state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case h.right.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
s : ℕ → Set α
⊢ ⨅ i, 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m) ≤ ⨅ i, 𝓟 (s i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rw [le_iInf_iff] | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) := by
use fun n => ⋂ m ≤ n, s m; constructor
· exact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl
apply le_antisymm <;> | Mathlib.Order.Filter.Bases.1027_0.YdUKAcRZtFgMABD | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) | Mathlib_Order_Filter_Bases |
case h.right.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
s : ℕ → Set α
⊢ ∀ (i : ℕ), ⨅ i, 𝓟 (s i) ≤ 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | intro i | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) := by
use fun n => ⋂ m ≤ n, s m; constructor
· exact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl
apply le_antisymm <;> rw [le_iInf_iff] <;> | Mathlib.Order.Filter.Bases.1027_0.YdUKAcRZtFgMABD | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) | Mathlib_Order_Filter_Bases |
case h.right.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
s : ℕ → Set α
⊢ ∀ (i : ℕ), ⨅ i, 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m) ≤ 𝓟 (s i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | intro i | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) := by
use fun n => ⋂ m ≤ n, s m; constructor
· exact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl
apply le_antisymm <;> rw [le_iInf_iff] <;> | Mathlib.Order.Filter.Bases.1027_0.YdUKAcRZtFgMABD | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) | Mathlib_Order_Filter_Bases |
case h.right.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
s : ℕ → Set α
i : ℕ
⊢ ⨅ i, 𝓟 (s i) ≤ 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rw [le_principal_iff] | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) := by
use fun n => ⋂ m ≤ n, s m; constructor
· exact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl
apply le_antisymm <;> rw [le_iInf_iff] <;> intro i
· | Mathlib.Order.Filter.Bases.1027_0.YdUKAcRZtFgMABD | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) | Mathlib_Order_Filter_Bases |
case h.right.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
s : ℕ → Set α
i : ℕ
⊢ ⋂ m, ⋂ (_ : m ≤ i), s m ∈ ⨅ i, 𝓟 (s i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | refine' (biInter_mem (finite_le_nat _)).2 fun j _ => _ | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) := by
use fun n => ⋂ m ≤ n, s m; constructor
· exact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl
apply le_antisymm <;> rw [le_iInf_iff] <;> intro i
· rw [le_principal_iff]
... | Mathlib.Order.Filter.Bases.1027_0.YdUKAcRZtFgMABD | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) | Mathlib_Order_Filter_Bases |
case h.right.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
s : ℕ → Set α
i j : ℕ
x✝ : j ∈ {i_1 | i_1 ≤ i}
⊢ s j ∈ ⨅ i, 𝓟 (s i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact mem_iInf_of_mem j (mem_principal_self _) | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) := by
use fun n => ⋂ m ≤ n, s m; constructor
· exact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl
apply le_antisymm <;> rw [le_iInf_iff] <;> intro i
· rw [le_principal_iff]
... | Mathlib.Order.Filter.Bases.1027_0.YdUKAcRZtFgMABD | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) | Mathlib_Order_Filter_Bases |
case h.right.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
s : ℕ → Set α
i : ℕ
⊢ ⨅ i, 𝓟 (⋂ m, ⋂ (_ : m ≤ i), s m) ≤ 𝓟 (s i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | refine iInf_le_of_le i (principal_mono.2 <| iInter₂_subset i ?_) | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) := by
use fun n => ⋂ m ≤ n, s m; constructor
· exact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl
apply le_antisymm <;> rw [le_iInf_iff] <;> intro i
· rw [le_principal_iff]
... | Mathlib.Order.Filter.Bases.1027_0.YdUKAcRZtFgMABD | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) | Mathlib_Order_Filter_Bases |
case h.right.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
s : ℕ → Set α
i : ℕ
⊢ i ≤ i | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rfl | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) := by
use fun n => ⋂ m ≤ n, s m; constructor
· exact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl
apply le_antisymm <;> rw [le_iInf_iff] <;> intro i
· rw [le_principal_iff]
... | Mathlib.Order.Filter.Bases.1027_0.YdUKAcRZtFgMABD | theorem antitone_seq_of_seq (s : ℕ → Set α) :
∃ t : ℕ → Set α, Antitone t ∧ ⨅ i, 𝓟 (s i) = ⨅ i, 𝓟 (t i) | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
inst✝ : CompleteLattice α
B : Set ι
Bcbl : Set.Countable B
f : ι → α
i₀ : ι
h : f i₀ = ⊤
⊢ ∃ x, ⨅ t ∈ B, f t = ⨅ i, f (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rcases B.eq_empty_or_nonempty with hB | Bnonempty | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) := by
| Mathlib.Order.Filter.Bases.1045_0.YdUKAcRZtFgMABD | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) | Mathlib_Order_Filter_Bases |
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
inst✝ : CompleteLattice α
B : Set ι
Bcbl : Set.Countable B
f : ι → α
i₀ : ι
h : f i₀ = ⊤
hB : B = ∅
⊢ ∃ x, ⨅ t ∈ B, f t = ⨅ i, f (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rw [hB, iInf_emptyset] | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) := by
rcases B.eq_empty_or_nonempty with hB | Bnonempty
· | Mathlib.Order.Filter.Bases.1045_0.YdUKAcRZtFgMABD | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) | Mathlib_Order_Filter_Bases |
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
inst✝ : CompleteLattice α
B : Set ι
Bcbl : Set.Countable B
f : ι → α
i₀ : ι
h : f i₀ = ⊤
hB : B = ∅
⊢ ∃ x, ⊤ = ⨅ i, f (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | use fun _ => i₀ | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) := by
rcases B.eq_empty_or_nonempty with hB | Bnonempty
· rw [hB, iInf_emptyset]
| Mathlib.Order.Filter.Bases.1045_0.YdUKAcRZtFgMABD | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) | Mathlib_Order_Filter_Bases |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
inst✝ : CompleteLattice α
B : Set ι
Bcbl : Set.Countable B
f : ι → α
i₀ : ι
h : f i₀ = ⊤
hB : B = ∅
⊢ ⊤ = ⨅ i, f i₀ | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | simp [h] | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) := by
rcases B.eq_empty_or_nonempty with hB | Bnonempty
· rw [hB, iInf_emptyset]
use fun _ => i₀
| Mathlib.Order.Filter.Bases.1045_0.YdUKAcRZtFgMABD | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) | Mathlib_Order_Filter_Bases |
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
inst✝ : CompleteLattice α
B : Set ι
Bcbl : Set.Countable B
f : ι → α
i₀ : ι
h : f i₀ = ⊤
Bnonempty : Set.Nonempty B
⊢ ∃ x, ⨅ t ∈ B, f t = ⨅ i, f (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact countable_biInf_eq_iInf_seq Bcbl Bnonempty f | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) := by
rcases B.eq_empty_or_nonempty with hB | Bnonempty
· rw [hB, iInf_emptyset]
use fun _ => i₀
simp [h]
· | Mathlib.Order.Filter.Bases.1045_0.YdUKAcRZtFgMABD | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
inst✝ : Preorder ι
l : Filter α
s : ι → Set α
hs : HasAntitoneBasis l s
t : Set α
⊢ (∃ i, True ∧ s i ⊆ t) ↔ ∃ i, s i ⊆ t | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | simp only [exists_prop, true_and] | protected theorem HasAntitoneBasis.mem_iff [Preorder ι] {l : Filter α} {s : ι → Set α}
(hs : l.HasAntitoneBasis s) {t : Set α} : t ∈ l ↔ ∃ i, s i ⊆ t :=
hs.toHasBasis.mem_iff.trans <| by | Mathlib.Order.Filter.Bases.1061_0.YdUKAcRZtFgMABD | protected theorem HasAntitoneBasis.mem_iff [Preorder ι] {l : Filter α} {s : ι → Set α}
(hs : l.HasAntitoneBasis s) {t : Set α} : t ∈ l ↔ ∃ i, s i ⊆ t | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
⊢ ∃ x, (∀ (i : ℕ), p (x i)) ∧ HasAntitoneBasis f fun i => s (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | obtain ⟨x', hx'⟩ : ∃ x : ℕ → Set α, f = ⨅ i, 𝓟 (x i) := by
rcases h with ⟨s, hsc, rfl⟩
rw [generate_eq_biInf]
exact countable_biInf_principal_eq_seq_iInf hsc | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
⊢ ∃ x, f = ⨅ i, 𝓟 (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rcases h with ⟨s, hsc, rfl⟩ | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case mk.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
p : ι' → Prop
s✝ : ι' → Set α
s : Set (Set α)
hsc : Set.Countable s
hs : HasBasis (generate s) p s✝
⊢ ∃ x, generate s = ⨅ i, 𝓟 (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rw [generate_eq_biInf] | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case mk.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
p : ι' → Prop
s✝ : ι' → Set α
s : Set (Set α)
hsc : Set.Countable s
hs : HasBasis (generate s) p s✝
⊢ ∃ x, ⨅ s_1 ∈ s, 𝓟 s_1 = ⨅ i, 𝓟 (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact countable_biInf_principal_eq_seq_iInf hsc | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
⊢ ∃ x, (∀ (i : ℕ), p (x i)) ∧ HasAntitoneBasis f fun i => s (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | have : ∀ i, x' i ∈ f := fun i => hx'.symm ▸ (iInf_le (fun i => 𝓟 (x' i)) i) (mem_principal_self _) | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
⊢ ∃ x, (∀ (i : ℕ), p (x i)) ∧ HasAntitoneBasis f fun i => s (x i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | let x : ℕ → { i : ι' // p i } := fun n =>
Nat.recOn n (hs.index _ <| this 0) fun n xn =>
hs.index _ <| inter_mem (this <| n + 1) (hs.mem_of_mem xn.2) | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | have x_anti : Antitone fun i => s (x i).1 :=
antitone_nat_of_succ_le fun i => (hs.set_index_subset _).trans (inter_subset_right _ _) | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | have x_subset : ∀ i, s (x i).1 ⊆ x' i := by
rintro (_ | i)
exacts [hs.set_index_subset _, (hs.set_index_subset _).trans (inter_subset_left _ _)] | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun n xn => in... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rintro (_ | i) | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case zero
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun ... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exacts [hs.set_index_subset _, (hs.set_index_subset _).trans (inter_subset_left _ _)] | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | refine' ⟨fun i => (x i).1, fun i => (x i).2, _⟩ | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | have : (⨅ i, 𝓟 (s (x i).1)).HasAntitoneBasis fun i => s (x i).1 := .iInf_principal x_anti | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this✝ : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fu... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | convert this | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case h.e'_4
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this✝ : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) f... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact
le_antisymm (le_iInf fun i => le_principal_iff.2 <| by cases i <;> apply hs.set_index_mem)
(hx'.symm ▸
le_iInf fun i => le_principal_iff.2 <| this.1.mem_iff.2 ⟨i, trivial, x_subset i⟩) | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this✝ : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun n xn => i... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | cases i | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case zero
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this✝ : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | apply hs.set_index_mem | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
case succ
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
p : ι' → Prop
s : ι' → Set α
hs : HasBasis f p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this✝ : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } :=
fun n =>
Nat.recOn n (index hs (x' 0) (_ : x' 0 ∈ f)) fun... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | apply hs.set_index_mem | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib.Order.Filter.Bases.1077_0.YdUKAcRZtFgMABD | /-- If `f` is countably generated and `f.HasBasis p s`, then `f` admits a decreasing basis
enumerated by natural numbers such that all sets have the form `s i`. More precisely, there is a
sequence `i n` such that `p (i n)` for all `n` and `s (i n)` is a decreasing sequence of sets which
forms a basis of `f`-/
theorem H... | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
inst✝ : IsCountablyGenerated f
x : ℕ → Set α
hx : HasAntitoneBasis f x
⊢ ∀ {s : Set α}, s ∈ f ↔ ∃ i, x i ⊆ s | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | simp [hx.1.mem_iff] | theorem exists_antitone_seq (f : Filter α) [f.IsCountablyGenerated] :
∃ x : ℕ → Set α, Antitone x ∧ ∀ {s}, s ∈ f ↔ ∃ i, x i ⊆ s :=
let ⟨x, hx⟩ := f.exists_antitone_basis
⟨x, hx.antitone, by | Mathlib.Order.Filter.Bases.1113_0.YdUKAcRZtFgMABD | theorem exists_antitone_seq (f : Filter α) [f.IsCountablyGenerated] :
∃ x : ℕ → Set α, Antitone x ∧ ∀ {s}, s ∈ f ↔ ∃ i, x i ⊆ s | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f g : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : IsCountablyGenerated g
⊢ IsCountablyGenerated (f ⊓ g) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rcases f.exists_antitone_basis with ⟨s, hs⟩ | instance Inf.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊓ g) := by
| Mathlib.Order.Filter.Bases.1119_0.YdUKAcRZtFgMABD | instance Inf.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊓ g) | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f g : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : IsCountablyGenerated g
s : ℕ → Set α
hs : HasAntitoneBasis f s
⊢ IsCountablyGenerated (f ⊓ g) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rcases g.exists_antitone_basis with ⟨t, ht⟩ | instance Inf.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊓ g) := by
rcases f.exists_antitone_basis with ⟨s, hs⟩
| Mathlib.Order.Filter.Bases.1119_0.YdUKAcRZtFgMABD | instance Inf.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊓ g) | Mathlib_Order_Filter_Bases |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f g : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : IsCountablyGenerated g
s : ℕ → Set α
hs : HasAntitoneBasis f s
t : ℕ → Set α
ht : HasAntitoneBasis g t
⊢ IsCountablyGenerated (f ⊓ g) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact HasCountableBasis.isCountablyGenerated ⟨hs.1.inf ht.1, Set.to_countable _⟩ | instance Inf.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊓ g) := by
rcases f.exists_antitone_basis with ⟨s, hs⟩
rcases g.exists_antitone_basis with ⟨t, ht⟩
| Mathlib.Order.Filter.Bases.1119_0.YdUKAcRZtFgMABD | instance Inf.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊓ g) | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f g : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : IsCountablyGenerated g
⊢ IsCountablyGenerated (f ⊔ g) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rcases f.exists_antitone_basis with ⟨s, hs⟩ | instance Sup.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊔ g) := by
| Mathlib.Order.Filter.Bases.1138_0.YdUKAcRZtFgMABD | instance Sup.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊔ g) | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f g : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : IsCountablyGenerated g
s : ℕ → Set α
hs : HasAntitoneBasis f s
⊢ IsCountablyGenerated (f ⊔ g) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rcases g.exists_antitone_basis with ⟨t, ht⟩ | instance Sup.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊔ g) := by
rcases f.exists_antitone_basis with ⟨s, hs⟩
| Mathlib.Order.Filter.Bases.1138_0.YdUKAcRZtFgMABD | instance Sup.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊔ g) | Mathlib_Order_Filter_Bases |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f g : Filter α
inst✝¹ : IsCountablyGenerated f
inst✝ : IsCountablyGenerated g
s : ℕ → Set α
hs : HasAntitoneBasis f s
t : ℕ → Set α
ht : HasAntitoneBasis g t
⊢ IsCountablyGenerated (f ⊔ g) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact
HasCountableBasis.isCountablyGenerated ⟨hs.1.sup ht.1, Set.to_countable _⟩ | instance Sup.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊔ g) := by
rcases f.exists_antitone_basis with ⟨s, hs⟩
rcases g.exists_antitone_basis with ⟨t, ht⟩
| Mathlib.Order.Filter.Bases.1138_0.YdUKAcRZtFgMABD | instance Sup.isCountablyGenerated (f g : Filter α) [IsCountablyGenerated f]
[IsCountablyGenerated g] : IsCountablyGenerated (f ⊔ g) | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
inst✝ : Countable β
x : β → Set α
⊢ IsCountablyGenerated (⨅ i, 𝓟 (x i)) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | use range x, countable_range x | theorem isCountablyGenerated_seq [Countable β] (x : β → Set α) :
IsCountablyGenerated (⨅ i, 𝓟 (x i)) := by
| Mathlib.Order.Filter.Bases.1158_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_seq [Countable β] (x : β → Set α) :
IsCountablyGenerated (⨅ i, 𝓟 (x i)) | Mathlib_Order_Filter_Bases |
case right
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
inst✝ : Countable β
x : β → Set α
⊢ ⨅ i, 𝓟 (x i) = generate (range x) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rw [generate_eq_biInf, iInf_range] | theorem isCountablyGenerated_seq [Countable β] (x : β → Set α) :
IsCountablyGenerated (⨅ i, 𝓟 (x i)) := by
use range x, countable_range x
| Mathlib.Order.Filter.Bases.1158_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_seq [Countable β] (x : β → Set α) :
IsCountablyGenerated (⨅ i, 𝓟 (x i)) | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : ∃ x, f = ⨅ i, 𝓟 (x i)
⊢ IsCountablyGenerated f | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rcases h with ⟨x, rfl⟩ | theorem isCountablyGenerated_of_seq {f : Filter α} (h : ∃ x : ℕ → Set α, f = ⨅ i, 𝓟 (x i)) :
f.IsCountablyGenerated := by
| Mathlib.Order.Filter.Bases.1164_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_of_seq {f : Filter α} (h : ∃ x : ℕ → Set α, f = ⨅ i, 𝓟 (x i)) :
f.IsCountablyGenerated | Mathlib_Order_Filter_Bases |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
x : ℕ → Set α
⊢ IsCountablyGenerated (⨅ i, 𝓟 (x i)) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | apply isCountablyGenerated_seq | theorem isCountablyGenerated_of_seq {f : Filter α} (h : ∃ x : ℕ → Set α, f = ⨅ i, 𝓟 (x i)) :
f.IsCountablyGenerated := by
rcases h with ⟨x, rfl⟩
| Mathlib.Order.Filter.Bases.1164_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_of_seq {f : Filter α} (h : ∃ x : ℕ → Set α, f = ⨅ i, 𝓟 (x i)) :
f.IsCountablyGenerated | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
⊢ IsCountablyGenerated f ↔ ∃ x, HasAntitoneBasis f x | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | constructor | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x := by
| Mathlib.Order.Filter.Bases.1175_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x | Mathlib_Order_Filter_Bases |
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
⊢ IsCountablyGenerated f → ∃ x, HasAntitoneBasis f x | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | intro h | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x := by
constructor
· | Mathlib.Order.Filter.Bases.1175_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x | Mathlib_Order_Filter_Bases |
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
h : IsCountablyGenerated f
⊢ ∃ x, HasAntitoneBasis f x | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact f.exists_antitone_basis | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x := by
constructor
· intro h
| Mathlib.Order.Filter.Bases.1175_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x | Mathlib_Order_Filter_Bases |
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
⊢ (∃ x, HasAntitoneBasis f x) → IsCountablyGenerated f | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rintro ⟨x, h⟩ | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x := by
constructor
· intro h
exact f.exists_antitone_basis
· | Mathlib.Order.Filter.Bases.1175_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x | Mathlib_Order_Filter_Bases |
case mpr.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
x : ℕ → Set α
h : HasAntitoneBasis f x
⊢ IsCountablyGenerated f | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rw [h.1.eq_iInf] | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x := by
constructor
· intro h
exact f.exists_antitone_basis
· rintro ⟨x, h⟩
| Mathlib.Order.Filter.Bases.1175_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x | Mathlib_Order_Filter_Bases |
case mpr.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
f : Filter α
x : ℕ → Set α
h : HasAntitoneBasis f x
⊢ IsCountablyGenerated (⨅ i, 𝓟 (x i)) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact isCountablyGenerated_seq x | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x := by
constructor
· intro h
exact f.exists_antitone_basis
· rintro ⟨x, h⟩
rw [h.1.eq_iInf]
| Mathlib.Order.Filter.Bases.1175_0.YdUKAcRZtFgMABD | theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
a : α
⊢ IsCountablyGenerated (pure a) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rw [← principal_singleton] | @[instance]
theorem isCountablyGenerated_pure (a : α) : IsCountablyGenerated (pure a) := by
| Mathlib.Order.Filter.Bases.1190_0.YdUKAcRZtFgMABD | @[instance]
theorem isCountablyGenerated_pure (a : α) : IsCountablyGenerated (pure a) | Mathlib_Order_Filter_Bases |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Sort u_5
a : α
⊢ IsCountablyGenerated (𝓟 {a}) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact isCountablyGenerated_principal _ | @[instance]
theorem isCountablyGenerated_pure (a : α) : IsCountablyGenerated (pure a) := by
rw [← principal_singleton]
| Mathlib.Order.Filter.Bases.1190_0.YdUKAcRZtFgMABD | @[instance]
theorem isCountablyGenerated_pure (a : α) : IsCountablyGenerated (pure a) | Mathlib_Order_Filter_Bases |
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι' : Sort u_5
ι : Sort u
α : Type v
inst✝¹ : Countable ι
f : ι → Filter α
inst✝ : ∀ (i : ι), IsCountablyGenerated (f i)
⊢ IsCountablyGenerated (⨅ i, f i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | choose s hs using fun i => exists_antitone_basis (f i) | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) := by
| Mathlib.Order.Filter.Bases.1209_0.YdUKAcRZtFgMABD | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) | Mathlib_Order_Filter_Bases |
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι' : Sort u_5
ι : Sort u
α : Type v
inst✝¹ : Countable ι
f : ι → Filter α
inst✝ : ∀ (i : ι), IsCountablyGenerated (f i)
s : ι → ℕ → Set α
hs : ∀ (i : ι), HasAntitoneBasis (f i) (s i)
⊢ IsCountablyGenerated (⨅ i, f i) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rw [← PLift.down_surjective.iInf_comp] | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) := by
choose s hs using fun i => exists_antitone_basis (f i)
| Mathlib.Order.Filter.Bases.1209_0.YdUKAcRZtFgMABD | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) | Mathlib_Order_Filter_Bases |
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι' : Sort u_5
ι : Sort u
α : Type v
inst✝¹ : Countable ι
f : ι → Filter α
inst✝ : ∀ (i : ι), IsCountablyGenerated (f i)
s : ι → ℕ → Set α
hs : ∀ (i : ι), HasAntitoneBasis (f i) (s i)
⊢ IsCountablyGenerated (⨅ x, f x.down) | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | refine' HasCountableBasis.isCountablyGenerated ⟨hasBasis_iInf fun n => (hs _).1, _⟩ | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) := by
choose s hs using fun i => exists_antitone_basis (f i)
rw [← PLift.down_surjective.iInf_comp]
| Mathlib.Order.Filter.Bases.1209_0.YdUKAcRZtFgMABD | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) | Mathlib_Order_Filter_Bases |
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι' : Sort u_5
ι : Sort u
α : Type v
inst✝¹ : Countable ι
f : ι → Filter α
inst✝ : ∀ (i : ι), IsCountablyGenerated (f i)
s : ι → ℕ → Set α
hs : ∀ (i : ι), HasAntitoneBasis (f i) (s i)
⊢ Set.Countable {If | Set.Finite If.fst ∧ (↑If.fst → True)} | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | refine' (countable_range <| Sigma.map ((↑) : Finset (PLift ι) → Set (PLift ι)) fun _ => id).mono _ | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) := by
choose s hs using fun i => exists_antitone_basis (f i)
rw [← PLift.down_surjective.iInf_comp]
refine' HasCountableBasis.isCountablyGenerated ⟨... | Mathlib.Order.Filter.Bases.1209_0.YdUKAcRZtFgMABD | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) | Mathlib_Order_Filter_Bases |
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι' : Sort u_5
ι : Sort u
α : Type v
inst✝¹ : Countable ι
f : ι → Filter α
inst✝ : ∀ (i : ι), IsCountablyGenerated (f i)
s : ι → ℕ → Set α
hs : ∀ (i : ι), HasAntitoneBasis (f i) (s i)
⊢ {If | Set.Finite If.fst ∧ (↑If.fst → True)} ⊆ range (Sigma.map Finset.toSet fun x... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | rintro ⟨I, f⟩ ⟨hI, -⟩ | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) := by
choose s hs using fun i => exists_antitone_basis (f i)
rw [← PLift.down_surjective.iInf_comp]
refine' HasCountableBasis.isCountablyGenerated ⟨... | Mathlib.Order.Filter.Bases.1209_0.YdUKAcRZtFgMABD | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) | Mathlib_Order_Filter_Bases |
case mk.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι' : Sort u_5
ι : Sort u
α : Type v
inst✝¹ : Countable ι
f✝ : ι → Filter α
inst✝ : ∀ (i : ι), IsCountablyGenerated (f✝ i)
s : ι → ℕ → Set α
hs : ∀ (i : ι), HasAntitoneBasis (f✝ i) (s i)
I : Set (PLift ι)
f : ↑I → ℕ
hI : Set.Finite { fst := I, snd := f ... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | lift I to Finset (PLift ι) using hI | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) := by
choose s hs using fun i => exists_antitone_basis (f i)
rw [← PLift.down_surjective.iInf_comp]
refine' HasCountableBasis.isCountablyGenerated ⟨... | Mathlib.Order.Filter.Bases.1209_0.YdUKAcRZtFgMABD | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) | Mathlib_Order_Filter_Bases |
case mk.intro.intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι' : Sort u_5
ι : Sort u
α : Type v
inst✝¹ : Countable ι
f✝ : ι → Filter α
inst✝ : ∀ (i : ι), IsCountablyGenerated (f✝ i)
s : ι → ℕ → Set α
hs : ∀ (i : ι), HasAntitoneBasis (f✝ i) (s i)
I : Finset (PLift ι)
f : ↑↑I → ℕ
⊢ { fst := ↑I, snd := f } ∈... | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
#align_import ord... | exact ⟨⟨I, f⟩, rfl⟩ | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) := by
choose s hs using fun i => exists_antitone_basis (f i)
rw [← PLift.down_surjective.iInf_comp]
refine' HasCountableBasis.isCountablyGenerated ⟨... | Mathlib.Order.Filter.Bases.1209_0.YdUKAcRZtFgMABD | instance iInf.isCountablyGenerated {ι : Sort u} {α : Type v} [Countable ι] (f : ι → Filter α)
[∀ i, IsCountablyGenerated (f i)] : IsCountablyGenerated (⨅ i, f i) | Mathlib_Order_Filter_Bases |
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α₀ α₁ α₂ : Type u₀
β : Type u₁
f : α₀ → α₁
f' : α₁ → α₂
x : F α₀ β
⊢ fst f' (fst f x) = fst (f' ∘ f) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [fst, bimap_bimap] | @[higher_order fst_comp_fst]
theorem comp_fst {α₀ α₁ α₂ β} (f : α₀ → α₁) (f' : α₁ → α₂) (x : F α₀ β) :
fst f' (fst f x) = fst (f' ∘ f) x := by | Mathlib.Control.Bifunctor.88_0.rLCDZq5jnVLHvgZ | @[higher_order fst_comp_fst]
theorem comp_fst {α₀ α₁ α₂ β} (f : α₀ → α₁) (f' : α₁ → α₂) (x : F α₀ β) :
fst f' (fst f x) = fst (f' ∘ f) x | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α₀ α₁ : Type u₀
β₀ β₁ : Type u₁
f : α₀ → α₁
f' : β₀ → β₁
x : F α₀ β₀
⊢ fst f (snd f' x) = bimap f f' x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [fst, bimap_bimap] | @[higher_order fst_comp_snd]
theorem fst_snd {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) :
fst f (snd f' x) = bimap f f' x := by | Mathlib.Control.Bifunctor.94_0.rLCDZq5jnVLHvgZ | @[higher_order fst_comp_snd]
theorem fst_snd {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) :
fst f (snd f' x) = bimap f f' x | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α₀ α₁ : Type u₀
β₀ β₁ : Type u₁
f : α₀ → α₁
f' : β₀ → β₁
x : F α₀ β₀
⊢ snd f' (fst f x) = bimap f f' x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [snd, bimap_bimap] | @[higher_order snd_comp_fst]
theorem snd_fst {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) :
snd f' (fst f x) = bimap f f' x := by | Mathlib.Control.Bifunctor.100_0.rLCDZq5jnVLHvgZ | @[higher_order snd_comp_fst]
theorem snd_fst {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) :
snd f' (fst f x) = bimap f f' x | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α : Type u₀
β₀ β₁ β₂ : Type u₁
g : β₀ → β₁
g' : β₁ → β₂
x : F α β₀
⊢ snd g' (snd g x) = snd (g' ∘ g) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [snd, bimap_bimap] | @[higher_order snd_comp_snd]
theorem comp_snd {α β₀ β₁ β₂} (g : β₀ → β₁) (g' : β₁ → β₂) (x : F α β₀) :
snd g' (snd g x) = snd (g' ∘ g) x := by | Mathlib.Control.Bifunctor.106_0.rLCDZq5jnVLHvgZ | @[higher_order snd_comp_snd]
theorem comp_snd {α β₀ β₁ β₂} (g : β₀ → β₁) (g' : β₁ → β₂) (x : F α β₀) :
snd g' (snd g x) = snd (g' ∘ g) x | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ LawfulBifunctor Prod | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | refine' { .. } | instance Prod.lawfulBifunctor : LawfulBifunctor Prod := by
| Mathlib.Control.Bifunctor.123_0.rLCDZq5jnVLHvgZ | instance Prod.lawfulBifunctor : LawfulBifunctor Prod | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ ∀ {α : Type ?u.3863} {β : Type ?u.3862} (x : α × β), bimap id id x = x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance Prod.lawfulBifunctor : LawfulBifunctor Prod := by
refine' { .. } <;> | Mathlib.Control.Bifunctor.123_0.rLCDZq5jnVLHvgZ | instance Prod.lawfulBifunctor : LawfulBifunctor Prod | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ ∀ {α₀ α₁ α₂ : Type ?u.3863} {β₀ β₁ β₂ : Type ?u.3862} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁) (g' : β₁ → β₂)
(x : α₀ × β₀), bimap f' g' (bimap f g x) = bimap (f' ∘ f) (g' ∘ g) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance Prod.lawfulBifunctor : LawfulBifunctor Prod := by
refine' { .. } <;> | Mathlib.Control.Bifunctor.123_0.rLCDZq5jnVLHvgZ | instance Prod.lawfulBifunctor : LawfulBifunctor Prod | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
α✝ : Type ?u.3863
β✝ : Type ?u.3862
x✝ : α✝ × β✝
⊢ bimap id id x✝ = x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | rfl | instance Prod.lawfulBifunctor : LawfulBifunctor Prod := by
refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.123_0.rLCDZq5jnVLHvgZ | instance Prod.lawfulBifunctor : LawfulBifunctor Prod | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
α₀✝ α₁✝ α₂✝ : Type ?u.3863
β₀✝ β₁✝ β₂✝ : Type ?u.3862
f✝ : α₀✝ → α₁✝
f'✝ : α₁✝ → α₂✝
g✝ : β₀✝ → β₁✝
g'✝ : β₁✝ → β₂✝
x✝ : α₀✝ × β₀✝
⊢ bimap f'✝ g'✝ (bimap f✝ g✝ x✝) = bimap (f'✝ ∘ f✝) (g'✝ ∘ g✝) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | rfl | instance Prod.lawfulBifunctor : LawfulBifunctor Prod := by
refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.123_0.rLCDZq5jnVLHvgZ | instance Prod.lawfulBifunctor : LawfulBifunctor Prod | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ LawfulBifunctor Const | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | refine' { .. } | instance LawfulBifunctor.const : LawfulBifunctor Const := by | Mathlib.Control.Bifunctor.130_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.const : LawfulBifunctor Const | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ ∀ {α : Type ?u.4164} {β : Type ?u.4165} (x : Const α β), bimap id id x = x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance LawfulBifunctor.const : LawfulBifunctor Const := by refine' { .. } <;> | Mathlib.Control.Bifunctor.130_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.const : LawfulBifunctor Const | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ ∀ {α₀ α₁ α₂ : Type ?u.4164} {β₀ β₁ β₂ : Type ?u.4165} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁) (g' : β₁ → β₂)
(x : Const α₀ β₀), bimap f' g' (bimap f g x) = bimap (f' ∘ f) (g' ∘ g) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance LawfulBifunctor.const : LawfulBifunctor Const := by refine' { .. } <;> | Mathlib.Control.Bifunctor.130_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.const : LawfulBifunctor Const | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
α✝ : Type ?u.4164
β✝ : Type ?u.4165
x✝ : Const α✝ β✝
⊢ bimap id id x✝ = x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | rfl | instance LawfulBifunctor.const : LawfulBifunctor Const := by refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.130_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.const : LawfulBifunctor Const | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
α₀✝ α₁✝ α₂✝ : Type ?u.4164
β₀✝ β₁✝ β₂✝ : Type ?u.4165
f✝ : α₀✝ → α₁✝
f'✝ : α₁✝ → α₂✝
g✝ : β₀✝ → β₁✝
g'✝ : β₁✝ → β₂✝
x✝ : Const α₀✝ β₀✝
⊢ bimap f'✝ g'✝ (bimap f✝ g✝ x✝) = bimap (f'✝ ∘ f✝) (g'✝ ∘ g✝) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | rfl | instance LawfulBifunctor.const : LawfulBifunctor Const := by refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.130_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.const : LawfulBifunctor Const | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
⊢ LawfulBifunctor (_root_.flip F) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | refine' { .. } | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) := by
| Mathlib.Control.Bifunctor.137_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
⊢ ∀ {α : Type u₁} {β : Type u₀} (x : _root_.flip F α β), bimap id id x = x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) := by
refine' { .. } <;> | Mathlib.Control.Bifunctor.137_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
⊢ ∀ {α₀ α₁ α₂ : Type u₁} {β₀ β₁ β₂ : Type u₀} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁) (g' : β₁ → β₂)
(x : _root_.flip F α₀ β₀), bimap f' g' (bimap f g x) = bimap (f' ∘ f) (g' ∘ g) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) := by
refine' { .. } <;> | Mathlib.Control.Bifunctor.137_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α✝ : Type u₁
β✝ : Type u₀
x✝ : _root_.flip F α✝ β✝
⊢ bimap id id x✝ = x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [bimap, functor_norm] | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) := by
refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.137_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α₀✝ α₁✝ α₂✝ : Type u₁
β₀✝ β₁✝ β₂✝ : Type u₀
f✝ : α₀✝ → α₁✝
f'✝ : α₁✝ → α₂✝
g✝ : β₀✝ → β₁✝
g'✝ : β₁✝ → β₂✝
x✝ : _root_.flip F α₀✝ β₀✝
⊢ bimap f'✝ g'✝ (bimap f✝ g✝ x✝) = bimap (f'✝ ∘ f✝) (g'✝ ∘ g✝) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [bimap, functor_norm] | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) := by
refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.137_0.rLCDZq5jnVLHvgZ | instance LawfulBifunctor.flip [LawfulBifunctor F] : LawfulBifunctor (flip F) | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ LawfulBifunctor Sum | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | refine' { .. } | instance Sum.lawfulBifunctor : LawfulBifunctor Sum := by
| Mathlib.Control.Bifunctor.144_0.rLCDZq5jnVLHvgZ | instance Sum.lawfulBifunctor : LawfulBifunctor Sum | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ ∀ {α : Type ?u.4972} {β : Type ?u.4971} (x : α ⊕ β), bimap id id x = x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | aesop | instance Sum.lawfulBifunctor : LawfulBifunctor Sum := by
refine' { .. } <;> | Mathlib.Control.Bifunctor.144_0.rLCDZq5jnVLHvgZ | instance Sum.lawfulBifunctor : LawfulBifunctor Sum | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝ : Bifunctor F
⊢ ∀ {α₀ α₁ α₂ : Type ?u.4972} {β₀ β₁ β₂ : Type ?u.4971} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁) (g' : β₁ → β₂)
(x : α₀ ⊕ β₀), bimap f' g' (bimap f g x) = bimap (f' ∘ f) (g' ∘ g) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | aesop | instance Sum.lawfulBifunctor : LawfulBifunctor Sum := by
refine' { .. } <;> | Mathlib.Control.Bifunctor.144_0.rLCDZq5jnVLHvgZ | instance Sum.lawfulBifunctor : LawfulBifunctor Sum | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α : Type u₀
⊢ LawfulFunctor (F α) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | refine' { .. } | instance (priority := 10) Bifunctor.lawfulFunctor [LawfulBifunctor F] {α} : LawfulFunctor (F α) :=
-- Porting note: `mapConst` is required to prove new theorem
by | Mathlib.Control.Bifunctor.153_0.rLCDZq5jnVLHvgZ | instance (priority | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α : Type u₀
⊢ ∀ {α_1 β : Type u₁}, mapConst = map ∘ Function.const β | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance (priority := 10) Bifunctor.lawfulFunctor [LawfulBifunctor F] {α} : LawfulFunctor (F α) :=
-- Porting note: `mapConst` is required to prove new theorem
by refine' { .. } <;> | Mathlib.Control.Bifunctor.153_0.rLCDZq5jnVLHvgZ | instance (priority | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α : Type u₀
⊢ ∀ {α_1 : Type u₁} (x : F α α_1), id <$> x = x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance (priority := 10) Bifunctor.lawfulFunctor [LawfulBifunctor F] {α} : LawfulFunctor (F α) :=
-- Porting note: `mapConst` is required to prove new theorem
by refine' { .. } <;> | Mathlib.Control.Bifunctor.153_0.rLCDZq5jnVLHvgZ | instance (priority | Mathlib_Control_Bifunctor |
case refine'_3
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α : Type u₀
⊢ ∀ {α_1 β γ : Type u₁} (g : α_1 → β) (h : β → γ) (x : F α α_1), (h ∘ g) <$> x = h <$> g <$> x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance (priority := 10) Bifunctor.lawfulFunctor [LawfulBifunctor F] {α} : LawfulFunctor (F α) :=
-- Porting note: `mapConst` is required to prove new theorem
by refine' { .. } <;> | Mathlib.Control.Bifunctor.153_0.rLCDZq5jnVLHvgZ | instance (priority | Mathlib_Control_Bifunctor |
case refine'_1
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α : Type u₀
α✝ β✝ : Type u₁
⊢ mapConst = map ∘ Function.const β✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [mapConst, Functor.map, functor_norm] | instance (priority := 10) Bifunctor.lawfulFunctor [LawfulBifunctor F] {α} : LawfulFunctor (F α) :=
-- Porting note: `mapConst` is required to prove new theorem
by refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.153_0.rLCDZq5jnVLHvgZ | instance (priority | Mathlib_Control_Bifunctor |
case refine'_2
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α : Type u₀
α✝ : Type u₁
x✝ : F α α✝
⊢ id <$> x✝ = x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [mapConst, Functor.map, functor_norm] | instance (priority := 10) Bifunctor.lawfulFunctor [LawfulBifunctor F] {α} : LawfulFunctor (F α) :=
-- Porting note: `mapConst` is required to prove new theorem
by refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.153_0.rLCDZq5jnVLHvgZ | instance (priority | Mathlib_Control_Bifunctor |
case refine'_3
F : Type u₀ → Type u₁ → Type u₂
inst✝¹ : Bifunctor F
inst✝ : LawfulBifunctor F
α : Type u₀
α✝ β✝ γ✝ : Type u₁
g✝ : α✝ → β✝
h✝ : β✝ → γ✝
x✝ : F α α✝
⊢ (h✝ ∘ g✝) <$> x✝ = h✝ <$> g✝ <$> x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [mapConst, Functor.map, functor_norm] | instance (priority := 10) Bifunctor.lawfulFunctor [LawfulBifunctor F] {α} : LawfulFunctor (F α) :=
-- Porting note: `mapConst` is required to prove new theorem
by refine' { .. } <;> intros <;> | Mathlib.Control.Bifunctor.153_0.rLCDZq5jnVLHvgZ | instance (priority | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝⁵ : Bifunctor F
G : Type u_1 → Type u₀
H : Type u_2 → Type u₁
inst✝⁴ : Functor G
inst✝³ : Functor H
inst✝² : LawfulFunctor G
inst✝¹ : LawfulFunctor H
inst✝ : LawfulBifunctor F
⊢ LawfulBifunctor (bicompl F G H) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | constructor | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) := by
| Mathlib.Control.Bifunctor.166_0.rLCDZq5jnVLHvgZ | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) | Mathlib_Control_Bifunctor |
case id_bimap
F : Type u₀ → Type u₁ → Type u₂
inst✝⁵ : Bifunctor F
G : Type u_1 → Type u₀
H : Type u_2 → Type u₁
inst✝⁴ : Functor G
inst✝³ : Functor H
inst✝² : LawfulFunctor G
inst✝¹ : LawfulFunctor H
inst✝ : LawfulBifunctor F
⊢ ∀ {α : Type u_1} {β : Type u_2} (x : bicompl F G H α β), bimap id id x = x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) := by
constructor <;> | Mathlib.Control.Bifunctor.166_0.rLCDZq5jnVLHvgZ | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) | Mathlib_Control_Bifunctor |
case bimap_bimap
F : Type u₀ → Type u₁ → Type u₂
inst✝⁵ : Bifunctor F
G : Type u_1 → Type u₀
H : Type u_2 → Type u₁
inst✝⁴ : Functor G
inst✝³ : Functor H
inst✝² : LawfulFunctor G
inst✝¹ : LawfulFunctor H
inst✝ : LawfulBifunctor F
⊢ ∀ {α₀ α₁ α₂ : Type u_1} {β₀ β₁ β₂ : Type u_2} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁)... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) := by
constructor <;> | Mathlib.Control.Bifunctor.166_0.rLCDZq5jnVLHvgZ | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) | Mathlib_Control_Bifunctor |
case id_bimap
F : Type u₀ → Type u₁ → Type u₂
inst✝⁵ : Bifunctor F
G : Type u_1 → Type u₀
H : Type u_2 → Type u₁
inst✝⁴ : Functor G
inst✝³ : Functor H
inst✝² : LawfulFunctor G
inst✝¹ : LawfulFunctor H
inst✝ : LawfulBifunctor F
α✝ : Type u_1
β✝ : Type u_2
x✝ : bicompl F G H α✝ β✝
⊢ bimap id id x✝ = x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [bimap, map_id, map_comp_map, functor_norm] | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) := by
constructor <;> intros <;> | Mathlib.Control.Bifunctor.166_0.rLCDZq5jnVLHvgZ | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) | Mathlib_Control_Bifunctor |
case bimap_bimap
F : Type u₀ → Type u₁ → Type u₂
inst✝⁵ : Bifunctor F
G : Type u_1 → Type u₀
H : Type u_2 → Type u₁
inst✝⁴ : Functor G
inst✝³ : Functor H
inst✝² : LawfulFunctor G
inst✝¹ : LawfulFunctor H
inst✝ : LawfulBifunctor F
α₀✝ α₁✝ α₂✝ : Type u_1
β₀✝ β₁✝ β₂✝ : Type u_2
f✝ : α₀✝ → α₁✝
f'✝ : α₁✝ → α₂✝
g✝ : β₀✝ → β₁... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [bimap, map_id, map_comp_map, functor_norm] | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) := by
constructor <;> intros <;> | Mathlib.Control.Bifunctor.166_0.rLCDZq5jnVLHvgZ | instance Function.bicompl.lawfulBifunctor [LawfulFunctor G] [LawfulFunctor H] [LawfulBifunctor F] :
LawfulBifunctor (bicompl F G H) | Mathlib_Control_Bifunctor |
F : Type u₀ → Type u₁ → Type u₂
inst✝³ : Bifunctor F
G : Type u₂ → Type u_1
inst✝² : Functor G
inst✝¹ : LawfulFunctor G
inst✝ : LawfulBifunctor F
⊢ LawfulBifunctor (bicompr G F) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | constructor | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) := by
| Mathlib.Control.Bifunctor.181_0.rLCDZq5jnVLHvgZ | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) | Mathlib_Control_Bifunctor |
case id_bimap
F : Type u₀ → Type u₁ → Type u₂
inst✝³ : Bifunctor F
G : Type u₂ → Type u_1
inst✝² : Functor G
inst✝¹ : LawfulFunctor G
inst✝ : LawfulBifunctor F
⊢ ∀ {α : Type u₀} {β : Type u₁} (x : bicompr G F α β), bimap id id x = x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) := by
constructor <;> | Mathlib.Control.Bifunctor.181_0.rLCDZq5jnVLHvgZ | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) | Mathlib_Control_Bifunctor |
case bimap_bimap
F : Type u₀ → Type u₁ → Type u₂
inst✝³ : Bifunctor F
G : Type u₂ → Type u_1
inst✝² : Functor G
inst✝¹ : LawfulFunctor G
inst✝ : LawfulBifunctor F
⊢ ∀ {α₀ α₁ α₂ : Type u₀} {β₀ β₁ β₂ : Type u₁} (f : α₀ → α₁) (f' : α₁ → α₂) (g : β₀ → β₁) (g' : β₁ → β₂)
(x : bicompr G F α₀ β₀), bimap f' g' (bimap f g x... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | intros | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) := by
constructor <;> | Mathlib.Control.Bifunctor.181_0.rLCDZq5jnVLHvgZ | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) | Mathlib_Control_Bifunctor |
case id_bimap
F : Type u₀ → Type u₁ → Type u₂
inst✝³ : Bifunctor F
G : Type u₂ → Type u_1
inst✝² : Functor G
inst✝¹ : LawfulFunctor G
inst✝ : LawfulBifunctor F
α✝ : Type u₀
β✝ : Type u₁
x✝ : bicompr G F α✝ β✝
⊢ bimap id id x✝ = x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [bimap, functor_norm] | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) := by
constructor <;> intros <;> | Mathlib.Control.Bifunctor.181_0.rLCDZq5jnVLHvgZ | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) | Mathlib_Control_Bifunctor |
case bimap_bimap
F : Type u₀ → Type u₁ → Type u₂
inst✝³ : Bifunctor F
G : Type u₂ → Type u_1
inst✝² : Functor G
inst✝¹ : LawfulFunctor G
inst✝ : LawfulBifunctor F
α₀✝ α₁✝ α₂✝ : Type u₀
β₀✝ β₁✝ β₂✝ : Type u₁
f✝ : α₀✝ → α₁✝
f'✝ : α₁✝ → α₂✝
g✝ : β₀✝ → β₁✝
g'✝ : β₁✝ → β₂✝
x✝ : bicompr G F α₀✝ β₀✝
⊢ bimap f'✝ g'✝ (bimap f✝ ... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Data.Sum.Basic
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6... | simp [bimap, functor_norm] | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) := by
constructor <;> intros <;> | Mathlib.Control.Bifunctor.181_0.rLCDZq5jnVLHvgZ | instance Function.bicompr.lawfulBifunctor [LawfulFunctor G] [LawfulBifunctor F] :
LawfulBifunctor (bicompr G F) | Mathlib_Control_Bifunctor |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ G.map (Fork.ι c) ≫ G.map f = 0 | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | rw [← G.map_comp, c.condition, G.map_zero] | @[reassoc (attr := simp)]
lemma map_condition : G.map c.ι ≫ G.map f = 0 := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.39_0.Ox2DGCW1z12SA2j | @[reassoc (attr | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ IsLimit (G.mapCone c) ≃ IsLimit (map c G) | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _) | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.51_0.Ox2DGCW1z12SA2j | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case refine'_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ parallelPair f 0 ⋙ G ≅ parallelPair (G.map f) 0 | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | refine' parallelPair.ext (Iso.refl _) (Iso.refl _) _ _ | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
| Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.51_0.Ox2DGCW1z12SA2j | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
case refine'_1.refine'_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : HasZeroMorphisms C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasZeroMorphisms D
X Y : C
f : X ⟶ Y
c : KernelFork f
G : C ⥤ D
inst✝ : Functor.PreservesZeroMorphisms G
⊢ (parallelPair f 0 ⋙ G).map WalkingParallelPairHom.left ≫
(Iso.r... | /-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | simp | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by
refine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)
refine' parallelPair.ext (Iso.refl _) (Is... | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels.51_0.Ox2DGCW1z12SA2j | /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if
the mapped kernel fork is limit. -/
def isLimitMapConeEquiv :
IsLimit (G.mapCone c) ≃ IsLimit (c.map G) | Mathlib_CategoryTheory_Limits_Preserves_Shapes_Kernels |
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