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
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x : ι → R ⧸ I
i : ι
⊢ AddHom.toFun
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | obtain ⟨_, _⟩ := @Quot.exists_rep _ _ (x i) | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
case h.intro
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x : ι → R ⧸ I
i : ι
w✝ : R
h✝ : Quot.mk Setoid.r w✝ = x i
⊢ AddHom.toFun
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
... | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | convert Quotient.out_eq' (x i) | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_... | Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX | /-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
⊢ f x i ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | classical
cases nonempty_fintype ι
rw [pi_eq_sum_univ x]
simp only [Finset.sum_apply, smul_eq_mul, map_sum, Pi.smul_apply, map_smul]
exact I.sum_mem fun j _ => I.mul_mem_right _ (hi j) | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
| Mathlib.RingTheory.Ideal.Quotient.377_0.TwNAv7Pc4PYOWjX | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I | Mathlib_RingTheory_Ideal_Quotient |
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
⊢ f x i ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | cases nonempty_fintype ι | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
classical
| Mathlib.RingTheory.Ideal.Quotient.377_0.TwNAv7Pc4PYOWjX | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I | Mathlib_RingTheory_Ideal_Quotient |
case intro
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
val✝ : Fintype ι
⊢ f x i ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | rw [pi_eq_sum_univ x] | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
classical
cases nonempty_fintype ι
| Mathlib.RingTheory.Ideal.Quotient.377_0.TwNAv7Pc4PYOWjX | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I | Mathlib_RingTheory_Ideal_Quotient |
case intro
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
val✝ : Fintype ι
⊢ f (∑ i : ι, x i • fun j => if i = j then 1 else 0) i ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | simp only [Finset.sum_apply, smul_eq_mul, map_sum, Pi.smul_apply, map_smul] | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
classical
cases nonempty_fintype ι
rw [pi_eq_... | Mathlib.RingTheory.Ideal.Quotient.377_0.TwNAv7Pc4PYOWjX | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I | Mathlib_RingTheory_Ideal_Quotient |
case intro
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
val✝ : Fintype ι
⊢ ∑ x_1 : ι, x x_1 * f (fun j => if x_1 = j then 1 else 0) i ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Cong... | exact I.sum_mem fun j _ => I.mul_mem_right _ (hi j) | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
classical
cases nonempty_fintype ι
rw [pi_eq_... | Mathlib.RingTheory.Ideal.Quotient.377_0.TwNAv7Pc4PYOWjX | /-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I | Mathlib_RingTheory_Ideal_Quotient |
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
⊢ natDegree (cancelLeads p q) < natDegree q | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | by_cases hp : p = 0 | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
| Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case pos
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : p = 0
⊢ natDegree (cancelLeads p q) < natDegree q | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | convert hq | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case h.e'_3
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : p = 0
⊢ natDegree (cancelLeads p q) = 0 | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | simp [hp, cancelLeads] | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
| Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
⊢ natDegree (cancelLeads p q) < natDegree q | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one] | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | by_cases h0 :
C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0 | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case pos
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C ... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C (leadingC... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | simp only [h0, natDegree_zero] | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : ¬C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | apply lt_of_le_of_ne | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : ¬C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | compute_degree! | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg.a.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : ¬C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree q - natDegree p + natDegr... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | rwa [Nat.sub_add_cancel] | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : ¬C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | contrapose! h0 | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q
⊢ C (leadingCoeff ... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | rw [← leadingCoeff_eq_zero, leadingCoeff, h0, mul_assoc, X_pow_mul, ← tsub_add_cancel_of_le h,
add_comm _ p.natDegree] | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q
⊢ coeff
(C (... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | simp only [coeff_mul_X_pow, coeff_neg, coeff_C_mul, add_tsub_cancel_left, coeff_add] | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
case neg.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q
⊢ leadingCoeff p *... | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.canc... | rw [add_comm p.natDegree, tsub_add_cancel_of_le h, ← leadingCoeff, ← leadingCoeff, comm,
add_right_neg] | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by
by_cases hp : p = 0
· convert hq
simp [hp, cancelLead... | Mathlib.Data.Polynomial.CancelLeads.52_0.8Zq2gl3suYkxnmI | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree | Mathlib_Data_Polynomial_CancelLeads |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
⊢ FactorsThruAlong S T (𝟙 X) ↔ FactorsThru S T | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp [FactorsThruAlong, FactorsThru] | @[simp]
lemma factorsThruAlong_id {X : C} (S T : Presieve X) :
S.FactorsThruAlong T (𝟙 X) ↔ S.FactorsThru T := by
| Mathlib.CategoryTheory.Sites.Coverage.81_0.qkZFgqEgDC2P633 | @[simp]
lemma factorsThruAlong_id {X : C} (S T : Presieve X) :
S.FactorsThruAlong T (𝟙 X) ↔ S.FactorsThru T | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
h : S ≤ T
Y : C
g : Y ⟶ X
hg : S g
⊢ 𝟙 Y ≫ g = g | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp | lemma factorsThru_of_le {X : C} (S T : Presieve X) (h : S ≤ T) :
S.FactorsThru T :=
fun Y g hg => ⟨Y, 𝟙 _, g, h _ hg, by | Mathlib.CategoryTheory.Sites.Coverage.86_0.qkZFgqEgDC2P633 | lemma factorsThru_of_le {X : C} (S T : Presieve X) (h : S ≤ T) :
S.FactorsThru T | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S : Presieve X
T : Sieve X
h : FactorsThru S T.arrows
⊢ S ≤ T.arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rintro Y f hf | lemma le_of_factorsThru_sieve {X : C} (S : Presieve X) (T : Sieve X) (h : S.FactorsThru T) :
S ≤ T := by
| Mathlib.CategoryTheory.Sites.Coverage.90_0.qkZFgqEgDC2P633 | lemma le_of_factorsThru_sieve {X : C} (S : Presieve X) (T : Sieve X) (h : S.FactorsThru T) :
S ≤ T | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S : Presieve X
T : Sieve X
h : FactorsThru S T.arrows
Y : C
f : Y ⟶ X
hf : f ∈ S
⊢ f ∈ T.arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨W, i, e, h1, rfl⟩ := h hf | lemma le_of_factorsThru_sieve {X : C} (S : Presieve X) (T : Sieve X) (h : S.FactorsThru T) :
S ≤ T := by
rintro Y f hf
| Mathlib.CategoryTheory.Sites.Coverage.90_0.qkZFgqEgDC2P633 | lemma le_of_factorsThru_sieve {X : C} (S : Presieve X) (T : Sieve X) (h : S.FactorsThru T) :
S ≤ T | Mathlib_CategoryTheory_Sites_Coverage |
case intro.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S : Presieve X
T : Sieve X
h : FactorsThru S T.arrows
Y W : C
i : Y ⟶ W
e : W ⟶ X
h1 : T.arrows e
hf : i ≫ e ∈ S
⊢ i ≫ e ∈ T.arrows | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact T.downward_closed h1 _ | lemma le_of_factorsThru_sieve {X : C} (S : Presieve X) (T : Sieve X) (h : S.FactorsThru T) :
S ≤ T := by
rintro Y f hf
obtain ⟨W, i, e, h1, rfl⟩ := h hf
| Mathlib.CategoryTheory.Sites.Coverage.90_0.qkZFgqEgDC2P633 | lemma le_of_factorsThru_sieve {X : C} (S : Presieve X) (T : Sieve X) (h : S.FactorsThru T) :
S ≤ T | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
H : FactorsThru S T
hS : IsSheafFor P S
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
⊢ IsSheafFor P T | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at * | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
| Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
H : FactorsThru S T
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | choose W i e h1 h2 using H | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine ⟨?_, fun x hx => ?_⟩ | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_1
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro x y₁ y₂ h₁ h₂ | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_1
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine hS.1.ext (fun Y g hg => ?_) | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_1
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [← h2 hg, op_comp, P.map_comp, types_comp_apply, h₁ _ (h1 _ ), h₂ _ (h1 _)] | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | let y : S.FamilyOfElements P := fun Y g hg => P.map (i _).op (x (e hg) (h1 _)) | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | have hy : y.Compatible := by
intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp]
apply hx
simp only [h2, h, Category.assoc] | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : ... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← types_comp_apply (P.map (i h₁).op) (P.map g₁.op),
← types_comp_apply (P.map (i h₂).op) (P.map g₂.op),
← P.map_comp, ← op_comp, ← P.map_comp, ← op_comp] | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ → ⦃g : ... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply hx | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case a
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z : C⦄ ... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [h2, h, Category.assoc] | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | let ⟨_, h2'⟩ := hS | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
W : ⦃Z... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨z, hz⟩ := h2' y hy | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine ⟨z, fun Y g hg => ?_⟩ | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalgamation x t
... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨R, hR1, hR2⟩ := h hg | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalg... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | choose WW ii ee hh1 hh2 using hR2 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalg... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine hR1.ext (fun Q t ht => ?_) | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalg... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← types_comp_apply (P.map g.op) (P.map t.op), ← P.map_comp, ← op_comp, ← hh2 ht,
op_comp, P.map_comp, types_comp_apply, hz _ (hh1 _),
← types_comp_apply _ (P.map (ii ht).op), ← P.map_comp, ← op_comp] | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2.intro.intro.intro
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAmalg... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply hx | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
case refine_2.intro.intro.intro.a
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X : C
S T : Presieve X
P : Cᵒᵖ ⥤ Type w
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ FactorsThruAlong R S f
hS :
IsSeparatedFor P S ∧
∀ (x : FamilyOfElements P S), FamilyOfElements.Compatible x → ∃ t, FamilyOfElements.IsAma... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [Category.assoc, h2, hh2] | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P := by
simp only [← Presieve.isSeparatedFor_and_exists_i... | Mathlib.CategoryTheory.Sites.Coverage.99_0.qkZFgqEgDC2P633 | lemma isSheafFor_of_factorsThru
{X : C} {S T : Presieve X}
(P : Cᵒᵖ ⥤ Type w)
(H : S.FactorsThru T) (hS : S.IsSheafFor P)
(h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ (R : Presieve Y),
R.IsSeparatedFor P ∧ R.FactorsThruAlong S f):
T.IsSheafFor P | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
⊢ ∀ ⦃X Y : C⦄ (f : Y ⟶ X),
∀ S ∈ (fun X => {S | Sieve.generate S ∈ GrothendieckTopology.sieves J X}) X,
∃ T ∈ (fun X => {S | Sieve.generate S ∈ GrothendieckTopology.sieves J X}) Y, Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro X Y f S (hS : Sieve.generate S ∈ J X) | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib.CategoryTheory.Sites.Coverage.158_0.qkZFgqEgDC2P633 | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Presieve X
hS : Sieve.generate S ∈ GrothendieckTopology.sieves J X
⊢ ∃ T ∈ (fun X => {S | Sieve.generate S ∈ GrothendieckTopology.sieves J X}) Y, Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine ⟨(Sieve.generate S).pullback f, ?_, fun Z g h => h⟩ | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib.CategoryTheory.Sites.Coverage.158_0.qkZFgqEgDC2P633 | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Presieve X
hS : Sieve.generate S ∈ GrothendieckTopology.sieves J X
⊢ (Sieve.pullback f (Sieve.generate S)).arrows ∈ (fun X => {S | Sieve.generate S ∈ GrothendieckTopology.sieves J X}) Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | dsimp | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib.CategoryTheory.Sites.Coverage.158_0.qkZFgqEgDC2P633 | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Presieve X
hS : Sieve.generate S ∈ GrothendieckTopology.sieves J X
⊢ Sieve.generate (Sieve.pullback f (Sieve.generate S)).arrows ∈ GrothendieckTopology.sieves J Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [Sieve.generate_sieve] | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib.CategoryTheory.Sites.Coverage.158_0.qkZFgqEgDC2P633 | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.23525
inst✝ : Category.{?u.23529, ?u.23525} C
J : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Presieve X
hS : Sieve.generate S ∈ GrothendieckTopology.sieves J X
⊢ Sieve.pullback f (Sieve.generate S) ∈ GrothendieckTopology.sieves J Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact J.pullback_stable _ hS | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib.CategoryTheory.Sites.Coverage.158_0.qkZFgqEgDC2P633 | variable (C) in
/--
Associate a coverage to any Grothendieck topology.
If `J` is a Grothendieck topology, and `K` is the associated coverage, then a presieve
`S` is a covering presieve for `K` if and only if the sieve that it generates is a
covering sieve for `J`.
-/
def ofGrothendieck (J : GrothendieckTopology C) : Co... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
⊢ Sieve.pullback f T = ⊤ | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | ext Z g | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ := by
| Mathlib.CategoryTheory.Sites.Coverage.189_0.qkZFgqEgDC2P633 | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ | Mathlib_CategoryTheory_Sites_Coverage |
case h
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ (Sieve.pullback f T).arrows g ↔ ⊤.arrows g | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true] | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ := by
ext Z g
| Mathlib.CategoryTheory.Sites.Coverage.189_0.qkZFgqEgDC2P633 | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ | Mathlib_CategoryTheory_Sites_Coverage |
case h
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ T.arrows (g ≫ f) | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply h | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ := by
ext Z g
simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true]
| Mathlib.CategoryTheory.Sites.Coverage.189_0.qkZFgqEgDC2P633 | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ | Mathlib_CategoryTheory_Sites_Coverage |
case h.a
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ S.arrows (g ≫ f) | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply S.downward_closed | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ := by
ext Z g
simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true]
apply h
| Mathlib.CategoryTheory.Sites.Coverage.189_0.qkZFgqEgDC2P633 | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ | Mathlib_CategoryTheory_Sites_Coverage |
case h.a.x
C : Type u_2
inst✝ : Category.{u_1, u_2} C
X Y : C
S T : Sieve X
h : S ≤ T
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ S.arrows f | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact hf | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ := by
ext Z g
simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true]
apply h
apply S.downward_closed
| Mathlib.CategoryTheory.Sites.Coverage.189_0.qkZFgqEgDC2P633 | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
⊢ saturate K X T | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply saturate.transitive _ _ _ hS | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T := by
| Mathlib.CategoryTheory.Sites.Coverage.197_0.qkZFgqEgDC2P633 | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → saturate K Y (Sieve.pullback f T) | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Y g hg | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T := by
apply saturate.transitive _ _ _ hS
| Mathlib.CategoryTheory.Sites.Coverage.197_0.qkZFgqEgDC2P633 | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
Y : C
g : Y ⟶ X
hg : S.arrows g
⊢ saturate K Y (Sieve.pullback g T) | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [eq_top_pullback (h := h)] | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T := by
apply saturate.transitive _ _ _ hS
intro Y g hg
| Mathlib.CategoryTheory.Sites.Coverage.197_0.qkZFgqEgDC2P633 | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
Y : C
g : Y ⟶ X
hg : S.arrows g
⊢ saturate K Y ⊤ | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply saturate.top | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T := by
apply saturate.transitive _ _ _ hS
intro Y g hg
rw [eq_top_pullback (h := h)]
· | Mathlib.CategoryTheory.Sites.Coverage.197_0.qkZFgqEgDC2P633 | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T | Mathlib_CategoryTheory_Sites_Coverage |
case hf
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
X : C
S T : Sieve X
h : S ≤ T
hS : saturate K X S
Y : C
g : Y ⟶ X
hg : S.arrows g
⊢ S.arrows g | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | assumption | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T := by
apply saturate.transitive _ _ _ hS
intro Y g hg
rw [eq_top_pullback (h := h)]
· apply saturate.top
· | Mathlib.CategoryTheory.Sites.Coverage.197_0.qkZFgqEgDC2P633 | lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)
(hS : saturate K X S) : saturate K X T | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
⊢ ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ saturate K X → Sieve.pullback f S ∈ saturate K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro X Y S f hS | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X Y : C
S : Sieve X
f : Y ⟶ X
hS : S ∈ saturate K X
⊢ Sieve.pullback f S ∈ saturate K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | induction hS generalizing Y with
| of X S hS =>
obtain ⟨R,hR1,hR2⟩ := K.pullback f S hS
suffices Sieve.generate R ≤ (Sieve.generate S).pullback f from
saturate_of_superset _ this (saturate.of _ _ hR1)
rintro Z g ⟨W, i, e, h1, h2⟩
obtain ⟨WW, ii, ee, hh1, hh2⟩ := hR2 h1
refine ⟨... | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X Y : C
S : Sieve X
f : Y ⟶ X
hS : S ∈ saturate K X
⊢ Sieve.pullback f S ∈ saturate K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | induction hS generalizing Y with
| of X S hS =>
obtain ⟨R,hR1,hR2⟩ := K.pullback f S hS
suffices Sieve.generate R ≤ (Sieve.generate S).pullback f from
saturate_of_superset _ this (saturate.of _ _ hR1)
rintro Z g ⟨W, i, e, h1, h2⟩
obtain ⟨WW, ii, ee, hh1, hh2⟩ := hR2 h1
refine ⟨... | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case of
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
⊢ Sieve.pullback f (Sieve.generate S) ∈ saturate K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | of X S hS =>
obtain ⟨R,hR1,hR2⟩ := K.pullback f S hS
suffices Sieve.generate R ≤ (Sieve.generate S).pullback f from
saturate_of_superset _ this (saturate.of _ _ hR1)
rintro Z g ⟨W, i, e, h1, h2⟩
obtain ⟨WW, ii, ee, hh1, hh2⟩ := hR2 h1
refine ⟨WW, i ≫ ii, ee, hh1, ?_⟩
simp o... | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case of
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
⊢ Sieve.pullback f (Sieve.generate S) ∈ saturate K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨R,hR1,hR2⟩ := K.pullback f S hS | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case of.intro.intro
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
R : Presieve Y
hR1 : R ∈ covering K Y
hR2 : Presieve.FactorsThruAlong R S f
⊢ Sieve.pullback f (Sieve.generate S) ∈ saturate K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | suffices Sieve.generate R ≤ (Sieve.generate S).pullback f from
saturate_of_superset _ this (saturate.of _ _ hR1) | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case of.intro.intro
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
R : Presieve Y
hR1 : R ∈ covering K Y
hR2 : Presieve.FactorsThruAlong R S f
⊢ Sieve.generate R ≤ Sieve.pullback f (Sieve.generate S) | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rintro Z g ⟨W, i, e, h1, h2⟩ | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case of.intro.intro.intro.intro.intro.intro
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
R : Presieve Y
hR1 : R ∈ covering K Y
hR2 : Presieve.FactorsThruAlong R S f
Z : C
g : Z ⟶ Y
W : C
i : Z ⟶ W
e : W ⟶ Y
h1 : ... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | obtain ⟨WW, ii, ee, hh1, hh2⟩ := hR2 h1 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case of.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
R : Presieve Y
hR1 : R ∈ covering K Y
hR2 : Presieve.FactorsThruAlong R S f
Z : C
g : Z ⟶ Y
W : C
i... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | refine ⟨WW, i ≫ ii, ee, hh1, ?_⟩ | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case of.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
Y : C
f : Y ⟶ X
R : Presieve Y
hR1 : R ∈ covering K Y
hR2 : Presieve.FactorsThruAlong R S f
Z : C
g : Z ⟶ Y
W : C
i... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | simp only [hh2, reassoc_of% h2, Category.assoc] | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case top
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S : Sieve X✝
X Y : C
f : Y ⟶ X
⊢ Sieve.pullback f ⊤ ∈ saturate K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | top X => apply saturate.top | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case top
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S : Sieve X✝
X Y : C
f : Y ⟶ X
⊢ Sieve.pullback f ⊤ ∈ saturate K Y | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply saturate.top | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case transitive
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝ : saturate K X R
hS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Sieve.pullback f R ∈ saturate K Y
a_ih✝ :
∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | transitive X R S _ hS H1 _ =>
apply saturate.transitive
apply H1 f
intro Z g hg
rw [← Sieve.pullback_comp]
exact hS hg | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case transitive
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝ : saturate K X R
hS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Sieve.pullback f R ∈ saturate K Y
a_ih✝ :
∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply saturate.transitive | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case transitive.a
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝ : saturate K X R
hS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Sieve.pullback f R ∈ saturate K Y
a_ih✝ :
∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply H1 f | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case transitive.a
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝ : saturate K X R
hS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Sieve.pullback f R ∈ saturate K Y
a_ih✝ :
∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro Z g hg | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case transitive.a
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝ : saturate K X R
hS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Sieve.pullback f R ∈ saturate K Y
a_ih✝ :
∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← Sieve.pullback_comp] | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
case transitive.a
C : Type ?u.27900
inst✝ : Category.{?u.27904, ?u.27900} C
K : Coverage C
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝ : saturate K X R
hS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Sieve.pullback f R ∈ saturate K Y
a_ih✝ :
∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact hS hg | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib.CategoryTheory.Sites.Coverage.205_0.qkZFgqEgDC2P633 | variable (C) in
/--
The Grothendieck topology associated to a coverage `K`.
It is defined *inductively* as follows:
1. If `S` is a covering presieve for `K`, then the sieve generated by `S` is a covering
sieve for the associated Grothendieck topology.
2. The top sieves are in the associated Grothendieck topology.
3. ... | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
⊢ toGrothendieck C K ≤ J ↔ K ≤ ofGrothendieck C J | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | constructor | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mp
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
⊢ toGrothendieck C K ≤ J → K ≤ ofGrothendieck C J | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro H X S hS | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mp
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : toGrothendieck C K ≤ J
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ S ∈ covering (ofGrothendieck C J) X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact H _ <| saturate.of _ _ hS | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
⊢ K ≤ ofGrothendieck C J → toGrothendieck C K ≤ J | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro H X S hS | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : K ≤ ofGrothendieck C J
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
⊢ S ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | induction hS with
| of X S hS => exact H _ hS
| top => apply J.top_mem
| transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : K ≤ ofGrothendieck C J
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves (toGrothendieck C K) X
⊢ S ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | induction hS with
| of X S hS => exact H _ hS
| top => apply J.top_mem
| transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : K ≤ ofGrothendieck C J
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ Sieve.generate S ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | of X S hS => exact H _ hS | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.of
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : K ≤ ofGrothendieck C J
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
hS : S ∈ covering K X
⊢ Sieve.generate S ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact H _ hS | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.top
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : K ≤ ofGrothendieck C J
X : C
S : Sieve X
X✝ : C
⊢ ⊤ ∈ GrothendieckTopology.sieves J X✝ | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | top => apply J.top_mem | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.top
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : K ≤ ofGrothendieck C J
X : C
S : Sieve X
X✝ : C
⊢ ⊤ ∈ GrothendieckTopology.sieves J X✝ | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply J.top_mem | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : K ≤ ofGrothendieck C J
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : R ∈ GrothendieckTopology.sieves... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | | transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case mpr.transitive
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
K : Coverage C
J : GrothendieckTopology C
H : K ≤ ofGrothendieck C J
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : saturate K X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → saturate K Y (Sieve.pullback f S)
H1 : R ∈ GrothendieckTopology.sieves... | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | exact J.transitive H1 _ H2 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
J : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
⊢ S ∈ GrothendieckTopology.sieves (toGrothendieck C (ofGrothendieck C J)) X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rw [← Sieve.generate_sieve S] | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
| Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
J : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
⊢ Sieve.generate S.arrows ∈ GrothendieckTopology.sieves (toGrothendieck C (ofGrothendieck C J)) X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply saturate.of | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case hS
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
J : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
⊢ S.arrows ∈ covering (ofGrothendieck C J) X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | dsimp [ofGrothendieck] | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
case hS
C : Type ?u.30764
inst✝ : Category.{?u.30768, ?u.30764} C
J : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ GrothendieckTopology.sieves J X
⊢ Sieve.generate S.arrows ∈ GrothendieckTopology.sieves J X | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | rwa [Sieve.generate_sieve S] | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ := toGrothendieck _ K
choice_eq := fun _ _ => rfl
le_l_u J X S hS := by
rw [← Sieve.generate_sieve S]
... | Mathlib.CategoryTheory.Sites.Coverage.248_0.qkZFgqEgDC2P633 | variable (C) in
/--
The two constructions `Coverage.toGrothendieck` and `Coverage.ofGrothendieck` form
a Galois insertion.
-/
def gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C) where
choice K _ | Mathlib_CategoryTheory_Sites_Coverage |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
⊢ toGrothendieck C K = sInf {J | K ≤ ofGrothendieck C J} | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply le_antisymm | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
| Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
⊢ toGrothendieck C K ≤ sInf {J | K ≤ ofGrothendieck C J} | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | apply le_sInf | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
⊢ ∀ b ∈ {J | K ≤ ofGrothendieck C J}, toGrothendieck C K ≤ b | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro J hJ | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
case a.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ ofGrothendieck C J}
⊢ toGrothendieck C K ≤ J | /-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "coveri... | intro X S hS | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } := by
apply le_antisy... | Mathlib.CategoryTheory.Sites.Coverage.271_0.qkZFgqEgDC2P633 | /--
An alternative characterization of the Grothendieck topology associated to a coverage `K`:
it is the infimum of all Grothendieck topologies whose associated coverage contains `K`.
-/
theorem toGrothendieck_eq_sInf (K : Coverage C) : toGrothendieck _ K =
sInf {J | K ≤ ofGrothendieck _ J } | Mathlib_CategoryTheory_Sites_Coverage |
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