File: Mathlib/Data/Finset/Interval.lean

full_name: Finset.instLocallyFiniteOrder
code: instance instLocallyFiniteOrder : LocallyFiniteOrder (Finset α) 
goal: LocallyFiniteOrder (Finset α)

full_name: Finset.Icc_eq_filter_powerset
code: theorem Icc_eq_filter_powerset : Icc s t = t.powerset.filter (s ⊆ ·) 
goal: Finset.Icc s t = Finset.filter (fun x => s ⊆ x) t.powerset

full_name: Finset.Ico_eq_filter_ssubsets
code: theorem Ico_eq_filter_ssubsets : Ico s t = t.ssubsets.filter (s ⊆ ·) 
goal: Finset.Ico s t = Finset.filter (fun x => s ⊆ x) t.ssubsets

full_name: Finset.Ioc_eq_filter_powerset
code: theorem Ioc_eq_filter_powerset : Ioc s t = t.powerset.filter (s ⊂ ·) 
goal: Finset.Ioc s t = Finset.filter (fun x => s ⊂ x) t.powerset

full_name: Finset.Ioo_eq_filter_ssubsets
code: theorem Ioo_eq_filter_ssubsets : Ioo s t = t.ssubsets.filter (s ⊂ ·) 
goal: Finset.Ioo s t = Finset.filter (fun x => s ⊂ x) t.ssubsets

full_name: Finset.Iic_eq_powerset
code: theorem Iic_eq_powerset : Iic s = s.powerset 
goal: Finset.Iic s = s.powerset

full_name: Finset.Iio_eq_ssubsets
code: theorem Iio_eq_ssubsets : Iio s = s.ssubsets 
goal: Finset.Iio s = s.ssubsets

full_name: Finset.Icc_eq_image_powerset
code: theorem Icc_eq_image_powerset (h : s ⊆ t) : Icc s t = (t \ s).powerset.image (s ∪ ·) 
goal: Finset.Icc s t = Finset.image (fun x => s ∪ x) (t \ s).powerset

full_name: Finset.Ico_eq_image_ssubsets
code: theorem Ico_eq_image_ssubsets (h : s ⊆ t) : Ico s t = (t \ s).ssubsets.image (s ∪ ·) 
goal: Finset.Ico s t = Finset.image (fun x => s ∪ x) (t \ s).ssubsets

full_name: Finset.card_Icc_finset
code: /-- Cardinality of a non-empty `Icc` of finsets. -/
theorem card_Icc_finset (h : s ⊆ t) : (Icc s t).card = 2 ^ (t.card - s.card) 
goal: (Finset.Icc s t).card = 2 ^ (t.card - s.card)

full_name: Finset.card_Ico_finset
code: /-- Cardinality of an `Ico` of finsets. -/
theorem card_Ico_finset (h : s ⊆ t) : (Ico s t).card = 2 ^ (t.card - s.card) - 1 
goal: (Finset.Ico s t).card = 2 ^ (t.card - s.card) - 1

full_name: Finset.card_Ioc_finset
code: /-- Cardinality of an `Ioc` of finsets. -/
theorem card_Ioc_finset (h : s ⊆ t) : (Ioc s t).card = 2 ^ (t.card - s.card) - 1 
goal: (Finset.Ioc s t).card = 2 ^ (t.card - s.card) - 1

full_name: Finset.card_Ioo_finset
code: /-- Cardinality of an `Ioo` of finsets. -/
theorem card_Ioo_finset (h : s ⊆ t) : (Ioo s t).card = 2 ^ (t.card - s.card) - 2 
goal: (Finset.Ioo s t).card = 2 ^ (t.card - s.card) - 2

full_name: Finset.card_Iic_finset
code: /-- Cardinality of an `Iic` of finsets. -/
theorem card_Iic_finset : (Iic s).card = 2 ^ s.card 
goal: (Finset.Iic s).card = 2 ^ s.card

full_name: Finset.card_Iio_finset
code: /-- Cardinality of an `Iio` of finsets. -/
theorem card_Iio_finset : (Iio s).card = 2 ^ s.card - 1 
goal: (Finset.Iio s).card = 2 ^ s.card - 1

full_name: Finset.monotone_iff_forall_le_cons
code: /-- A function `f` from `Finset α` is monotone if and only if `f s ≤ f (cons a s ha)` for all `s`
and `a ∉ s`. -/
lemma monotone_iff_forall_le_cons : Monotone f ↔ ∀ s, ∀ ⦃a⦄ (ha), f s ≤ f (cons a s ha) 
goal: Monotone f ↔ ∀ (s : Finset α) ⦃a : α⦄ (ha : a ∉ s), f s ≤ f (Finset.cons a s ha)

full_name: Finset.antitone_iff_forall_cons_le
code: /-- A function `f` from `Finset α` is antitone if and only if `f (cons a s ha) ≤ f s` for all
`s` and `a ∉ s`. -/
lemma antitone_iff_forall_cons_le : Antitone f ↔ ∀ s ⦃a⦄ ha, f (cons a s ha) ≤ f s 
goal: Antitone f ↔ ∀ (s : Finset α) ⦃a : α⦄ (ha : a ∉ s), f (Finset.cons a s ha) ≤ f s

full_name: Finset.strictMono_iff_forall_lt_cons
code: /-- A function `f` from `Finset α` is strictly monotone if and only if `f s < f (cons a s ha)` for
all `s` and `a ∉ s`. -/
lemma strictMono_iff_forall_lt_cons : StrictMono f ↔ ∀ s ⦃a⦄ ha, f s < f (cons a s ha) 
goal: StrictMono f ↔ ∀ (s : Finset α) ⦃a : α⦄ (ha : a ∉ s), f s < f (Finset.cons a s ha)

full_name: Finset.strictAnti_iff_forall_cons_lt
code: /-- A function `f` from `Finset α` is strictly antitone if and only if `f (cons a s ha) < f s` for
all `s` and `a ∉ s`. -/
lemma strictAnti_iff_forall_cons_lt : StrictAnti f ↔ ∀ s ⦃a⦄ ha, f (cons a s ha) < f s 
goal: StrictAnti f ↔ ∀ (s : Finset α) ⦃a : α⦄ (ha : a ∉ s), f (Finset.cons a s ha) < f s

full_name: Finset.monotone_iff_forall_le_insert
code: /-- A function `f` from `Finset α` is monotone if and only if `f s ≤ f (insert a s)` for all `s` and
`a ∉ s`. -/
lemma monotone_iff_forall_le_insert : Monotone f ↔ ∀ s ⦃a⦄, a ∉ s → f s ≤ f (insert a s) 
goal: Monotone f ↔ ∀ (s : Finset α) ⦃a : α⦄, a ∉ s → f s ≤ f (insert a s)

full_name: Finset.antitone_iff_forall_insert_le
code: /-- A function `f` from `Finset α` is antitone if and only if `f (insert a s) ≤ f s` for all
`s` and `a ∉ s`. -/
lemma antitone_iff_forall_insert_le : Antitone f ↔ ∀ s ⦃a⦄, a ∉ s → f (insert a s) ≤ f s 
goal: Antitone f ↔ ∀ (s : Finset α) ⦃a : α⦄, a ∉ s → f (insert a s) ≤ f s

full_name: Finset.strictMono_iff_forall_lt_insert
code: /-- A function `f` from `Finset α` is strictly monotone if and only if `f s < f (insert a s)` for
all `s` and `a ∉ s`. -/
lemma strictMono_iff_forall_lt_insert : StrictMono f ↔ ∀ s ⦃a⦄, a ∉ s → f s < f (insert a s) 
goal: StrictMono f ↔ ∀ (s : Finset α) ⦃a : α⦄, a ∉ s → f s < f (insert a s)

full_name: Finset.strictAnti_iff_forall_lt_insert
code: /-- A function `f` from `Finset α` is strictly antitone if and only if `f (insert a s) < f s` for
all `s` and `a ∉ s`. -/
lemma strictAnti_iff_forall_lt_insert : StrictAnti f ↔ ∀ s ⦃a⦄, a ∉ s → f (insert a s) < f s 
goal: StrictAnti f ↔ ∀ (s : Finset α) ⦃a : α⦄, a ∉ s → f (insert a s) < f s