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 |
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import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 165 | 165 | theorem nonneg_iff_ne_neg_one {a : SignType} : 0 ≤ a ↔ a ≠ -1 := by | cases a <;> decide
| 345 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 168 | 168 | theorem neg_one_lt_iff {a : SignType} : -1 < a ↔ 0 ≤ a := by | cases a <;> decide
| 345 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 171 | 171 | theorem nonpos_iff {a : SignType} : a ≤ 0 ↔ a = -1 ∨ a = 0 := by | cases a <;> decide
| 345 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 174 | 174 | theorem nonpos_iff_ne_one {a : SignType} : a ≤ 0 ↔ a ≠ 1 := by | cases a <;> decide
| 345 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 177 | 177 | theorem lt_one_iff {a : SignType} : a < 1 ↔ a ≤ 0 := by | cases a <;> decide
| 345 |
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
instance : TopologicalSpace SignType :=
⊥
instance : DiscreteTopology SignType :=
⟨rfl⟩
variable {α : Type*} [Zero α] [Topological... | Mathlib/Topology/Instances/Sign.lean | 32 | 35 | theorem continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a := by |
refine (continuousAt_const : ContinuousAt (fun _ => (1 : SignType)) a).congr ?_
rw [Filter.EventuallyEq, eventually_nhds_iff]
exact ⟨{ x | 0 < x }, fun x hx => (sign_pos hx).symm, isOpen_lt' 0, h⟩
| 346 |
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
instance : TopologicalSpace SignType :=
⊥
instance : DiscreteTopology SignType :=
⟨rfl⟩
variable {α : Type*} [Zero α] [Topological... | Mathlib/Topology/Instances/Sign.lean | 38 | 41 | theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a := by |
refine (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr ?_
rw [Filter.EventuallyEq, eventually_nhds_iff]
exact ⟨{ x | x < 0 }, fun x hx => (sign_neg hx).symm, isOpen_gt' 0, h⟩
| 346 |
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Rat.Lemmas
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e... | Mathlib/Data/Rat/Cast/Defs.lean | 120 | 121 | theorem cast_natCast (n : ℕ) : ((n : ℚ) : α) = n := by |
rw [← Int.cast_natCast, cast_intCast, Int.cast_natCast]
| 347 |
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Rat.Lemmas
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e... | Mathlib/Data/Rat/Cast/Defs.lean | 143 | 144 | theorem cast_commute (r : ℚ) (a : α) : Commute (↑r) a := by |
simpa only [cast_def] using (r.1.cast_commute a).div_left (r.2.cast_commute a)
| 347 |
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Rat.Lemmas
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e... | Mathlib/Data/Rat/Cast/Defs.lean | 237 | 238 | theorem map_ratCast [DivisionRing α] [DivisionRing β] [RingHomClass F α β] (f : F) (q : ℚ) :
f q = q := by | rw [cast_def, map_div₀, map_intCast, map_natCast, cast_def]
| 347 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 296 | 298 | theorem gluedCoverT'_fst_fst (x y z : 𝒰.J) :
𝒰.gluedCoverT' x y z ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd := by |
delta gluedCoverT'; simp
| 348 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 302 | 304 | theorem gluedCoverT'_fst_snd (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ pullback.fst ≫ pullback.snd = pullback.snd ≫ pullback.snd := by |
delta gluedCoverT'; simp
| 348 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 308 | 310 | theorem gluedCoverT'_snd_fst (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ pullback.snd := by |
delta gluedCoverT'; simp
| 348 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 314 | 316 | theorem gluedCoverT'_snd_snd (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ pullback.snd ≫ pullback.snd = pullback.fst ≫ pullback.fst := by |
delta gluedCoverT'; simp
| 348 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 319 | 322 | theorem glued_cover_cocycle_fst (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.fst =
pullback.fst := by |
apply pullback.hom_ext <;> simp
| 348 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 325 | 328 | theorem glued_cover_cocycle_snd (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.snd =
pullback.snd := by |
apply pullback.hom_ext <;> simp [pullback.condition]
| 348 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 331 | 335 | theorem glued_cover_cocycle (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y = 𝟙 _ := by |
apply pullback.hom_ext <;> simp_rw [Category.id_comp, Category.assoc]
· apply glued_cover_cocycle_fst
· apply glued_cover_cocycle_snd
| 348 |
import Mathlib.Probability.Kernel.Disintegration.Basic
open MeasureTheory ProbabilityTheory MeasurableSpace
open scoped ENNReal
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω]
namespace MeasureTheory... | Mathlib/Probability/Kernel/Disintegration/Integral.lean | 261 | 267 | theorem AEStronglyMeasurable.ae_integrable_condKernel_iff {f : α × Ω → F}
(hf : AEStronglyMeasurable f ρ) :
(∀ᵐ a ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a)) ∧
Integrable (fun a ↦ ∫ ω, ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔ Integrable f ρ := by |
rw [← ρ.compProd_fst_condKernel] at hf
conv_rhs => rw [← ρ.compProd_fst_condKernel]
rw [Measure.integrable_compProd_iff hf]
| 349 |
import Mathlib.Probability.Kernel.Disintegration.Integral
open MeasureTheory Set Filter MeasurableSpace
open scoped ENNReal MeasureTheory Topology ProbabilityTheory
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] ... | Mathlib/Probability/Kernel/Disintegration/Unique.lean | 47 | 56 | theorem eq_condKernel_of_measure_eq_compProd' (κ : kernel α Ω) [IsSFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) {s : Set Ω} (hs : MeasurableSet s) :
∀ᵐ x ∂ρ.fst, κ x s = ρ.condKernel x s := by |
refine ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite
(kernel.measurable_coe κ hs) (kernel.measurable_coe ρ.condKernel hs) (fun t ht _ ↦ ?_)
conv_rhs => rw [Measure.set_lintegral_condKernel_eq_measure_prod ht hs, hκ]
simp only [Measure.compProd_apply (ht.prod hs), Set.mem_prod, ← lintegral_indicator _ ht]
... | 350 |
import Mathlib.Probability.Kernel.Disintegration.Integral
open MeasureTheory Set Filter MeasurableSpace
open scoped ENNReal MeasureTheory Topology ProbabilityTheory
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] ... | Mathlib/Probability/Kernel/Disintegration/Unique.lean | 81 | 124 | theorem eq_condKernel_of_measure_eq_compProd (κ : kernel α Ω) [IsFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) :
∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x := by |
-- The idea is to transport the question to `ℝ` from `Ω` using `embeddingReal`
-- and then construct a measure on `α × ℝ`
let f := embeddingReal Ω
have hf := measurableEmbedding_embeddingReal Ω
set ρ' : Measure (α × ℝ) := ρ.map (Prod.map id f) with hρ'def
have hρ' : ρ'.fst = ρ.fst := by
ext s hs
rw... | 350 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 88 | 93 | theorem _root_.MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
(hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
(∀ᵐ a ∂μ.map X, Integrable (fun ω => f (a, ω)) (condDistrib Y X μ a)) ∧
Integrable (fun a => ∫ ω, ‖f (a, ω)‖ ∂condDistrib Y X μ a) (μ.map ... |
rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prod_mk₀ hY]
| 351 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 98 | 101 | theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map
(hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂condDistrib Y X μ x) (μ.map X) := by |
rw [← Measure.fst_map_prod_mk₀ hY, condDistrib]; exact hf.integral_condKernel
| 351 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 120 | 130 | theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)
(κ : kernel β Ω) [IsFiniteKernel κ] (hκ : μ.map (fun x => (X x, Y x)) = μ.map X ⊗ₘ κ) :
∀ᵐ x ∂μ.map X, κ x = condDistrib Y X μ x := by |
have heq : μ.map X = (μ.map (fun x ↦ (X x, Y x))).fst := by
ext s hs
rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply]
exacts [rfl, Measurable.prod hX hY, measurable_fst hs]
rw [heq, condDistrib]
refine eq_condKernel_of_measure_eq_compProd _ ?_
convert hκ
exact heq.symm
| 351 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 134 | 142 | theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableSet s) :
Integrable (fun a => (condDistrib Y X μ (X a) s).toReal) μ := by |
refine integrable_toReal_of_lintegral_ne_top ?_ ?_
· exact Measurable.comp_aemeasurable (kernel.measurable_coe _ hs) hX
· refine ne_of_lt ?_
calc
∫⁻ a, condDistrib Y X μ (X a) s ∂μ ≤ ∫⁻ _, 1 ∂μ := lintegral_mono fun a => prob_le_one
_ = μ univ := lintegral_one
_ < ∞ := measure_lt_top _ _
| 351 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 145 | 148 | theorem _root_.MeasureTheory.Integrable.condDistrib_ae_map
(hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
∀ᵐ b ∂μ.map X, Integrable (fun ω => f (b, ω)) (condDistrib Y X μ b) := by |
rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.condKernel_ae
| 351 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 157 | 160 | theorem _root_.MeasureTheory.Integrable.integral_norm_condDistrib_map
(hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂condDistrib Y X μ x) (μ.map X) := by |
rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.integral_norm_condKernel
| 351 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 171 | 174 | theorem _root_.MeasureTheory.Integrable.norm_integral_condDistrib_map
(hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
Integrable (fun x => ‖∫ y, f (x, y) ∂condDistrib Y X μ x‖) (μ.map X) := by |
rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.norm_integral_condKernel
| 351 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 198 | 206 | theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurable Y μ)
(hs : MeasurableSet s) (ht : MeasurableSet t) :
∫⁻ a in X ⁻¹' t, condDistrib Y X μ (X a) s ∂μ = μ (X ⁻¹' t ∩ Y ⁻¹' s) := by |
-- Porting note: need to massage the LHS integrand into the form accepted by `lintegral_comp`
-- (`rw` does not see that the two forms are defeq)
conv_lhs => arg 2; change (fun a => ((condDistrib Y X μ) a) s) ∘ X
rw [lintegral_comp (kernel.measurable_coe _ hs) hX, condDistrib, ← Measure.restrict_map hX ht, ←
... | 351 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 209 | 213 | theorem set_lintegral_condDistrib_of_measurableSet (hX : Measurable X) (hY : AEMeasurable Y μ)
(hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet[mβ.comap X] t) :
∫⁻ a in t, condDistrib Y X μ (X a) s ∂μ = μ (t ∩ Y ⁻¹' s) := by |
obtain ⟨t', ht', rfl⟩ := ht
rw [set_lintegral_preimage_condDistrib hX hY hs ht']
| 351 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 87 | 88 | theorem Rel.addLeft (a : lib R X) {b c : lib R X} (h : Rel R X b c) : Rel R X (a + b) (a + c) := by |
rw [add_comm _ b, add_comm _ c]; exact h.add_right _
| 352 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 91 | 92 | theorem Rel.neg {a b : lib R X} (h : Rel R X a b) : Rel R X (-a) (-b) := by |
simpa only [neg_one_smul] using h.smul (-1)
| 352 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 95 | 96 | theorem Rel.subLeft (a : lib R X) {b c : lib R X} (h : Rel R X b c) : Rel R X (a - b) (a - c) := by |
simpa only [sub_eq_add_neg] using h.neg.addLeft a
| 352 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 99 | 100 | theorem Rel.subRight {a b : lib R X} (c : lib R X) (h : Rel R X a b) : Rel R X (a - c) (b - c) := by |
simpa only [sub_eq_add_neg] using h.add_right (-c)
| 352 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 103 | 106 | theorem Rel.smulOfTower {S : Type*} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (t : S)
(a b : lib R X) (h : Rel R X a b) : Rel R X (t • a) (t • b) := by |
rw [← smul_one_smul R t a, ← smul_one_smul R t b]
exact h.smul _
| 352 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 32 | 34 | theorem Coprime.dvd_of_dvd_mul_right (H1 : Coprime k n) (H2 : k ∣ m * n) : k ∣ m := by |
let t := dvd_gcd (Nat.dvd_mul_left k m) H2
rwa [gcd_mul_left, H1.gcd_eq_one, Nat.mul_one] at t
| 353 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 39 | 44 | theorem Coprime.gcd_mul_left_cancel (m : Nat) (H : Coprime k n) : gcd (k * m) n = gcd m n :=
have H1 : Coprime (gcd (k * m) n) k := by |
rw [Coprime, Nat.gcd_assoc, H.symm.gcd_eq_one, gcd_one_right]
Nat.dvd_antisymm
(dvd_gcd (H1.dvd_of_dvd_mul_left (gcd_dvd_left _ _)) (gcd_dvd_right _ _))
(gcd_dvd_gcd_mul_left _ _ _)
| 353 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 46 | 47 | theorem Coprime.gcd_mul_right_cancel (m : Nat) (H : Coprime k n) : gcd (m * k) n = gcd m n := by |
rw [Nat.mul_comm m k, H.gcd_mul_left_cancel m]
| 353 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 49 | 51 | theorem Coprime.gcd_mul_left_cancel_right (n : Nat)
(H : Coprime k m) : gcd m (k * n) = gcd m n := by |
rw [gcd_comm m n, gcd_comm m (k * n), H.gcd_mul_left_cancel n]
| 353 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 53 | 55 | theorem Coprime.gcd_mul_right_cancel_right (n : Nat)
(H : Coprime k m) : gcd m (n * k) = gcd m n := by |
rw [Nat.mul_comm n k, H.gcd_mul_left_cancel_right n]
| 353 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 57 | 59 | theorem coprime_div_gcd_div_gcd
(H : 0 < gcd m n) : Coprime (m / gcd m n) (n / gcd m n) := by |
rw [coprime_iff_gcd_eq_one, gcd_div (gcd_dvd_left m n) (gcd_dvd_right m n), Nat.div_self H]
| 353 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 65 | 75 | theorem exists_coprime (m n : Nat) :
∃ m' n', Coprime m' n' ∧ m = m' * gcd m n ∧ n = n' * gcd m n := by |
cases eq_zero_or_pos (gcd m n) with
| inl h0 =>
rw [gcd_eq_zero_iff] at h0
refine ⟨1, 1, gcd_one_left 1, ?_⟩
simp [h0]
| inr hpos =>
exact ⟨_, _, coprime_div_gcd_div_gcd hpos,
(Nat.div_mul_cancel (gcd_dvd_left m n)).symm,
(Nat.div_mul_cancel (gcd_dvd_right m n)).symm⟩
| 353 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 87 | 91 | theorem Coprime.coprime_dvd_left (H1 : m ∣ k) (H2 : Coprime k n) : Coprime m n := by |
apply eq_one_of_dvd_one
rw [Coprime] at H2
have := Nat.gcd_dvd_gcd_of_dvd_left n H1
rwa [← H2]
| 353 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 108 | 118 | theorem Coprime.coprime_div_left (cmn : Coprime m n) (dvd : a ∣ m) : Coprime (m / a) n := by |
match eq_zero_or_pos a with
| .inl h0 =>
rw [h0] at dvd
rw [Nat.eq_zero_of_zero_dvd dvd] at cmn ⊢
simp; assumption
| .inr hpos =>
let ⟨k, hk⟩ := dvd
rw [hk, Nat.mul_div_cancel_left _ hpos]
rw [hk] at cmn
exact cmn.coprime_mul_left
| 353 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 25 | 28 | theorem add_tsub_cancel_of_le (h : a ≤ b) : a + (b - a) = b := by |
refine le_antisymm ?_ le_add_tsub
obtain ⟨c, rfl⟩ := exists_add_of_le h
exact add_le_add_left add_tsub_le_left a
| 354 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 31 | 33 | theorem tsub_add_cancel_of_le (h : a ≤ b) : b - a + a = b := by |
rw [add_comm]
exact add_tsub_cancel_of_le h
| 354 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 44 | 45 | theorem tsub_le_tsub_iff_right (h : c ≤ b) : a - c ≤ b - c ↔ a ≤ b := by |
rw [tsub_le_iff_right, tsub_add_cancel_of_le h]
| 354 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 48 | 49 | theorem tsub_left_inj (h1 : c ≤ a) (h2 : c ≤ b) : a - c = b - c ↔ a = b := by |
simp_rw [le_antisymm_iff, tsub_le_tsub_iff_right h1, tsub_le_tsub_iff_right h2]
| 354 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 57 | 60 | theorem lt_of_tsub_lt_tsub_right_of_le (h : c ≤ b) (h2 : a - c < b - c) : a < b := by |
refine ((tsub_le_tsub_iff_right h).mp h2.le).lt_of_ne ?_
rintro rfl
exact h2.false
| 354 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 63 | 65 | theorem tsub_add_tsub_cancel (hab : b ≤ a) (hcb : c ≤ b) : a - b + (b - c) = a - c := by |
convert tsub_add_cancel_of_le (tsub_le_tsub_right hab c) using 2
rw [tsub_tsub, add_tsub_cancel_of_le hcb]
| 354 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 68 | 69 | theorem tsub_tsub_tsub_cancel_right (h : c ≤ b) : a - c - (b - c) = a - b := by |
rw [tsub_tsub, add_tsub_cancel_of_le h]
| 354 |
import Mathlib.Logic.Relation
import Mathlib.Order.GaloisConnection
#align_import data.setoid.basic from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
variable {α : Type*} {β : Type*}
def Setoid.Rel (r : Setoid α) : α → α → Prop :=
@Setoid.r _ r
#align setoid.rel Setoid.Rel
instanc... | Mathlib/Data/Setoid/Basic.lean | 155 | 158 | theorem sInf_def {s : Set (Setoid α)} : (sInf s).Rel = sInf (Rel '' s) := by |
ext
simp only [sInf_image, iInf_apply, iInf_Prop_eq]
rfl
| 355 |
import Mathlib.Geometry.Manifold.Sheaf.Smooth
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
noncomputable section
universe u
variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
{EM : Type*} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM]
{HM : Type*} [TopologicalSpace HM] (IM : ModelWit... | Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean | 43 | 98 | theorem smoothSheafCommRing.isUnit_stalk_iff {x : M}
(f : (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x) :
IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) := by |
constructor
· rintro ⟨⟨f, g, hf, hg⟩, rfl⟩ (h' : smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x f = 0)
simpa [h'] using congr_arg (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) hf
· let S := (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf
-- Suppose that `f`, in the stalk at `x`, is nonzero at `x`
rintro (hf :... | 356 |
import Mathlib.Geometry.Manifold.Sheaf.Smooth
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
noncomputable section
universe u
variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
{EM : Type*} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM]
{HM : Type*} [TopologicalSpace HM] (IM : ModelWit... | Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean | 102 | 107 | theorem smoothSheafCommRing.nonunits_stalk (x : M) :
nonunits ((smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x)
= RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) := by |
ext1 f
rw [mem_nonunits_iff, not_iff_comm, Iff.comm]
apply smoothSheafCommRing.isUnit_stalk_iff
| 356 |
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem getElem_eq_data_getElem (a : ByteArray) (h : i < a.size) : a[i] = a.data[i] := rfl
@[simp] theorem uset_eq_set (a : ByteArray) {i : USize} (h : i.toNat < a.size... | .lake/packages/batteries/Batteries/Data/ByteArray.lean | 76 | 77 | theorem size_append (a b : ByteArray) : (a ++ b).size = a.size + b.size := by |
simp only [size, append_eq, append_data]; exact Array.size_append ..
| 357 |
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem getElem_eq_data_getElem (a : ByteArray) (h : i < a.size) : a[i] = a.data[i] := rfl
@[simp] theorem uset_eq_set (a : ByteArray) {i : USize} (h : i.toNat < a.size... | .lake/packages/batteries/Batteries/Data/ByteArray.lean | 79 | 82 | theorem get_append_left {a b : ByteArray} (hlt : i < a.size)
(h : i < (a ++ b).size := size_append .. ▸ Nat.lt_of_lt_of_le hlt (Nat.le_add_right ..)) :
(a ++ b)[i] = a[i] := by |
simp [getElem_eq_data_getElem]; exact Array.get_append_left hlt
| 357 |
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem getElem_eq_data_getElem (a : ByteArray) (h : i < a.size) : a[i] = a.data[i] := rfl
@[simp] theorem uset_eq_set (a : ByteArray) {i : USize} (h : i.toNat < a.size... | .lake/packages/batteries/Batteries/Data/ByteArray.lean | 84 | 87 | theorem get_append_right {a b : ByteArray} (hle : a.size ≤ i) (h : i < (a ++ b).size)
(h' : i - a.size < b.size := Nat.sub_lt_left_of_lt_add hle (size_append .. ▸ h)) :
(a ++ b)[i] = b[i - a.size] := by |
simp [getElem_eq_data_getElem]; exact Array.get_append_right hle
| 357 |
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem getElem_eq_data_getElem (a : ByteArray) (h : i < a.size) : a[i] = a.data[i] := rfl
@[simp] theorem uset_eq_set (a : ByteArray) {i : USize} (h : i.toNat < a.size... | .lake/packages/batteries/Batteries/Data/ByteArray.lean | 102 | 105 | theorem get_extract_aux {a : ByteArray} {start stop} (h : i < (a.extract start stop).size) :
start + i < a.size := by |
apply Nat.add_lt_of_lt_sub'; apply Nat.lt_of_lt_of_le h
rw [size_extract, ← Nat.sub_min_sub_right]; exact Nat.min_le_right ..
| 357 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 144 | 147 | theorem mapRange_id (h : ∀ i, id (0 : β₁ i) = 0 := fun i => rfl) (g : Π₀ i : ι, β₁ i) :
mapRange (fun i => (id : β₁ i → β₁ i)) h g = g := by |
ext
rfl
| 358 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 150 | 154 | theorem mapRange_comp (f : ∀ i, β₁ i → β₂ i) (f₂ : ∀ i, β i → β₁ i) (hf : ∀ i, f i 0 = 0)
(hf₂ : ∀ i, f₂ i 0 = 0) (h : ∀ i, (f i ∘ f₂ i) 0 = 0) (g : Π₀ i : ι, β i) :
mapRange (fun i => f i ∘ f₂ i) h g = mapRange f hf (mapRange f₂ hf₂ g) := by |
ext
simp only [mapRange_apply]; rfl
| 358 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 158 | 161 | theorem mapRange_zero (f : ∀ i, β₁ i → β₂ i) (hf : ∀ i, f i 0 = 0) :
mapRange f hf (0 : Π₀ i, β₁ i) = 0 := by |
ext
simp only [mapRange_apply, coe_zero, Pi.zero_apply, hf]
| 358 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α : Type*} {N : α → Type*}
namespace DFinsupp
variable [DecidableEq α]
section NHasZero
variable [∀ a, DecidableEq (N a)] [∀ a, Zero (N a)] (f g : Π₀... | Mathlib/Data/DFinsupp/NeLocus.lean | 41 | 43 | theorem mem_neLocus {f g : Π₀ a, N a} {a : α} : a ∈ f.neLocus g ↔ f a ≠ g a := by |
simpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff,
and_iff_right_iff_imp] using Ne.ne_or_ne _
| 359 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α : Type*} {N : α → Type*}
namespace DFinsupp
variable [DecidableEq α]
section NHasZero
variable [∀ a, DecidableEq (N a)] [∀ a, Zero (N a)] (f g : Π₀... | Mathlib/Data/DFinsupp/NeLocus.lean | 67 | 68 | theorem neLocus_comm : f.neLocus g = g.neLocus f := by |
simp_rw [neLocus, Finset.union_comm, ne_comm]
| 359 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α : Type*} {N : α → Type*}
namespace DFinsupp
variable [DecidableEq α]
section NHasZero
variable [∀ a, DecidableEq (N a)] [∀ a, Zero (N a)] (f g : Π₀... | Mathlib/Data/DFinsupp/NeLocus.lean | 72 | 74 | theorem neLocus_zero_right : f.neLocus 0 = f.support := by |
ext
rw [mem_neLocus, mem_support_iff, coe_zero, Pi.zero_apply]
| 359 |
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.DFinsupp.Basic
#align_import algebra.direct_sum.basic from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Function
universe u v w u₁
variable (ι : Type v) [dec_ι : DecidableEq ι] (β : ι → Type w)
def DirectSum... | Mathlib/Algebra/DirectSum/Basic.lean | 155 | 159 | theorem sum_univ_of [Fintype ι] (x : ⨁ i, β i) :
∑ i ∈ Finset.univ, of β i (x i) = x := by |
apply DFinsupp.ext (fun i ↦ ?_)
rw [DFinsupp.finset_sum_apply]
simp [of_apply]
| 360 |
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
#align_import algebra.star.unitary from "leanprover-community/mathlib"@"247a102b14f3cebfee126293341af5f6bed00237"
def unitary (R : Type*) [Monoid R] [StarMul R] : Submonoid R where
carrier := { U | star U * U = 1 ∧ U * st... | Mathlib/Algebra/Star/Unitary.lean | 141 | 145 | theorem _root_.IsUnit.mem_unitary_of_star_mul_self {u : R} (hu : IsUnit u)
(h_mul : star u * u = 1) : u ∈ unitary R := by |
refine unitary.mem_iff.mpr ⟨h_mul, ?_⟩
lift u to Rˣ using hu
exact left_inv_eq_right_inv h_mul u.mul_inv ▸ u.mul_inv
| 361 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 90 | 94 | theorem lt_or_lt_of_mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂ := by |
contrapose!
exact fun h => mul_le_mul' h.1 h.2
| 362 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 99 | 103 | theorem le_or_lt_of_mul_le_mul [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂ := by |
contrapose!
exact fun h => mul_lt_mul_of_lt_of_le h.1 h.2
| 362 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 108 | 112 | theorem lt_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ := by |
contrapose!
exact fun h => mul_lt_mul_of_le_of_lt h.1 h.2
| 362 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 117 | 121 | theorem le_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂ := by |
contrapose!
exact fun h => mul_lt_mul_of_lt_of_lt h.1 h.2
| 362 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 126 | 133 | theorem mul_lt_mul_iff_of_le_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂)
(hb : b₁ ≤ b₂) : a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂ := by |
refine ⟨lt_or_lt_of_mul_lt_mul, fun h => ?_⟩
cases' h with ha' hb'
· exact mul_lt_mul_of_lt_of_le ha' hb
· exact mul_lt_mul_of_le_of_lt ha hb'
| 362 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 22 | 23 | theorem max_one_div_max_inv_one_eq_self (a : α) : max a 1 / max a⁻¹ 1 = a := by |
rcases le_total a 1 with (h | h) <;> simp [h]
| 363 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 57 | 58 | theorem min_div_div_right' (a b c : α) : min (a / c) (b / c) = min a b / c := by |
simpa only [div_eq_mul_inv] using min_mul_mul_right a b c⁻¹
| 363 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 63 | 64 | theorem max_div_div_right' (a b c : α) : max (a / c) (b / c) = max a b / c := by |
simpa only [div_eq_mul_inv] using max_mul_mul_right a b c⁻¹
| 363 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 69 | 70 | theorem min_div_div_left' (a b c : α) : min (a / b) (a / c) = a / max b c := by |
simp only [div_eq_mul_inv, min_mul_mul_left, min_inv_inv']
| 363 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 75 | 76 | theorem max_div_div_left' (a b c : α) : max (a / b) (a / c) = a / min b c := by |
simp only [div_eq_mul_inv, max_mul_mul_left, max_inv_inv']
| 363 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 86 | 93 | theorem max_sub_max_le_max (a b c d : α) : max a b - max c d ≤ max (a - c) (b - d) := by |
simp only [sub_le_iff_le_add, max_le_iff]; constructor
· calc
a = a - c + c := (sub_add_cancel a c).symm
_ ≤ max (a - c) (b - d) + max c d := add_le_add (le_max_left _ _) (le_max_left _ _)
· calc
b = b - d + d := (sub_add_cancel b d).symm
_ ≤ max (a - c) (b - d) + max c d := add_le_add (le_max_ri... | 363 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 96 | 100 | theorem abs_max_sub_max_le_max (a b c d : α) : |max a b - max c d| ≤ max |a - c| |b - d| := by |
refine abs_sub_le_iff.2 ⟨?_, ?_⟩
· exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _))
· rw [abs_sub_comm a c, abs_sub_comm b d]
exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _))
| 363 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 57 | 57 | theorem dichotomy (b : Bool) : b = false ∨ b = true := by | cases b <;> simp
| 364 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 99 | 99 | theorem or_inl {a b : Bool} (H : a) : a || b := by | simp [H]
| 364 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 102 | 102 | theorem or_inr {a b : Bool} (H : b) : a || b := by | cases a <;> simp [H]
| 364 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 109 | 109 | theorem and_elim_left : ∀ {a b : Bool}, a && b → a := by | decide
| 364 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 112 | 112 | theorem and_intro : ∀ {a b : Bool}, a → b → a && b := by | decide
| 364 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 115 | 115 | theorem and_elim_right : ∀ {a b : Bool}, a && b → b := by | decide
| 364 |
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Subtype
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.countable.defs from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
open Function
universe u v
variable {α : Sort u} {β : Sort v}
... | Mathlib/Data/Countable/Defs.lean | 159 | 161 | theorem uncountable_iff_forall_not_surjective [Nonempty α] :
Uncountable α ↔ ∀ f : ℕ → α, ¬Surjective f := by |
rw [← not_countable_iff, countable_iff_exists_surjective, not_exists]
| 365 |
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists OrderedCommMonoid
assert_not_exists MonoidWithZero
assert_not_exists MulA... | Mathlib/Data/Finset/Image.lean | 81 | 86 | theorem mem_map_equiv {f : α ≃ β} {b : β} : b ∈ s.map f.toEmbedding ↔ f.symm b ∈ s := by |
rw [mem_map]
exact
⟨by
rintro ⟨a, H, rfl⟩
simpa, fun h => ⟨_, h, by simp⟩⟩
| 366 |
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists OrderedCommMonoid
assert_not_exists MonoidWithZero
assert_not_exists MulA... | Mathlib/Data/Finset/Image.lean | 141 | 144 | theorem map_cast_heq {α β} (h : α = β) (s : Finset α) :
HEq (s.map (Equiv.cast h).toEmbedding) s := by |
subst h
simp
| 366 |
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists OrderedCommMonoid
assert_not_exists MonoidWithZero
assert_not_exists MulA... | Mathlib/Data/Finset/Image.lean | 151 | 153 | theorem map_comm {β'} {f : β ↪ γ} {g : α ↪ β} {f' : α ↪ β'} {g' : β' ↪ γ}
(h_comm : ∀ a, f (g a) = g' (f' a)) : (s.map g).map f = (s.map f').map g' := by |
simp_rw [map_map, Embedding.trans, Function.comp, h_comm]
| 366 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 69 | 69 | theorem card_mono : Monotone (@card α) := by | apply card_le_card
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 91 | 94 | theorem card_singleton_inter [DecidableEq α] : ({a} ∩ s).card ≤ 1 := by |
cases' Finset.decidableMem a s with h h
· simp [Finset.singleton_inter_of_not_mem h]
· simp [Finset.singleton_inter_of_mem h]
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 107 | 108 | theorem card_insert_of_not_mem (h : a ∉ s) : (insert a s).card = s.card + 1 := by |
rw [← cons_eq_insert _ _ h, card_cons]
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 111 | 111 | theorem card_insert_of_mem (h : a ∈ s) : card (insert a s) = s.card := by | rw [insert_eq_of_mem h]
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 114 | 118 | theorem card_insert_le (a : α) (s : Finset α) : card (insert a s) ≤ s.card + 1 := by |
by_cases h : a ∈ s
· rw [insert_eq_of_mem h]
exact Nat.le_succ _
· rw [card_insert_of_not_mem h]
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 143 | 146 | theorem card_insert_eq_ite : card (insert a s) = if a ∈ s then s.card else s.card + 1 := by |
by_cases h : a ∈ s
· rw [card_insert_of_mem h, if_pos h]
· rw [card_insert_of_not_mem h, if_neg h]
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 150 | 152 | theorem card_pair_eq_one_or_two : ({a,b} : Finset α).card = 1 ∨ ({a,b} : Finset α).card = 2 := by |
simp [card_insert_eq_ite]
tauto
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 155 | 156 | theorem card_pair (h : a ≠ b) : ({a, b} : Finset α).card = 2 := by |
rw [card_insert_of_not_mem (not_mem_singleton.2 h), card_singleton]
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 171 | 175 | theorem cast_card_erase_of_mem {R} [AddGroupWithOne R] {s : Finset α} (hs : a ∈ s) :
((s.erase a).card : R) = s.card - 1 := by |
rw [card_erase_of_mem hs, Nat.cast_sub, Nat.cast_one]
rw [Nat.add_one_le_iff, Finset.card_pos]
exact ⟨a, hs⟩
| 367 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 259 | 260 | theorem length_toList (s : Finset α) : s.toList.length = s.card := by |
rw [toList, ← Multiset.coe_card, Multiset.coe_toList, card_def]
| 367 |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 76 | 78 | theorem product_image_fst [DecidableEq α] (ht : t.Nonempty) : (s ×ˢ t).image Prod.fst = s := by |
ext i
simp [mem_image, ht.exists_mem]
| 368 |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 81 | 83 | theorem product_image_snd [DecidableEq β] (ht : s.Nonempty) : (s ×ˢ t).image Prod.snd = t := by |
ext i
simp [mem_image, ht.exists_mem]
| 368 |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 137 | 139 | theorem product_biUnion [DecidableEq γ] (s : Finset α) (t : Finset β) (f : α × β → Finset γ) :
(s ×ˢ t).biUnion f = s.biUnion fun a => t.biUnion fun b => f (a, b) := by |
classical simp_rw [product_eq_biUnion, biUnion_biUnion, image_biUnion]
| 368 |
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