Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 145 | 146 | theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ := by |
simp only [laverage_eq, lintegral_congr_ae h]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 149 | 150 | theorem setLaverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ := by |
simp only [setLaverage_eq, set_lintegral_congr h, measure_congr h]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 153 | 155 | theorem setLaverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
⨍⁻ x in s, f x ∂μ = ⨍⁻ x in s, g x ∂μ := by |
simp only [laverage_eq, set_lintegral_congr_fun hs h]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 158 | 162 | theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by |
obtain rfl | hμ := eq_or_ne μ 0
· simp
· rw [laverage_eq]
exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 169 | 180 | theorem laverage_add_measure :
⨍⁻ x, f x ∂(μ + ν) =
μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν := by |
by_cases hμ : IsFiniteMeasure μ; swap
· rw [not_isFiniteMeasure_iff] at hμ
simp [laverage_eq, hμ]
by_cases hν : IsFiniteMeasure ν; swap
· rw [not_isFiniteMeasure_iff] at hν
simp [laverage_eq, hν]
haveI := hμ; haveI := hν
simp only [← ENNReal.mul_div_right_comm, measure_mul_laverage, ← ENNReal.add_d... | 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 315 | 315 | theorem average_zero : ⨍ _, (0 : E) ∂μ = 0 := by | rw [average, integral_zero]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 319 | 320 | theorem average_zero_measure (f : α → E) : ⨍ x, f x ∂(0 : Measure α) = 0 := by |
rw [average, smul_zero, integral_zero_measure]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 332 | 333 | theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by |
rw [average_eq', integral_smul_measure, ENNReal.toReal_inv]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 336 | 337 | theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by |
rw [average, measure_univ, inv_one, one_smul]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 341 | 347 | theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
(μ univ).toReal • ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by |
rcases eq_or_ne μ 0 with hμ | hμ
· rw [hμ, integral_zero_measure, average_zero_measure, smul_zero]
· rw [average_eq, smul_inv_smul₀]
refine (ENNReal.toReal_pos ?_ <| measure_ne_top _ _).ne'
rwa [Ne, measure_univ_eq_zero]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 350 | 351 | theorem setAverage_eq (f : α → E) (s : Set α) :
⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by | rw [average_eq, restrict_apply_univ]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 354 | 356 | theorem setAverage_eq' (f : α → E) (s : Set α) :
⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by |
simp only [average_eq', restrict_apply_univ]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 361 | 362 | theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ := by |
simp only [average_eq, integral_congr_ae h]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 365 | 366 | theorem setAverage_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by |
simp only [setAverage_eq, setIntegral_congr_set_ae h, measure_congr h]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 369 | 370 | theorem setAverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ := by | simp only [average_eq, setIntegral_congr_ae hs h]
| 215 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 373 | 380 | theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
(hμ : Integrable f μ) (hν : Integrable f ν) :
⨍ x, f x ∂(μ + ν) =
((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ +
((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f... |
simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add,
← integral_add_measure hμ hν, ← ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)]
rw [average_eq, Measure.add_apply]
| 215 |
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 82 | 92 | theorem map_smul (c : 𝕜) (x : V) : e (c • x) = ‖c‖₊ * e x := by |
apply le_antisymm (e.map_smul_le' c x)
by_cases hc : c = 0
· simp [hc]
calc
(‖c‖₊ : ℝ≥0∞) * e x = ‖c‖₊ * e (c⁻¹ • c • x) := by rw [inv_smul_smul₀ hc]
_ ≤ ‖c‖₊ * (‖c⁻¹‖₊ * e (c • x)) := mul_le_mul_left' (e.map_smul_le' _ _) _
_ = e (c • x) := by
rw [← mul_assoc, nnnorm_inv, ENNReal.coe_inv, EN... | 216 |
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 96 | 98 | theorem map_zero : e 0 = 0 := by |
rw [← zero_smul 𝕜 (0 : V), e.map_smul]
norm_num
| 216 |
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 107 | 110 | theorem map_neg (x : V) : e (-x) = e x :=
calc
e (-x) = ‖(-1 : 𝕜)‖₊ * e x := by | rw [← map_smul, neg_one_smul]
_ = e x := by simp
| 216 |
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 113 | 113 | theorem map_sub_rev (x y : V) : e (x - y) = e (y - x) := by | rw [← neg_sub, e.map_neg]
| 216 |
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 120 | 124 | theorem map_sub_le (x y : V) : e (x - y) ≤ e x + e y :=
calc
e (x - y) = e (x + -y) := by | rw [sub_eq_add_neg]
_ ≤ e x + e (-y) := e.map_add_le x (-y)
_ = e x + e y := by rw [e.map_neg]
| 216 |
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed_space.int from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
namespace Int
| Mathlib/Analysis/NormedSpace/Int.lean | 24 | 26 | theorem nnnorm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖₊ = 1 := by |
obtain rfl | rfl := units_eq_one_or e <;>
simp only [Units.coe_neg_one, Units.val_one, nnnorm_neg, nnnorm_one]
| 217 |
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed_space.int from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
namespace Int
theorem nnnorm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖₊ = 1 := by
obtain rfl | rfl := units_eq_one_or e <;>
simp only [Units.coe_neg_one, Un... | Mathlib/Analysis/NormedSpace/Int.lean | 29 | 30 | theorem norm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖ = 1 := by |
rw [← coe_nnnorm, nnnorm_coe_units, NNReal.coe_one]
| 217 |
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed_space.int from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
namespace Int
theorem nnnorm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖₊ = 1 := by
obtain rfl | rfl := units_eq_one_or e <;>
simp only [Units.coe_neg_one, Un... | Mathlib/Analysis/NormedSpace/Int.lean | 41 | 42 | theorem toNat_add_toNat_neg_eq_nnnorm (n : ℤ) : ↑n.toNat + ↑(-n).toNat = ‖n‖₊ := by |
rw [← Nat.cast_add, toNat_add_toNat_neg_eq_natAbs, NNReal.natCast_natAbs]
| 217 |
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed_space.int from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
namespace Int
theorem nnnorm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖₊ = 1 := by
obtain rfl | rfl := units_eq_one_or e <;>
simp only [Units.coe_neg_one, Un... | Mathlib/Analysis/NormedSpace/Int.lean | 46 | 48 | theorem toNat_add_toNat_neg_eq_norm (n : ℤ) : ↑n.toNat + ↑(-n).toNat = ‖n‖ := by |
simpa only [NNReal.coe_natCast, NNReal.coe_add] using
congrArg NNReal.toReal (toNat_add_toNat_neg_eq_nnnorm n)
| 217 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 48 | 48 | theorem true_eq_false_eq_False : ¬true = false := by | decide
| 218 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 51 | 51 | theorem false_eq_true_eq_False : ¬false = true := by | decide
| 218 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 54 | 54 | theorem eq_false_eq_not_eq_true (b : Bool) : (¬b = true) = (b = false) := by | simp
| 218 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 57 | 57 | theorem eq_true_eq_not_eq_false (b : Bool) : (¬b = false) = (b = true) := by | simp
| 218 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 68 | 69 | theorem and_eq_true_eq_eq_true_and_eq_true (a b : Bool) :
((a && b) = true) = (a = true ∧ b = true) := by | simp
| 218 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 72 | 73 | theorem or_eq_true_eq_eq_true_or_eq_true (a b : Bool) :
((a || b) = true) = (a = true ∨ b = true) := by | simp
| 218 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 76 | 76 | theorem not_eq_true_eq_eq_false (a : Bool) : (not a = true) = (a = false) := by | cases a <;> simp
| 218 |
import Mathlib.Order.Disjoint
#align_import order.prop_instances from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
instance Prop.instDistribLattice : DistribLattice Prop where
sup := Or
le_sup_left := @Or.inl
le_sup_right := @Or.inr
sup_le := fun _ _ _ => Or.rec
inf := And
... | Mathlib/Order/PropInstances.lean | 72 | 80 | theorem disjoint_iff [∀ i, OrderBot (α' i)] {f g : ∀ i, α' i} :
Disjoint f g ↔ ∀ i, Disjoint (f i) (g i) := by |
classical
constructor
· intro h i x hf hg
exact (update_le_iff.mp <| h (update_le_iff.mpr ⟨hf, fun _ _ => bot_le⟩)
(update_le_iff.mpr ⟨hg, fun _ _ => bot_le⟩)).1
· intro h x hf hg i
apply h i (hf i) (hg i)
| 219 |
import Mathlib.Order.Disjoint
#align_import order.prop_instances from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
instance Prop.instDistribLattice : DistribLattice Prop where
sup := Or
le_sup_left := @Or.inl
le_sup_right := @Or.inr
sup_le := fun _ _ _ => Or.rec
inf := And
... | Mathlib/Order/PropInstances.lean | 88 | 90 | theorem isCompl_iff [∀ i, BoundedOrder (α' i)] {f g : ∀ i, α' i} :
IsCompl f g ↔ ∀ i, IsCompl (f i) (g i) := by |
simp_rw [_root_.isCompl_iff, disjoint_iff, codisjoint_iff, forall_and]
| 219 |
import Mathlib.Order.Disjoint
#align_import order.prop_instances from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
instance Prop.instDistribLattice : DistribLattice Prop where
sup := Or
le_sup_left := @Or.inl
le_sup_right := @Or.inr
sup_le := fun _ _ _ => Or.rec
inf := And
... | Mathlib/Order/PropInstances.lean | 106 | 108 | theorem Prop.isCompl_iff {P Q : Prop} : IsCompl P Q ↔ ¬(P ↔ Q) := by |
rw [_root_.isCompl_iff, Prop.disjoint_iff, Prop.codisjoint_iff, not_iff]
by_cases P <;> by_cases Q <;> simp [*]
| 219 |
import Mathlib.Data.Part
import Mathlib.Data.Nat.Upto
import Mathlib.Data.Stream.Defs
import Mathlib.Tactic.Common
#align_import control.fix from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v
open scoped Classical
variable {α : Type*} {β : α → Type*}
class Fix (α : Typ... | Mathlib/Control/Fix.lean | 111 | 113 | theorem fix_def' {x : α} (h' : ¬∃ i, (Fix.approx f i x).Dom) : Part.fix f x = none := by |
dsimp [Part.fix]
rw [assert_neg h']
| 220 |
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | Mathlib/Topology/Order/LeftRightNhds.lean | 40 | 60 | theorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[>] a,
s ∈ 𝓝[Ioc a b] a,
s ∈ 𝓝[Ioo a b] a,
∃ u ∈ Ioc a b, Ioo a u ⊆ s,
∃ u ∈ Ioi a, Ioo a u ⊆ s] := by |
tfae_have 1 ↔ 2
· rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]
tfae_have 1 ↔ 3
· rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
tfae_have 4 → 5
· exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
tfae_have 5 → 1
· rintro ⟨u, hau, hu⟩
exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu
tfae_have 1... | 221 |
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | Mathlib/Topology/Order/LeftRightNhds.lean | 82 | 87 | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by |
by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· simp only [ha, false_or]
rw [isTop_iff_isMax, not_isMax_iff] at ha
simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covBy_iff_Ioo_eq]
| 221 |
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | Mathlib/Topology/Order/LeftRightNhds.lean | 99 | 102 | theorem countable_setOf_isolated_right [SecondCountableTopology α] :
{ x : α | 𝓝[>] x = ⊥ }.Countable := by |
simp only [nhdsWithin_Ioi_eq_bot_iff, setOf_or]
exact (subsingleton_isTop α).countable.union countable_setOf_covBy_right
| 221 |
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | Mathlib/Topology/Order/LeftRightNhds.lean | 112 | 120 | theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by |
rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]
constructor
· rintro ⟨u, au, as⟩
rcases exists_between au with ⟨v, hv⟩
exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩
· rintro ⟨u, au, as⟩
exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩
| 221 |
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | Mathlib/Topology/Order/LeftRightNhds.lean | 131 | 138 | theorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`
s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`
s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`
∃ l ∈ Ico a b... | -- 4 : `s` includes `(l, b)` for some `l < b`
simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using
TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)
| 221 |
import Mathlib.Topology.Order.LeftRightNhds
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section OrderTopology
variable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]
[OrderTopology β]
| Mathlib/Topology/Order/IsLUB.lean | 24 | 32 | theorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝[≤] a, x ∈ s := by |
rcases hs with ⟨a', ha'⟩
intro h
rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)
· exact h.self_of_nhdsWithin le_rfl ha'
· rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩
rcases ha.exists_between hba with ⟨b', hb's, hb'⟩
exact hb hb' hb's
| 222 |
import Mathlib.Topology.Order.LeftRightNhds
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section OrderTopology
variable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]
[OrderTopology β]
theorem IsLUB.fr... | Mathlib/Topology/Order/IsLUB.lean | 77 | 80 | theorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :
IsLUB s a := by |
rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf
exact isLUB_of_mem_nhds hsa (mem_principal_self s)
| 222 |
import Mathlib.Topology.Order.LeftRightNhds
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section OrderTopology
variable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]
[OrderTopology β]
theorem IsLUB.fr... | Mathlib/Topology/Order/IsLUB.lean | 93 | 100 | theorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by |
rintro _ ⟨x, hx, rfl⟩
replace ha := ha.inter_Ici_of_mem hx
haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩
refine ge_of_tendsto (hb.mono_left (nhdsWithin_mono a (inter_subset_left (t := Ici x)))) ?_
exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2
| 222 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
| Mathlib/Topology/Order/DenselyOrdered.lean | 25 | 29 | theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by |
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]
exact isGLB_Ioi.mem_closure h
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 52 | 61 | theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ioo_subset_Icc_self isClosed_Icc
· cases' hab.lt_or_lt with hab hab
· rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]
have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab
simp only [insert_subset_iff, singleton_subset_iff]
exact ⟨(isGLB... | 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 66 | 70 | theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
rw [closure_Ioo hab]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 75 | 79 | theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ico_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)
rw [closure_Ioo hab]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 83 | 84 | theorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by |
rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 101 | 102 | theorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by |
rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 106 | 108 | theorem Icc_mem_nhds_iff [NoMinOrder α] [NoMaxOrder α] {a b x : α} :
Icc a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by |
rw [← interior_Icc, mem_interior_iff_mem_nhds]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 111 | 112 | theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by |
rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 116 | 117 | theorem Ico_mem_nhds_iff [NoMinOrder α] {a b x : α} : Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by |
rw [← interior_Ico, mem_interior_iff_mem_nhds]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 120 | 121 | theorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by |
rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 125 | 126 | theorem Ioc_mem_nhds_iff [NoMaxOrder α] {a b x : α} : Ioc a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by |
rw [← interior_Ioc, mem_interior_iff_mem_nhds]
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 136 | 143 | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by |
rcases eq_or_ne a b with (rfl | h)
· simp
· calc
Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self
_ = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Ioc a b)) :=
closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)
| 223 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 146 | 147 | theorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by |
simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a)
| 223 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 32 | 36 | theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit (cospan f f) F]
[Mono f] : Mono (F.map f) := by |
have := isLimitPullbackConeMapOfIsLimit F _ (PullbackCone.isLimitMkIdId f)
simp_rw [F.map_id] at this
apply PullbackCone.mono_of_isLimitMkIdId _ this
| 224 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 45 | 49 | theorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (cospan f f) F]
[Mono (F.map f)] : Mono f := by |
have := PullbackCone.isLimitMkIdId (F.map f)
simp_rw [← F.map_id] at this
apply PullbackCone.mono_of_isLimitMkIdId _ (isLimitOfIsLimitPullbackConeMap F _ this)
| 224 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 58 | 62 | theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColimit (span f f) F]
[Epi f] : Epi (F.map f) := by |
have := isColimitPushoutCoconeMapOfIsColimit F _ (PushoutCocone.isColimitMkIdId f)
simp_rw [F.map_id] at this
apply PushoutCocone.epi_of_isColimitMkIdId _ this
| 224 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 71 | 77 | theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit (span f f) F]
[Epi (F.map f)] : Epi f := by |
have := PushoutCocone.isColimitMkIdId (F.map f)
simp_rw [← F.map_id] at this
apply
PushoutCocone.epi_of_isColimitMkIdId _
(isColimitOfIsColimitPushoutCoconeMap F _ this)
| 224 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ u₁ u₂ u... | Mathlib/CategoryTheory/Functor/Category.lean | 68 | 68 | theorem congr_app {α β : F ⟶ G} (h : α = β) (X : C) : α.app X = β.app X := by | rw [h]
| 225 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ u₁ u₂ u... | Mathlib/CategoryTheory/Functor/Category.lean | 121 | 122 | theorem hcomp_id_app {H : D ⥤ E} (α : F ⟶ G) (X : C) : (α ◫ 𝟙 H).app X = H.map (α.app X) := by |
simp
| 225 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ u₁ u₂ u... | Mathlib/CategoryTheory/Functor/Category.lean | 125 | 125 | theorem id_hcomp_app {H : E ⥤ C} (α : F ⟶ G) (X : E) : (𝟙 H ◫ α).app X = α.app _ := by | simp
| 225 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ u₁ u₂ u... | Mathlib/CategoryTheory/Functor/Category.lean | 132 | 134 | theorem exchange {I J K : D ⥤ E} (α : F ⟶ G) (β : G ⟶ H) (γ : I ⟶ J) (δ : J ⟶ K) :
(α ≫ β) ◫ (γ ≫ δ) = (α ◫ γ) ≫ β ◫ δ := by |
aesop_cat
| 225 |
import Mathlib.Topology.Category.Profinite.Basic
universe u
namespace Profinite
variable {ι : Type u} {X : ι → Type} [∀ i, TopologicalSpace (X i)] (C : Set ((i : ι) → X i))
(J K : ι → Prop)
namespace IndexFunctor
open ContinuousMap
def obj : Set ((i : {i : ι // J i}) → X i) := ContinuousMap.precomp (Subty... | Mathlib/Topology/Category/Profinite/Product.lean | 58 | 62 | theorem surjective_π_app :
Function.Surjective (π_app C J) := by |
intro x
obtain ⟨y, hy⟩ := x.prop
exact ⟨⟨y, hy.1⟩, Subtype.ext hy.2⟩
| 226 |
import Mathlib.Topology.Category.Profinite.Basic
universe u
namespace Profinite
variable {ι : Type u} {X : ι → Type} [∀ i, TopologicalSpace (X i)] (C : Set ((i : ι) → X i))
(J K : ι → Prop)
namespace IndexFunctor
open ContinuousMap
def obj : Set ((i : {i : ι // J i}) → X i) := ContinuousMap.precomp (Subty... | Mathlib/Topology/Category/Profinite/Product.lean | 68 | 75 | theorem eq_of_forall_π_app_eq (a b : C)
(h : ∀ (J : Finset ι), π_app C (· ∈ J) a = π_app C (· ∈ J) b) : a = b := by |
ext i
specialize h ({i} : Finset ι)
rw [Subtype.ext_iff] at h
simp only [π_app, ContinuousMap.precomp, ContinuousMap.coe_mk,
Set.MapsTo.val_restrict_apply] at h
exact congr_fun h ⟨i, Finset.mem_singleton.mpr rfl⟩
| 226 |
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units.Hom
#align_import algebra.group.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
assert_not_exists MonoidWithZero
-- TODO:
-- assert_not_exists AddMonoidWithOne
assert_not_exists DenselyOrdered
variable {A ... | Mathlib/Algebra/Group/Prod.lean | 79 | 81 | theorem one_mk_mul_one_mk [Monoid M] [Mul N] (b₁ b₂ : N) :
((1 : M), b₁) * (1, b₂) = (1, b₁ * b₂) := by |
rw [mk_mul_mk, mul_one]
| 227 |
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units.Hom
#align_import algebra.group.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
assert_not_exists MonoidWithZero
-- TODO:
-- assert_not_exists AddMonoidWithOne
assert_not_exists DenselyOrdered
variable {A ... | Mathlib/Algebra/Group/Prod.lean | 86 | 88 | theorem mk_one_mul_mk_one [Mul M] [Monoid N] (a₁ a₂ : M) :
(a₁, (1 : N)) * (a₂, 1) = (a₁ * a₂, 1) := by |
rw [mk_mul_mk, mul_one]
| 227 |
import Mathlib.Algebra.Group.Prod
#align_import data.nat.cast.prod from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
assert_not_exists MonoidWithZero
variable {α β : Type*}
namespace Prod
variable [AddMonoidWithOne α] [AddMonoidWithOne β]
instance instAddMonoidWithOne : AddMonoidWi... | Mathlib/Data/Nat/Cast/Prod.lean | 29 | 29 | theorem fst_natCast (n : ℕ) : (n : α × β).fst = n := by | induction n <;> simp [*]
| 228 |
import Mathlib.Algebra.Group.Prod
#align_import data.nat.cast.prod from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
assert_not_exists MonoidWithZero
variable {α β : Type*}
namespace Prod
variable [AddMonoidWithOne α] [AddMonoidWithOne β]
instance instAddMonoidWithOne : AddMonoidWi... | Mathlib/Data/Nat/Cast/Prod.lean | 39 | 39 | theorem snd_natCast (n : ℕ) : (n : α × β).snd = n := by | induction n <;> simp [*]
| 228 |
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.BoundedOrder
import Mathlib.Mathport.Notation
import Mathlib.Data.Sigma.Basic
#align_import data.sigma.order from "leanprover-community/mathlib"@"1fc36cc9c8264e6e81253f88be7fb2cb6c92d76a"
namespace Sigma
variable {ι : Type*} {α : ι → Type*}
-- Porting note: I... | Mathlib/Data/Sigma/Order.lean | 79 | 86 | theorem le_def [∀ i, LE (α i)] {a b : Σi, α i} : a ≤ b ↔ ∃ h : a.1 = b.1, h.rec a.2 ≤ b.2 := by |
constructor
· rintro ⟨i, a, b, h⟩
exact ⟨rfl, h⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
rintro ⟨rfl : i = j, h⟩
exact le.fiber _ _ _ h
| 229 |
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.BoundedOrder
import Mathlib.Mathport.Notation
import Mathlib.Data.Sigma.Basic
#align_import data.sigma.order from "leanprover-community/mathlib"@"1fc36cc9c8264e6e81253f88be7fb2cb6c92d76a"
namespace Sigma
variable {ι : Type*} {α : ι → Type*}
-- Porting note: I... | Mathlib/Data/Sigma/Order.lean | 89 | 96 | theorem lt_def [∀ i, LT (α i)] {a b : Σi, α i} : a < b ↔ ∃ h : a.1 = b.1, h.rec a.2 < b.2 := by |
constructor
· rintro ⟨i, a, b, h⟩
exact ⟨rfl, h⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
rintro ⟨rfl : i = j, h⟩
exact lt.fiber _ _ _ h
| 229 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 94 | 95 | theorem distrib_three_right [Mul R] [Add R] [RightDistribClass R] (a b c d : R) :
(a + b + c) * d = a * d + b * d + c * d := by | simp [right_distrib]
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 156 | 157 | theorem add_one_mul [RightDistribClass α] (a b : α) : (a + 1) * b = a * b + b := by |
rw [add_mul, one_mul]
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 160 | 161 | theorem mul_add_one [LeftDistribClass α] (a b : α) : a * (b + 1) = a * b + a := by |
rw [mul_add, mul_one]
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 164 | 165 | theorem one_add_mul [RightDistribClass α] (a b : α) : (1 + a) * b = b + a * b := by |
rw [add_mul, one_mul]
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 168 | 169 | theorem mul_one_add [LeftDistribClass α] (a b : α) : a * (1 + b) = a + a * b := by |
rw [mul_add, mul_one]
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 197 | 198 | theorem mul_ite {α} [Mul α] (P : Prop) [Decidable P] (a b c : α) :
(a * if P then b else c) = if P then a * b else a * c := by | split_ifs <;> rfl
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 203 | 204 | theorem ite_mul {α} [Mul α] (P : Prop) [Decidable P] (a b c : α) :
(if P then a else b) * c = if P then a * c else b * c := by | split_ifs <;> rfl
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 218 | 221 | theorem ite_sub_ite {α} [Sub α] (P : Prop) [Decidable P] (a b c d : α) :
((if P then a else b) - if P then c else d) = if P then a - c else b - d := by |
split
repeat rfl
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 223 | 226 | theorem ite_add_ite {α} [Add α] (P : Prop) [Decidable P] (a b c d : α) :
((if P then a else b) + if P then c else d) = if P then a + c else b + d := by |
split
repeat rfl
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 244 | 245 | theorem mul_boole {α} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) :
(a * if P then 1 else 0) = if P then a else 0 := by | simp
| 230 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 249 | 250 | theorem boole_mul {α} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) :
(if P then 1 else 0) * a = if P then a else 0 := by | simp
| 230 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.ring.divisibility from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
variable {α β : Type*}
section Semigroup
variable [Semigroup α] [Semigroup β]... | Mathlib/Algebra/Ring/Divisibility/Basic.lean | 31 | 38 | theorem MulEquiv.decompositionMonoid [DecompositionMonoid β] : DecompositionMonoid α where
primal a b c h := by |
rw [← map_dvd_iff f, map_mul] at h
obtain ⟨a₁, a₂, h⟩ := DecompositionMonoid.primal _ h
refine ⟨symm f a₁, symm f a₂, ?_⟩
simp_rw [← map_dvd_iff f, ← map_mul, eq_symm_apply]
iterate 2 erw [(f : α ≃* β).apply_symm_apply]
exact h
| 231 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 84 | 86 | theorem IsCoprime.ne_zero [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1)) : p ≠ 0 := by |
rintro rfl
exact not_isCoprime_zero_zero h
| 232 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 89 | 92 | theorem IsCoprime.ne_zero_or_ne_zero [Nontrivial R] (h : IsCoprime x y) : x ≠ 0 ∨ y ≠ 0 := by |
apply not_or_of_imp
rintro rfl rfl
exact not_isCoprime_zero_zero h
| 232 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 102 | 105 | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y := by |
let ⟨a, b, H⟩ := H1
rw [← mul_one y, ← H, mul_add, ← mul_assoc, mul_left_comm]
exact dvd_add (dvd_mul_left _ _) (H2.mul_left _)
| 232 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 108 | 111 | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z := by |
let ⟨a, b, H⟩ := H1
rw [← one_mul z, ← H, add_mul, mul_right_comm, mul_assoc b]
exact dvd_add (dvd_mul_left _ _) (H2.mul_left _)
| 232 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 114 | 121 | theorem IsCoprime.mul_left (H1 : IsCoprime x z) (H2 : IsCoprime y z) : IsCoprime (x * y) z :=
let ⟨a, b, h1⟩ := H1
let ⟨c, d, h2⟩ := H2
⟨a * c, a * x * d + b * c * y + b * d * z,
calc a * c * (x * y) + (a * x * d + b * c * y + b * d * z) * z
_ = (a * x + b * z) * (c * y + d * z) := by | ring
_ = 1 := by rw [h1, h2, mul_one]
⟩
| 232 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 124 | 126 | theorem IsCoprime.mul_right (H1 : IsCoprime x y) (H2 : IsCoprime x z) : IsCoprime x (y * z) := by |
rw [isCoprime_comm] at H1 H2 ⊢
exact H1.mul_left H2
| 232 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 129 | 136 | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by |
obtain ⟨a, b, h⟩ := H
rw [← mul_one z, ← h, mul_add]
apply dvd_add
· rw [mul_comm z, mul_assoc]
exact (mul_dvd_mul_left _ H2).mul_left _
· rw [mul_comm b, ← mul_assoc]
exact (mul_dvd_mul_right H1 _).mul_right _
| 232 |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.ring.abs from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
#align_import data.nat.parity from "leanpr... | Mathlib/Algebra/Order/Ring/Abs.lean | 192 | 193 | theorem abs_dvd (a b : α) : |a| ∣ b ↔ a ∣ b := by |
cases' abs_choice a with h h <;> simp only [h, neg_dvd]
| 233 |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.ring.abs from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
#align_import data.nat.parity from "leanpr... | Mathlib/Algebra/Order/Ring/Abs.lean | 201 | 202 | theorem dvd_abs (a b : α) : a ∣ |b| ↔ a ∣ b := by |
cases' abs_choice b with h h <;> simp only [h, dvd_neg]
| 233 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
| Mathlib/Data/Int/Order/Units.lean | 17 | 18 | theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by |
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
| 234 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 21 | 21 | theorem isUnit_sq {a : ℤ} (ha : IsUnit a) : a ^ 2 = 1 := by | rw [sq, isUnit_mul_self ha]
| 234 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 25 | 26 | theorem units_sq (u : ℤˣ) : u ^ 2 = 1 := by |
rw [Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one, isUnit_sq u.isUnit]
| 234 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 33 | 33 | theorem units_mul_self (u : ℤˣ) : u * u = 1 := by | rw [← sq, units_sq]
| 234 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 37 | 37 | theorem units_inv_eq_self (u : ℤˣ) : u⁻¹ = u := by | rw [inv_eq_iff_mul_eq_one, units_mul_self]
| 234 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 40 | 41 | theorem units_div_eq_mul (u₁ u₂ : ℤˣ) : u₁ / u₂ = u₁ * u₂ := by |
rw [div_eq_mul_inv, units_inv_eq_self]
| 234 |
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