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 ⌀ |
|---|---|---|---|---|---|---|
gc_upperPolar_lowerPolar :
GaloisConnection (toDual ∘ upperPolar r) (lowerPolar r ∘ ofDual) := fun _ _ =>
subset_upperPolar_iff_subset_lowerPolar
@[deprecated (since := "2025-07-10")]
alias gc_intentClosure_extentClosure := gc_upperPolar_lowerPolar | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | gc_upperPolar_lowerPolar | null |
upperPolar_swap (t : Set β) : upperPolar (swap r) t = lowerPolar r t :=
rfl
@[deprecated (since := "2025-07-10")]
alias intentClosure_swap := upperPolar_swap | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | upperPolar_swap | null |
lowerPolar_swap (s : Set α) : lowerPolar (swap r) s = upperPolar r s :=
rfl
@[deprecated (since := "2025-07-10")]
alias extentClosure_swap := lowerPolar_swap
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | lowerPolar_swap | null |
upperPolar_empty : upperPolar r ∅ = univ :=
eq_univ_of_forall fun _ _ => False.elim
@[deprecated (since := "2025-07-10")]
alias intentClosure_empty := upperPolar_empty
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | upperPolar_empty | null |
lowerPolar_empty : lowerPolar r ∅ = univ :=
upperPolar_empty _
@[deprecated (since := "2025-07-10")]
alias extentClosure_empty := lowerPolar_empty
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | lowerPolar_empty | null |
upperPolar_union (s₁ s₂ : Set α) :
upperPolar r (s₁ ∪ s₂) = upperPolar r s₁ ∩ upperPolar r s₂ :=
ext fun _ => forall₂_or_left
@[deprecated (since := "2025-07-10")]
alias intentClosure_union := upperPolar_union
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | upperPolar_union | null |
lowerPolar_union (t₁ t₂ : Set β) :
lowerPolar r (t₁ ∪ t₂) = lowerPolar r t₁ ∩ lowerPolar r t₂ :=
upperPolar_union ..
@[deprecated (since := "2025-07-10")]
alias extentClosure_union := lowerPolar_union
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | lowerPolar_union | null |
upperPolar_iUnion (f : ι → Set α) :
upperPolar r (⋃ i, f i) = ⋂ i, upperPolar r (f i) :=
(gc_upperPolar_lowerPolar r).l_iSup
@[deprecated (since := "2025-07-10")]
alias intentClosure_iUnion := upperPolar_iUnion
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | upperPolar_iUnion | null |
lowerPolar_iUnion (f : ι → Set β) :
lowerPolar r (⋃ i, f i) = ⋂ i, lowerPolar r (f i) :=
upperPolar_iUnion ..
@[deprecated (since := "2025-07-10")]
alias extentClosure_iUnion := lowerPolar_iUnion | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | lowerPolar_iUnion | null |
upperPolar_iUnion₂ (f : ∀ i, κ i → Set α) :
upperPolar r (⋃ (i) (j), f i j) = ⋂ (i) (j), upperPolar r (f i j) :=
(gc_upperPolar_lowerPolar r).l_iSup₂
@[deprecated (since := "2025-07-10")]
alias intentClosure_iUnion₂ := upperPolar_iUnion₂ | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | upperPolar_iUnion₂ | null |
lowerPolar_iUnion₂ (f : ∀ i, κ i → Set β) :
lowerPolar r (⋃ (i) (j), f i j) = ⋂ (i) (j), lowerPolar r (f i j) :=
upperPolar_iUnion₂ ..
@[deprecated (since := "2025-07-10")]
alias extentClosure_iUnion₂ := lowerPolar_iUnion₂ | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | lowerPolar_iUnion₂ | null |
subset_lowerPolar_upperPolar (s : Set α) :
s ⊆ lowerPolar r (upperPolar r s) :=
(gc_upperPolar_lowerPolar r).le_u_l _
@[deprecated (since := "2025-07-10")]
alias subset_extentClosure_intentClosure := subset_lowerPolar_upperPolar | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | subset_lowerPolar_upperPolar | null |
subset_upperPolar_lowerPolar (t : Set β) :
t ⊆ upperPolar r (lowerPolar r t) :=
subset_lowerPolar_upperPolar _ t
@[deprecated (since := "2025-07-10")]
alias subset_intentClosure_extentClosure := subset_upperPolar_lowerPolar
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | subset_upperPolar_lowerPolar | null |
upperPolar_lowerPolar_upperPolar (s : Set α) :
upperPolar r (lowerPolar r <| upperPolar r s) = upperPolar r s :=
(gc_upperPolar_lowerPolar r).l_u_l_eq_l _
@[deprecated (since := "2025-07-10")]
alias intentClosure_extentClosure_intentClosure := upperPolar_lowerPolar_upperPolar
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | upperPolar_lowerPolar_upperPolar | null |
lowerPolar_upperPolar_lowerPolar (t : Set β) :
lowerPolar r (upperPolar r <| lowerPolar r t) = lowerPolar r t :=
upperPolar_lowerPolar_upperPolar _ t
@[deprecated (since := "2025-07-10")]
alias extentClosure_intentClosure_extentClosure := lowerPolar_upperPolar_lowerPolar | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | lowerPolar_upperPolar_lowerPolar | null |
upperPolar_anti : Antitone (upperPolar r) :=
(gc_upperPolar_lowerPolar r).monotone_l
@[deprecated (since := "2025-07-10")]
alias intentClosure_anti := upperPolar_anti | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | upperPolar_anti | null |
lowerPolar_anti : Antitone (lowerPolar r) :=
upperPolar_anti _
@[deprecated (since := "2025-07-10")]
alias extentClosure_anti := lowerPolar_anti
/-! ### Concepts -/
variable (α β) | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | lowerPolar_anti | null |
Concept where
/-- The extent of a concept. -/
extent : Set α
/-- The intent of a concept. -/
intent : Set β
/-- The intent consists of all elements related to all elements of the extent. -/
upperPolar_extent : upperPolar r extent = intent
/-- The extent consists of all elements related to all elements of ... | structure | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | Concept | The formal concepts of a relation. A concept of `r : α → β → Prop` is a pair of sets `s`, `t`
such that `s` is the set of all elements that are `r`-related to all of `t` and `t` is the set of
all elements that are `r`-related to all of `s`. |
@[ext]
ext (h : c.extent = d.extent) : c = d := by
obtain ⟨s₁, t₁, rfl, _⟩ := c
obtain ⟨s₂, t₂, rfl, _⟩ := d
substs h
rfl | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | ext | null |
ext' (h : c.intent = d.intent) : c = d := by
obtain ⟨s₁, t₁, _, rfl⟩ := c
obtain ⟨s₂, t₂, _, rfl⟩ := d
substs h
rfl | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | ext' | null |
extent_injective : Injective (@extent α β r) := fun _ _ => ext
@[deprecated (since := "2025-07-10")]
alias fst_injective := extent_injective | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_injective | null |
intent_injective : Injective (@intent α β r) := fun _ _ => ext'
@[deprecated (since := "2025-07-10")]
alias snd_injective := intent_injective | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_injective | null |
rel_extent_intent {x y} (hx : x ∈ c.extent) (hy : y ∈ c.intent) : r x y := by
rw [← c.upperPolar_extent] at hy
exact hy hx | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | rel_extent_intent | null |
disjoint_extent_intent [IsIrrefl α r'] : Disjoint c'.extent c'.intent := by
rw [disjoint_iff_forall_ne]
rintro x hx _ hx' rfl
exact irrefl x (rel_extent_intent hx hx') | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | disjoint_extent_intent | Note that if `r'` is the `≤` relation, this theorem will often not be true! |
mem_extent_of_rel_extent [IsTrans α r'] {x y} (hy : r' y x) (hx : x ∈ c'.extent) :
y ∈ c'.extent := by
rw [← lowerPolar_intent]
exact fun z hz ↦ _root_.trans hy (rel_extent_intent hx hz) | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | mem_extent_of_rel_extent | null |
mem_intent_of_intent_rel [IsTrans α r'] {x y} (hy : r' x y) (hx : x ∈ c'.intent) :
y ∈ c'.intent := by
rw [← upperPolar_extent]
exact fun z hz ↦ _root_.trans (rel_extent_intent hz hx) hy | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | mem_intent_of_intent_rel | null |
codisjoint_extent_intent [IsTrichotomous α r'] [IsTrans α r'] :
Codisjoint c'.extent c'.intent := by
rw [codisjoint_iff_le_sup]
refine fun x _ ↦ or_iff_not_imp_left.2 fun hx ↦ ?_
rw [← upperPolar_extent]
intro y hy
obtain h | rfl | h := trichotomous_of r' x y
· cases hx <| mem_extent_of_rel_extent h hy
... | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | codisjoint_extent_intent | null |
instSupConcept : Max (Concept α β r) :=
⟨fun c d =>
{ extent := lowerPolar r (c.intent ∩ d.intent)
intent := c.intent ∩ d.intent
upperPolar_extent := by
rw [← c.upperPolar_extent, ← d.upperPolar_extent, ← upperPolar_union,
upperPolar_lowerPolar_upperPolar]
lowerPolar_intent := ... | instance | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | instSupConcept | null |
instInfConcept : Min (Concept α β r) :=
⟨fun c d =>
{ extent := c.extent ∩ d.extent
intent := upperPolar r (c.extent ∩ d.extent)
upperPolar_extent := rfl
lowerPolar_intent := by
rw [← c.lowerPolar_intent, ← d.lowerPolar_intent, ← lowerPolar_union,
lowerPolar_upperPolar_lowerPol... | instance | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | instInfConcept | null |
instSemilatticeInfConcept : SemilatticeInf (Concept α β r) :=
(extent_injective.semilatticeInf _) fun _ _ => rfl
@[simp] | instance | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | instSemilatticeInfConcept | null |
extent_subset_extent_iff : c.extent ⊆ d.extent ↔ c ≤ d :=
Iff.rfl
@[deprecated (since := "2025-07-10")]
alias fst_subset_fst_iff := extent_subset_extent_iff
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_subset_extent_iff | null |
extent_ssubset_extent_iff : c.extent ⊂ d.extent ↔ c < d :=
Iff.rfl
@[deprecated (since := "2025-07-10")]
alias fst_ssubset_fst_iff := extent_ssubset_extent_iff
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_ssubset_extent_iff | null |
intent_subset_intent_iff : c.intent ⊆ d.intent ↔ d ≤ c := by
refine ⟨fun h => ?_, fun h => ?_⟩
· rw [← extent_subset_extent_iff, ← c.lowerPolar_intent, ← d.lowerPolar_intent]
exact lowerPolar_anti _ h
· rw [← c.upperPolar_extent, ← d.upperPolar_extent]
exact upperPolar_anti _ h
@[deprecated (since := "202... | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_subset_intent_iff | null |
intent_ssubset_intent_iff : c.intent ⊂ d.intent ↔ d < c := by
rw [ssubset_iff_subset_not_subset, lt_iff_le_not_ge,
intent_subset_intent_iff, intent_subset_intent_iff]
@[deprecated (since := "2025-07-10")]
alias snd_ssubset_snd_iff := intent_ssubset_intent_iff | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_ssubset_intent_iff | null |
strictMono_extent : StrictMono (@extent α β r) := fun _ _ =>
extent_ssubset_extent_iff.2
@[deprecated (since := "2025-07-10")]
alias strictMono_fst := strictMono_extent | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | strictMono_extent | null |
strictAnti_intent : StrictAnti (@intent α β r) := fun _ _ =>
intent_ssubset_intent_iff.2
@[deprecated (since := "2025-07-10")]
alias strictMono_snd := strictAnti_intent | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | strictAnti_intent | null |
instLatticeConcept : Lattice (Concept α β r) :=
{ Concept.instSemilatticeInfConcept with
sup := (· ⊔ ·)
le_sup_left := fun _ _ => intent_subset_intent_iff.1 inter_subset_left
le_sup_right := fun _ _ => intent_subset_intent_iff.1 inter_subset_right
sup_le := fun c d e => by
simp_rw [← intent_subs... | instance | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | instLatticeConcept | null |
instBoundedOrderConcept : BoundedOrder (Concept α β r) where
top := ⟨univ, upperPolar r univ, rfl, eq_univ_of_forall fun _ _ hb => hb trivial⟩
le_top _ := subset_univ _
bot := ⟨lowerPolar r univ, univ, eq_univ_of_forall fun _ _ ha => ha trivial, rfl⟩
bot_le _ := intent_subset_intent_iff.1 <| subset_univ _ | instance | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | instBoundedOrderConcept | null |
@[simp]
extent_top : (⊤ : Concept α β r).extent = univ :=
rfl
@[deprecated (since := "2025-07-10")]
alias top_fst := extent_top
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_top | null |
intent_top : (⊤ : Concept α β r).intent = upperPolar r univ :=
rfl
@[deprecated (since := "2025-07-10")]
alias top_snd := intent_top
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_top | null |
extent_bot : (⊥ : Concept α β r).extent = lowerPolar r univ :=
rfl
@[deprecated (since := "2025-07-10")]
alias bot_fst := extent_bot
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_bot | null |
intent_bot : (⊥ : Concept α β r).intent = univ :=
rfl
@[deprecated (since := "2025-07-10")]
alias bot_snd := intent_bot
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_bot | null |
extent_sup (c d : Concept α β r) : (c ⊔ d).extent = lowerPolar r (c.intent ∩ d.intent) :=
rfl
@[deprecated (since := "2025-07-10")]
alias sup_fst := extent_top
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_sup | null |
intent_sup (c d : Concept α β r) : (c ⊔ d).intent = c.intent ∩ d.intent :=
rfl
@[deprecated (since := "2025-07-10")]
alias sup_snd := intent_sup
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_sup | null |
extent_inf (c d : Concept α β r) : (c ⊓ d).extent = c.extent ∩ d.extent :=
rfl
@[deprecated (since := "2025-07-10")]
alias inf_fst := extent_inf
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_inf | null |
intent_inf (c d : Concept α β r) : (c ⊓ d).intent = upperPolar r (c.extent ∩ d.extent) :=
rfl
@[deprecated (since := "2025-07-10")]
alias inf_snd := intent_inf
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_inf | null |
extent_sSup (S : Set (Concept α β r)) :
(sSup S).extent = lowerPolar r (⋂ c ∈ S, intent c) :=
rfl
@[deprecated (since := "2025-07-10")]
alias sSup_fst := extent_sSup
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_sSup | null |
intent_sSup (S : Set (Concept α β r)) : (sSup S).intent = ⋂ c ∈ S, intent c :=
rfl
@[deprecated (since := "2025-07-10")]
alias sSup_snd := intent_sSup
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_sSup | null |
extent_sInf (S : Set (Concept α β r)) : (sInf S).extent = ⋂ c ∈ S, extent c :=
rfl
@[deprecated (since := "2025-07-10")]
alias sInf_fst := extent_sInf
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | extent_sInf | null |
intent_sInf (S : Set (Concept α β r)) :
(sInf S).intent = upperPolar r (⋂ c ∈ S, extent c) :=
rfl
@[deprecated (since := "2025-07-10")]
alias sInf_snd := intent_sInf | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | intent_sInf | null |
@[simps]
swap (c : Concept α β r) : Concept β α (swap r) :=
⟨c.intent, c.extent, c.lowerPolar_intent, c.upperPolar_extent⟩
@[simp] | def | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | swap | Swap the sets of a concept to make it a concept of the dual context. |
swap_swap (c : Concept α β r) : c.swap.swap = c :=
ext rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | swap_swap | null |
swap_le_swap_iff : c.swap ≤ d.swap ↔ d ≤ c :=
intent_subset_intent_iff
@[simp] | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | swap_le_swap_iff | null |
swap_lt_swap_iff : c.swap < d.swap ↔ d < c :=
intent_ssubset_intent_iff | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | swap_lt_swap_iff | null |
@[simps]
swapEquiv : (Concept α β r)ᵒᵈ ≃o Concept β α (Function.swap r) where
toFun := swap ∘ ofDual
invFun := toDual ∘ swap
left_inv := swap_swap
right_inv := swap_swap
map_rel_iff' := swap_le_swap_iff | def | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | swapEquiv | The dual of a concept lattice is isomorphic to the concept lattice of the dual context. |
OrderTop.copy {h : LE α} {h' : LE α} (c : @OrderTop α h')
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(le_eq : ∀ x y : α, (@LE.le α h) x y ↔ x ≤ y) : @OrderTop α h :=
@OrderTop.mk α h { top := top } fun _ ↦ by simp [eq_top, le_eq] | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | OrderTop.copy | A function to create a provable equal copy of a top order
with possibly different definitional equalities. |
OrderBot.copy {h : LE α} {h' : LE α} (c : @OrderBot α h')
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(le_eq : ∀ x y : α, (@LE.le α h) x y ↔ x ≤ y) : @OrderBot α h :=
@OrderBot.mk α h { bot := bot } fun _ ↦ by simp [eq_bot, le_eq] | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | OrderBot.copy | A function to create a provable equal copy of a bottom order
with possibly different definitional equalities. |
BoundedOrder.copy {h : LE α} {h' : LE α} (c : @BoundedOrder α h')
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(le_eq : ∀ x y : α, (@LE.le α h) x y ↔ x ≤ y) : @BoundedOrder α h :=
@BoundedOrder.mk α h (@OrderTop.mk α h { top := top }... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | BoundedOrder.copy | A function to create a provable equal copy of a bounded order
with possibly different definitional equalities. |
Lattice.copy (c : Lattice α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf : α → α → α) (eq_inf : inf = (by infer_instance : Min α).min) : Lattice α where
le := le
sup := sup
inf := inf
lt := fun a b ↦ le a b ... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | Lattice.copy | A function to create a provable equal copy of a lattice
with possibly different definitional equalities. |
DistribLattice.copy (c : DistribLattice α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf : α → α → α) (eq_inf : inf = (by infer_instance : Min α).min) : DistribLattice α where
toLattice := Lattice.copy (@DistribLatt... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | DistribLattice.copy | A function to create a provable equal copy of a distributive lattice
with possibly different definitional equalities. |
GeneralizedHeytingAlgebra.copy (c : GeneralizedHeytingAlgebra α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf : α → α → α) (eq_inf : inf = (by infer_inst... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | GeneralizedHeytingAlgebra.copy | A function to create a provable equal copy of a generalised heyting algebra
with possibly different definitional equalities. |
GeneralizedCoheytingAlgebra.copy (c : GeneralizedCoheytingAlgebra α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf : α → α → α) (eq_inf : inf = (by infer_... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | GeneralizedCoheytingAlgebra.copy | A function to create a provable equal copy of a generalised co-Heyting algebra
with possibly different definitional equalities. |
HeytingAlgebra.copy (c : HeytingAlgebra α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf : ... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | HeytingAlgebra.copy | A function to create a provable equal copy of a heyting algebra
with possibly different definitional equalities. |
CoheytingAlgebra.copy (c : CoheytingAlgebra α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(in... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | CoheytingAlgebra.copy | A function to create a provable equal copy of a co-Heyting algebra
with possibly different definitional equalities. |
BiheytingAlgebra.copy (c : BiheytingAlgebra α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(in... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | BiheytingAlgebra.copy | A function to create a provable equal copy of a bi-Heyting algebra
with possibly different definitional equalities. |
CompleteLattice.copy (c : CompleteLattice α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf ... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | CompleteLattice.copy | A function to create a provable equal copy of a complete lattice
with possibly different definitional equalities. |
Frame.copy (c : Frame α) (le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf : α → α → α) (eq_inf : i... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | Frame.copy | A function to create a provable equal copy of a frame with possibly different definitional
equalities. |
Coframe.copy (c : Coframe α) (le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf : α → α → α) (eq_inf... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | Coframe.copy | A function to create a provable equal copy of a coframe with possibly different definitional
equalities. |
CompleteDistribLattice.copy (c : CompleteDistribLattice α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(top : α) (eq_top : top = (by infer_instance : Top α).top)
(bot : α) (eq_bot : bot = (by infer_instance : Bot α).bot)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | CompleteDistribLattice.copy | A function to create a provable equal copy of a complete distributive lattice
with possibly different definitional equalities. |
ConditionallyCompleteLattice.copy (c : ConditionallyCompleteLattice α)
(le : α → α → Prop) (eq_le : le = (by infer_instance : LE α).le)
(sup : α → α → α) (eq_sup : sup = (by infer_instance : Max α).max)
(inf : α → α → α) (eq_inf : inf = (by infer_instance : Min α).min)
(sSup : Set α → α) (eq_sSup : sSup... | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic"
] | Mathlib/Order/Copy.lean | ConditionallyCompleteLattice.copy | A function to create a provable equal copy of a conditionally complete lattice
with possibly different definitional equalities. |
exists_between_finsets [DenselyOrdered α] [NoMinOrder α]
[NoMaxOrder α] [nonem : Nonempty α] (lo hi : Finset α) (lo_lt_hi : ∀ x ∈ lo, ∀ y ∈ hi, x < y) :
∃ m : α, (∀ x ∈ lo, x < m) ∧ ∀ y ∈ hi, m < y :=
if nlo : lo.Nonempty then
if nhi : hi.Nonempty then
Exists.elim
(exists_between (lo_lt_hi... | theorem | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | exists_between_finsets | Suppose `α` is a nonempty dense linear order without endpoints, and
suppose `lo`, `hi`, are finite subsets with all of `lo` strictly before `hi`.
Then there is an element of `α` strictly between `lo` and `hi`. |
exists_orderEmbedding_insert [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β]
[nonem : Nonempty β] (S : Finset α) (f : S ↪o β) (a : α) :
∃ (g : (insert a S : Finset α) ↪o β),
g ∘ (Set.inclusion ((S.subset_insert a) : ↑S ⊆ ↑(insert a S))) = f := by
let Slt := {x ∈ S.attach | x.val < a}.image f
let Sgt ... | lemma | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | exists_orderEmbedding_insert | null |
PartialIso : Type _ :=
{ f : Finset (α × β) //
∀ p ∈ f, ∀ q ∈ f,
cmp (Prod.fst p) (Prod.fst q) = cmp (Prod.snd p) (Prod.snd q) } | def | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | PartialIso | The type of partial order isomorphisms between `α` and `β` defined on finite subsets.
A partial order isomorphism is encoded as a finite subset of `α × β`, consisting
of pairs which should be identified. |
exists_across [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β]
(f : PartialIso α β) (a : α) :
∃ b : β, ∀ p ∈ f.val, cmp (Prod.fst p) a = cmp (Prod.snd p) b := by
by_cases h : ∃ b, (a, b) ∈ f.val
· obtain ⟨b, hb⟩ := h
exact ⟨b, fun p hp ↦ f.prop _ hp _ hb⟩
have :
∀ x ∈ {p ∈ f.val | p.f... | theorem | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | exists_across | For each `a`, we can find a `b` in the codomain, such that `a`'s relation to
the domain of `f` is `b`'s relation to the image of `f`.
Thus, if `a` is not already in `f`, then we can extend `f` by sending `a` to `b`. |
protected comm : PartialIso α β → PartialIso β α :=
Subtype.map (Finset.image (Equiv.prodComm _ _)) fun f hf p hp q hq ↦
Eq.symm <|
hf ((Equiv.prodComm α β).symm p)
(by
rw [← Finset.mem_coe, Finset.coe_image, Equiv.image_eq_preimage] at hp
rwa [← Finset.mem_coe])
((Equiv.... | def | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | comm | A partial isomorphism between `α` and `β` is also a partial isomorphism between `β` and `α`. |
definedAtLeft [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β] (a : α) :
Cofinal (PartialIso α β) where
carrier := {f | ∃ b : β, (a, b) ∈ f.val}
isCofinal f := by
obtain ⟨b, a_b⟩ := exists_across f a
refine
⟨⟨insert (a, b) f.val, fun p hp q hq ↦ ?_⟩, ⟨b, Finset.mem_insert_self _ _⟩,
... | def | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | definedAtLeft | The set of partial isomorphisms defined at `a : α`, together with a proof that any
partial isomorphism can be extended to one defined at `a`. |
definedAtRight [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α] [Nonempty α] (b : β) :
Cofinal (PartialIso α β) where
carrier := {f | ∃ a, (a, b) ∈ f.val}
isCofinal f := by
rcases (definedAtLeft α b).isCofinal f.comm with ⟨f', ⟨a, ha⟩, hl⟩
refine ⟨f'.comm, ⟨a, ?_⟩, ?_⟩
· change (a, b) ∈ f'.val.image... | def | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | definedAtRight | The set of partial isomorphisms defined at `b : β`, together with a proof that any
partial isomorphism can be extended to include `b`. We prove this by symmetry. |
funOfIdeal [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β] (a : α)
(I : Ideal (PartialIso α β)) :
(∃ f, f ∈ definedAtLeft β a ∧ f ∈ I) → { b // ∃ f ∈ I, (a, b) ∈ Subtype.val f } :=
Classical.indefiniteDescription _ ∘ fun ⟨f, ⟨b, hb⟩, hf⟩ ↦ ⟨b, f, hf, hb⟩ | def | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | funOfIdeal | Given an ideal which intersects `definedAtLeft β a`, pick `b : β` such that
some partial function in the ideal maps `a` to `b`. |
invOfIdeal [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α] [Nonempty α] (b : β)
(I : Ideal (PartialIso α β)) :
(∃ f, f ∈ definedAtRight α b ∧ f ∈ I) → { a // ∃ f ∈ I, (a, b) ∈ Subtype.val f } :=
Classical.indefiniteDescription _ ∘ fun ⟨f, ⟨a, ha⟩, hf⟩ ↦ ⟨a, f, hf, ha⟩ | def | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | invOfIdeal | Given an ideal which intersects `definedAtRight α b`, pick `a : α` such that
some partial function in the ideal maps `a` to `b`. |
embedding_from_countable_to_dense [Countable α] [DenselyOrdered β] [Nontrivial β] :
Nonempty (α ↪o β) := by
cases nonempty_encodable α
rcases exists_pair_lt β with ⟨x, y, hxy⟩
obtain ⟨a, ha⟩ := exists_between hxy
haveI : Nonempty (Set.Ioo x y) := ⟨⟨a, ha⟩⟩
let our_ideal : Ideal (PartialIso α _) :=
ide... | theorem | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | embedding_from_countable_to_dense | Any countable linear order embeds in any nontrivial dense linear order. |
iso_of_countable_dense [Countable α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α]
[Nonempty α] [Countable β] [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β] :
Nonempty (α ≃o β) := by
cases nonempty_encodable α
cases nonempty_encodable β
let to_cofinal : α ⊕ β → Cofinal (PartialIso α β) :... | theorem | Order | [
"Mathlib.Order.Ideal",
"Mathlib.Data.Finset.Max"
] | Mathlib/Order/CountableDenseLinearOrder.lean | iso_of_countable_dense | Any two countable dense, nonempty linear orders without endpoints are order isomorphic. This is
also known as **Cantor's isomorphism theorem**. |
WCovBy.le (h : a ⩿ b) : a ≤ b :=
h.1 | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.le | null |
WCovBy.refl (a : α) : a ⩿ a :=
⟨le_rfl, fun _ hc => hc.not_gt⟩
@[simp] lemma WCovBy.rfl : a ⩿ a := WCovBy.refl a | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.refl | null |
protected Eq.wcovBy (h : a = b) : a ⩿ b :=
h ▸ WCovBy.rfl | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | Eq.wcovBy | null |
wcovBy_of_le_of_le (h1 : a ≤ b) (h2 : b ≤ a) : a ⩿ b :=
⟨h1, fun _ hac hcb => (hac.trans hcb).not_ge h2⟩
alias LE.le.wcovBy_of_le := wcovBy_of_le_of_le | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_of_le_of_le | null |
AntisymmRel.wcovBy (h : AntisymmRel (· ≤ ·) a b) : a ⩿ b :=
wcovBy_of_le_of_le h.1 h.2 | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | AntisymmRel.wcovBy | null |
WCovBy.wcovBy_iff_le (hab : a ⩿ b) : b ⩿ a ↔ b ≤ a :=
⟨fun h => h.le, fun h => h.wcovBy_of_le hab.le⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.wcovBy_iff_le | null |
wcovBy_of_eq_or_eq (hab : a ≤ b) (h : ∀ c, a ≤ c → c ≤ b → c = a ∨ c = b) : a ⩿ b :=
⟨hab, fun c ha hb => (h c ha.le hb.le).elim ha.ne' hb.ne⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_of_eq_or_eq | null |
AntisymmRel.trans_wcovBy (hab : AntisymmRel (· ≤ ·) a b) (hbc : b ⩿ c) : a ⩿ c :=
⟨hab.1.trans hbc.le, fun _ had hdc => hbc.2 (hab.2.trans_lt had) hdc⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | AntisymmRel.trans_wcovBy | null |
wcovBy_congr_left (hab : AntisymmRel (· ≤ ·) a b) : a ⩿ c ↔ b ⩿ c :=
⟨hab.symm.trans_wcovBy, hab.trans_wcovBy⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_congr_left | null |
WCovBy.trans_antisymm_rel (hab : a ⩿ b) (hbc : AntisymmRel (· ≤ ·) b c) : a ⩿ c :=
⟨hab.le.trans hbc.1, fun _ had hdc => hab.2 had <| hdc.trans_le hbc.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.trans_antisymm_rel | null |
wcovBy_congr_right (hab : AntisymmRel (· ≤ ·) a b) : c ⩿ a ↔ c ⩿ b :=
⟨fun h => h.trans_antisymm_rel hab, fun h => h.trans_antisymm_rel hab.symm⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_congr_right | null |
not_wcovBy_iff (h : a ≤ b) : ¬a ⩿ b ↔ ∃ c, a < c ∧ c < b := by
simp_rw [WCovBy, h, true_and, not_forall, exists_prop, not_not] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | not_wcovBy_iff | If `a ≤ b`, then `b` does not cover `a` iff there's an element in between. |
WCovBy.isRefl : IsRefl α (· ⩿ ·) :=
⟨WCovBy.refl⟩ | instance | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.isRefl | null |
WCovBy.Ioo_eq (h : a ⩿ b) : Ioo a b = ∅ :=
eq_empty_iff_forall_notMem.2 fun _ hx => h.2 hx.1 hx.2 | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.Ioo_eq | null |
wcovBy_iff_Ioo_eq : a ⩿ b ↔ a ≤ b ∧ Ioo a b = ∅ :=
and_congr_right' <| by simp [eq_empty_iff_forall_notMem] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_iff_Ioo_eq | null |
WCovBy.of_le_of_le (hac : a ⩿ c) (hab : a ≤ b) (hbc : b ≤ c) : b ⩿ c :=
⟨hbc, fun _x hbx hxc ↦ hac.2 (hab.trans_lt hbx) hxc⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.of_le_of_le | null |
WCovBy.of_le_of_le' (hac : a ⩿ c) (hab : a ≤ b) (hbc : b ≤ c) : a ⩿ b :=
⟨hab, fun _x hax hxb ↦ hac.2 hax <| hxb.trans_le hbc⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.of_le_of_le' | null |
WCovBy.of_image (f : α ↪o β) (h : f a ⩿ f b) : a ⩿ b :=
⟨f.le_iff_le.mp h.le, fun _ hac hcb => h.2 (f.lt_iff_lt.mpr hac) (f.lt_iff_lt.mpr hcb)⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.of_image | null |
WCovBy.image (f : α ↪o β) (hab : a ⩿ b) (h : (range f).OrdConnected) : f a ⩿ f b := by
refine ⟨f.monotone hab.le, fun c ha hb => ?_⟩
obtain ⟨c, rfl⟩ := h.out (mem_range_self _) (mem_range_self _) ⟨ha.le, hb.le⟩
rw [f.lt_iff_lt] at ha hb
exact hab.2 ha hb | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.image | null |
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