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
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (I... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have I : Icc a x ⊆ Icc a b := Icc_subset_Icc_right hx.2 | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (I... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact (has_deriv_within_taylorWithinEval_at_Icc x h (I ht) hf.of_succ hf').mono I | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWith... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have := norm_image_sub_le_of_norm_deriv_le_segment' A h' x (right_mem_Icc.2 hx.1) | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWith... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp only [taylorWithinEval_self] at this | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWith... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | refine' this.trans_eq _ | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWith... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [abs_of_nonneg (sub_nonneg.mpr hx.1)] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C x : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
h : a < b
hf' : DifferentiableOn ℝ (iteratedDerivWith... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | ring | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rcases eq_or_lt_of_le hab with (rfl | h) | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
⊢ ∃ C, ∀ x ∈ Icc a a, ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ C * (x - a) ^ (n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | refine' ⟨0, fun x hx => _⟩ | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
x : ℝ
hx : x ∈ Icc a a
⊢ ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ 0 * (x - a) ^ (n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | have : x = a := by simpa [← le_antisymm_iff] using hx | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
x : ℝ
hx : x ∈ Icc a a
⊢ x = a | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simpa [← le_antisymm_iff] using hx | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
x : ℝ
hx : x ∈ Icc a a
this : x = a
⊢ ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ 0 * (x - a) ^ (n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | simp [← this] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
h : a < b
⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | let g : ℝ → ℝ := fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
h : a < b
g : ℝ → ℝ := fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖
⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | use SupSet.sSup (g '' Icc a b) / (n !) | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
h : a < b
g : ℝ → ℝ := fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖
⊢ ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ sSup (... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | intro x hx | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
h : a < b
g : ℝ → ℝ := fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖
x : ℝ
hx : x ∈ Icc a b
⊢ ‖f x - taylorWithinEval f n (Icc a b) a x‖ ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | rw [div_mul_eq_mul_div₀] | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
h : a < b
g : ℝ → ℝ := fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖
x : ℝ
hx : x ∈ Icc a b
⊢ ‖f x - taylorWithinEval f n (Icc a b) a x‖ ... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | refine' taylor_mean_remainder_bound hab hf hx fun y => _ | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
h : a < b
g : ℝ → ℝ := fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖
x : ℝ
hx : x ∈ Icc a b
y : ℝ
⊢ y ∈ Icc a b → ‖iteratedDerivWithin (n... | /-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Polynomial.Module
... | exact (hf.continuousOn_iteratedDerivWithin rfl.le <| uniqueDiffOn_Icc h).norm.le_sSup_image_Icc | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib.Analysis.Calculus.Taylor.355_0.INXnr4jrmq9RIjK | /-- **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. -/
the... | Mathlib_Analysis_Calculus_Taylor |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Equiv.Perm α
f : α ↪ β
x✝ : α
⊢ invOfMemRange f ((fun a => { val := f a, property := (_ : f a ∈ Set.range ⇑f) }) x✝) = x✝ | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | simp | /-- Computably turn an embedding `f : α ↪ β` into an equiv `α ≃ Set.range f`,
if `α` is a `Fintype`. Has poor computational performance, due to exhaustive searching in
constructed inverse. When a better inverse is known, use `Equiv.ofLeftInverse'` or
`Equiv.ofLeftInverse` instead. This is the computable version of `Equ... | Mathlib.Logic.Equiv.Fintype.33_0.fUeUwBtkCE24P9E | /-- Computably turn an embedding `f : α ↪ β` into an equiv `α ≃ Set.range f`,
if `α` is a `Fintype`. Has poor computational performance, due to exhaustive searching in
constructed inverse. When a better inverse is known, use `Equiv.ofLeftInverse'` or
`Equiv.ofLeftInverse` instead. This is the computable version of `Equ... | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Equiv.Perm α
f : α ↪ β
x✝ : ↑(Set.range ⇑f)
⊢ (fun a => { val := f a, property := (_ : f a ∈ Set.range ⇑f) }) (invOfMemRange f x✝) = x✝ | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | simp | /-- Computably turn an embedding `f : α ↪ β` into an equiv `α ≃ Set.range f`,
if `α` is a `Fintype`. Has poor computational performance, due to exhaustive searching in
constructed inverse. When a better inverse is known, use `Equiv.ofLeftInverse'` or
`Equiv.ofLeftInverse` instead. This is the computable version of `Equ... | Mathlib.Logic.Equiv.Fintype.33_0.fUeUwBtkCE24P9E | /-- Computably turn an embedding `f : α ↪ β` into an equiv `α ≃ Set.range f`,
if `α` is a `Fintype`. Has poor computational performance, due to exhaustive searching in
constructed inverse. When a better inverse is known, use `Equiv.ofLeftInverse'` or
`Equiv.ofLeftInverse` instead. This is the computable version of `Equ... | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Equiv.Perm α
f : α ↪ β
a : α
⊢ (toEquivRange f).symm { val := f a, property := (_ : f a ∈ Set.range ⇑f) } = a | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | simp [Equiv.symm_apply_eq] | @[simp]
theorem Function.Embedding.toEquivRange_symm_apply_self (a : α) :
f.toEquivRange.symm ⟨f a, Set.mem_range_self a⟩ = a := by | Mathlib.Logic.Equiv.Fintype.48_0.fUeUwBtkCE24P9E | @[simp]
theorem Function.Embedding.toEquivRange_symm_apply_self (a : α) :
f.toEquivRange.symm ⟨f a, Set.mem_range_self a⟩ = a | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Equiv.Perm α
f : α ↪ β
⊢ toEquivRange f = Equiv.ofInjective ⇑f (_ : Injective ⇑f) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | ext | theorem Function.Embedding.toEquivRange_eq_ofInjective :
f.toEquivRange = Equiv.ofInjective f f.injective := by
| Mathlib.Logic.Equiv.Fintype.53_0.fUeUwBtkCE24P9E | theorem Function.Embedding.toEquivRange_eq_ofInjective :
f.toEquivRange = Equiv.ofInjective f f.injective | Mathlib_Logic_Equiv_Fintype |
case H.a
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Equiv.Perm α
f : α ↪ β
x✝ : α
⊢ ↑((toEquivRange f) x✝) = ↑((Equiv.ofInjective ⇑f (_ : Injective ⇑f)) x✝) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | simp | theorem Function.Embedding.toEquivRange_eq_ofInjective :
f.toEquivRange = Equiv.ofInjective f f.injective := by
ext
| Mathlib.Logic.Equiv.Fintype.53_0.fUeUwBtkCE24P9E | theorem Function.Embedding.toEquivRange_eq_ofInjective :
f.toEquivRange = Equiv.ofInjective f f.injective | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Perm α
f : α ↪ β
a : α
⊢ (viaFintypeEmbedding e f) (f a) = f (e a) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | rw [Equiv.Perm.viaFintypeEmbedding] | @[simp]
theorem Equiv.Perm.viaFintypeEmbedding_apply_image (a : α) :
e.viaFintypeEmbedding f (f a) = f (e a) := by
| Mathlib.Logic.Equiv.Fintype.70_0.fUeUwBtkCE24P9E | @[simp]
theorem Equiv.Perm.viaFintypeEmbedding_apply_image (a : α) :
e.viaFintypeEmbedding f (f a) = f (e a) | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Perm α
f : α ↪ β
a : α
⊢ (extendDomain e (Function.Embedding.toEquivRange f)) (f a) = f (e a) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | convert Equiv.Perm.extendDomain_apply_image e (Function.Embedding.toEquivRange f) a | @[simp]
theorem Equiv.Perm.viaFintypeEmbedding_apply_image (a : α) :
e.viaFintypeEmbedding f (f a) = f (e a) := by
rw [Equiv.Perm.viaFintypeEmbedding]
| Mathlib.Logic.Equiv.Fintype.70_0.fUeUwBtkCE24P9E | @[simp]
theorem Equiv.Perm.viaFintypeEmbedding_apply_image (a : α) :
e.viaFintypeEmbedding f (f a) = f (e a) | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Perm α
f : α ↪ β
b : β
h : b ∈ Set.range ⇑f
⊢ (viaFintypeEmbedding e f) b = f (e (Function.Embedding.invOfMemRange f { val := b, property := h })) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | simp only [viaFintypeEmbedding, Function.Embedding.invOfMemRange] | theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : β} (h : b ∈ Set.range f) :
e.viaFintypeEmbedding f b = f (e (f.invOfMemRange ⟨b, h⟩)) := by
| Mathlib.Logic.Equiv.Fintype.77_0.fUeUwBtkCE24P9E | theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : β} (h : b ∈ Set.range f) :
e.viaFintypeEmbedding f b = f (e (f.invOfMemRange ⟨b, h⟩)) | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Perm α
f : α ↪ β
b : β
h : b ∈ Set.range ⇑f
⊢ (extendDomain e (Function.Embedding.toEquivRange f)) b =
f (e (Function.Injective.invOfMemRange (_ : Function.Injective ⇑f) { val := b, property := h })) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | rw [Equiv.Perm.extendDomain_apply_subtype] | theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : β} (h : b ∈ Set.range f) :
e.viaFintypeEmbedding f b = f (e (f.invOfMemRange ⟨b, h⟩)) := by
simp only [viaFintypeEmbedding, Function.Embedding.invOfMemRange]
| Mathlib.Logic.Equiv.Fintype.77_0.fUeUwBtkCE24P9E | theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : β} (h : b ∈ Set.range f) :
e.viaFintypeEmbedding f b = f (e (f.invOfMemRange ⟨b, h⟩)) | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Perm α
f : α ↪ β
b : β
h : b ∈ Set.range ⇑f
⊢ ↑((Function.Embedding.toEquivRange f) (e ((Function.Embedding.toEquivRange f).symm { val := b, property := ?h }))) =
f (e (Function.Injective.invOfMemRange (_ : Function.Injective ⇑f) { val := b, pro... | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | congr | theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : β} (h : b ∈ Set.range f) :
e.viaFintypeEmbedding f b = f (e (f.invOfMemRange ⟨b, h⟩)) := by
simp only [viaFintypeEmbedding, Function.Embedding.invOfMemRange]
rw [Equiv.Perm.extendDomain_apply_subtype]
| Mathlib.Logic.Equiv.Fintype.77_0.fUeUwBtkCE24P9E | theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : β} (h : b ∈ Set.range f) :
e.viaFintypeEmbedding f b = f (e (f.invOfMemRange ⟨b, h⟩)) | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq β
e : Perm α
f : α ↪ β
b : β
h : b ∉ Set.range ⇑f
⊢ (viaFintypeEmbedding e f) b = b | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | rwa [Equiv.Perm.viaFintypeEmbedding, Equiv.Perm.extendDomain_apply_not_subtype] | theorem Equiv.Perm.viaFintypeEmbedding_apply_not_mem_range {b : β} (h : b ∉ Set.range f) :
e.viaFintypeEmbedding f b = b := by
| Mathlib.Logic.Equiv.Fintype.84_0.fUeUwBtkCE24P9E | theorem Equiv.Perm.viaFintypeEmbedding_apply_not_mem_range {b : β} (h : b ∉ Set.range f) :
e.viaFintypeEmbedding f b = b | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e : Perm α
f : α ↪ β
inst✝¹ : DecidableEq α
inst✝ : Fintype β
⊢ sign (viaFintypeEmbedding e f) = sign e | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | simp [Equiv.Perm.viaFintypeEmbedding] | @[simp]
theorem Equiv.Perm.viaFintypeEmbedding_sign [DecidableEq α] [Fintype β] :
Equiv.Perm.sign (e.viaFintypeEmbedding f) = Equiv.Perm.sign e := by
| Mathlib.Logic.Equiv.Fintype.89_0.fUeUwBtkCE24P9E | @[simp]
theorem Equiv.Perm.viaFintypeEmbedding_sign [DecidableEq α] [Fintype β] :
Equiv.Perm.sign (e.viaFintypeEmbedding f) = Equiv.Perm.sign e | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : p x
⊢ (extendSubtype e) x = ↑(e { val := x, property := hx }) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | dsimp only [extendSubtype] | theorem extendSubtype_apply_of_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
e.extendSubtype x = e ⟨x, hx⟩ := by
| Mathlib.Logic.Equiv.Fintype.117_0.fUeUwBtkCE24P9E | theorem extendSubtype_apply_of_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
e.extendSubtype x = e ⟨x, hx⟩ | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : p x
⊢ (subtypeCongr e (toCompl e)) x = ↑(e { val := x, property := hx }) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | simp only [subtypeCongr, Equiv.trans_apply, Equiv.sumCongr_apply] | theorem extendSubtype_apply_of_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
e.extendSubtype x = e ⟨x, hx⟩ := by
dsimp only [extendSubtype]
| Mathlib.Logic.Equiv.Fintype.117_0.fUeUwBtkCE24P9E | theorem extendSubtype_apply_of_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
e.extendSubtype x = e ⟨x, hx⟩ | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : p x
⊢ (sumCompl fun x => q x) (Sum.map (⇑e) (⇑(toCompl e)) ((sumCompl fun x => p x).symm x)) =
↑(e { val := x, property... | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | rw [sumCompl_apply_symm_of_pos _ _ hx, Sum.map_inl, sumCompl_apply_inl] | theorem extendSubtype_apply_of_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
e.extendSubtype x = e ⟨x, hx⟩ := by
dsimp only [extendSubtype]
simp only [subtypeCongr, Equiv.trans_apply, Equiv.sumCongr_apply]
| Mathlib.Logic.Equiv.Fintype.117_0.fUeUwBtkCE24P9E | theorem extendSubtype_apply_of_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
e.extendSubtype x = e ⟨x, hx⟩ | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : p x
⊢ q ((extendSubtype e) x) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | convert (e ⟨x, hx⟩).2 | theorem extendSubtype_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
q (e.extendSubtype x) := by
| Mathlib.Logic.Equiv.Fintype.124_0.fUeUwBtkCE24P9E | theorem extendSubtype_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
q (e.extendSubtype x) | Mathlib_Logic_Equiv_Fintype |
case h.e'_1
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : p x
⊢ (extendSubtype e) x = ↑(e { val := x, property := hx }) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | rw [e.extendSubtype_apply_of_mem _ hx] | theorem extendSubtype_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
q (e.extendSubtype x) := by
convert (e ⟨x, hx⟩).2
| Mathlib.Logic.Equiv.Fintype.124_0.fUeUwBtkCE24P9E | theorem extendSubtype_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
q (e.extendSubtype x) | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ (extendSubtype e) x = ↑((toCompl e) { val := x, property := hx }) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | dsimp only [extendSubtype] | theorem extendSubtype_apply_of_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
e.extendSubtype x = e.toCompl ⟨x, hx⟩ := by
| Mathlib.Logic.Equiv.Fintype.130_0.fUeUwBtkCE24P9E | theorem extendSubtype_apply_of_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
e.extendSubtype x = e.toCompl ⟨x, hx⟩ | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ (subtypeCongr e (toCompl e)) x = ↑((toCompl e) { val := x, property := hx }) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | simp only [subtypeCongr, Equiv.trans_apply, Equiv.sumCongr_apply] | theorem extendSubtype_apply_of_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
e.extendSubtype x = e.toCompl ⟨x, hx⟩ := by
dsimp only [extendSubtype]
| Mathlib.Logic.Equiv.Fintype.130_0.fUeUwBtkCE24P9E | theorem extendSubtype_apply_of_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
e.extendSubtype x = e.toCompl ⟨x, hx⟩ | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ (sumCompl fun x => q x) (Sum.map (⇑e) (⇑(toCompl e)) ((sumCompl fun x => p x).symm x)) =
↑((toCompl e) { val := ... | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | rw [sumCompl_apply_symm_of_neg _ _ hx, Sum.map_inr, sumCompl_apply_inr] | theorem extendSubtype_apply_of_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
e.extendSubtype x = e.toCompl ⟨x, hx⟩ := by
dsimp only [extendSubtype]
simp only [subtypeCongr, Equiv.trans_apply, Equiv.sumCongr_apply]
| Mathlib.Logic.Equiv.Fintype.130_0.fUeUwBtkCE24P9E | theorem extendSubtype_apply_of_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
e.extendSubtype x = e.toCompl ⟨x, hx⟩ | Mathlib_Logic_Equiv_Fintype |
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ ¬q ((extendSubtype e) x) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | convert (e.toCompl ⟨x, hx⟩).2 | theorem extendSubtype_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
¬q (e.extendSubtype x) := by
| Mathlib.Logic.Equiv.Fintype.137_0.fUeUwBtkCE24P9E | theorem extendSubtype_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
¬q (e.extendSubtype x) | Mathlib_Logic_Equiv_Fintype |
case h.e'_1.h.e'_1
α : Type u_1
β : Type u_2
inst✝³ : Fintype α
inst✝² : DecidableEq β
e✝ : Perm α
f : α ↪ β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ (extendSubtype e) x = ↑((toCompl e) { val := x, property := hx }) | /-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | rw [e.extendSubtype_apply_of_not_mem _ hx] | theorem extendSubtype_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
¬q (e.extendSubtype x) := by
convert (e.toCompl ⟨x, hx⟩).2
| Mathlib.Logic.Equiv.Fintype.137_0.fUeUwBtkCE24P9E | theorem extendSubtype_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
¬q (e.extendSubtype x) | Mathlib_Logic_Equiv_Fintype |
𝕜 : Type u_1
inst✝ : IsROrC 𝕜
⊢ Tendsto (fun n => (↑n)⁻¹) atTop (nhds 0) | /-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, Patrick Massot
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.Complex.ReImTopology
/-!
# A collection of specific limit computations for `Is... | convert tendsto_algebraMap_inverse_atTop_nhds_0_nat 𝕜 | theorem IsROrC.tendsto_inverse_atTop_nhds_0_nat :
Tendsto (fun n : ℕ => (n : 𝕜)⁻¹) atTop (nhds 0) := by
| Mathlib.Analysis.SpecificLimits.IsROrC.18_0.GxvrNOjFh6rYkey | theorem IsROrC.tendsto_inverse_atTop_nhds_0_nat :
Tendsto (fun n : ℕ => (n : 𝕜)⁻¹) atTop (nhds 0) | Mathlib_Analysis_SpecificLimits_IsROrC |
case h.e'_3.h
𝕜 : Type u_1
inst✝ : IsROrC 𝕜
x✝ : ℕ
⊢ (↑x✝)⁻¹ = (⇑(algebraMap ℝ 𝕜) ∘ fun n => (↑n)⁻¹) x✝ | /-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, Patrick Massot
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.Complex.ReImTopology
/-!
# A collection of specific limit computations for `Is... | simp | theorem IsROrC.tendsto_inverse_atTop_nhds_0_nat :
Tendsto (fun n : ℕ => (n : 𝕜)⁻¹) atTop (nhds 0) := by
convert tendsto_algebraMap_inverse_atTop_nhds_0_nat 𝕜
| Mathlib.Analysis.SpecificLimits.IsROrC.18_0.GxvrNOjFh6rYkey | theorem IsROrC.tendsto_inverse_atTop_nhds_0_nat :
Tendsto (fun n : ℕ => (n : 𝕜)⁻¹) atTop (nhds 0) | Mathlib_Analysis_SpecificLimits_IsROrC |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
r : HomRel C
a b : C
m₁ m₂ : a ⟶ b
h : r m₁ m₂
⊢ CompClosure r m₁ m₂ | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simpa using CompClosure.intro (𝟙 _) m₁ m₂ (𝟙 _) h | theorem CompClosure.of {a b : C} (m₁ m₂ : a ⟶ b) (h : r m₁ m₂) : CompClosure r m₁ m₂ := by
| Mathlib.CategoryTheory.Quotient.65_0.34bZdkqpf1A9Wub | theorem CompClosure.of {a b : C} (m₁ m₂ : a ⟶ b) (h : r m₁ m₂) : CompClosure r m₁ m₂ | Mathlib_CategoryTheory_Quotient |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
r : HomRel C
a b c : C
f : a ⟶ b
a✝ b✝ : C
x : b ⟶ a✝
m₁ m₂ : a✝ ⟶ b✝
y : b✝ ⟶ c
h : r m₁ m₂
⊢ CompClosure r (f ≫ x ≫ m₁ ≫ y) (f ≫ x ≫ m₂ ≫ y) | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simpa using CompClosure.intro (f ≫ x) m₁ m₂ y h | theorem comp_left {a b c : C} (f : a ⟶ b) :
∀ (g₁ g₂ : b ⟶ c) (_ : CompClosure r g₁ g₂), CompClosure r (f ≫ g₁) (f ≫ g₂)
| _, _, ⟨x, m₁, m₂, y, h⟩ => by | Mathlib.CategoryTheory.Quotient.69_0.34bZdkqpf1A9Wub | theorem comp_left {a b c : C} (f : a ⟶ b) :
∀ (g₁ g₂ : b ⟶ c) (_ : CompClosure r g₁ g₂), CompClosure r (f ≫ g₁) (f ≫ g₂)
| _, _, ⟨x, m₁, m₂, y, h⟩ => by simpa using CompClosure.intro (f ≫ x) m₁ m₂ y h | Mathlib_CategoryTheory_Quotient |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
r : HomRel C
a b c : C
g : b ⟶ c
a✝ b✝ : C
x : a ⟶ a✝
m₁ m₂ : a✝ ⟶ b✝
y : b✝ ⟶ b
h : r m₁ m₂
⊢ CompClosure r ((x ≫ m₁ ≫ y) ≫ g) ((x ≫ m₂ ≫ y) ≫ g) | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simpa using CompClosure.intro x m₁ m₂ (y ≫ g) h | theorem comp_right {a b c : C} (g : b ⟶ c) :
∀ (f₁ f₂ : a ⟶ b) (_ : CompClosure r f₁ f₂), CompClosure r (f₁ ≫ g) (f₂ ≫ g)
| _, _, ⟨x, m₁, m₂, y, h⟩ => by | Mathlib.CategoryTheory.Quotient.74_0.34bZdkqpf1A9Wub | theorem comp_right {a b c : C} (g : b ⟶ c) :
∀ (f₁ f₂ : a ⟶ b) (_ : CompClosure r f₁ f₂), CompClosure r (f₁ ≫ g) (f₂ ≫ g)
| _, _, ⟨x, m₁, m₂, y, h⟩ => by simpa using CompClosure.intro x m₁ m₂ (y ≫ g) h | Mathlib_CategoryTheory_Quotient |
C : Type ?u.6549
inst✝ : Category.{?u.6553, ?u.6549} C
r : HomRel C
X✝ Y✝ : Quotient r
f : X✝ ⟶ Y✝
⊢ ∀ (a : X✝.as ⟶ Y✝.as), 𝟙 X✝ ≫ Quot.mk (CompClosure r) a = Quot.mk (CompClosure r) a | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp | instance category : Category (Quotient r) where
Hom := Hom r
id a := Quot.mk _ (𝟙 a.as)
comp := @comp _ _ r
comp_id f := Quot.inductionOn f $ by simp
id_comp f := Quot.inductionOn f $ by | Mathlib.CategoryTheory.Quotient.103_0.34bZdkqpf1A9Wub | instance category : Category (Quotient r) where
Hom | Mathlib_CategoryTheory_Quotient |
C : Type ?u.6549
inst✝ : Category.{?u.6553, ?u.6549} C
r : HomRel C
X✝ Y✝ : Quotient r
f : X✝ ⟶ Y✝
⊢ ∀ (a : X✝.as ⟶ Y✝.as), Quot.mk (CompClosure r) a ≫ 𝟙 Y✝ = Quot.mk (CompClosure r) a | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp | instance category : Category (Quotient r) where
Hom := Hom r
id a := Quot.mk _ (𝟙 a.as)
comp := @comp _ _ r
comp_id f := Quot.inductionOn f $ by | Mathlib.CategoryTheory.Quotient.103_0.34bZdkqpf1A9Wub | instance category : Category (Quotient r) where
Hom | Mathlib_CategoryTheory_Quotient |
C : Type ?u.6549
inst✝ : Category.{?u.6553, ?u.6549} C
r : HomRel C
W✝ X✝ Y✝ Z✝ : Quotient r
f : W✝ ⟶ X✝
g : X✝ ⟶ Y✝
h : Y✝ ⟶ Z✝
⊢ ∀ (a : Y✝.as ⟶ Z✝.as) (a_1 : X✝.as ⟶ Y✝.as) (a_2 : W✝.as ⟶ X✝.as),
(Quot.mk (CompClosure r) a_2 ≫ Quot.mk (CompClosure r) a_1) ≫ Quot.mk (CompClosure r) a =
Quot.mk (CompClosure r... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp | instance category : Category (Quotient r) where
Hom := Hom r
id a := Quot.mk _ (𝟙 a.as)
comp := @comp _ _ r
comp_id f := Quot.inductionOn f $ by simp
id_comp f := Quot.inductionOn f $ by simp
assoc f g h := Quot.inductionOn f $ Quot.inductionOn g $ Quot.inductionOn h $ by | Mathlib.CategoryTheory.Quotient.103_0.34bZdkqpf1A9Wub | instance category : Category (Quotient r) where
Hom | Mathlib_CategoryTheory_Quotient |
C : Type ?u.9520
inst✝ : Category.{?u.9524, ?u.9520} C
r : HomRel C
X✝ Y✝ : C
f : (functor r).obj X✝ ⟶ (functor r).obj Y✝
⊢ (functor r).map ((fun X Y f => Quot.out f) X✝ Y✝ f) = f | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | dsimp [functor] | noncomputable instance fullFunctor : Full (functor r) where
preimage := @fun X Y f ↦ Quot.out f
witness f := by
| Mathlib.CategoryTheory.Quotient.118_0.34bZdkqpf1A9Wub | noncomputable instance fullFunctor : Full (functor r) where
preimage | Mathlib_CategoryTheory_Quotient |
C : Type ?u.9520
inst✝ : Category.{?u.9524, ?u.9520} C
r : HomRel C
X✝ Y✝ : C
f : (functor r).obj X✝ ⟶ (functor r).obj Y✝
⊢ Quot.mk (CompClosure r) (Quot.out f) = f | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp | noncomputable instance fullFunctor : Full (functor r) where
preimage := @fun X Y f ↦ Quot.out f
witness f := by
dsimp [functor]
| Mathlib.CategoryTheory.Quotient.118_0.34bZdkqpf1A9Wub | noncomputable instance fullFunctor : Full (functor r) where
preimage | Mathlib_CategoryTheory_Quotient |
C : Type ?u.9843
inst✝ : Category.{?u.9847, ?u.9843} C
r : HomRel C
Y : Quotient r
⊢ (functor r).obj Y.as = Y | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | ext | instance essSurj_functor : EssSurj (functor r) where
mem_essImage Y :=
⟨Y.as, ⟨eqToIso (by
| Mathlib.CategoryTheory.Quotient.124_0.34bZdkqpf1A9Wub | instance essSurj_functor : EssSurj (functor r) where
mem_essImage Y | Mathlib_CategoryTheory_Quotient |
case as
C : Type ?u.9843
inst✝ : Category.{?u.9847, ?u.9843} C
r : HomRel C
Y : Quotient r
⊢ ((functor r).obj Y.as).as = Y.as | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rfl | instance essSurj_functor : EssSurj (functor r) where
mem_essImage Y :=
⟨Y.as, ⟨eqToIso (by
ext
| Mathlib.CategoryTheory.Quotient.124_0.34bZdkqpf1A9Wub | instance essSurj_functor : EssSurj (functor r) where
mem_essImage Y | Mathlib_CategoryTheory_Quotient |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
P : {a b : Quotient r} → (a ⟶ b) → Prop
h : ∀ {x y : C} (f : x ⟶ y), P ((functor r).map f)
⊢ ∀ {a b : Quotient r} (f : a ⟶ b), P f | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rintro ⟨x⟩ ⟨y⟩ ⟨f⟩ | protected theorem induction {P : ∀ {a b : Quotient r}, (a ⟶ b) → Prop}
(h : ∀ {x y : C} (f : x ⟶ y), P ((functor r).map f)) :
∀ {a b : Quotient r} (f : a ⟶ b), P f := by
| Mathlib.CategoryTheory.Quotient.130_0.34bZdkqpf1A9Wub | protected theorem induction {P : ∀ {a b : Quotient r}, (a ⟶ b) → Prop}
(h : ∀ {x y : C} (f : x ⟶ y), P ((functor r).map f)) :
∀ {a b : Quotient r} (f : a ⟶ b), P f | Mathlib_CategoryTheory_Quotient |
case mk.mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
P : {a b : Quotient r} → (a ⟶ b) → Prop
h : ∀ {x y : C} (f : x ⟶ y), P ((functor r).map f)
x y : C
f✝ : { as := x } ⟶ { as := y }
f : { as := x }.as ⟶ { as := y }.as
⊢ P (Quot.mk (CompClosure r) f) | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | exact h f | protected theorem induction {P : ∀ {a b : Quotient r}, (a ⟶ b) → Prop}
(h : ∀ {x y : C} (f : x ⟶ y), P ((functor r).map f)) :
∀ {a b : Quotient r} (f : a ⟶ b), P f := by
rintro ⟨x⟩ ⟨y⟩ ⟨f⟩
| Mathlib.CategoryTheory.Quotient.130_0.34bZdkqpf1A9Wub | protected theorem induction {P : ∀ {a b : Quotient r}, (a ⟶ b) → Prop}
(h : ∀ {x y : C} (f : x ⟶ y), P ((functor r).map f)) :
∀ {a b : Quotient r} (f : a ⟶ b), P f | Mathlib_CategoryTheory_Quotient |
C : Type u_2
inst✝ : Category.{u_1, u_2} C
r : HomRel C
a b : C
f₁ f₂ : a ⟶ b
h : r f₁ f₂
⊢ (functor r).map f₁ = (functor r).map f₂ | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simpa using Quot.sound (CompClosure.intro (𝟙 a) f₁ f₂ (𝟙 b) h) | protected theorem sound {a b : C} {f₁ f₂ : a ⟶ b} (h : r f₁ f₂) :
(functor r).map f₁ = (functor r).map f₂ := by
| Mathlib.CategoryTheory.Quotient.137_0.34bZdkqpf1A9Wub | protected theorem sound {a b : C} {f₁ f₂ : a ⟶ b} (h : r f₁ f₂) :
(functor r).map f₁ = (functor r).map f₂ | Mathlib_CategoryTheory_Quotient |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f g : X ⟶ Y
⊢ CompClosure r f g ↔ r f g | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | constructor | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g := by
| Mathlib.CategoryTheory.Quotient.142_0.34bZdkqpf1A9Wub | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g | Mathlib_CategoryTheory_Quotient |
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f g : X ⟶ Y
⊢ CompClosure r f g → r f g | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | intro hfg | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g := by
constructor
· | Mathlib.CategoryTheory.Quotient.142_0.34bZdkqpf1A9Wub | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g | Mathlib_CategoryTheory_Quotient |
case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f g : X ⟶ Y
hfg : CompClosure r f g
⊢ r f g | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | induction' hfg with m m' hm | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g := by
constructor
· intro hfg
| Mathlib.CategoryTheory.Quotient.142_0.34bZdkqpf1A9Wub | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g | Mathlib_CategoryTheory_Quotient |
case mp.intro
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f g : X ⟶ Y
m m' : C
hm : X ⟶ m
m₁✝ m₂✝ : m ⟶ m'
g✝ : m' ⟶ Y
h✝ : r m₁✝ m₂✝
⊢ r (hm ≫ m₁✝ ≫ g✝) (hm ≫ m₂✝ ≫ g✝) | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | exact Congruence.compLeft _ (Congruence.compRight _ (by assumption)) | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g := by
constructor
· intro hfg
induction' hfg with m m' hm
| Mathlib.CategoryTheory.Quotient.142_0.34bZdkqpf1A9Wub | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g | Mathlib_CategoryTheory_Quotient |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f g : X ⟶ Y
m m' : C
hm : X ⟶ m
m₁✝ m₂✝ : m ⟶ m'
g✝ : m' ⟶ Y
h✝ : r m₁✝ m₂✝
⊢ r m₁✝ m₂✝ | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | assumption | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g := by
constructor
· intro hfg
induction' hfg with m m' hm
exact Congruence.compLeft _ (Congruence.compRight _ (by | Mathlib.CategoryTheory.Quotient.142_0.34bZdkqpf1A9Wub | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g | Mathlib_CategoryTheory_Quotient |
case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f g : X ⟶ Y
⊢ r f g → CompClosure r f g | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | exact CompClosure.of _ _ _ | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g := by
constructor
· intro hfg
induction' hfg with m m' hm
exact Congruence.compLeft _ (Congruence.compRight _ (by assumption))
· | Mathlib.CategoryTheory.Quotient.142_0.34bZdkqpf1A9Wub | lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :
CompClosure r f g ↔ r f g | Mathlib_CategoryTheory_Quotient |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
⊢ CompClosure r = r | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | ext | @[simp]
theorem compClosure_eq_self [h : Congruence r] :
CompClosure r = r := by
| Mathlib.CategoryTheory.Quotient.150_0.34bZdkqpf1A9Wub | @[simp]
theorem compClosure_eq_self [h : Congruence r] :
CompClosure r = r | Mathlib_CategoryTheory_Quotient |
case h.h.h.h.a
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ CompClosure r x✝¹ x✝ ↔ r x✝¹ x✝ | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp only [compClosure_iff_self] | @[simp]
theorem compClosure_eq_self [h : Congruence r] :
CompClosure r = r := by
ext
| Mathlib.CategoryTheory.Quotient.150_0.34bZdkqpf1A9Wub | @[simp]
theorem compClosure_eq_self [h : Congruence r] :
CompClosure r = r | Mathlib_CategoryTheory_Quotient |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f f' : X ⟶ Y
⊢ (functor r).map f = (functor r).map f' ↔ r f f' | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | dsimp [functor] | theorem functor_map_eq_iff [h : Congruence r] {X Y : C} (f f' : X ⟶ Y) :
(functor r).map f = (functor r).map f' ↔ r f f' := by
| Mathlib.CategoryTheory.Quotient.156_0.34bZdkqpf1A9Wub | theorem functor_map_eq_iff [h : Congruence r] {X Y : C} (f f' : X ⟶ Y) :
(functor r).map f = (functor r).map f' ↔ r f f' | Mathlib_CategoryTheory_Quotient |
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f f' : X ⟶ Y
⊢ Quot.mk (CompClosure r) f = Quot.mk (CompClosure r) f' ↔ r f f' | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rw [Equivalence.quot_mk_eq_iff, compClosure_eq_self r] | theorem functor_map_eq_iff [h : Congruence r] {X Y : C} (f f' : X ⟶ Y) :
(functor r).map f = (functor r).map f' ↔ r f f' := by
dsimp [functor]
| Mathlib.CategoryTheory.Quotient.156_0.34bZdkqpf1A9Wub | theorem functor_map_eq_iff [h : Congruence r] {X Y : C} (f f' : X ⟶ Y) :
(functor r).map f = (functor r).map f' ↔ r f f' | Mathlib_CategoryTheory_Quotient |
case h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f f' : X ⟶ Y
⊢ _root_.Equivalence (CompClosure r) | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simpa only [compClosure_eq_self r] using h.equivalence | theorem functor_map_eq_iff [h : Congruence r] {X Y : C} (f f' : X ⟶ Y) :
(functor r).map f = (functor r).map f' ↔ r f f' := by
dsimp [functor]
rw [Equivalence.quot_mk_eq_iff, compClosure_eq_self r]
| Mathlib.CategoryTheory.Quotient.156_0.34bZdkqpf1A9Wub | theorem functor_map_eq_iff [h : Congruence r] {X Y : C} (f f' : X ⟶ Y) :
(functor r).map f = (functor r).map f' ↔ r f f' | Mathlib_CategoryTheory_Quotient |
C : Type ?u.12785
inst✝¹ : Category.{?u.12789, ?u.12785} C
r : HomRel C
D : Type ?u.12817
inst✝ : Category.{?u.12821, ?u.12817} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
a b : Quotient r
hf : a ⟶ b
⊢ ∀ (a_1 b_1 : a.as ⟶ b.as), CompClosure r a_1 b_1 → (fun f => F.map f) a_1 = (fun f => F... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rintro _ _ ⟨_, _, _, _, h⟩ | /-- The induced functor on the quotient category. -/
def lift : Quotient r ⥤ D where
obj a := F.obj a.as
map := @fun a b hf ↦
Quot.liftOn hf (fun f ↦ F.map f)
(by
| Mathlib.CategoryTheory.Quotient.166_0.34bZdkqpf1A9Wub | /-- The induced functor on the quotient category. -/
def lift : Quotient r ⥤ D where
obj a | Mathlib_CategoryTheory_Quotient |
case intro
C : Type ?u.12785
inst✝¹ : Category.{?u.12789, ?u.12785} C
r : HomRel C
D : Type ?u.12817
inst✝ : Category.{?u.12821, ?u.12817} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
a b : Quotient r
hf : a ⟶ b
a✝ b✝ : C
f✝ : a.as ⟶ a✝
m₁✝ m₂✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ b.as
h : r m₁✝ m₂✝
⊢ (fun... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp [H _ _ _ _ h] | /-- The induced functor on the quotient category. -/
def lift : Quotient r ⥤ D where
obj a := F.obj a.as
map := @fun a b hf ↦
Quot.liftOn hf (fun f ↦ F.map f)
(by
rintro _ _ ⟨_, _, _, _, h⟩
| Mathlib.CategoryTheory.Quotient.166_0.34bZdkqpf1A9Wub | /-- The induced functor on the quotient category. -/
def lift : Quotient r ⥤ D where
obj a | Mathlib_CategoryTheory_Quotient |
C : Type ?u.12785
inst✝¹ : Category.{?u.12789, ?u.12785} C
r : HomRel C
D : Type ?u.12817
inst✝ : Category.{?u.12821, ?u.12817} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
⊢ ∀ {X Y Z : Quotient r} (f : X ⟶ Y) (g : Y ⟶ Z),
{ obj := fun a => F.obj a.as,
map := fun a b hf =>
... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rintro a b c ⟨f⟩ ⟨g⟩ | /-- The induced functor on the quotient category. -/
def lift : Quotient r ⥤ D where
obj a := F.obj a.as
map := @fun a b hf ↦
Quot.liftOn hf (fun f ↦ F.map f)
(by
rintro _ _ ⟨_, _, _, _, h⟩
simp [H _ _ _ _ h])
map_id a := F.map_id a.as
map_comp := by
| Mathlib.CategoryTheory.Quotient.166_0.34bZdkqpf1A9Wub | /-- The induced functor on the quotient category. -/
def lift : Quotient r ⥤ D where
obj a | Mathlib_CategoryTheory_Quotient |
case mk.mk
C : Type ?u.12785
inst✝¹ : Category.{?u.12789, ?u.12785} C
r : HomRel C
D : Type ?u.12817
inst✝ : Category.{?u.12821, ?u.12817} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
a b c : Quotient r
f✝ : a ⟶ b
f : a.as ⟶ b.as
g✝ : b ⟶ c
g : b.as ⟶ c.as
⊢ { obj := fun a => F.obj a.as,
... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | exact F.map_comp f g | /-- The induced functor on the quotient category. -/
def lift : Quotient r ⥤ D where
obj a := F.obj a.as
map := @fun a b hf ↦
Quot.liftOn hf (fun f ↦ F.map f)
(by
rintro _ _ ⟨_, _, _, _, h⟩
simp [H _ _ _ _ h])
map_id a := F.map_id a.as
map_comp := by
rintro a b c ⟨f⟩ ⟨g⟩
| Mathlib.CategoryTheory.Quotient.166_0.34bZdkqpf1A9Wub | /-- The induced functor on the quotient category. -/
def lift : Quotient r ⥤ D where
obj a | Mathlib_CategoryTheory_Quotient |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
⊢ functor r ⋙ lift r F H = F | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | apply Functor.ext | theorem lift_spec : functor r ⋙ lift r F H = F := by
| Mathlib.CategoryTheory.Quotient.180_0.34bZdkqpf1A9Wub | theorem lift_spec : functor r ⋙ lift r F H = F | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
⊢ autoParam
(∀ (X Y : C) (f : X ⟶ Y),
(functor r ⋙ lift r F H).map f =
eqToHom (_ : ?F.obj X = ?G.obj X) ≫ F.map f ≫ e... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rotate_left | theorem lift_spec : functor r ⋙ lift r F H = F := by
apply Functor.ext; | Mathlib.CategoryTheory.Quotient.180_0.34bZdkqpf1A9Wub | theorem lift_spec : functor r ⋙ lift r F H = F | Mathlib_CategoryTheory_Quotient |
case h_obj
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
⊢ ∀ (X : C), (functor r ⋙ lift r F H).obj X = F.obj X | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rintro X | theorem lift_spec : functor r ⋙ lift r F H = F := by
apply Functor.ext; rotate_left
· | Mathlib.CategoryTheory.Quotient.180_0.34bZdkqpf1A9Wub | theorem lift_spec : functor r ⋙ lift r F H = F | Mathlib_CategoryTheory_Quotient |
case h_obj
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X : C
⊢ (functor r ⋙ lift r F H).obj X = F.obj X | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rfl | theorem lift_spec : functor r ⋙ lift r F H = F := by
apply Functor.ext; rotate_left
· rintro X
| Mathlib.CategoryTheory.Quotient.180_0.34bZdkqpf1A9Wub | theorem lift_spec : functor r ⋙ lift r F H = F | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
⊢ autoParam
(∀ (X Y : C) (f : X ⟶ Y),
(functor r ⋙ lift r F H).map f =
eqToHom (_ : (functor r ⋙ lift r F H).obj X = (... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rintro X Y f | theorem lift_spec : functor r ⋙ lift r F H = F := by
apply Functor.ext; rotate_left
· rintro X
rfl
· | Mathlib.CategoryTheory.Quotient.180_0.34bZdkqpf1A9Wub | theorem lift_spec : functor r ⋙ lift r F H = F | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X Y : C
f : X ⟶ Y
⊢ (functor r ⋙ lift r F H).map f =
eqToHom (_ : (functor r ⋙ lift r F H).obj X = (functor r ⋙ lift r F H).obj X) ≫... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | dsimp [lift, functor] | theorem lift_spec : functor r ⋙ lift r F H = F := by
apply Functor.ext; rotate_left
· rintro X
rfl
· rintro X Y f
| Mathlib.CategoryTheory.Quotient.180_0.34bZdkqpf1A9Wub | theorem lift_spec : functor r ⋙ lift r F H = F | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X Y : C
f : X ⟶ Y
⊢ Quot.liftOn (Quot.mk (CompClosure r) f) (fun f => F.map f)
(_ : ∀ (a b : { as := X }.as ⟶ { as := Y }.as), Com... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp | theorem lift_spec : functor r ⋙ lift r F H = F := by
apply Functor.ext; rotate_left
· rintro X
rfl
· rintro X Y f
dsimp [lift, functor]
| Mathlib.CategoryTheory.Quotient.180_0.34bZdkqpf1A9Wub | theorem lift_spec : functor r ⋙ lift r F H = F | Mathlib_CategoryTheory_Quotient |
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
Φ : Quotient r ⥤ D
hΦ : functor r ⋙ Φ = F
⊢ Φ = lift r F H | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | subst_vars | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
| Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
⊢ Φ = lift r (functor r ⋙ Φ) H | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | fapply Functor.hext | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
| Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
case h_obj
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
⊢ ∀ (X : Quotient r), Φ.obj X = (lift r (functor r ⋙ Φ) H).obj X | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rintro X | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
fapply Functor.hext
· | Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
case h_obj
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
X : Quotient r
⊢ Φ.obj X = (lift r (functor r ⋙ Φ) H).obj X | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | dsimp [lift, Functor] | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
fapply Functor.hext
· rintro X
| Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
case h_obj
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
X : Quotient r
⊢ Φ.obj X = Φ.obj ((functor r).obj X.as) | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | congr | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
fapply Functor.hext
· rintro X
dsimp [lift, Functor]
| Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
⊢ ∀ (X Y : Quotient r) (f : X ⟶ Y), HEq (Φ.map f) ((lift r (functor r ⋙ Φ) H).map f) | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rintro _ _ f | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
fapply Functor.hext
· rintro X
dsimp [lift, Functor]
congr
· | Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
X✝ Y✝ : Quotient r
f : X✝ ⟶ Y✝
⊢ HEq (Φ.map f) ((lift r (functor r ⋙ Φ) H).map f) | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | dsimp [lift, Functor] | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
fapply Functor.hext
· rintro X
dsimp [lift, Functor]
congr
· rintro _ _ f
| Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
X✝ Y✝ : Quotient r
f : X✝ ⟶ Y✝
⊢ HEq (Φ.map f)
(Quot.liftOn f (fun f => Φ.map ((functor r).map ... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | refine Quot.inductionOn f (fun _ ↦ ?_) | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
fapply Functor.hext
· rintro X
dsimp [lift, Functor]
congr
· rintro _ _ f
dsimp [lift, Functor]
| Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
X✝ Y✝ : Quotient r
f : X✝ ⟶ Y✝
x✝ : X✝.as ⟶ Y✝.as
⊢ HEq (Φ.map (Quot.mk (CompClosure r) x✝))
(Q... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp only [Quot.liftOn_mk, Functor.comp_map] | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
fapply Functor.hext
· rintro X
dsimp [lift, Functor]
congr
· rintro _ _ f
dsimp [lift, Functor]
refine Quot.inductionOn f (fun _ ↦ ?_) -- porting note: this line was originally an `apply`
| Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
case h_map
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
Φ : Quotient r ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → (functor r ⋙ Φ).map f₁ = (functor r ⋙ Φ).map f₂
X✝ Y✝ : Quotient r
f : X✝ ⟶ Y✝
x✝ : X✝.as ⟶ Y✝.as
⊢ HEq (Φ.map (Quot.mk (CompClosure r) x✝)) (Φ.map... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | congr | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H := by
subst_vars
fapply Functor.hext
· rintro X
dsimp [lift, Functor]
congr
· rintro _ _ f
dsimp [lift, Functor]
refine Quot.inductionOn f (fun _ ↦ ?_) -- porting note: this line was originally an `apply`
simp... | Mathlib.CategoryTheory.Quotient.189_0.34bZdkqpf1A9Wub | theorem lift_unique (Φ : Quotient r ⥤ D) (hΦ : functor r ⋙ Φ = F) : Φ = lift r F H | Mathlib_CategoryTheory_Quotient |
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_3, u_4} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X Y : C
f : X ⟶ Y
⊢ (lift r F H).map ((functor r).map f) = F.map f | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rw [← NatIso.naturality_1 (lift.isLift r F H)] | theorem lift_map_functor_map {X Y : C} (f : X ⟶ Y) :
(lift r F H).map ((functor r).map f) = F.map f := by
| Mathlib.CategoryTheory.Quotient.217_0.34bZdkqpf1A9Wub | theorem lift_map_functor_map {X Y : C} (f : X ⟶ Y) :
(lift r F H).map ((functor r).map f) = F.map f | Mathlib_CategoryTheory_Quotient |
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_3, u_4} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X Y : C
f : X ⟶ Y
⊢ (lift r F H).map ((functor r).map f) =
(lift.isLift r F H).inv.app X ≫ (functor r ⋙ lift r F H).map f ≫ (lift.isLift r F H)... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | dsimp [lift, functor] | theorem lift_map_functor_map {X Y : C} (f : X ⟶ Y) :
(lift r F H).map ((functor r).map f) = F.map f := by
rw [← NatIso.naturality_1 (lift.isLift r F H)]
| Mathlib.CategoryTheory.Quotient.217_0.34bZdkqpf1A9Wub | theorem lift_map_functor_map {X Y : C} (f : X ⟶ Y) :
(lift r F H).map ((functor r).map f) = F.map f | Mathlib_CategoryTheory_Quotient |
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_3, u_4} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X Y : C
f : X ⟶ Y
⊢ Quot.liftOn (Quot.mk (CompClosure r) f) (fun f => F.map f)
(_ : ∀ (a b : { as := X }.as ⟶ { as := Y }.as), CompClosure r ... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | simp | theorem lift_map_functor_map {X Y : C} (f : X ⟶ Y) :
(lift r F H).map ((functor r).map f) = F.map f := by
rw [← NatIso.naturality_1 (lift.isLift r F H)]
dsimp [lift, functor]
| Mathlib.CategoryTheory.Quotient.217_0.34bZdkqpf1A9Wub | theorem lift_map_functor_map {X Y : C} (f : X ⟶ Y) :
(lift r F H).map ((functor r).map f) = F.map f | Mathlib_CategoryTheory_Quotient |
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
F✝ : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F G : Quotient r ⥤ D
τ₁ τ₂ : F ⟶ G
h : whiskerLeft (functor r) τ₁ = whiskerLeft (functor r) τ₂
⊢ τ₁.app = τ₂.app | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | ext1 ⟨X⟩ | lemma natTrans_ext {F G : Quotient r ⥤ D} (τ₁ τ₂ : F ⟶ G)
(h : whiskerLeft (Quotient.functor r) τ₁ = whiskerLeft (Quotient.functor r) τ₂) : τ₁ = τ₂ :=
NatTrans.ext _ _ (by | Mathlib.CategoryTheory.Quotient.226_0.34bZdkqpf1A9Wub | lemma natTrans_ext {F G : Quotient r ⥤ D} (τ₁ τ₂ : F ⟶ G)
(h : whiskerLeft (Quotient.functor r) τ₁ = whiskerLeft (Quotient.functor r) τ₂) : τ₁ = τ₂ | Mathlib_CategoryTheory_Quotient |
case h.mk
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
F✝ : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F G : Quotient r ⥤ D
τ₁ τ₂ : F ⟶ G
h : whiskerLeft (functor r) τ₁ = whiskerLeft (functor r) τ₂
X : C
⊢ τ₁.app { as := X } = τ₂.app { a... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | exact NatTrans.congr_app h X | lemma natTrans_ext {F G : Quotient r ⥤ D} (τ₁ τ₂ : F ⟶ G)
(h : whiskerLeft (Quotient.functor r) τ₁ = whiskerLeft (Quotient.functor r) τ₂) : τ₁ = τ₂ :=
NatTrans.ext _ _ (by ext1 ⟨X⟩; | Mathlib.CategoryTheory.Quotient.226_0.34bZdkqpf1A9Wub | lemma natTrans_ext {F G : Quotient r ⥤ D} (τ₁ τ₂ : F ⟶ G)
(h : whiskerLeft (Quotient.functor r) τ₁ = whiskerLeft (Quotient.functor r) τ₂) : τ₁ = τ₂ | Mathlib_CategoryTheory_Quotient |
C : Type ?u.21219
inst✝¹ : Category.{?u.21223, ?u.21219} C
r : HomRel C
D : Type ?u.21251
inst✝ : Category.{?u.21255, ?u.21251} D
F✝ : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F G : Quotient r ⥤ D
τ : functor r ⋙ F ⟶ functor r ⋙ G
x✝¹ x✝ : Quotient r
X Y : C
⊢ ∀ (f : { as := X } ⟶ { as := ... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rintro ⟨f⟩ | /-- In order to define a natural transformation `F ⟶ G` with `F G : Quotient r ⥤ D`, it suffices
to do so after precomposing with `Quotient.functor r`. -/
def natTransLift {F G : Quotient r ⥤ D} (τ : Quotient.functor r ⋙ F ⟶ Quotient.functor r ⋙ G) :
F ⟶ G where
app := fun ⟨X⟩ => τ.app X
naturality := fun ⟨X⟩ ⟨... | Mathlib.CategoryTheory.Quotient.232_0.34bZdkqpf1A9Wub | /-- In order to define a natural transformation `F ⟶ G` with `F G : Quotient r ⥤ D`, it suffices
to do so after precomposing with `Quotient.functor r`. -/
def natTransLift {F G : Quotient r ⥤ D} (τ : Quotient.functor r ⋙ F ⟶ Quotient.functor r ⋙ G) :
F ⟶ G where
app | Mathlib_CategoryTheory_Quotient |
case mk
C : Type ?u.21219
inst✝¹ : Category.{?u.21223, ?u.21219} C
r : HomRel C
D : Type ?u.21251
inst✝ : Category.{?u.21255, ?u.21251} D
F✝ : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F G : Quotient r ⥤ D
τ : functor r ⋙ F ⟶ functor r ⋙ G
x✝¹ x✝ : Quotient r
X Y : C
f✝ : { as := X } ⟶ { as... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | exact τ.naturality f | /-- In order to define a natural transformation `F ⟶ G` with `F G : Quotient r ⥤ D`, it suffices
to do so after precomposing with `Quotient.functor r`. -/
def natTransLift {F G : Quotient r ⥤ D} (τ : Quotient.functor r ⋙ F ⟶ Quotient.functor r ⋙ G) :
F ⟶ G where
app := fun ⟨X⟩ => τ.app X
naturality := fun ⟨X⟩ ⟨... | Mathlib.CategoryTheory.Quotient.232_0.34bZdkqpf1A9Wub | /-- In order to define a natural transformation `F ⟶ G` with `F G : Quotient r ⥤ D`, it suffices
to do so after precomposing with `Quotient.functor r`. -/
def natTransLift {F G : Quotient r ⥤ D} (τ : Quotient.functor r ⋙ F ⟶ Quotient.functor r ⋙ G) :
F ⟶ G where
app | Mathlib_CategoryTheory_Quotient |
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
F✝ : C ⥤ D
H✝ : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F G H : Quotient r ⥤ D
τ : functor r ⋙ F ⟶ functor r ⋙ G
τ' : functor r ⋙ G ⟶ functor r ⋙ H
⊢ natTransLift r τ ≫ natTransLift r τ' = natTransLi... | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | aesop_cat | @[reassoc]
lemma comp_natTransLift {F G H : Quotient r ⥤ D}
(τ : Quotient.functor r ⋙ F ⟶ Quotient.functor r ⋙ G)
(τ' : Quotient.functor r ⋙ G ⟶ Quotient.functor r ⋙ H) :
natTransLift r τ ≫ natTransLift r τ' = natTransLift r (τ ≫ τ') := by | Mathlib.CategoryTheory.Quotient.246_0.34bZdkqpf1A9Wub | @[reassoc]
lemma comp_natTransLift {F G H : Quotient r ⥤ D}
(τ : Quotient.functor r ⋙ F ⟶ Quotient.functor r ⋙ G)
(τ' : Quotient.functor r ⋙ G ⟶ Quotient.functor r ⋙ H) :
natTransLift r τ ≫ natTransLift r τ' = natTransLift r (τ ≫ τ') | Mathlib_CategoryTheory_Quotient |
C : Type u_3
inst✝¹ : Category.{u_1, u_3} C
r : HomRel C
D : Type u_4
inst✝ : Category.{u_2, u_4} D
F✝ : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F : Quotient r ⥤ D
⊢ natTransLift r (𝟙 (functor r ⋙ F)) = 𝟙 F | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | aesop_cat | @[simp]
lemma natTransLift_id (F : Quotient r ⥤ D) :
natTransLift r (𝟙 (Quotient.functor r ⋙ F)) = 𝟙 _ := by | Mathlib.CategoryTheory.Quotient.252_0.34bZdkqpf1A9Wub | @[simp]
lemma natTransLift_id (F : Quotient r ⥤ D) :
natTransLift r (𝟙 (Quotient.functor r ⋙ F)) = 𝟙 _ | Mathlib_CategoryTheory_Quotient |
C : Type ?u.27722
inst✝¹ : Category.{?u.27726, ?u.27722} C
r : HomRel C
D : Type ?u.27754
inst✝ : Category.{?u.27758, ?u.27754} D
F✝ : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F G : Quotient r ⥤ D
τ : functor r ⋙ F ≅ functor r ⋙ G
⊢ natTransLift r τ.hom ≫ natTransLift r τ.inv = 𝟙 F | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rw [comp_natTransLift, τ.hom_inv_id, natTransLift_id] | /-- In order to define a natural isomorphism `F ≅ G` with `F G : Quotient r ⥤ D`, it suffices
to do so after precomposing with `Quotient.functor r`. -/
@[simps]
def natIsoLift {F G : Quotient r ⥤ D} (τ : Quotient.functor r ⋙ F ≅ Quotient.functor r ⋙ G) :
F ≅ G where
hom := natTransLift _ τ.hom
inv := natTransLi... | Mathlib.CategoryTheory.Quotient.256_0.34bZdkqpf1A9Wub | /-- In order to define a natural isomorphism `F ≅ G` with `F G : Quotient r ⥤ D`, it suffices
to do so after precomposing with `Quotient.functor r`. -/
@[simps]
def natIsoLift {F G : Quotient r ⥤ D} (τ : Quotient.functor r ⋙ F ≅ Quotient.functor r ⋙ G) :
F ≅ G where
hom | Mathlib_CategoryTheory_Quotient |
C : Type ?u.27722
inst✝¹ : Category.{?u.27726, ?u.27722} C
r : HomRel C
D : Type ?u.27754
inst✝ : Category.{?u.27758, ?u.27754} D
F✝ : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F✝.map f₁ = F✝.map f₂
F G : Quotient r ⥤ D
τ : functor r ⋙ F ≅ functor r ⋙ G
⊢ natTransLift r τ.inv ≫ natTransLift r τ.hom = 𝟙 G | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | rw [comp_natTransLift, τ.inv_hom_id, natTransLift_id] | /-- In order to define a natural isomorphism `F ≅ G` with `F G : Quotient r ⥤ D`, it suffices
to do so after precomposing with `Quotient.functor r`. -/
@[simps]
def natIsoLift {F G : Quotient r ⥤ D} (τ : Quotient.functor r ⋙ F ≅ Quotient.functor r ⋙ G) :
F ≅ G where
hom := natTransLift _ τ.hom
inv := natTransLi... | Mathlib.CategoryTheory.Quotient.256_0.34bZdkqpf1A9Wub | /-- In order to define a natural isomorphism `F ≅ G` with `F G : Quotient r ⥤ D`, it suffices
to do so after precomposing with `Quotient.functor r`. -/
@[simps]
def natIsoLift {F G : Quotient r ⥤ D} (τ : Quotient.functor r ⋙ F ≅ Quotient.functor r ⋙ G) :
F ≅ G where
hom | Mathlib_CategoryTheory_Quotient |
C : Type ?u.32590
inst✝¹ : Category.{?u.32594, ?u.32590} C
r : HomRel C
D : Type ?u.32622
inst✝ : Category.{?u.32626, ?u.32622} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
⊢ ∀ {X Y : Quotient r ⥤ D}, Function.Injective ((whiskeringLeft C (Quotient r) D).obj (functor r)).map | /-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a... | apply natTrans_ext | instance faithful_whiskeringLeft_functor :
Faithful ((whiskeringLeft C _ D).obj (functor r)) := ⟨by | Mathlib.CategoryTheory.Quotient.272_0.34bZdkqpf1A9Wub | instance faithful_whiskeringLeft_functor :
Faithful ((whiskeringLeft C _ D).obj (functor r)) | Mathlib_CategoryTheory_Quotient |
a b : ℝ
n : ℕ
f : ℝ → ℝ
μ ν : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
c d r : ℝ
h : -1 < r
⊢ IntervalIntegrable (fun x => x ^ r) volume a b | /-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by
exact IntervalIntegrable.trans (this a).symm (this b) | /-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a
stronger hypothesis on `r`. -/
theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) :
IntervalIntegrable (fun x => x ^ r) volume a b := by
| Mathlib.Analysis.SpecialFunctions.Integrals.70_0.61KIRGZyZnxsI7s | /-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a
stronger hypothesis on `r`. -/
theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) :
IntervalIntegrable (fun x => x ^ r) volume a b | Mathlib_Analysis_SpecialFunctions_Integrals |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.