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 |
|---|---|---|---|---|---|---|
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
ppredB predB : K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_denom_eq : denominators g n = ppredB
succ_nth_denom_eq : denominators g (n + 1) = predB
⊢ denominators g (n + 2) = gp.b * predB + gp.a * ppredB | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.continuants n = conts ∧ conts.b = ppredB | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.62_0.nOytPSFGrohRR6p | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
ppredB predB : K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
nth_denom_eq : denominators g n = ppredB
succ_nth_denom_eq : denominators g (n + 1) = predB
⊢ ∃ conts, continuants g n = conts ∧ conts.b = ppredB
case intro.in... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | exact exists_conts_b_of_denom nth_denom_eq | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.62_0.nOytPSFGrohRR6p | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
case intro.intro.refl
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
predB : K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
succ_nth_denom_eq : denominators g (n + 1) = predB
ppredConts : Pair K
nth_conts_eq : continuants g n = ppredConts
nth_denom_eq : denominators g... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | obtain ⟨predConts, succ_nth_conts_eq, ⟨rfl⟩⟩ :
∃ conts, g.continuants (n + 1) = conts ∧ conts.b = predB | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.62_0.nOytPSFGrohRR6p | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
predB : K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
succ_nth_denom_eq : denominators g (n + 1) = predB
ppredConts : Pair K
nth_conts_eq : continuants g n = ppredConts
nth_denom_eq : denominators g n = ppredConts.b
⊢ ∃ ... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | exact exists_conts_b_of_denom succ_nth_denom_eq | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.62_0.nOytPSFGrohRR6p | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
case intro.intro.refl.intro.intro.refl
K : Type u_1
g : GeneralizedContinuedFraction K
n : ℕ
inst✝ : DivisionRing K
gp : Pair K
succ_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp
ppredConts : Pair K
nth_conts_eq : continuants g n = ppredConts
nth_denom_eq : denominators g n = ppredConts.b
predConts : Pair K
succ_nt... | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | rw [denom_eq_conts_b, continuants_recurrence succ_nth_s_eq nth_conts_eq succ_nth_conts_eq] | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence.62_0.nOytPSFGrohRR6p | /-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -/
theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppred... | Mathlib_Algebra_ContinuedFractions_ContinuantsRecurrence |
p : ℝ[X]
⊢ Tendsto (fun x => eval x p / rexp x) atTop (𝓝 0) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Polynomial.Eval
/-!
# Limits of `P(x) / e ^ x` for a polynomial `P`
In this file we prove that $\... | induction p using Polynomial.induction_on' with
| h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n)
| h_add p q hp hq => simpa [add_div] using hp.add hq | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by
| Mathlib.Analysis.SpecialFunctions.PolynomialExp.27_0.0J48LYNgQ4iVYMM | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) | Mathlib_Analysis_SpecialFunctions_PolynomialExp |
p : ℝ[X]
⊢ Tendsto (fun x => eval x p / rexp x) atTop (𝓝 0) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Polynomial.Eval
/-!
# Limits of `P(x) / e ^ x` for a polynomial `P`
In this file we prove that $\... | induction p using Polynomial.induction_on' with
| h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n)
| h_add p q hp hq => simpa [add_div] using hp.add hq | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by
| Mathlib.Analysis.SpecialFunctions.PolynomialExp.27_0.0J48LYNgQ4iVYMM | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) | Mathlib_Analysis_SpecialFunctions_PolynomialExp |
case h_monomial
n : ℕ
c : ℝ
⊢ Tendsto (fun x => eval x ((monomial n) c) / rexp x) atTop (𝓝 0) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Polynomial.Eval
/-!
# Limits of `P(x) / e ^ x` for a polynomial `P`
In this file we prove that $\... | | h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n) | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by
induction p using Polynomial.induction_on' with
| Mathlib.Analysis.SpecialFunctions.PolynomialExp.27_0.0J48LYNgQ4iVYMM | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) | Mathlib_Analysis_SpecialFunctions_PolynomialExp |
case h_monomial
n : ℕ
c : ℝ
⊢ Tendsto (fun x => eval x ((monomial n) c) / rexp x) atTop (𝓝 0) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Polynomial.Eval
/-!
# Limits of `P(x) / e ^ x` for a polynomial `P`
In this file we prove that $\... | simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n) | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by
induction p using Polynomial.induction_on' with
| h_monomial n c => | Mathlib.Analysis.SpecialFunctions.PolynomialExp.27_0.0J48LYNgQ4iVYMM | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) | Mathlib_Analysis_SpecialFunctions_PolynomialExp |
case h_add
p q : ℝ[X]
hp : Tendsto (fun x => eval x p / rexp x) atTop (𝓝 0)
hq : Tendsto (fun x => eval x q / rexp x) atTop (𝓝 0)
⊢ Tendsto (fun x => eval x (p + q) / rexp x) atTop (𝓝 0) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Polynomial.Eval
/-!
# Limits of `P(x) / e ^ x` for a polynomial `P`
In this file we prove that $\... | | h_add p q hp hq => simpa [add_div] using hp.add hq | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by
induction p using Polynomial.induction_on' with
| h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n)
| Mathlib.Analysis.SpecialFunctions.PolynomialExp.27_0.0J48LYNgQ4iVYMM | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) | Mathlib_Analysis_SpecialFunctions_PolynomialExp |
case h_add
p q : ℝ[X]
hp : Tendsto (fun x => eval x p / rexp x) atTop (𝓝 0)
hq : Tendsto (fun x => eval x q / rexp x) atTop (𝓝 0)
⊢ Tendsto (fun x => eval x (p + q) / rexp x) atTop (𝓝 0) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Polynomial.Eval
/-!
# Limits of `P(x) / e ^ x` for a polynomial `P`
In this file we prove that $\... | simpa [add_div] using hp.add hq | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by
induction p using Polynomial.induction_on' with
| h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n)
| h_add p q hp hq => | Mathlib.Analysis.SpecialFunctions.PolynomialExp.27_0.0J48LYNgQ4iVYMM | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) | Mathlib_Analysis_SpecialFunctions_PolynomialExp |
R : Type u
inst✝ : Ring R
M N : ModuleCat R
f : M ⟶ N
hf : Mono f
⊢ (LinearEquiv.toModuleIso'
(LinearEquiv.symm (Submodule.quotEquivOfEqBot (LinearMap.ker f) (_ : LinearMap.ker f = ⊥)) ≪≫ₗ
(LinearMap.quotKerEquivRange f ≪≫ₗ
LinearEquiv.ofEq (LinearMap.range f) (LinearMap.ker (Submodu... | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheo... | ext | /-- In the category of modules, every monomorphism is normal. -/
def normalMono (hf : Mono f) : NormalMono f where
Z := of R (N ⧸ LinearMap.range f)
g := f.range.mkQ
w := LinearMap.range_mkQ_comp _
isLimit :=
/- The following [invalid Lean code](https://github.com/leanprover-community/lean/issues/341)
... | Mathlib.Algebra.Category.ModuleCat.Abelian.32_0.YQEqfRpdC3YQKHM | /-- In the category of modules, every monomorphism is normal. -/
def normalMono (hf : Mono f) : NormalMono f where
Z | Mathlib_Algebra_Category_ModuleCat_Abelian |
case h
R : Type u
inst✝ : Ring R
M N : ModuleCat R
f : M ⟶ N
hf : Mono f
x✝ : ↑M
⊢ ((LinearEquiv.toModuleIso'
(LinearEquiv.symm (Submodule.quotEquivOfEqBot (LinearMap.ker f) (_ : LinearMap.ker f = ⊥)) ≪≫ₗ
(LinearMap.quotKerEquivRange f ≪≫ₗ
LinearEquiv.ofEq (LinearMap.range f) (... | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheo... | rfl | /-- In the category of modules, every monomorphism is normal. -/
def normalMono (hf : Mono f) : NormalMono f where
Z := of R (N ⧸ LinearMap.range f)
g := f.range.mkQ
w := LinearMap.range_mkQ_comp _
isLimit :=
/- The following [invalid Lean code](https://github.com/leanprover-community/lean/issues/341)
... | Mathlib.Algebra.Category.ModuleCat.Abelian.32_0.YQEqfRpdC3YQKHM | /-- In the category of modules, every monomorphism is normal. -/
def normalMono (hf : Mono f) : NormalMono f where
Z | Mathlib_Algebra_Category_ModuleCat_Abelian |
R : Type u
inst✝ : Ring R
M N : ModuleCat R
f : M ⟶ N
hf : Epi f
⊢ Cofork.π (cokernelCocone (Submodule.subtype (LinearMap.ker f))) ≫
(LinearEquiv.toModuleIso'
(Submodule.quotEquivOfEq (LinearMap.range (Submodule.subtype (LinearMap.ker f))) (LinearMap.ker f)
(_ : LinearMap.range (Submodul... | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheo... | ext | /-- In the category of modules, every epimorphism is normal. -/
def normalEpi (hf : Epi f) : NormalEpi f where
W := of R (LinearMap.ker f)
g := (LinearMap.ker f).subtype
w := LinearMap.comp_ker_subtype _
isColimit :=
/- The following invalid Lean code might help you understand what's going on here:
... | Mathlib.Algebra.Category.ModuleCat.Abelian.55_0.YQEqfRpdC3YQKHM | /-- In the category of modules, every epimorphism is normal. -/
def normalEpi (hf : Epi f) : NormalEpi f where
W | Mathlib_Algebra_Category_ModuleCat_Abelian |
case h
R : Type u
inst✝ : Ring R
M N : ModuleCat R
f : M ⟶ N
hf : Epi f
x✝ : ↑((parallelPair (Submodule.subtype (LinearMap.ker f)) 0).obj WalkingParallelPair.one)
⊢ (Cofork.π (cokernelCocone (Submodule.subtype (LinearMap.ker f))) ≫
(LinearEquiv.toModuleIso'
(Submodule.quotEquivOfEq (LinearMap.range ... | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheo... | rfl | /-- In the category of modules, every epimorphism is normal. -/
def normalEpi (hf : Epi f) : NormalEpi f where
W := of R (LinearMap.ker f)
g := (LinearMap.ker f).subtype
w := LinearMap.comp_ker_subtype _
isColimit :=
/- The following invalid Lean code might help you understand what's going on here:
... | Mathlib.Algebra.Category.ModuleCat.Abelian.55_0.YQEqfRpdC3YQKHM | /-- In the category of modules, every epimorphism is normal. -/
def normalEpi (hf : Epi f) : NormalEpi f where
W | Mathlib_Algebra_Category_ModuleCat_Abelian |
R : Type u
inst✝ : Ring R
M N : ModuleCat R
f : M ⟶ N
O : ModuleCat R
g : N ⟶ O
⊢ Exact f g ↔ range f = ker g | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheo... | rw [abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)] | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g := by
| Mathlib.Algebra.Category.ModuleCat.Abelian.122_0.YQEqfRpdC3YQKHM | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g | Mathlib_Algebra_Category_ModuleCat_Abelian |
R : Type u
inst✝ : Ring R
M N : ModuleCat R
f : M ⟶ N
O : ModuleCat R
g : N ⟶ O
⊢ f ≫ g = 0 ∧ Fork.ι (kernelCone g) ≫ Cofork.π (cokernelCocone f) = 0 ↔ range f = ker g | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheo... | exact
⟨fun h => le_antisymm (range_le_ker_iff.2 h.1) (ker_le_range_iff.2 h.2), fun h =>
⟨range_le_ker_iff.1 <| le_of_eq h, ker_le_range_iff.1 <| le_of_eq h.symm⟩⟩ | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g := by
rw [abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
| Mathlib.Algebra.Category.ModuleCat.Abelian.122_0.YQEqfRpdC3YQKHM | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g | Mathlib_Algebra_Category_ModuleCat_Abelian |
α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
x y : ToType m
⊢ x.fst = y.fst ↔ x.fst = y.fst | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | cases x | @[simp]
theorem Multiset.coe_eq {x y : m} : (x : α) = (y : α) ↔ x.1 = y.1 := by
| Mathlib.Data.Multiset.Fintype.75_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.coe_eq {x y : m} : (x : α) = (y : α) ↔ x.1 = y.1 | Mathlib_Data_Multiset_Fintype |
case mk
α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
y : ToType m
fst✝ : α
snd✝ : Fin (count fst✝ m)
⊢ { fst := fst✝, snd := snd✝ }.fst = y.fst ↔ { fst := fst✝, snd := snd✝ }.fst = y.fst | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | cases y | @[simp]
theorem Multiset.coe_eq {x y : m} : (x : α) = (y : α) ↔ x.1 = y.1 := by
cases x
| Mathlib.Data.Multiset.Fintype.75_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.coe_eq {x y : m} : (x : α) = (y : α) ↔ x.1 = y.1 | Mathlib_Data_Multiset_Fintype |
case mk.mk
α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
fst✝¹ : α
snd✝¹ : Fin (count fst✝¹ m)
fst✝ : α
snd✝ : Fin (count fst✝ m)
⊢ { fst := fst✝¹, snd := snd✝¹ }.fst = { fst := fst✝, snd := snd✝ }.fst ↔
{ fst := fst✝¹, snd := snd✝¹ }.fst = { fst := fst✝, snd := snd✝ }.fst | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rfl | @[simp]
theorem Multiset.coe_eq {x y : m} : (x : α) = (y : α) ↔ x.1 = y.1 := by
cases x
cases y
| Mathlib.Data.Multiset.Fintype.75_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.coe_eq {x y : m} : (x : α) = (y : α) ↔ x.1 = y.1 | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
⊢ ∀ (x : α × ℕ),
(x ∈
Finset.biUnion (Multiset.toFinset m) fun x =>
Finset.map { toFun := Prod.mk x, inj' := (_ : Function.Injective (Prod.mk x)) }
(Finset.range (Multiset.count x m))) ↔
x ∈ {p | p.2 < Multiset.count p.1 m} | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rintro ⟨x, i⟩ | instance : Fintype { p : α × ℕ | p.2 < m.count p.1 } :=
Fintype.ofFinset
(m.toFinset.biUnion fun x ↦ (Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩)
(by
| Mathlib.Data.Multiset.Fintype.104_0.gXgzg6nY9WO2bG2 | instance : Fintype { p : α × ℕ | p.2 < m.count p.1 } | Mathlib_Data_Multiset_Fintype |
case mk
α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
x : α
i : ℕ
⊢ ((x, i) ∈
Finset.biUnion (Multiset.toFinset m) fun x =>
Finset.map { toFun := Prod.mk x, inj' := (_ : Function.Injective (Prod.mk x)) }
(Finset.range (Multiset.count x m))) ↔
(x, i) ∈ {p | p.2 < Multiset.count p.1 m} | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp only [Finset.mem_biUnion, Multiset.mem_toFinset, Finset.mem_map, Finset.mem_range,
Function.Embedding.coeFn_mk, Prod.mk.inj_iff, Set.mem_setOf_eq] | instance : Fintype { p : α × ℕ | p.2 < m.count p.1 } :=
Fintype.ofFinset
(m.toFinset.biUnion fun x ↦ (Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩)
(by
rintro ⟨x, i⟩
| Mathlib.Data.Multiset.Fintype.104_0.gXgzg6nY9WO2bG2 | instance : Fintype { p : α × ℕ | p.2 < m.count p.1 } | Mathlib_Data_Multiset_Fintype |
case mk
α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
x : α
i : ℕ
⊢ (∃ a ∈ m, ∃ a_1 < Multiset.count a m, a = x ∧ a_1 = i) ↔ i < Multiset.count x m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp only [← and_assoc, exists_eq_right, and_iff_right_iff_imp] | instance : Fintype { p : α × ℕ | p.2 < m.count p.1 } :=
Fintype.ofFinset
(m.toFinset.biUnion fun x ↦ (Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩)
(by
rintro ⟨x, i⟩
simp only [Finset.mem_biUnion, Multiset.mem_toFinset, Finset.mem_map, Finset.mem_range,
Function.Embedding.... | Mathlib.Data.Multiset.Fintype.104_0.gXgzg6nY9WO2bG2 | instance : Fintype { p : α × ℕ | p.2 < m.count p.1 } | Mathlib_Data_Multiset_Fintype |
case mk
α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
x : α
i : ℕ
⊢ i < Multiset.count x m → x ∈ m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | exact fun h ↦ Multiset.count_pos.mp (pos_of_gt h) | instance : Fintype { p : α × ℕ | p.2 < m.count p.1 } :=
Fintype.ofFinset
(m.toFinset.biUnion fun x ↦ (Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩)
(by
rintro ⟨x, i⟩
simp only [Finset.mem_biUnion, Multiset.mem_toFinset, Finset.mem_map, Finset.mem_range,
Function.Embedding.... | Mathlib.Data.Multiset.Fintype.104_0.gXgzg6nY9WO2bG2 | instance : Fintype { p : α × ℕ | p.2 < m.count p.1 } | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : m₁ ≤ m₂
⊢ toEnumFinset m₁ ⊆ toEnumFinset m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | intro p | @[mono]
theorem Multiset.toEnumFinset_mono {m₁ m₂ : Multiset α} (h : m₁ ≤ m₂) :
m₁.toEnumFinset ⊆ m₂.toEnumFinset := by
| Mathlib.Data.Multiset.Fintype.131_0.gXgzg6nY9WO2bG2 | @[mono]
theorem Multiset.toEnumFinset_mono {m₁ m₂ : Multiset α} (h : m₁ ≤ m₂) :
m₁.toEnumFinset ⊆ m₂.toEnumFinset | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : m₁ ≤ m₂
p : α × ℕ
⊢ p ∈ toEnumFinset m₁ → p ∈ toEnumFinset m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp only [Multiset.mem_toEnumFinset] | @[mono]
theorem Multiset.toEnumFinset_mono {m₁ m₂ : Multiset α} (h : m₁ ≤ m₂) :
m₁.toEnumFinset ⊆ m₂.toEnumFinset := by
intro p
| Mathlib.Data.Multiset.Fintype.131_0.gXgzg6nY9WO2bG2 | @[mono]
theorem Multiset.toEnumFinset_mono {m₁ m₂ : Multiset α} (h : m₁ ≤ m₂) :
m₁.toEnumFinset ⊆ m₂.toEnumFinset | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : m₁ ≤ m₂
p : α × ℕ
⊢ p.2 < count p.1 m₁ → p.2 < count p.1 m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | exact gt_of_ge_of_gt (Multiset.le_iff_count.mp h p.1) | @[mono]
theorem Multiset.toEnumFinset_mono {m₁ m₂ : Multiset α} (h : m₁ ≤ m₂) :
m₁.toEnumFinset ⊆ m₂.toEnumFinset := by
intro p
simp only [Multiset.mem_toEnumFinset]
| Mathlib.Data.Multiset.Fintype.131_0.gXgzg6nY9WO2bG2 | @[mono]
theorem Multiset.toEnumFinset_mono {m₁ m₂ : Multiset α} (h : m₁ ≤ m₂) :
m₁.toEnumFinset ⊆ m₂.toEnumFinset | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
⊢ toEnumFinset m₁ ⊆ toEnumFinset m₂ ↔ m₁ ≤ m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩ | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
| Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
⊢ m₁ ≤ m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [Multiset.le_iff_count] | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
| Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
⊢ ∀ (a : α), count a m₁ ≤ count a m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | intro x | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
| Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
x : α
⊢ count x m₁ ≤ count x m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | by_cases hx : x ∈ m₁ | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
| Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
case pos
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
x : α
hx : x ∈ m₁
⊢ count x m₁ ≤ count x m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | apply Nat.le_of_pred_lt | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
by_cases hx : x ∈ m₁
· | Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
case pos.a
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
x : α
hx : x ∈ m₁
⊢ Nat.pred (count x m₁) < count x m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | have : (x, m₁.count x - 1) ∈ m₁.toEnumFinset := by
rw [Multiset.mem_toEnumFinset]
exact Nat.pred_lt (ne_of_gt (Multiset.count_pos.mpr hx)) | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
by_cases hx : x ∈ m₁
· apply Nat.le_of_pred_lt
| Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
x : α
hx : x ∈ m₁
⊢ (x, count x m₁ - 1) ∈ toEnumFinset m₁ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [Multiset.mem_toEnumFinset] | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
by_cases hx : x ∈ m₁
· apply Nat.le_of_pred_lt
have : (x, m₁.count x - 1) ∈ m₁.toEnumFinset := by... | Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
x : α
hx : x ∈ m₁
⊢ (x, count x m₁ - 1).2 < count (x, count x m₁ - 1).1 m₁ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | exact Nat.pred_lt (ne_of_gt (Multiset.count_pos.mpr hx)) | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
by_cases hx : x ∈ m₁
· apply Nat.le_of_pred_lt
have : (x, m₁.count x - 1) ∈ m₁.toEnumFinset := by... | Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
case pos.a
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
x : α
hx : x ∈ m₁
this : (x, count x m₁ - 1) ∈ toEnumFinset m₁
⊢ Nat.pred (count x m₁) < count x m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simpa only [Multiset.mem_toEnumFinset] using h this | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
by_cases hx : x ∈ m₁
· apply Nat.le_of_pred_lt
have : (x, m₁.count x - 1) ∈ m₁.toEnumFinset := by... | Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
case neg
α : Type u_1
inst✝ : DecidableEq α
m m₁ m₂ : Multiset α
h : toEnumFinset m₁ ⊆ toEnumFinset m₂
x : α
hx : x ∉ m₁
⊢ count x m₁ ≤ count x m₂ | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp [hx] | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ := by
refine' ⟨fun h ↦ _, Multiset.toEnumFinset_mono⟩
rw [Multiset.le_iff_count]
intro x
by_cases hx : x ∈ m₁
· apply Nat.le_of_pred_lt
have : (x, m₁.count x - 1) ∈ m₁.toEnumFinset := by... | Mathlib.Data.Multiset.Fintype.139_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEnumFinset_subset_iff {m₁ m₂ : Multiset α} :
m₁.toEnumFinset ⊆ m₂.toEnumFinset ↔ m₁ ≤ m₂ | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Function.Injective fun x => (x.fst, ↑x.snd) | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | intro ⟨x, i, hi⟩ ⟨y, j, hj⟩ | /-- The embedding from a multiset into `α × ℕ` where the second coordinate enumerates repeats.
If you are looking for the function `m → α`, that would be plain `(↑)`. -/
@[simps]
def Multiset.coeEmbedding (m : Multiset α) : m ↪ α × ℕ
where
toFun x := (x, x.2)
inj' := by
| Mathlib.Data.Multiset.Fintype.154_0.gXgzg6nY9WO2bG2 | /-- The embedding from a multiset into `α × ℕ` where the second coordinate enumerates repeats.
If you are looking for the function `m → α`, that would be plain `(↑)`. -/
@[simps]
def Multiset.coeEmbedding (m : Multiset α) : m ↪ α × ℕ
where
toFun x | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : α
i : ℕ
hi : i < count x m
y : α
j : ℕ
hj : j < count y m
⊢ (fun x => (x.fst, ↑x.snd)) { fst := x, snd := { val := i, isLt := hi } } =
(fun x => (x.fst, ↑x.snd)) { fst := y, snd := { val := j, isLt := hj } } →
{ fst := x, snd := { val := i, isLt := hi }... | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rintro ⟨⟩ | /-- The embedding from a multiset into `α × ℕ` where the second coordinate enumerates repeats.
If you are looking for the function `m → α`, that would be plain `(↑)`. -/
@[simps]
def Multiset.coeEmbedding (m : Multiset α) : m ↪ α × ℕ
where
toFun x := (x, x.2)
inj' := by
intro ⟨x, i, hi⟩ ⟨y, j, hj⟩
| Mathlib.Data.Multiset.Fintype.154_0.gXgzg6nY9WO2bG2 | /-- The embedding from a multiset into `α × ℕ` where the second coordinate enumerates repeats.
If you are looking for the function `m → α`, that would be plain `(↑)`. -/
@[simps]
def Multiset.coeEmbedding (m : Multiset α) : m ↪ α × ℕ
where
toFun x | Mathlib_Data_Multiset_Fintype |
case refl
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : α
i : ℕ
hi hj : i < count x m
⊢ { fst := x, snd := { val := i, isLt := hi } } = { fst := x, snd := { val := i, isLt := hj } } | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rfl | /-- The embedding from a multiset into `α × ℕ` where the second coordinate enumerates repeats.
If you are looking for the function `m → α`, that would be plain `(↑)`. -/
@[simps]
def Multiset.coeEmbedding (m : Multiset α) : m ↪ α × ℕ
where
toFun x := (x, x.2)
inj' := by
intro ⟨x, i, hi⟩ ⟨y, j, hj⟩
rintr... | Mathlib.Data.Multiset.Fintype.154_0.gXgzg6nY9WO2bG2 | /-- The embedding from a multiset into `α × ℕ` where the second coordinate enumerates repeats.
If you are looking for the function `m → α`, that would be plain `(↑)`. -/
@[simps]
def Multiset.coeEmbedding (m : Multiset α) : m ↪ α × ℕ
where
toFun x | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : ToType m
⊢ (coeEmbedding m) x ∈ toEnumFinset m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [Multiset.mem_toEnumFinset] | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x :=
⟨m.coeEmbedding x, by
| Mathlib.Data.Multiset.Fintype.166_0.gXgzg6nY9WO2bG2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : ToType m
⊢ ((coeEmbedding m) x).2 < count ((coeEmbedding m) x).1 m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | exact x.2.2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x :=
⟨m.coeEmbedding x, by
rw [Multiset.mem_toEnumFinset]
| Mathlib.Data.Multiset.Fintype.166_0.gXgzg6nY9WO2bG2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : { x // x ∈ toEnumFinset m }
⊢ (↑x).2 < count (↑x).1 m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [← Multiset.mem_toEnumFinset] | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x :=
⟨m.coeEmbedding x, by
rw [Multiset.mem_toEnumFinset]
exact x.2.2⟩
invFun x :=
⟨x.1.1... | Mathlib.Data.Multiset.Fintype.166_0.gXgzg6nY9WO2bG2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : { x // x ∈ toEnumFinset m }
⊢ ↑x ∈ toEnumFinset m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | exact x.2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x :=
⟨m.coeEmbedding x, by
rw [Multiset.mem_toEnumFinset]
exact x.2.2⟩
invFun x :=
⟨x.1.1... | Mathlib.Data.Multiset.Fintype.166_0.gXgzg6nY9WO2bG2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Function.LeftInverse (fun x => { fst := (↑x).1, snd := { val := (↑x).2, isLt := (_ : (↑x).2 < count (↑x).1 m) } })
fun x => { val := (coeEmbedding m) x, property := (_ : (coeEmbedding m) x ∈ toEnumFinset m) } | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rintro ⟨x, i, h⟩ | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x :=
⟨m.coeEmbedding x, by
rw [Multiset.mem_toEnumFinset]
exact x.2.2⟩
invFun x :=
⟨x.1.1... | Mathlib.Data.Multiset.Fintype.166_0.gXgzg6nY9WO2bG2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x | Mathlib_Data_Multiset_Fintype |
case mk.mk
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : α
i : ℕ
h : i < count x m
⊢ (fun x => { fst := (↑x).1, snd := { val := (↑x).2, isLt := (_ : (↑x).2 < count (↑x).1 m) } })
((fun x => { val := (coeEmbedding m) x, property := (_ : (coeEmbedding m) x ∈ toEnumFinset m) })
{ fst := x, snd := ... | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rfl | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x :=
⟨m.coeEmbedding x, by
rw [Multiset.mem_toEnumFinset]
exact x.2.2⟩
invFun x :=
⟨x.1.1... | Mathlib.Data.Multiset.Fintype.166_0.gXgzg6nY9WO2bG2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Function.RightInverse (fun x => { fst := (↑x).1, snd := { val := (↑x).2, isLt := (_ : (↑x).2 < count (↑x).1 m) } })
fun x => { val := (coeEmbedding m) x, property := (_ : (coeEmbedding m) x ∈ toEnumFinset m) } | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rintro ⟨⟨x, i⟩, h⟩ | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x :=
⟨m.coeEmbedding x, by
rw [Multiset.mem_toEnumFinset]
exact x.2.2⟩
invFun x :=
⟨x.1.1... | Mathlib.Data.Multiset.Fintype.166_0.gXgzg6nY9WO2bG2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x | Mathlib_Data_Multiset_Fintype |
case mk.mk
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : α
i : ℕ
h : (x, i) ∈ toEnumFinset m
⊢ (fun x => { val := (coeEmbedding m) x, property := (_ : (coeEmbedding m) x ∈ toEnumFinset m) })
((fun x => { fst := (↑x).1, snd := { val := (↑x).2, isLt := (_ : (↑x).2 < count (↑x).1 m) } })
{ val := ... | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rfl | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x :=
⟨m.coeEmbedding x, by
rw [Multiset.mem_toEnumFinset]
exact x.2.2⟩
invFun x :=
⟨x.1.1... | Mathlib.Data.Multiset.Fintype.166_0.gXgzg6nY9WO2bG2 | /-- Another way to coerce a `Multiset` to a type is to go through `m.toEnumFinset` and coerce
that `Finset` to a type. -/
@[simps]
def Multiset.coeEquiv (m : Multiset α) : m ≃ m.toEnumFinset
where
toFun x | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Function.Embedding.trans (Equiv.toEmbedding (coeEquiv m)) (Function.Embedding.subtype fun x => x ∈ toEnumFinset m) =
coeEmbedding m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | ext | @[simp]
theorem Multiset.toEmbedding_coeEquiv_trans (m : Multiset α) :
m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype _) = m.coeEmbedding := by | Mathlib.Data.Multiset.Fintype.187_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEmbedding_coeEquiv_trans (m : Multiset α) :
m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype _) = m.coeEmbedding | Mathlib_Data_Multiset_Fintype |
case h.a
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x✝ : ToType m
⊢ ((Function.Embedding.trans (Equiv.toEmbedding (coeEquiv m)) (Function.Embedding.subtype fun x => x ∈ toEnumFinset m))
x✝).1 =
((coeEmbedding m) x✝).1 | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rfl | @[simp]
theorem Multiset.toEmbedding_coeEquiv_trans (m : Multiset α) :
m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype _) = m.coeEmbedding := by ext <;> | Mathlib.Data.Multiset.Fintype.187_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEmbedding_coeEquiv_trans (m : Multiset α) :
m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype _) = m.coeEmbedding | Mathlib_Data_Multiset_Fintype |
case h.a
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x✝ : ToType m
⊢ ((Function.Embedding.trans (Equiv.toEmbedding (coeEquiv m)) (Function.Embedding.subtype fun x => x ∈ toEnumFinset m))
x✝).2 =
((coeEmbedding m) x✝).2 | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rfl | @[simp]
theorem Multiset.toEmbedding_coeEquiv_trans (m : Multiset α) :
m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype _) = m.coeEmbedding := by ext <;> | Mathlib.Data.Multiset.Fintype.187_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.toEmbedding_coeEquiv_trans (m : Multiset α) :
m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype _) = m.coeEmbedding | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Finset.map (coeEmbedding m) Finset.univ = toEnumFinset m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | ext ⟨x, i⟩ | theorem Multiset.map_univ_coeEmbedding (m : Multiset α) :
(Finset.univ : Finset m).map m.coeEmbedding = m.toEnumFinset := by
| Mathlib.Data.Multiset.Fintype.197_0.gXgzg6nY9WO2bG2 | theorem Multiset.map_univ_coeEmbedding (m : Multiset α) :
(Finset.univ : Finset m).map m.coeEmbedding = m.toEnumFinset | Mathlib_Data_Multiset_Fintype |
case a.mk
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : α
i : ℕ
⊢ (x, i) ∈ Finset.map (coeEmbedding m) Finset.univ ↔ (x, i) ∈ toEnumFinset m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp only [Fin.exists_iff, Finset.mem_map, Finset.mem_univ, Multiset.coeEmbedding_apply,
Prod.mk.inj_iff, exists_true_left, Multiset.exists_coe, Multiset.coe_mk, Fin.val_mk,
exists_prop, exists_eq_right_right, exists_eq_right, Multiset.mem_toEnumFinset, iff_self_iff,
true_and_iff] | theorem Multiset.map_univ_coeEmbedding (m : Multiset α) :
(Finset.univ : Finset m).map m.coeEmbedding = m.toEnumFinset := by
ext ⟨x, i⟩
| Mathlib.Data.Multiset.Fintype.197_0.gXgzg6nY9WO2bG2 | theorem Multiset.map_univ_coeEmbedding (m : Multiset α) :
(Finset.univ : Finset m).map m.coeEmbedding = m.toEnumFinset | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : α
⊢ Finset.filter (fun p => x = p.1) (toEnumFinset m) =
Finset.map { toFun := Prod.mk x, inj' := (_ : Function.Injective (Prod.mk x)) } (Finset.range (count x m)) | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | ext ⟨y, i⟩ | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ := by
| Mathlib.Data.Multiset.Fintype.206_0.gXgzg6nY9WO2bG2 | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ | Mathlib_Data_Multiset_Fintype |
case a.mk
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x y : α
i : ℕ
⊢ (y, i) ∈ Finset.filter (fun p => x = p.1) (toEnumFinset m) ↔
(y, i) ∈ Finset.map { toFun := Prod.mk x, inj' := (_ : Function.Injective (Prod.mk x)) } (Finset.range (count x m)) | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp only [eq_comm, Finset.mem_filter, Multiset.mem_toEnumFinset, Finset.mem_map,
Finset.mem_range, Function.Embedding.coeFn_mk, Prod.mk.inj_iff, exists_prop,
exists_eq_right_right', and_congr_left_iff] | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ := by
ext ⟨y, i⟩
| Mathlib.Data.Multiset.Fintype.206_0.gXgzg6nY9WO2bG2 | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ | Mathlib_Data_Multiset_Fintype |
case a.mk
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x y : α
i : ℕ
⊢ x = y → (i < count y m ↔ i < count x m) | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rintro rfl | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ := by
ext ⟨y, i⟩
simp only [eq_comm, Finset.mem_filter, Multiset.mem_toEnumFinset, Finset.mem_map,
Finset.mem_range, Function.Embe... | Mathlib.Data.Multiset.Fintype.206_0.gXgzg6nY9WO2bG2 | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ | Mathlib_Data_Multiset_Fintype |
case a.mk
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : α
i : ℕ
⊢ i < count x m ↔ i < count x m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rfl | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ := by
ext ⟨y, i⟩
simp only [eq_comm, Finset.mem_filter, Multiset.mem_toEnumFinset, Finset.mem_map,
Finset.mem_range, Function.Embe... | Mathlib.Data.Multiset.Fintype.206_0.gXgzg6nY9WO2bG2 | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ map Prod.fst (toEnumFinset m).val = m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | ext x | @[simp]
theorem Multiset.map_toEnumFinset_fst (m : Multiset α) : m.toEnumFinset.val.map Prod.fst = m := by
| Mathlib.Data.Multiset.Fintype.217_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.map_toEnumFinset_fst (m : Multiset α) : m.toEnumFinset.val.map Prod.fst = m | Mathlib_Data_Multiset_Fintype |
case a
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
x : α
⊢ count x (map Prod.fst (toEnumFinset m).val) = count x m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp only [Multiset.count_map, ← Finset.filter_val, Multiset.toEnumFinset_filter_eq,
Finset.map_val, Finset.range_val, Multiset.card_map, Multiset.card_range] | @[simp]
theorem Multiset.map_toEnumFinset_fst (m : Multiset α) : m.toEnumFinset.val.map Prod.fst = m := by
ext x
| Mathlib.Data.Multiset.Fintype.217_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.map_toEnumFinset_fst (m : Multiset α) : m.toEnumFinset.val.map Prod.fst = m | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Finset.image Prod.fst (toEnumFinset m) = toFinset m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [Finset.image, Multiset.map_toEnumFinset_fst] | @[simp]
theorem Multiset.image_toEnumFinset_fst (m : Multiset α) :
m.toEnumFinset.image Prod.fst = m.toFinset := by
| Mathlib.Data.Multiset.Fintype.224_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.image_toEnumFinset_fst (m : Multiset α) :
m.toEnumFinset.image Prod.fst = m.toFinset | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ map (fun x => x.fst) Finset.univ.val = m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | have := m.map_toEnumFinset_fst | @[simp]
theorem Multiset.map_univ_coe (m : Multiset α) :
(Finset.univ : Finset m).val.map (fun x : m ↦ (x : α)) = m := by
| Mathlib.Data.Multiset.Fintype.230_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.map_univ_coe (m : Multiset α) :
(Finset.univ : Finset m).val.map (fun x : m ↦ (x : α)) = m | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
this : map Prod.fst (toEnumFinset m).val = m
⊢ map (fun x => x.fst) Finset.univ.val = m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [← m.map_univ_coeEmbedding] at this | @[simp]
theorem Multiset.map_univ_coe (m : Multiset α) :
(Finset.univ : Finset m).val.map (fun x : m ↦ (x : α)) = m := by
have := m.map_toEnumFinset_fst
| Mathlib.Data.Multiset.Fintype.230_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.map_univ_coe (m : Multiset α) :
(Finset.univ : Finset m).val.map (fun x : m ↦ (x : α)) = m | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
this : map Prod.fst (Finset.map (coeEmbedding m) Finset.univ).val = m
⊢ map (fun x => x.fst) Finset.univ.val = m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simpa only [Finset.map_val, Multiset.coeEmbedding_apply, Multiset.map_map,
Function.comp_apply] using this | @[simp]
theorem Multiset.map_univ_coe (m : Multiset α) :
(Finset.univ : Finset m).val.map (fun x : m ↦ (x : α)) = m := by
have := m.map_toEnumFinset_fst
rw [← m.map_univ_coeEmbedding] at this
| Mathlib.Data.Multiset.Fintype.230_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.map_univ_coe (m : Multiset α) :
(Finset.univ : Finset m).val.map (fun x : m ↦ (x : α)) = m | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ : Multiset α
β : Type u_2
m : Multiset α
f : α → β
⊢ map (fun x => f x.fst) Finset.univ.val = map f m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | erw [← Multiset.map_map] | @[simp]
theorem Multiset.map_univ {β : Type*} (m : Multiset α) (f : α → β) :
((Finset.univ : Finset m).val.map fun (x : m) ↦ f (x : α)) = m.map f := by
| Mathlib.Data.Multiset.Fintype.239_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.map_univ {β : Type*} (m : Multiset α) (f : α → β) :
((Finset.univ : Finset m).val.map fun (x : m) ↦ f (x : α)) = m.map f | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ : Multiset α
β : Type u_2
m : Multiset α
f : α → β
⊢ map f (map (fun x => x.fst) Finset.univ.val) = map f m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [Multiset.map_univ_coe] | @[simp]
theorem Multiset.map_univ {β : Type*} (m : Multiset α) (f : α → β) :
((Finset.univ : Finset m).val.map fun (x : m) ↦ f (x : α)) = m.map f := by
erw [← Multiset.map_map]
| Mathlib.Data.Multiset.Fintype.239_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.map_univ {β : Type*} (m : Multiset α) (f : α → β) :
((Finset.univ : Finset m).val.map fun (x : m) ↦ f (x : α)) = m.map f | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Finset.card (toEnumFinset m) = card m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [Finset.card, ← Multiset.card_map Prod.fst m.toEnumFinset.val] | @[simp]
theorem Multiset.card_toEnumFinset (m : Multiset α) : m.toEnumFinset.card = Multiset.card m := by
| Mathlib.Data.Multiset.Fintype.246_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.card_toEnumFinset (m : Multiset α) : m.toEnumFinset.card = Multiset.card m | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ card (map Prod.fst (toEnumFinset m).val) = card m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | congr | @[simp]
theorem Multiset.card_toEnumFinset (m : Multiset α) : m.toEnumFinset.card = Multiset.card m := by
rw [Finset.card, ← Multiset.card_map Prod.fst m.toEnumFinset.val]
| Mathlib.Data.Multiset.Fintype.246_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.card_toEnumFinset (m : Multiset α) : m.toEnumFinset.card = Multiset.card m | Mathlib_Data_Multiset_Fintype |
case h.e_6.h
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ map Prod.fst (toEnumFinset m).val = m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | exact m.map_toEnumFinset_fst | @[simp]
theorem Multiset.card_toEnumFinset (m : Multiset α) : m.toEnumFinset.card = Multiset.card m := by
rw [Finset.card, ← Multiset.card_map Prod.fst m.toEnumFinset.val]
congr
| Mathlib.Data.Multiset.Fintype.246_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.card_toEnumFinset (m : Multiset α) : m.toEnumFinset.card = Multiset.card m | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Fintype.card (ToType m) = card m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [Fintype.card_congr m.coeEquiv] | @[simp]
theorem Multiset.card_coe (m : Multiset α) : Fintype.card m = Multiset.card m := by
| Mathlib.Data.Multiset.Fintype.253_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.card_coe (m : Multiset α) : Fintype.card m = Multiset.card m | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝ : DecidableEq α
m✝ m : Multiset α
⊢ Fintype.card { x // x ∈ toEnumFinset m } = card m | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp only [Fintype.card_coe, card_toEnumFinset] | @[simp]
theorem Multiset.card_coe (m : Multiset α) : Fintype.card m = Multiset.card m := by
rw [Fintype.card_congr m.coeEquiv]
| Mathlib.Data.Multiset.Fintype.253_0.gXgzg6nY9WO2bG2 | @[simp]
theorem Multiset.card_coe (m : Multiset α) : Fintype.card m = Multiset.card m | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝¹ : DecidableEq α
m✝ : Multiset α
inst✝ : CommMonoid α
m : Multiset α
⊢ prod m = ∏ x : ToType m, x.fst | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | congr | @[to_additive]
theorem Multiset.prod_eq_prod_coe [CommMonoid α] (m : Multiset α) : m.prod = ∏ x : m, (x : α) := by
| Mathlib.Data.Multiset.Fintype.259_0.gXgzg6nY9WO2bG2 | @[to_additive]
theorem Multiset.prod_eq_prod_coe [CommMonoid α] (m : Multiset α) : m.prod = ∏ x : m, (x : α) | Mathlib_Data_Multiset_Fintype |
case e_a
α : Type u_1
inst✝¹ : DecidableEq α
m✝ : Multiset α
inst✝ : CommMonoid α
m : Multiset α
⊢ m = map (fun x => x.fst) Finset.univ.val | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | erw [map_univ_coe] | @[to_additive]
theorem Multiset.prod_eq_prod_coe [CommMonoid α] (m : Multiset α) : m.prod = ∏ x : m, (x : α) := by
congr
-- Porting note: `simp` fails with "maximum recursion depth has been reached"
| Mathlib.Data.Multiset.Fintype.259_0.gXgzg6nY9WO2bG2 | @[to_additive]
theorem Multiset.prod_eq_prod_coe [CommMonoid α] (m : Multiset α) : m.prod = ∏ x : m, (x : α) | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝¹ : DecidableEq α
m✝ : Multiset α
inst✝ : CommMonoid α
m : Multiset α
⊢ prod m = ∏ x in toEnumFinset m, x.1 | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | congr | @[to_additive]
theorem Multiset.prod_eq_prod_toEnumFinset [CommMonoid α] (m : Multiset α) :
m.prod = ∏ x in m.toEnumFinset, x.1 := by
| Mathlib.Data.Multiset.Fintype.267_0.gXgzg6nY9WO2bG2 | @[to_additive]
theorem Multiset.prod_eq_prod_toEnumFinset [CommMonoid α] (m : Multiset α) :
m.prod = ∏ x in m.toEnumFinset, x.1 | Mathlib_Data_Multiset_Fintype |
case e_a
α : Type u_1
inst✝¹ : DecidableEq α
m✝ : Multiset α
inst✝ : CommMonoid α
m : Multiset α
⊢ m = map (fun x => x.1) (toEnumFinset m).val | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | simp | @[to_additive]
theorem Multiset.prod_eq_prod_toEnumFinset [CommMonoid α] (m : Multiset α) :
m.prod = ∏ x in m.toEnumFinset, x.1 := by
congr
| Mathlib.Data.Multiset.Fintype.267_0.gXgzg6nY9WO2bG2 | @[to_additive]
theorem Multiset.prod_eq_prod_toEnumFinset [CommMonoid α] (m : Multiset α) :
m.prod = ∏ x in m.toEnumFinset, x.1 | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝¹ : DecidableEq α
m✝ : Multiset α
β : Type u_2
inst✝ : CommMonoid β
m : Multiset α
f : α → ℕ → β
⊢ ∏ x in toEnumFinset m, f x.1 x.2 = ∏ x : ToType m, f x.fst ↑x.snd | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [Fintype.prod_equiv m.coeEquiv (fun x ↦ f x x.2) fun x ↦ f x.1.1 x.1.2] | @[to_additive]
theorem Multiset.prod_toEnumFinset {β : Type*} [CommMonoid β] (m : Multiset α) (f : α → ℕ → β) :
∏ x in m.toEnumFinset, f x.1 x.2 = ∏ x : m, f x x.2 := by
| Mathlib.Data.Multiset.Fintype.275_0.gXgzg6nY9WO2bG2 | @[to_additive]
theorem Multiset.prod_toEnumFinset {β : Type*} [CommMonoid β] (m : Multiset α) (f : α → ℕ → β) :
∏ x in m.toEnumFinset, f x.1 x.2 = ∏ x : m, f x x.2 | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝¹ : DecidableEq α
m✝ : Multiset α
β : Type u_2
inst✝ : CommMonoid β
m : Multiset α
f : α → ℕ → β
⊢ ∏ x in toEnumFinset m, f x.1 x.2 = ∏ x : { x // x ∈ toEnumFinset m }, f (↑x).1 (↑x).2 | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rw [← m.toEnumFinset.prod_coe_sort fun x ↦ f x.1 x.2] | @[to_additive]
theorem Multiset.prod_toEnumFinset {β : Type*} [CommMonoid β] (m : Multiset α) (f : α → ℕ → β) :
∏ x in m.toEnumFinset, f x.1 x.2 = ∏ x : m, f x x.2 := by
rw [Fintype.prod_equiv m.coeEquiv (fun x ↦ f x x.2) fun x ↦ f x.1.1 x.1.2]
· | Mathlib.Data.Multiset.Fintype.275_0.gXgzg6nY9WO2bG2 | @[to_additive]
theorem Multiset.prod_toEnumFinset {β : Type*} [CommMonoid β] (m : Multiset α) (f : α → ℕ → β) :
∏ x in m.toEnumFinset, f x.1 x.2 = ∏ x : m, f x x.2 | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝¹ : DecidableEq α
m✝ : Multiset α
β : Type u_2
inst✝ : CommMonoid β
m : Multiset α
f : α → ℕ → β
⊢ ∀ (x : ToType m), f x.fst ↑x.snd = f (↑((coeEquiv m) x)).1 (↑((coeEquiv m) x)).2 | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | intro x | @[to_additive]
theorem Multiset.prod_toEnumFinset {β : Type*} [CommMonoid β] (m : Multiset α) (f : α → ℕ → β) :
∏ x in m.toEnumFinset, f x.1 x.2 = ∏ x : m, f x x.2 := by
rw [Fintype.prod_equiv m.coeEquiv (fun x ↦ f x x.2) fun x ↦ f x.1.1 x.1.2]
· rw [← m.toEnumFinset.prod_coe_sort fun x ↦ f x.1 x.2]
· | Mathlib.Data.Multiset.Fintype.275_0.gXgzg6nY9WO2bG2 | @[to_additive]
theorem Multiset.prod_toEnumFinset {β : Type*} [CommMonoid β] (m : Multiset α) (f : α → ℕ → β) :
∏ x in m.toEnumFinset, f x.1 x.2 = ∏ x : m, f x x.2 | Mathlib_Data_Multiset_Fintype |
α : Type u_1
inst✝¹ : DecidableEq α
m✝ : Multiset α
β : Type u_2
inst✝ : CommMonoid β
m : Multiset α
f : α → ℕ → β
x : ToType m
⊢ f x.fst ↑x.snd = f (↑((coeEquiv m) x)).1 (↑((coeEquiv m) x)).2 | /-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Prod.Lex
#align_import data.multiset.fintype from "leanprover-community/math... | rfl | @[to_additive]
theorem Multiset.prod_toEnumFinset {β : Type*} [CommMonoid β] (m : Multiset α) (f : α → ℕ → β) :
∏ x in m.toEnumFinset, f x.1 x.2 = ∏ x : m, f x x.2 := by
rw [Fintype.prod_equiv m.coeEquiv (fun x ↦ f x x.2) fun x ↦ f x.1.1 x.1.2]
· rw [← m.toEnumFinset.prod_coe_sort fun x ↦ f x.1 x.2]
· intro x... | Mathlib.Data.Multiset.Fintype.275_0.gXgzg6nY9WO2bG2 | @[to_additive]
theorem Multiset.prod_toEnumFinset {β : Type*} [CommMonoid β] (m : Multiset α) (f : α → ℕ → β) :
∏ x in m.toEnumFinset, f x.1 x.2 = ∏ x : m, f x x.2 | Mathlib_Data_Multiset_Fintype |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Precoherent C
inst✝ : HasFiniteCoproducts C
X Y Z : C
f : X ⟶ Y
g : Z ⟶ Y
x✝ : EffectiveEpi g
⊢ ∃ W h, ∃ (_ : EffectiveEpi h), ∃ i, i ≫ g = h ≫ f | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have hp := Precoherent.pullback f PUnit (fun () ↦ Z) (fun () ↦ g) | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.77_0.rkSRr0zuqme90Yu | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Precoherent C
inst✝ : HasFiniteCoproducts C
X Y Z : C
f : X ⟶ Y
g : Z ⟶ Y
x✝ : EffectiveEpi g
hp :
(EffectiveEpiFamily (fun x => Z) fun x =>
match x with
| PUnit.unit => g) →
∃ β x X₂ π₂,
EffectiveEpiFamily X₂ π₂ ∧
∃ i ι,
∀ (b : β)... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simp only [exists_const] at hp | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ := by
have hp := Precoherent.pullback f PUnit (fun () ↦ Z) (fun () ↦ g)
| Mathlib.CategoryTheory.Sites.RegularExtensive.77_0.rkSRr0zuqme90Yu | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Precoherent C
inst✝ : HasFiniteCoproducts C
X Y Z : C
f : X ⟶ Y
g : Z ⟶ Y
x✝ : EffectiveEpi g
hp :
(EffectiveEpiFamily (fun x => Z) fun x => g) →
∃ β x X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f
⊢ ∃ W h, ∃ (_ : EffectiveEpi h), ∃ i, i ≫ g = ... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [← effectiveEpi_iff_effectiveEpiFamily g] at hp | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ := by
have hp := Precoherent.pullback f PUnit (fun () ↦ Z) (fun () ↦ g)
simp only [exists_const] at hp
| Mathlib.CategoryTheory.Sites.RegularExtensive.77_0.rkSRr0zuqme90Yu | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Precoherent C
inst✝ : HasFiniteCoproducts C
X Y Z : C
f : X ⟶ Y
g : Z ⟶ Y
x✝ : EffectiveEpi g
hp : EffectiveEpi g → ∃ β x X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f
⊢ ∃ W h, ∃ (_ : EffectiveEpi h), ∃ i, i ≫ g = h ≫ f | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨β, _, X₂, π₂, h, ι, hι⟩ := hp inferInstance | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ := by
have hp := Precoherent.pullback f PUnit (fun () ↦ Z) (fun () ↦ g)
simp only [exists_const] at hp
rw [← effectiveEpi_iff_effectiveEpiFamily g] at hp
| Mathlib.CategoryTheory.Sites.RegularExtensive.77_0.rkSRr0zuqme90Yu | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Precoherent C
inst✝ : HasFiniteCoproducts C
X Y Z : C
f : X ⟶ Y
g : Z ⟶ Y
x✝ : EffectiveEpi g
hp : EffectiveEpi g → ∃ β x X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f
β : Type
w✝ : Fintype β
X₂ : β → C
π₂ : (... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨∐ X₂, Sigma.desc π₂, inferInstance, Sigma.desc ι, ?_⟩ | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ := by
have hp := Precoherent.pullback f PUnit (fun () ↦ Z) (fun () ↦ g)
simp only [exists_const] at hp
rw [← effectiveEpi_iff_effectiveEpiFamily g] at hp
obtain ⟨β, _, X₂, π₂, h, ι, hι⟩ := hp inferInstance
... | Mathlib.CategoryTheory.Sites.RegularExtensive.77_0.rkSRr0zuqme90Yu | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Precoherent C
inst✝ : HasFiniteCoproducts C
X Y Z : C
f : X ⟶ Y
g : Z ⟶ Y
x✝ : EffectiveEpi g
hp : EffectiveEpi g → ∃ β x X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f
β : Type
w✝ : Fintype β
X₂ : β → C
π₂ : (... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | ext b | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ := by
have hp := Precoherent.pullback f PUnit (fun () ↦ Z) (fun () ↦ g)
simp only [exists_const] at hp
rw [← effectiveEpi_iff_effectiveEpiFamily g] at hp
obtain ⟨β, _, X₂, π₂, h, ι, hι⟩ := hp inferInstance
... | Mathlib.CategoryTheory.Sites.RegularExtensive.77_0.rkSRr0zuqme90Yu | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.intro.intro.h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Precoherent C
inst✝ : HasFiniteCoproducts C
X Y Z : C
f : X ⟶ Y
g : Z ⟶ Y
x✝ : EffectiveEpi g
hp : EffectiveEpi g → ∃ β x X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f
β : Type
w✝ : Fintype β
X₂ : β → C
π₂ :... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | simpa using hι b | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ := by
have hp := Precoherent.pullback f PUnit (fun () ↦ Z) (fun () ↦ g)
simp only [exists_const] at hp
rw [← effectiveEpi_iff_effectiveEpiFamily g] at hp
obtain ⟨β, _, X₂, π₂, h, ι, hι⟩ := hp inferInstance
... | Mathlib.CategoryTheory.Sites.RegularExtensive.77_0.rkSRr0zuqme90Yu | instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C where
exists_fac {X Y Z} f g _ | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
⊢ ∀ ⦃X Y : C⦄ (f : Y ⟶ X),
∀ S ∈ (fun B => {S | ∃ X f, (S = Presieve.ofArrows (fun x => X) fun x => f) ∧ EffectiveEpi f}) X,
∃ T ∈ (fun B => {S | ∃ X f, (S = Presieve.ofArrows (fun x => X) fun x => f) ∧ EffectiveEpi f}) Y,
Presieve.FactorsThru... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro X Y f S ⟨Z, π, hπ, h_epi⟩ | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
| Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
⊢ ∃ T ∈ (fun B => {S | ∃ X f, (S = Presieve.ofArrows (fun x => X) fun x => f) ∧ EffectiveEpi f}) Y,
Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | have := Preregular.exists_fac f π | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
| Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
this : ∃ W h, ∃ (_ : EffectiveEpi h), ∃ i, i ≫ π = h ≫ f
⊢ ∃ T ∈ (fun B => {S | ∃ X f, (S = Presieve.ofArrows (fun x => X) fun x => f) ∧ E... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | obtain ⟨W, h, _, i, this⟩ := this | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ ∃ T ∈ (fun B => {S | ∃ X f, (S = Presiev... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨Presieve.singleton h, ⟨?_, ?_⟩⟩ | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ Presieve.singleton h ∈ (fun B =... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | exact ⟨W, h, by {rw [Presieve.ofArrows_pUnit h]}, inferInstance⟩ | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ Presieve.singleton h = Presieve.ofArrows (fun x => W) fun x => h | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | {rw [Presieve.ofArrows_pUnit h]} | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ Presieve.singleton h = Presieve.ofArrows (fun x => W) fun x => h | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [Presieve.ofArrows_pUnit h] | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ Presieve.FactorsThruAlong (Pres... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro W g hg | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
W✝ : C
h : W✝ ⟶ Y
w✝ : EffectiveEpi h
i : W✝ ⟶ Z
this : i ≫ π = h ≫ f
W : C
g : W ⟶ Y
hg : Presieve.... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | cases hg | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ ∃ W_1 i e, S e ∧ i ≫ e = h ≫... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | refine ⟨Z, i, π, ⟨?_, this⟩⟩ | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
S : Presieve X
Z : C
π : Z ⟶ X
hπ : S = Presieve.ofArrows (fun x => Z) fun x => π
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ S π | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | cases hπ | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2.mk.refl
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
Z : C
π : Z ⟶ X
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ Presieve.ofArrows (fun x => Z) (fun x => π) π | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | rw [Presieve.ofArrows_pUnit] | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
case intro.intro.intro.intro.refine_2.mk.refl
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preregular C
X Y : C
f : Y ⟶ X
Z : C
π : Z ⟶ X
h_epi : EffectiveEpi π
W : C
h : W ⟶ Y
w✝ : EffectiveEpi h
i : W ⟶ Z
this : i ≫ π = h ≫ f
⊢ Presieve.singleton π π | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | exact Presieve.singleton.mk | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B := { S | ∃ (X : C) (f : X ⟶ B), S = Presieve.ofArrows (fun (_ : Unit) ↦ X)
(fun (_ : Unit) ↦ f) ∧ EffectiveEpi f }
pullback := by
intro X Y f S ⟨Z, π, hπ, h_epi⟩
have := Preregular.exi... | Mathlib.CategoryTheory.Sites.RegularExtensive.87_0.rkSRr0zuqme90Yu | /--
The regular coverage on a regular category `C`.
-/
def regularCoverage [Preregular C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
⊢ ∀ ⦃X Y : C⦄ (f : Y ⟶ X),
∀ S ∈ (fun B => {S | ∃ α x X π, S = Presieve.ofArrows X π ∧ IsIso (Sigma.desc π)}) X,
∃ T ∈ (fun B => {S | ∃ α x X π, S = Presieve.ofArrows X π ∧ IsIso (Sigma.desc π)}) Y,
Presieve.FactorsThruAlong T S f | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | intro X Y f S ⟨α, hα, Z, π, hS, h_iso⟩ | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : FinitaryPreExtensive C
X Y : C
f : Y ⟶ X
S : Presieve X
α : Type
hα : Fintype α
Z : α → C
π : (a : α) → Z a ⟶ X
hS : S = Presieve.ofArrows Z π
h_iso : IsIso (Sigma.desc π)
⊢ ∃ T ∈ (fun B => {S | ∃ α x X π, S = Presieve.ofArrows X π ∧ IsIso (Sigma.desc π)}) Y, Presieve.Facto... | /-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.... | let Z' : α → C := fun a ↦ pullback f (π a) | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B := { S | ∃ (α : Type) (_ : Fintype α) (X : α → C) (π : (a : α) → (X a ⟶ B)),
S = Presieve.ofArrows X π ∧ IsIso (S... | Mathlib.CategoryTheory.Sites.RegularExtensive.106_0.rkSRr0zuqme90Yu | /--
The extensive coverage on an extensive category `C`
TODO: use general colimit API instead of `IsIso (Sigma.desc π)`
-/
def extensiveCoverage [FinitaryPreExtensive C] : Coverage C where
covering B | Mathlib_CategoryTheory_Sites_RegularExtensive |
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