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 ⌀ |
|---|---|---|---|---|---|---|
SemilatticeInf.toCompleteSemilatticeInf [SemilatticeInf α] (sInf : Set α → α)
(h : ∀ s, InfClosed s → IsGLB s (sInf s)) : CompleteSemilatticeInf α where
sInf := fun s => sInf (infClosure s)
sInf_le _ _ ha := (h _ infClosed_infClosure).1 <| subset_infClosure ha
le_sInf s a ha := (le_isGLB_iff <| h _ infClosed_... | def | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | SemilatticeInf.toCompleteSemilatticeInf | A meet-semilattice where every inf-closed set has a greatest lower bound is automatically
complete. |
SupClosed.iSup_mem_of_nonempty [Finite ι] [Nonempty ι] (hs : SupClosed s)
(hf : ∀ i, f i ∈ s) : ⨆ i, f i ∈ s := by
cases nonempty_fintype (PLift ι)
rw [← iSup_plift_down, ← Finset.sup'_univ_eq_ciSup]
exact hs.finsetSup'_mem Finset.univ_nonempty fun _ _ ↦ hf _ | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | SupClosed.iSup_mem_of_nonempty | null |
InfClosed.iInf_mem_of_nonempty [Finite ι] [Nonempty ι] (hs : InfClosed s)
(hf : ∀ i, f i ∈ s) : ⨅ i, f i ∈ s := hs.dual.iSup_mem_of_nonempty hf | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | InfClosed.iInf_mem_of_nonempty | null |
SupClosed.sSup_mem_of_nonempty (hs : SupClosed s) (ht : t.Finite) (ht' : t.Nonempty)
(hts : t ⊆ s) : sSup t ∈ s := by
have := ht.to_subtype
have := ht'.to_subtype
rw [sSup_eq_iSup']
exact hs.iSup_mem_of_nonempty (by simpa) | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | SupClosed.sSup_mem_of_nonempty | null |
InfClosed.sInf_mem_of_nonempty (hs : InfClosed s) (ht : t.Finite) (ht' : t.Nonempty)
(hts : t ⊆ s) : sInf t ∈ s := hs.dual.sSup_mem_of_nonempty ht ht' hts | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | InfClosed.sInf_mem_of_nonempty | null |
compl_image_latticeClosure (s : Set α) :
compl '' latticeClosure s = latticeClosure (compl '' s) :=
image_latticeClosure' s _ compl_sup_distrib (fun _ _ => compl_inf) | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | compl_image_latticeClosure | null |
compl_image_latticeClosure_eq_of_compl_image_eq_self (hs : compl '' s = s) :
compl '' latticeClosure s = latticeClosure s :=
compl_image_latticeClosure s ▸ hs.symm ▸ rfl | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | compl_image_latticeClosure_eq_of_compl_image_eq_self | null |
SupClosed.biSup_mem_of_nonempty {ι : Type*} {t : Set ι} {f : ι → α} (hs : SupClosed s)
(ht : t.Finite) (ht' : t.Nonempty) (hf : ∀ i ∈ t, f i ∈ s) : ⨆ i ∈ t, f i ∈ s := by
rw [← sSup_image]
exact hs.sSup_mem_of_nonempty (ht.image _) (by simpa) (by simpa) | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | SupClosed.biSup_mem_of_nonempty | null |
InfClosed.biInf_mem_of_nonempty {ι : Type*} {t : Set ι} {f : ι → α} (hs : InfClosed s)
(ht : t.Finite) (ht' : t.Nonempty) (hf : ∀ i ∈ t, f i ∈ s) : ⨅ i ∈ t, f i ∈ s :=
hs.dual.biSup_mem_of_nonempty ht ht' hf | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | InfClosed.biInf_mem_of_nonempty | null |
SupClosed.iSup_mem [Finite ι] (hs : SupClosed s) (hbot : ⊥ ∈ s) (hf : ∀ i, f i ∈ s) :
⨆ i, f i ∈ s := by
cases isEmpty_or_nonempty ι
· simpa [iSup_of_empty]
· exact hs.iSup_mem_of_nonempty hf | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | SupClosed.iSup_mem | null |
InfClosed.iInf_mem [Finite ι] (hs : InfClosed s) (htop : ⊤ ∈ s) (hf : ∀ i, f i ∈ s) :
⨅ i, f i ∈ s := hs.dual.iSup_mem htop hf | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | InfClosed.iInf_mem | null |
SupClosed.sSup_mem (hs : SupClosed s) (ht : t.Finite) (hbot : ⊥ ∈ s) (hts : t ⊆ s) :
sSup t ∈ s := by
have := ht.to_subtype
rw [sSup_eq_iSup']
exact hs.iSup_mem hbot (by simpa) | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | SupClosed.sSup_mem | null |
InfClosed.sInf_mem (hs : InfClosed s) (ht : t.Finite) (htop : ⊤ ∈ s) (hts : t ⊆ s) :
sInf t ∈ s := hs.dual.sSup_mem ht htop hts | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | InfClosed.sInf_mem | null |
SupClosed.biSup_mem {ι : Type*} {t : Set ι} {f : ι → α} (hs : SupClosed s)
(ht : t.Finite) (hbot : ⊥ ∈ s) (hf : ∀ i ∈ t, f i ∈ s) : ⨆ i ∈ t, f i ∈ s := by
rw [← sSup_image]
exact hs.sSup_mem (ht.image _) hbot (by simpa) | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | SupClosed.biSup_mem | null |
InfClosed.biInf_mem {ι : Type*} {t : Set ι} {f : ι → α} (hs : InfClosed s)
(ht : t.Finite) (htop : ⊤ ∈ s) (hf : ∀ i ∈ t, f i ∈ s) : ⨅ i ∈ t, f i ∈ s :=
hs.dual.biSup_mem ht htop hf | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Prod",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Set.Finite.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.ConditionallyCompleteLattice.Finset"
] | Mathlib/Order/SupClosed.lean | InfClosed.biInf_mem | null |
SupIndep (s : Finset ι) (f : ι → α) : Prop :=
∀ ⦃t⦄, t ⊆ s → ∀ ⦃i⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f)
variable {s t : Finset ι} {f : ι → α} {i : ι} | def | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | SupIndep | Supremum independence of finite sets. We avoid the "obvious" definition using `s.erase i`
because `erase` would require decidable equality on `ι`. |
supIndep_iff_disjoint_erase [DecidableEq ι] :
s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) :=
⟨fun hs _ hi => hs (erase_subset _ _) hi (notMem_erase _ _), fun hs _ ht i hi hit =>
(hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩ | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | supIndep_iff_disjoint_erase | The RHS looks like the definition of `iSupIndep`. |
protected SupIndep.sup [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α}
(hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) :
(s.sup g).SupIndep f := by
simp_rw [supIndep_iff_pairwiseDisjoint] at hs hg ⊢
rw [sup_eq_biUnion, coe_biUnion]
exact hs.biUnion_finset hg | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | SupIndep.sup | If both the index type and the lattice have decidable equality,
then the `SupIndep` predicate is decidable.
TODO: speedup the definition and drop the `[DecidableEq ι]` assumption
by iterating over the pairs `(a, t)` such that `s = Finset.cons a t _`
using something like `List.eraseIdx`
or by generating both `f i` and ... |
protected SupIndep.biUnion [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α}
(hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) :
(s.biUnion g).SupIndep f := by
rw [← sup_eq_biUnion]
exact hs.sup hg | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | SupIndep.biUnion | Bind operation for `SupIndep`. |
protected SupIndep.sigma {β : ι → Type*} {s : Finset ι} {g : ∀ i, Finset (β i)}
{f : Sigma β → α} (hs : s.SupIndep fun i => (g i).sup fun b => f ⟨i, b⟩)
(hg : ∀ i ∈ s, (g i).SupIndep fun b => f ⟨i, b⟩) : (s.sigma g).SupIndep f := by
rintro t ht ⟨i, b⟩ hi hit
rw [Finset.disjoint_sup_right]
rintro ⟨j, c⟩ hj... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | SupIndep.sigma | Bind operation for `SupIndep`. |
protected SupIndep.product {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α}
(hs : s.SupIndep fun i => t.sup fun i' => f (i, i'))
(ht : t.SupIndep fun i' => s.sup fun i => f (i, i')) : (s ×ˢ t).SupIndep f := by
rintro u hu ⟨i, i'⟩ hi hiu
rw [Finset.disjoint_sup_right]
rintro ⟨j, j'⟩ hj
have hij := (ne_of_... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | SupIndep.product | null |
supIndep_product_iff {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} :
(s.product t).SupIndep f ↔ (s.SupIndep fun i => t.sup fun i' => f (i, i'))
∧ t.SupIndep fun i' => s.sup fun i => f (i, i') := by
refine ⟨?_, fun h => h.1.product h.2⟩
simp_rw [supIndep_iff_pairwiseDisjoint]
refine fun h => ⟨fun i hi ... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | supIndep_product_iff | null |
sSupIndep (s : Set α) : Prop :=
∀ ⦃a⦄, a ∈ s → Disjoint a (sSup (s \ {a}))
variable {s : Set α} (hs : sSupIndep s)
@[simp] | def | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep | An independent set of elements in a complete lattice is one in which every element is disjoint
from the `Sup` of the rest. |
sSupIndep_empty : sSupIndep (∅ : Set α) := fun x hx =>
(Set.notMem_empty x hx).elim
include hs in | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep_empty | null |
sSupIndep.mono {t : Set α} (hst : t ⊆ s) : sSupIndep t := fun _ ha =>
(hs (hst ha)).mono_right (sSup_le_sSup (diff_subset_diff_left hst))
include hs in | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep.mono | null |
sSupIndep.pairwiseDisjoint : s.PairwiseDisjoint id := fun _ hx y hy h =>
disjoint_sSup_right (hs hx) ((mem_diff y).mpr ⟨hy, h.symm⟩) | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep.pairwiseDisjoint | If the elements of a set are independent, then any pair within that set is disjoint. |
sSupIndep_singleton (a : α) : sSupIndep ({a} : Set α) := fun i hi ↦ by
simp_all | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep_singleton | null |
sSupIndep_pair {a b : α} (hab : a ≠ b) :
sSupIndep ({a, b} : Set α) ↔ Disjoint a b := by
constructor
· intro h
exact h.pairwiseDisjoint (mem_insert _ _) (mem_insert_of_mem _ (mem_singleton _)) hab
· rintro h c ((rfl : c = a) | (rfl : c = b))
· convert h using 1
simp [hab, sSup_singleton]
· c... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep_pair | null |
sSupIndep.disjoint_sSup {x : α} {y : Set α} (hx : x ∈ s) (hy : y ⊆ s) (hxy : x ∉ y) :
Disjoint x (sSup y) := by
have := (hs.mono <| insert_subset_iff.mpr ⟨hx, hy⟩) (mem_insert x _)
rw [insert_diff_of_mem _ (mem_singleton _), diff_singleton_eq_self hxy] at this
exact this | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep.disjoint_sSup | If the elements of a set are independent, then any element is disjoint from the `sSup` of some
subset of the rest. |
iSupIndep {ι : Sort*} {α : Type*} [CompleteLattice α] (t : ι → α) : Prop :=
∀ i : ι, Disjoint (t i) (⨆ (j) (_ : j ≠ i), t j) | def | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep | An independent indexed family of elements in a complete lattice is one in which every element
is disjoint from the `iSup` of the rest.
Example: an indexed family of non-zero elements in a
vector space is linearly independent iff the indexed family of subspaces they generate is
independent in this sense.
Exa... |
sSupIndep_iff {α : Type*} [CompleteLattice α] (s : Set α) :
sSupIndep s ↔ iSupIndep ((↑) : s → α) := by
simp_rw [iSupIndep, sSupIndep, SetCoe.forall, sSup_eq_iSup]
refine forall₂_congr fun a ha => ?_
simp [iSup_subtype, iSup_and]
variable {t : ι → α} (ht : iSupIndep t) | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep_iff | null |
iSupIndep_def : iSupIndep t ↔ ∀ i, Disjoint (t i) (⨆ (j) (_ : j ≠ i), t j) :=
Iff.rfl | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_def | null |
iSupIndep_def' : iSupIndep t ↔ ∀ i, Disjoint (t i) (sSup (t '' { j | j ≠ i })) := by
simp_rw [sSup_image]
rfl | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_def' | null |
iSupIndep_def'' :
iSupIndep t ↔ ∀ i, Disjoint (t i) (sSup { a | ∃ j ≠ i, t j = a }) := by
rw [iSupIndep_def']
aesop
@[simp] | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_def'' | null |
iSupIndep_empty (t : Empty → α) : iSupIndep t :=
nofun
@[simp] | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_empty | null |
iSupIndep_pempty (t : PEmpty → α) : iSupIndep t :=
nofun
include ht in | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_pempty | null |
iSupIndep.pairwiseDisjoint : Pairwise (Disjoint on t) := fun x y h =>
disjoint_sSup_right (ht x) ⟨y, iSup_pos h.symm⟩ | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.pairwiseDisjoint | If the elements of a set are independent, then any pair within that set is disjoint. |
iSupIndep.mono {s t : ι → α} (hs : iSupIndep s) (hst : t ≤ s) : iSupIndep t :=
fun i => (hs i).mono (hst i) <| iSup₂_mono fun j _ => hst j | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.mono | null |
iSupIndep.comp {ι ι' : Sort*} {t : ι → α} {f : ι' → ι} (ht : iSupIndep t)
(hf : Injective f) : iSupIndep (t ∘ f) := fun i =>
(ht (f i)).mono_right <| by
refine (iSup_mono fun i => ?_).trans (iSup_comp_le _ f)
exact iSup_const_mono hf.ne | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.comp | Composing an independent indexed family with an injective function on the index results in
another independent indexed family. |
iSupIndep.comp' {ι ι' : Sort*} {t : ι → α} {f : ι' → ι} (ht : iSupIndep <| t ∘ f)
(hf : Surjective f) : iSupIndep t := by
intro i
obtain ⟨i', rfl⟩ := hf i
rw [← hf.iSup_comp]
exact (ht i').mono_right (biSup_mono fun j' hij => mt (congr_arg f) hij) | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.comp' | null |
iSupIndep.sSupIndep_range (ht : iSupIndep t) : sSupIndep <| range t := by
rw [sSupIndep_iff]
rw [← coe_comp_rangeFactorization t] at ht
exact ht.comp' rangeFactorization_surjective
@[simp] | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.sSupIndep_range | null |
iSupIndep_ne_bot :
iSupIndep (fun i : {i // t i ≠ ⊥} ↦ t i) ↔ iSupIndep t := by
refine ⟨fun h ↦ ?_, fun h ↦ h.comp Subtype.val_injective⟩
simp only [iSupIndep_def] at h ⊢
intro i
cases eq_or_ne (t i) ⊥ with
| inl hi => simp [hi]
| inr hi => ?_
convert h ⟨i, hi⟩
have : ∀ j, ⨆ (_ : t j = ⊥), t j = ⊥ :... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_ne_bot | null |
iSupIndep.injOn (ht : iSupIndep t) : InjOn t {i | t i ≠ ⊥} := by
rintro i _ j (hj : t j ≠ ⊥) h
by_contra! contra
apply hj
suffices t j ≤ ⨆ (k) (_ : k ≠ i), t k by
replace ht := (ht i).mono_right this
rwa [h, disjoint_self] at ht
replace contra : j ≠ i := Ne.symm contra
exact le_iSup₂ (f := fun x _ ↦... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.injOn | null |
iSupIndep.injective (ht : iSupIndep t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Injective t := by
suffices univ = {i | t i ≠ ⊥} by rw [injective_iff_injOn_univ, this]; exact ht.injOn
simp_all | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.injective | null |
iSupIndep_pair {i j : ι} (hij : i ≠ j) (huniv : ∀ k, k = i ∨ k = j) :
iSupIndep t ↔ Disjoint (t i) (t j) := by
constructor
· exact fun h => h.pairwiseDisjoint hij
· rintro h k
obtain rfl | rfl := huniv k
· refine h.mono_right (iSup_le fun i => iSup_le fun hi => Eq.le ?_)
rw [(huniv i).resolve_le... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_pair | null |
iSupIndep.map_orderIso {ι : Sort*} {α β : Type*} [CompleteLattice α]
[CompleteLattice β] (f : α ≃o β) {a : ι → α} (ha : iSupIndep a) : iSupIndep (f ∘ a) :=
fun i => ((ha i).map_orderIso f).mono_right (f.monotone.le_map_iSup₂ _)
@[simp] | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.map_orderIso | Composing an independent indexed family with an order isomorphism on the elements results in
another independent indexed family. |
iSupIndep_map_orderIso_iff {ι : Sort*} {α β : Type*} [CompleteLattice α]
[CompleteLattice β] (f : α ≃o β) {a : ι → α} : iSupIndep (f ∘ a) ↔ iSupIndep a :=
⟨fun h =>
have hf : f.symm ∘ f ∘ a = a := congr_arg (· ∘ a) f.left_inv.comp_eq_id
hf ▸ h.map_orderIso f.symm,
fun h => h.map_orderIso f⟩ | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_map_orderIso_iff | null |
iSupIndep.disjoint_biSup {ι : Type*} {α : Type*} [CompleteLattice α] {t : ι → α}
(ht : iSupIndep t) {x : ι} {y : Set ι} (hx : x ∉ y) : Disjoint (t x) (⨆ i ∈ y, t i) :=
Disjoint.mono_right (biSup_mono fun _ hi => (ne_of_mem_of_not_mem hi hx :)) (ht x) | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.disjoint_biSup | If the elements of a set are independent, then any element is disjoint from the `iSup` of some
subset of the rest. |
iSupIndep.of_coe_Iic_comp {ι : Sort*} {a : α} {t : ι → Set.Iic a}
(ht : iSupIndep ((↑) ∘ t : ι → α)) : iSupIndep t := by
intro i x
specialize ht i
simp_rw [Function.comp_apply, ← Set.Iic.coe_iSup] at ht
exact @ht x | lemma | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.of_coe_Iic_comp | null |
iSupIndep_iff_supIndep {s : Finset ι} {f : ι → α} :
iSupIndep (f ∘ ((↑) : s → ι)) ↔ s.SupIndep f := by
classical
rw [Finset.supIndep_iff_disjoint_erase]
refine Subtype.forall.trans (forall₂_congr fun a b => ?_)
rw [Finset.sup_eq_iSup]
congr! 1
refine iSup_subtype.trans ?_
congr! 1
simp... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_iff_supIndep | null |
iSupIndep.supIndep' {f : ι → α} (s : Finset ι) (h : iSupIndep f) : s.SupIndep f :=
iSupIndep.supIndep (h.comp Subtype.coe_injective) | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep.supIndep' | null |
iSupIndep_iff_supIndep_univ [Fintype ι] {f : ι → α} :
iSupIndep f ↔ Finset.univ.SupIndep f := by
classical
simp [Finset.supIndep_iff_disjoint_erase, iSupIndep, Finset.sup_eq_iSup]
alias ⟨iSupIndep.sup_indep_univ, Finset.SupIndep.iSupIndep_of_univ⟩ := iSupIndep_iff_supIndep_univ | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_iff_supIndep_univ | A variant of `CompleteLattice.iSupIndep_iff_supIndep` for `Fintype`s. |
sSupIndep_iff_pairwiseDisjoint {s : Set α} : sSupIndep s ↔ s.PairwiseDisjoint id :=
⟨sSupIndep.pairwiseDisjoint, fun hs _ hi =>
disjoint_sSup_iff.2 fun _ hj => hs hi hj.1 <| Ne.symm hj.2⟩
alias ⟨_, _root_.Set.PairwiseDisjoint.sSupIndep⟩ := sSupIndep_iff_pairwiseDisjoint
open scoped Function in -- required for sco... | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | sSupIndep_iff_pairwiseDisjoint | null |
iSupIndep_iff_pairwiseDisjoint {f : ι → α} : iSupIndep f ↔ Pairwise (Disjoint on f) :=
⟨iSupIndep.pairwiseDisjoint, fun hs _ =>
disjoint_iSup_iff.2 fun _ => disjoint_iSup_iff.2 fun hij => hs hij.symm⟩ | theorem | Order | [
"Mathlib.Data.Finset.Lattice.Union",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Order.CompleteLatticeIntervals"
] | Mathlib/Order/SupIndep.lean | iSupIndep_iff_pairwiseDisjoint | null |
symmDiff [Max α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a | def | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff | The symmetric difference operator on a type with `⊔` and `\` is `(A \ B) ⊔ (B \ A)`. |
bihimp [Min α] [HImp α] (a b : α) : α :=
(b ⇨ a) ⊓ (a ⇨ b) | def | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp | The Heyting bi-implication is `(b ⇨ a) ⊓ (a ⇨ b)`. This generalizes equivalence of
propositions. |
symmDiff_def [Max α] [SDiff α] (a b : α) : a ∆ b = a \ b ⊔ b \ a :=
rfl | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_def | null |
bihimp_def [Min α] [HImp α] (a b : α) : a ⇔ b = (b ⇨ a) ⊓ (a ⇨ b) :=
rfl | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_def | null |
symmDiff_eq_Xor' (p q : Prop) : p ∆ q = Xor' p q :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq_Xor' | null |
bihimp_iff_iff {p q : Prop} : p ⇔ q ↔ (p ↔ q) :=
iff_iff_implies_and_implies.symm.trans Iff.comm
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_iff_iff | null |
Bool.symmDiff_eq_xor : ∀ p q : Bool, p ∆ q = xor p q := by decide | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | Bool.symmDiff_eq_xor | null |
@[simp]
toDual_symmDiff : toDual (a ∆ b) = toDual a ⇔ toDual b :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | toDual_symmDiff | null |
ofDual_bihimp (a b : αᵒᵈ) : ofDual (a ⇔ b) = ofDual a ∆ ofDual b :=
rfl | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | ofDual_bihimp | null |
symmDiff_comm : a ∆ b = b ∆ a := by simp only [symmDiff, sup_comm] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_comm | null |
symmDiff_isCommutative : Std.Commutative (α := α) (· ∆ ·) :=
⟨symmDiff_comm⟩
@[simp] | instance | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_isCommutative | null |
symmDiff_self : a ∆ a = ⊥ := by rw [symmDiff, sup_idem, sdiff_self]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_self | null |
symmDiff_bot : a ∆ ⊥ = a := by rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_bot | null |
bot_symmDiff : ⊥ ∆ a = a := by rw [symmDiff_comm, symmDiff_bot]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bot_symmDiff | null |
symmDiff_eq_bot {a b : α} : a ∆ b = ⊥ ↔ a = b := by
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq_bot | null |
symmDiff_of_le {a b : α} (h : a ≤ b) : a ∆ b = b \ a := by
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_of_le | null |
symmDiff_of_ge {a b : α} (h : b ≤ a) : a ∆ b = a \ b := by
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_of_ge | null |
symmDiff_le {a b c : α} (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ∆ b ≤ c :=
sup_le (sdiff_le_iff.2 ha) <| sdiff_le_iff.2 hb | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_le | null |
symmDiff_le_iff {a b c : α} : a ∆ b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c := by
simp_rw [symmDiff, sup_le_iff, sdiff_le_iff]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_le_iff | null |
symmDiff_le_sup {a b : α} : a ∆ b ≤ a ⊔ b :=
sup_le_sup sdiff_le sdiff_le | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_le_sup | null |
symmDiff_eq_sup_sdiff_inf : a ∆ b = (a ⊔ b) \ (a ⊓ b) := by simp [sup_sdiff, symmDiff] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_eq_sup_sdiff_inf | null |
Disjoint.symmDiff_eq_sup {a b : α} (h : Disjoint a b) : a ∆ b = a ⊔ b := by
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | Disjoint.symmDiff_eq_sup | null |
symmDiff_sdiff : a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) := by
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_sdiff | null |
symmDiff_sdiff_inf : a ∆ b \ (a ⊓ b) = a ∆ b := by
rw [symmDiff_sdiff]
simp [symmDiff]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_sdiff_inf | null |
symmDiff_sdiff_eq_sup : a ∆ (b \ a) = a ⊔ b := by
rw [symmDiff, sdiff_idem]
exact
le_antisymm (sup_le_sup sdiff_le sdiff_le)
(sup_le le_sdiff_sup <| le_sdiff_sup.trans <| sup_le le_sup_right le_sdiff_sup)
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_sdiff_eq_sup | null |
sdiff_symmDiff_eq_sup : (a \ b) ∆ b = a ⊔ b := by
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | sdiff_symmDiff_eq_sup | null |
symmDiff_sup_inf : a ∆ b ⊔ a ⊓ b = a ⊔ b := by
refine le_antisymm (sup_le symmDiff_le_sup inf_le_sup) ?_
rw [sup_inf_left, symmDiff]
refine sup_le (le_inf le_sup_right ?_) (le_inf ?_ le_sup_right)
· rw [sup_right_comm]
exact le_sup_of_le_left le_sdiff_sup
· rw [sup_assoc]
exact le_sup_of_le_right le_s... | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_sup_inf | null |
inf_sup_symmDiff : a ⊓ b ⊔ a ∆ b = a ⊔ b := by rw [sup_comm, symmDiff_sup_inf]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | inf_sup_symmDiff | null |
symmDiff_symmDiff_inf : a ∆ b ∆ (a ⊓ b) = a ⊔ b := by
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_symmDiff_inf | null |
inf_symmDiff_symmDiff : (a ⊓ b) ∆ (a ∆ b) = a ⊔ b := by
rw [symmDiff_comm, symmDiff_symmDiff_inf] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | inf_symmDiff_symmDiff | null |
symmDiff_triangle : a ∆ c ≤ a ∆ b ⊔ b ∆ c := by
refine (sup_le_sup (sdiff_triangle a b c) <| sdiff_triangle _ b _).trans_eq ?_
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | symmDiff_triangle | null |
le_symmDiff_sup_right (a b : α) : a ≤ (a ∆ b) ⊔ b := by
convert symmDiff_triangle a b ⊥ <;> rw [symmDiff_bot] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | le_symmDiff_sup_right | null |
le_symmDiff_sup_left (a b : α) : b ≤ (a ∆ b) ⊔ a :=
symmDiff_comm a b ▸ le_symmDiff_sup_right .. | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | le_symmDiff_sup_left | null |
@[simp]
toDual_bihimp : toDual (a ⇔ b) = toDual a ∆ toDual b :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | toDual_bihimp | null |
ofDual_symmDiff (a b : αᵒᵈ) : ofDual (a ∆ b) = ofDual a ⇔ ofDual b :=
rfl | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | ofDual_symmDiff | null |
bihimp_comm : a ⇔ b = b ⇔ a := by simp only [(· ⇔ ·), inf_comm] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_comm | null |
bihimp_isCommutative : Std.Commutative (α := α) (· ⇔ ·) :=
⟨bihimp_comm⟩
@[simp] | instance | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_isCommutative | null |
bihimp_self : a ⇔ a = ⊤ := by rw [bihimp, inf_idem, himp_self]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_self | null |
bihimp_top : a ⇔ ⊤ = a := by rw [bihimp, himp_top, top_himp, inf_top_eq]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_top | null |
top_bihimp : ⊤ ⇔ a = a := by rw [bihimp_comm, bihimp_top]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | top_bihimp | null |
bihimp_eq_top {a b : α} : a ⇔ b = ⊤ ↔ a = b :=
@symmDiff_eq_bot αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_eq_top | null |
bihimp_of_le {a b : α} (h : a ≤ b) : a ⇔ b = b ⇨ a := by
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_of_le | null |
bihimp_of_ge {a b : α} (h : b ≤ a) : a ⇔ b = a ⇨ b := by
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | bihimp_of_ge | null |
le_bihimp {a b c : α} (hb : a ⊓ b ≤ c) (hc : a ⊓ c ≤ b) : a ≤ b ⇔ c :=
le_inf (le_himp_iff.2 hc) <| le_himp_iff.2 hb | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | le_bihimp | null |
le_bihimp_iff {a b c : α} : a ≤ b ⇔ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b := by
simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm]
@[simp] | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | le_bihimp_iff | null |
inf_le_bihimp {a b : α} : a ⊓ b ≤ a ⇔ b :=
inf_le_inf le_himp le_himp | theorem | Order | [
"Mathlib.Order.BooleanAlgebra.Basic",
"Mathlib.Logic.Equiv.Basic"
] | Mathlib/Order/SymmDiff.lean | inf_le_bihimp | null |
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