases"
] | Mathlib/Order/KrullDimension.lean | height_int | null |
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