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 | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 167 | 167 | theorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by | rw [add_comm, map_add_one, add_comm 1]
| 1 | 2.718282 | 0 | 0 | 3 | 35 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 213 | 214 | theorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :
(f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by | simp only [← mul_apply, f.inv_mul, coe_one, id]
| 1 | 2.718282 | 0 | 0 | 3 | 35 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 218 | 219 | theorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :
f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by | simp only [← mul_apply, f.mul_inv, coe_one, id]
| 1 | 2.718282 | 0 | 0 | 3 | 35 |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
open Set
variable {α β γ δ : Type*} {l : Filter α} {f : α → β}
namespace Filter
def EventuallyConst (f : α → β) (l : Filter α) : Prop := (map f l).Subsingleton
theorem HasBasis.eventuallyConst_iff {ι : Sort*} {p : ι → Prop} {s : ι → S... | Mathlib/Order/Filter/EventuallyConst.lean | 57 | 59 | theorem eventuallyConst_pred' {p : α → Prop} :
EventuallyConst p l ↔ (p =ᶠ[l] fun _ ↦ False) ∨ (p =ᶠ[l] fun _ ↦ True) := by |
simp only [eventuallyConst_iff_exists_eventuallyEq, Prop.exists_iff]
| 1 | 2.718282 | 0 | 0 | 4 | 36 |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
open Set
variable {α β γ δ : Type*} {l : Filter α} {f : α → β}
namespace Filter
def EventuallyConst (f : α → β) (l : Filter α) : Prop := (map f l).Subsingleton
theorem HasBasis.eventuallyConst_iff {ι : Sort*} {p : ι → Prop} {s : ι → S... | Mathlib/Order/Filter/EventuallyConst.lean | 61 | 63 | theorem eventuallyConst_pred {p : α → Prop} :
EventuallyConst p l ↔ (∀ᶠ x in l, p x) ∨ (∀ᶠ x in l, ¬p x) := by |
simp [eventuallyConst_pred', or_comm, EventuallyEq]
| 1 | 2.718282 | 0 | 0 | 4 | 36 |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
open Set
variable {α β γ δ : Type*} {l : Filter α} {f : α → β}
namespace Filter
def EventuallyConst (f : α → β) (l : Filter α) : Prop := (map f l).Subsingleton
theorem HasBasis.eventuallyConst_iff {ι : Sort*} {p : ι → Prop} {s : ι → S... | Mathlib/Order/Filter/EventuallyConst.lean | 73 | 75 | theorem EventuallyEq.eventuallyConst_iff {g : α → β} (h : f =ᶠ[l] g) :
EventuallyConst f l ↔ EventuallyConst g l := by |
simp only [EventuallyConst, map_congr h]
| 1 | 2.718282 | 0 | 0 | 4 | 36 |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
open Set
variable {α β γ δ : Type*} {l : Filter α} {f : α → β}
namespace Filter
def EventuallyConst (f : α → β) (l : Filter α) : Prop := (map f l).Subsingleton
theorem HasBasis.eventuallyConst_iff {ι : Sort*} {p : ι → Prop} {s : ι → S... | Mathlib/Order/Filter/EventuallyConst.lean | 145 | 147 | theorem mulIndicator_const_iff :
EventuallyConst (s.mulIndicator fun _ ↦ c) l ↔ c = 1 ∨ EventuallyConst s l := by |
rcases eq_or_ne c 1 with rfl | hc <;> simp [mulIndicator_const_iff_of_ne, *]
| 1 | 2.718282 | 0 | 0 | 4 | 36 |
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.Cat... | Mathlib/Topology/Sheaves/Stalks.lean | 113 | 114 | theorem germ_res_apply (F : X.Presheaf C) {U V : Opens X} (i : U ⟶ V) (x : U) [ConcreteCategory C]
(s) : germ F x (F.map i.op s) = germ F (i x) s := by | rw [← comp_apply, germ_res]
| 1 | 2.718282 | 0 | 0 | 2 | 37 |
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.Cat... | Mathlib/Topology/Sheaves/Stalks.lean | 150 | 153 | theorem stalkPushforward_germ (f : X ⟶ Y) (F : X.Presheaf C) (U : Opens Y)
(x : (Opens.map f).obj U) :
(f _* F).germ ⟨(f : X → Y) (x : X), x.2⟩ ≫ F.stalkPushforward C f x = F.germ x := by |
simp [germ, stalkPushforward]
| 1 | 2.718282 | 0 | 0 | 2 | 37 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 69 | 70 | theorem cast_to_int [AddGroupWithOne α] (n : PosNum) : ((n : ℤ) : α) = n := by |
rw [← to_nat_to_int, Int.cast_natCast, cast_to_nat]
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 81 | 81 | theorem one_add (n : PosNum) : 1 + n = succ n := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 84 | 84 | theorem add_one (n : PosNum) : n + 1 = succ n := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 210 | 210 | theorem add_zero (n : Num) : n + 0 = n := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 213 | 213 | theorem zero_add (n : Num) : 0 + n = n := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 712 | 712 | theorem bit_to_nat (b n) : (bit b n : ℕ) = Nat.bit b n := by | cases b <;> cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 1,053 | 1,053 | theorem zneg_zneg (n : ZNum) : - -n = n := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 1,056 | 1,056 | theorem zneg_bit1 (n : ZNum) : -n.bit1 = (-n).bitm1 := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 1,059 | 1,059 | theorem zneg_bitm1 (n : ZNum) : -n.bitm1 = (-n).bit1 := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 9 | 38 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.Index
#align_import group_theory.commensurable from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
variable {G : Type*} [Group G]
def Commensurable (H K : Subgroup... | Mathlib/GroupTheory/Commensurable.lean | 81 | 82 | theorem commensurable_inv (H : Subgroup G) (g : ConjAct G) :
Commensurable (g • H) H ↔ Commensurable H (g⁻¹ • H) := by | rw [commensurable_conj, inv_smul_smul]
| 1 | 2.718282 | 0 | 0 | 1 | 39 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
universe u v
namespace SimpleGraph
@[ext]
structure Subgraph {V : Type u} (G : SimpleGra... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 177 | 178 | theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by |
simp [Subgraph.spanningCoe]
| 1 | 2.718282 | 0 | 0 | 1 | 40 |
import Mathlib.CategoryTheory.Functor.Basic
import Mathlib.Util.AddRelatedDecl
import Mathlib.Lean.Meta.Simp
open Lean Meta Elab Tactic
open Mathlib.Tactic
namespace CategoryTheory
variable {C : Type*} [Category C]
| Mathlib/Tactic/CategoryTheory/Reassoc.lean | 34 | 35 | theorem eq_whisker' {X Y : C} {f g : X ⟶ Y} (w : f = g) {Z : C} (h : Y ⟶ Z) :
f ≫ h = g ≫ h := by | rw [w]
| 1 | 2.718282 | 0 | 0 | 1 | 41 |
import Mathlib.Tactic.Linarith.Datatypes
import Mathlib.Tactic.Zify
import Mathlib.Tactic.CancelDenoms.Core
import Batteries.Data.RBMap.Basic
import Mathlib.Data.HashMap
import Mathlib.Control.Basic
set_option autoImplicit true
namespace Linarith
open Lean hiding Rat
open Elab Tactic Meta
open Qq
partial def ... | Mathlib/Tactic/Linarith/Preprocessing.lean | 273 | 273 | theorem without_one_mul [MulOneClass M] {a b : M} (h : 1 * a = b) : a = b := by | rwa [one_mul] at h
| 1 | 2.718282 | 0 | 0 | 1 | 42 |
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.hom.units from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u v w
namespace Units
variable {α : Ty... | Mathlib/Algebra/Group/Units/Hom.lean | 94 | 94 | theorem map_id : map (MonoidHom.id M) = MonoidHom.id Mˣ := by | ext; rfl
| 1 | 2.718282 | 0 | 0 | 4 | 43 |
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.hom.units from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u v w
namespace Units
variable {α : Ty... | Mathlib/Algebra/Group/Units/Hom.lean | 150 | 152 | theorem mul_liftRight_inv {f : M →* N} {g : M → Nˣ} (h : ∀ x, ↑(g x) = f x) (x) :
f x * ↑(liftRight f g h x)⁻¹ = 1 := by |
rw [Units.mul_inv_eq_iff_eq_mul, one_mul, coe_liftRight]
| 1 | 2.718282 | 0 | 0 | 4 | 43 |
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.hom.units from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u v w
namespace Units
variable {α : Ty... | Mathlib/Algebra/Group/Units/Hom.lean | 157 | 159 | theorem liftRight_inv_mul {f : M →* N} {g : M → Nˣ} (h : ∀ x, ↑(g x) = f x) (x) :
↑(liftRight f g h x)⁻¹ * f x = 1 := by |
rw [Units.inv_mul_eq_iff_eq_mul, mul_one, coe_liftRight]
| 1 | 2.718282 | 0 | 0 | 4 | 43 |
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.hom.units from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u v w
namespace Units
variable {α : Ty... | Mathlib/Algebra/Group/Units/Hom.lean | 198 | 199 | theorem map [MonoidHomClass F M N] (f : F) {x : M} (h : IsUnit x) : IsUnit (f x) := by |
rcases h with ⟨y, rfl⟩; exact (Units.map (f : M →* N) y).isUnit
| 1 | 2.718282 | 0 | 0 | 4 | 43 |
import Mathlib.Order.Sublattice
import Mathlib.Order.Hom.CompleteLattice
open Function Set
variable (α β : Type*) [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
structure CompleteSublattice extends Sublattice α where
sSupClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sSup s ∈ carrier
sInfClosed... | Mathlib/Order/CompleteSublattice.lean | 84 | 85 | theorem coe_sSup' (S : Set L) : (↑(sSup S) : α) = ⨆ N ∈ S, (N : α) := by |
rw [coe_sSup, ← Set.image, sSup_image]
| 1 | 2.718282 | 0 | 0 | 2 | 44 |
import Mathlib.Order.Sublattice
import Mathlib.Order.Hom.CompleteLattice
open Function Set
variable (α β : Type*) [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
structure CompleteSublattice extends Sublattice α where
sSupClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sSup s ∈ carrier
sInfClosed... | Mathlib/Order/CompleteSublattice.lean | 89 | 90 | theorem coe_sInf' (S : Set L) : (↑(sInf S) : α) = ⨅ N ∈ S, (N : α) := by |
rw [coe_sInf, ← Set.image, sInf_image]
| 1 | 2.718282 | 0 | 0 | 2 | 44 |
import Mathlib.Geometry.Manifold.Algebra.Monoid
#align_import geometry.manifold.algebra.lie_group from "leanprover-community/mathlib"@"f9ec187127cc5b381dfcf5f4a22dacca4c20b63d"
noncomputable section
open scoped Manifold
-- See note [Design choices about smooth algebraic structures]
class LieAddGroup {𝕜 : Type*... | Mathlib/Geometry/Manifold/Algebra/LieGroup.lean | 171 | 174 | theorem ContMDiffWithinAt.div {f g : M → G} {s : Set M} {x₀ : M}
(hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀) :
ContMDiffWithinAt I' I n (fun x => f x / g x) s x₀ := by |
simp_rw [div_eq_mul_inv]; exact hf.mul hg.inv
| 1 | 2.718282 | 0 | 0 | 1 | 45 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity.Core
#align_import data.nat.factorial.double_factorial from "leanprover-community/mathlib"@"7daeaf3072304c498b653628add84a88d0e78767"
open Nat
namespace Nat
@[sim... | Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | 48 | 48 | theorem doubleFactorial_add_one (n : ℕ) : (n + 1)‼ = (n + 1) * (n - 1)‼ := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 0 | 1 | 46 |
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.LinearAlgebra.Ray
import Mathlib.Tactic.GCongr
#align_import analysis.convex.segment from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
... | Mathlib/Analysis/Convex/Segment.lean | 62 | 65 | theorem segment_eq_image₂ (x y : E) :
[x -[𝕜] y] =
(fun p : 𝕜 × 𝕜 => p.1 • x + p.2 • y) '' { p | 0 ≤ p.1 ∧ 0 ≤ p.2 ∧ p.1 + p.2 = 1 } := by |
simp only [segment, image, Prod.exists, mem_setOf_eq, exists_prop, and_assoc]
| 1 | 2.718282 | 0 | 0 | 2 | 47 |
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.LinearAlgebra.Ray
import Mathlib.Tactic.GCongr
#align_import analysis.convex.segment from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
... | Mathlib/Analysis/Convex/Segment.lean | 68 | 71 | theorem openSegment_eq_image₂ (x y : E) :
openSegment 𝕜 x y =
(fun p : 𝕜 × 𝕜 => p.1 • x + p.2 • y) '' { p | 0 < p.1 ∧ 0 < p.2 ∧ p.1 + p.2 = 1 } := by |
simp only [openSegment, image, Prod.exists, mem_setOf_eq, exists_prop, and_assoc]
| 1 | 2.718282 | 0 | 0 | 2 | 47 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 66 | 67 | theorem coe_injective ⦃f g : α →ₛ β⦄ (H : (f : α → β) = g) : f = g := by |
cases f; cases g; congr
| 1 | 2.718282 | 0 | 0 | 3 | 48 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 125 | 126 | theorem forall_mem_range {f : α →ₛ β} {p : β → Prop} : (∀ y ∈ f.range, p y) ↔ ∀ x, p (f x) := by |
simp only [mem_range, Set.forall_mem_range]
| 1 | 2.718282 | 0 | 0 | 3 | 48 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 129 | 130 | theorem exists_range_iff {f : α →ₛ β} {p : β → Prop} : (∃ y ∈ f.range, p y) ↔ ∃ x, p (f x) := by |
simpa only [mem_range, exists_prop] using Set.exists_range_iff
| 1 | 2.718282 | 0 | 0 | 3 | 48 |
import Mathlib.Algebra.Group.WithOne.Defs
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
import Mathlib.Algebra.Order.Group.Units
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Monoid.Basic
imp... | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 128 | 129 | theorem le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b := by |
simpa only [mul_inv_cancel_right₀ h] using mul_le_mul_right' hab c⁻¹
| 1 | 2.718282 | 0 | 0 | 1 | 49 |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 65 | 66 | theorem normSq_eq_norm_mul_self (a : ℍ) : normSq a = ‖a‖ * ‖a‖ := by |
rw [← inner_self, real_inner_self_eq_norm_mul_norm]
| 1 | 2.718282 | 0 | 0 | 6 | 50 |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 73 | 74 | theorem norm_coe (a : ℝ) : ‖(a : ℍ)‖ = ‖a‖ := by |
rw [norm_eq_sqrt_real_inner, inner_self, normSq_coe, Real.sqrt_sq_eq_abs, Real.norm_eq_abs]
| 1 | 2.718282 | 0 | 0 | 6 | 50 |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 83 | 84 | theorem norm_star (a : ℍ) : ‖star a‖ = ‖a‖ := by |
simp_rw [norm_eq_sqrt_real_inner, inner_self, normSq_star]
| 1 | 2.718282 | 0 | 0 | 6 | 50 |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 132 | 132 | theorem coeComplex_add (z w : ℂ) : ↑(z + w) = (z + w : ℍ) := by | ext <;> simp
| 1 | 2.718282 | 0 | 0 | 6 | 50 |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 136 | 136 | theorem coeComplex_mul (z w : ℂ) : ↑(z * w) = (z * w : ℍ) := by | ext <;> simp
| 1 | 2.718282 | 0 | 0 | 6 | 50 |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 150 | 150 | theorem coe_real_complex_mul (r : ℝ) (z : ℂ) : (r • z : ℍ) = ↑r * ↑z := by | ext <;> simp
| 1 | 2.718282 | 0 | 0 | 6 | 50 |
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]
| 1 | 2.718282 | 0 | 0 | 1 | 51 |
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.Algebra.Star
import Mathlib.Topology.Instances.Int
import Mathlib.Topology.Order.Bornology
#align_import topology.instances.real fro... | Mathlib/Topology/Instances/Real.lean | 77 | 78 | theorem Real.cobounded_eq : cobounded ℝ = atBot ⊔ atTop := by |
simp only [← comap_dist_right_atTop (0 : ℝ), Real.dist_eq, sub_zero, comap_abs_atTop]
| 1 | 2.718282 | 0 | 0 | 2 | 52 |
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.Algebra.Star
import Mathlib.Topology.Instances.Int
import Mathlib.Topology.Order.Bornology
#align_import topology.instances.real fro... | Mathlib/Topology/Instances/Real.lean | 92 | 94 | theorem Real.mem_closure_iff {s : Set ℝ} {x : ℝ} :
x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, |y - x| < ε := by |
simp [mem_closure_iff_nhds_basis nhds_basis_ball, Real.dist_eq]
| 1 | 2.718282 | 0 | 0 | 2 | 52 |
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.biproducts from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w₁ w₂ v₁ v₂ u₁ u₂
noncomputable section
open ... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | 349 | 351 | theorem biprodComparison'_comp_biprodComparison :
biprodComparison' F X Y ≫ biprodComparison F X Y = 𝟙 (F.obj X ⊞ F.obj Y) := by |
ext <;> simp [← Functor.map_comp]
| 1 | 2.718282 | 0 | 0 | 1 | 53 |
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 181 | 181 | theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by | simp [lt_irrefl]
| 1 | 2.718282 | 0 | 0 | 4 | 54 |
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 186 | 186 | theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by | simp [le_refl]
| 1 | 2.718282 | 0 | 0 | 4 | 54 |
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 191 | 191 | theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by | simp [le_refl]
| 1 | 2.718282 | 0 | 0 | 4 | 54 |
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 196 | 196 | theorem left_mem_Ioc : a ∈ Ioc a b ↔ False := by | simp [lt_irrefl]
| 1 | 2.718282 | 0 | 0 | 4 | 54 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
| Mathlib/Algebra/Ring/Identities.lean | 24 | 26 | theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 + y₂ ^ 2) = (x₁ * y₁ - x₂ * y₂) ^ 2 + (x₁ * y₂ + x₂ * y₁) ^ 2 := by |
ring
| 1 | 2.718282 | 0 | 0 | 6 | 55 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 31 | 34 | theorem sq_add_mul_sq_mul_sq_add_mul_sq :
(x₁ ^ 2 + n * x₂ ^ 2) * (y₁ ^ 2 + n * y₂ ^ 2) =
(x₁ * y₁ - n * x₂ * y₂) ^ 2 + n * (x₁ * y₂ + x₂ * y₁) ^ 2 := by |
ring
| 1 | 2.718282 | 0 | 0 | 6 | 55 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 39 | 41 | theorem pow_four_add_four_mul_pow_four :
a ^ 4 + 4 * b ^ 4 = ((a - b) ^ 2 + b ^ 2) * ((a + b) ^ 2 + b ^ 2) := by |
ring
| 1 | 2.718282 | 0 | 0 | 6 | 55 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 46 | 48 | theorem pow_four_add_four_mul_pow_four' :
a ^ 4 + 4 * b ^ 4 = (a ^ 2 - 2 * a * b + 2 * b ^ 2) * (a ^ 2 + 2 * a * b + 2 * b ^ 2) := by |
ring
| 1 | 2.718282 | 0 | 0 | 6 | 55 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 55 | 60 | theorem sum_four_sq_mul_sum_four_sq :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2) * (y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄) ^ 2 + (x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁) ... |
ring
| 1 | 2.718282 | 0 | 0 | 6 | 55 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 67 | 78 | theorem sum_eight_sq_mul_sum_eight_sq :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 + x₈ ^ 2) *
(y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2 + y₅ ^ 2 + y₆ ^ 2 + y₇ ^ 2 + y₈ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄ - x₅ * y₅ - x₆ * y₆ - x₇ * y₇ - x₈ * y₈) ^ 2 +
(x₁ * y₂ + x₂ * y₁ + x₃ * ... |
ring
| 1 | 2.718282 | 0 | 0 | 6 | 55 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.ZornAtoms
#align_import order.filter.ultrafilter from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v
variable {α : Type u} {β : Type v} {γ : Type*}
open Set Filter Function
open scoped Classical
open Filter
inst... | Mathlib/Order/Filter/Ultrafilter.lean | 115 | 116 | theorem disjoint_iff_not_le {f : Ultrafilter α} {g : Filter α} : Disjoint (↑f) g ↔ ¬↑f ≤ g := by |
rw [← inf_neBot_iff, neBot_iff, Ne, not_not, disjoint_iff]
| 1 | 2.718282 | 0 | 0 | 2 | 56 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.ZornAtoms
#align_import order.filter.ultrafilter from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v
variable {α : Type u} {β : Type v} {γ : Type*}
open Set Filter Function
open scoped Classical
open Filter
inst... | Mathlib/Order/Filter/Ultrafilter.lean | 135 | 135 | theorem compl_mem_iff_not_mem : sᶜ ∈ f ↔ s ∉ f := by | rw [← compl_not_mem_iff, compl_compl]
| 1 | 2.718282 | 0 | 0 | 2 | 56 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure
open Filter MeasureTheory Topology
variable {α : Type*}... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 226 | 228 | theorem _root_.Filter.HasBasis.vitaliFamily {ι : Sort*} {p : ι → Prop} {s : ι → Set α} {x : α}
(h : (𝓝 x).HasBasis p s) : (v.filterAt x).HasBasis p (fun i ↦ {t ∈ v.setsAt x | t ⊆ s i}) := by |
simpa only [← Set.setOf_inter_eq_sep] using h.smallSets.inf_principal _
| 1 | 2.718282 | 0 | 0 | 2 | 57 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure
open Filter MeasureTheory Topology
variable {α : Type*}... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 234 | 236 | theorem mem_filterAt_iff {x : α} {s : Set (Set α)} :
s ∈ v.filterAt x ↔ ∃ ε > (0 : ℝ), ∀ a ∈ v.setsAt x, a ⊆ closedBall x ε → a ∈ s := by |
simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_setOf]
| 1 | 2.718282 | 0 | 0 | 2 | 57 |
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Cases
import Mathlib.Algebra.NeZero
import Mathlib.Logic.Function.Basic
#align_import algebra.char_zero.defs from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d"
class CharZero (R) [AddMonoidWithOne R] : Prop where
cast_injecti... | Mathlib/Algebra/CharZero/Defs.lean | 79 | 79 | theorem cast_eq_zero {n : ℕ} : (n : R) = 0 ↔ n = 0 := by | rw [← cast_zero, cast_inj]
| 1 | 2.718282 | 0 | 0 | 2 | 58 |
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Cases
import Mathlib.Algebra.NeZero
import Mathlib.Logic.Function.Basic
#align_import algebra.char_zero.defs from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d"
class CharZero (R) [AddMonoidWithOne R] : Prop where
cast_injecti... | Mathlib/Algebra/CharZero/Defs.lean | 92 | 92 | theorem cast_eq_one {n : ℕ} : (n : R) = 1 ↔ n = 1 := by | rw [← cast_one, cast_inj]
| 1 | 2.718282 | 0 | 0 | 2 | 58 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Dynamics.FixedPoints.Basic
open Finset Function
section AddCommMonoid
variable {α M : Type*} [AddCommMonoid M]
def birkhoffSum (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := ∑ k ∈ range n, g (f^[k] x)
theorem birkhoffSum_zero (f : α → α) (g : α → ... | Mathlib/Dynamics/BirkhoffSum/Basic.lean | 51 | 53 | theorem birkhoffSum_add (f : α → α) (g : α → M) (m n : ℕ) (x : α) :
birkhoffSum f g (m + n) x = birkhoffSum f g m x + birkhoffSum f g n (f^[m] x) := by |
simp_rw [birkhoffSum, sum_range_add, add_comm m, iterate_add_apply]
| 1 | 2.718282 | 0 | 0 | 2 | 59 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Dynamics.FixedPoints.Basic
open Finset Function
section AddCommMonoid
variable {α M : Type*} [AddCommMonoid M]
def birkhoffSum (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := ∑ k ∈ range n, g (f^[k] x)
theorem birkhoffSum_zero (f : α → α) (g : α → ... | Mathlib/Dynamics/BirkhoffSum/Basic.lean | 55 | 57 | theorem Function.IsFixedPt.birkhoffSum_eq {f : α → α} {x : α} (h : IsFixedPt f x) (g : α → M)
(n : ℕ) : birkhoffSum f g n x = n • g x := by |
simp [birkhoffSum, (h.iterate _).eq]
| 1 | 2.718282 | 0 | 0 | 2 | 59 |
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import group_theory.group_action.group from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
universe u v w
... | Mathlib/GroupTheory/GroupAction/Group.lean | 30 | 30 | theorem inv_smul_smul (c : α) (x : β) : c⁻¹ • c • x = x := by | rw [smul_smul, mul_left_inv, one_smul]
| 1 | 2.718282 | 0 | 0 | 2 | 60 |
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import group_theory.group_action.group from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
universe u v w
... | Mathlib/GroupTheory/GroupAction/Group.lean | 35 | 36 | theorem smul_inv_smul (c : α) (x : β) : c • c⁻¹ • x = x := by |
rw [smul_smul, mul_right_inv, one_smul]
| 1 | 2.718282 | 0 | 0 | 2 | 60 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.free_monoid.basic from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
variable {α : Type*} {β : Type*} {γ : Type*} {M : Type*} [Monoid M] {N :... | Mathlib/Algebra/FreeMonoid/Basic.lean | 111 | 112 | theorem toList_prod (xs : List (FreeMonoid α)) : toList xs.prod = (xs.map toList).join := by |
induction xs <;> simp [*, List.join]
| 1 | 2.718282 | 0 | 0 | 1 | 61 |
import Batteries.Data.RBMap.Basic
import Batteries.Tactic.SeqFocus
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] All
theorem All.trivial (H : ∀ {x : α}, p x) : ∀ {t : RBNode α}, t.All p
| nil => _root_.trivial
| node .. => ⟨H, All.trivial H, All.trivial H⟩
| .lake/packages/batteries/Batteries/Data/RBMap/WF.lean | 27 | 28 | theorem All_and {t : RBNode α} : t.All (fun a => p a ∧ q a) ↔ t.All p ∧ t.All q := by |
induction t <;> simp [*, and_assoc, and_left_comm]
| 1 | 2.718282 | 0 | 0 | 2 | 62 |
import Batteries.Data.RBMap.Basic
import Batteries.Tactic.SeqFocus
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] All
theorem All.trivial (H : ∀ {x : α}, p x) : ∀ {t : RBNode α}, t.All p
| nil => _root_.trivial
| node .. => ⟨H, All.trivial H, All.trivial H⟩
theorem All_and {t : RBNode α}... | .lake/packages/batteries/Batteries/Data/RBMap/WF.lean | 51 | 52 | theorem reverse_eq_iff {t t' : RBNode α} : t.reverse = t' ↔ t = t'.reverse := by |
constructor <;> rintro rfl <;> simp
| 1 | 2.718282 | 0 | 0 | 2 | 62 |
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.CategoryTheory.Elementwise
import Ma... | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | 66 | 69 | theorem pi_lift_π_apply' {β : Type v} (f : β → Type v) {P : Type v}
(s : ∀ b, P ⟶ f b) (b : β) (x : P) :
(Pi.π f b : (piObj f) → f b) (@Pi.lift β _ _ f _ P s x) = s b x := by |
simp
| 1 | 2.718282 | 0 | 0 | 2 | 63 |
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.CategoryTheory.Elementwise
import Ma... | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | 82 | 84 | theorem pi_map_π_apply' {β : Type v} {f g : β → Type v} (α : ∀ j, f j ⟶ g j) (b : β) (x) :
(Pi.π g b : ∏ᶜ g → g b) (Pi.map α x) = α b ((Pi.π f b : ∏ᶜ f → f b) x) := by |
simp
| 1 | 2.718282 | 0 | 0 | 2 | 63 |
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.natural_transformation from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄
variable {C :... | Mathlib/CategoryTheory/NatTrans.lean | 63 | 64 | theorem congr_app {F G : C ⥤ D} {α β : NatTrans F G} (h : α = β) (X : C) : α.app X = β.app X := by |
aesop_cat
| 1 | 2.718282 | 0 | 0 | 1 | 64 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
#align_import group_theory.perm.basic from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
universe u v
nam... | Mathlib/GroupTheory/Perm/Basic.lean | 125 | 127 | theorem zpow_apply_comm {α : Type*} (σ : Perm α) (m n : ℤ) {x : α} :
(σ ^ m) ((σ ^ n) x) = (σ ^ n) ((σ ^ m) x) := by |
rw [← Equiv.Perm.mul_apply, ← Equiv.Perm.mul_apply, zpow_mul_comm]
| 1 | 2.718282 | 0 | 0 | 1 | 65 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subsemiring.Basic
#align_import ring_theory.subring.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
universe u v w
variable {R : Type u} {S : Type v} {T : Type w} [Ring R]
section SubringClass
class Su... | Mathlib/Algebra/Ring/Subring/Basic.lean | 88 | 88 | theorem intCast_mem (n : ℤ) : (n : R) ∈ s := by | simp only [← zsmul_one, zsmul_mem, one_mem]
| 1 | 2.718282 | 0 | 0 | 1 | 66 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 42 | 43 | theorem congrArg₂ (f : α → β → γ) {x x' : α} {y y' : β}
(hx : x = x') (hy : y = y') : f x y = f x' y' := by | subst hx hy; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 72 | 72 | theorem Eq.congr_left {x y z : α} (h : x = y) : x = z ↔ y = z := by | rw [h]
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 74 | 74 | theorem Eq.congr_right {x y z : α} (h : x = y) : z = x ↔ z = y := by | rw [h]
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 88 | 91 | theorem eqRec_eq_cast {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') :
@Eq.rec α a motive x a' e = cast (e ▸ rfl) x := by |
subst e; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 94 | 97 | theorem eqRec_heq_self {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') :
HEq (@Eq.rec α a motive x a' e) x := by |
subst e; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 100 | 103 | theorem eqRec_heq_iff_heq {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') {β : Sort _} (y : β) :
HEq (@Eq.rec α a motive x a' e) y ↔ HEq x y := by |
subst e; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 106 | 109 | theorem heq_eqRec_iff_heq {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') {β : Sort _} (y : β) :
HEq y (@Eq.rec α a motive x a' e) ↔ HEq y x := by |
subst e; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 142 | 143 | theorem congr_eqRec {β : α → Sort _} (f : (x : α) → β x → γ) (h : x = x') (y : β x) :
f x' (Eq.rec y h) = f x y := by | cases h; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 67 |
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Ring.Hom.Defs
#align_import algebra.ring.units from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
open Funct... | Mathlib/Algebra/Ring/Units.lean | 50 | 50 | theorem neg_divp (a : α) (u : αˣ) : -(a /ₚ u) = -a /ₚ u := by | simp only [divp, neg_mul]
| 1 | 2.718282 | 0 | 0 | 2 | 68 |
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Ring.Hom.Defs
#align_import algebra.ring.units from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
open Funct... | Mathlib/Algebra/Ring/Units.lean | 61 | 62 | theorem divp_add_divp_same (a b : α) (u : αˣ) : a /ₚ u + b /ₚ u = (a + b) /ₚ u := by |
simp only [divp, add_mul]
| 1 | 2.718282 | 0 | 0 | 2 | 68 |
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
namespace CategoryTheory
open CategoryTheory.Limits Preadditive
variable {C D : Type*} [Category C] [Category D] [Preadditive D... | Mathlib/CategoryTheory/Preadditive/FunctorCategory.lean | 123 | 124 | theorem app_units_zsmul (X : C) (α : F ⟶ G) (n : ℤˣ) : (n • α).app X = n • α.app X := by |
apply app_zsmul
| 1 | 2.718282 | 0 | 0 | 2 | 69 |
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
namespace CategoryTheory
open CategoryTheory.Limits Preadditive
variable {C D : Type*} [Category C] [Category D] [Preadditive D... | Mathlib/CategoryTheory/Preadditive/FunctorCategory.lean | 127 | 129 | theorem app_sum {ι : Type*} (s : Finset ι) (X : C) (α : ι → (F ⟶ G)) :
(∑ i ∈ s, α i).app X = ∑ i ∈ s, (α i).app X := by |
simp only [← appHom_apply, map_sum]
| 1 | 2.718282 | 0 | 0 | 2 | 69 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 56 | 57 | theorem Discrete.mk_as {α : Type u₁} (X : Discrete α) : Discrete.mk X.as = X := by |
rfl
| 1 | 2.718282 | 0 | 0 | 2 | 70 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 186 | 187 | theorem functor_map {I : Type u₁} (F : I → C) {i : Discrete I} (f : i ⟶ i) :
(Discrete.functor F).map f = 𝟙 (F i.as) := by | aesop_cat
| 1 | 2.718282 | 0 | 0 | 2 | 70 |
import Mathlib.Data.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
#align_import data.analysis.topology from "leanprover-community/mathlib"@"55d771df074d0dd020139ee1cd4b95521422df9f"
open Set
open Filter hiding Realizer
open Topology
structure Ctop (α σ : Type*) where
f ... | Mathlib/Data/Analysis/Topology.lean | 79 | 80 | theorem ofEquiv_val (E : σ ≃ τ) (F : Ctop α σ) (a : τ) : F.ofEquiv E a = F (E.symm a) := by |
cases F; rfl
| 1 | 2.718282 | 0 | 0 | 1 | 71 |
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.modeq from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
namespace AddCommGroup
variable {α : Type*}
section AddCommGroup
variable [AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}
... | Mathlib/Algebra/ModEq.lean | 102 | 102 | theorem modEq_zero : a ≡ b [PMOD 0] ↔ a = b := by | simp [ModEq, sub_eq_zero, eq_comm]
| 1 | 2.718282 | 0 | 0 | 3 | 72 |
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.modeq from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
namespace AddCommGroup
variable {α : Type*}
section AddCommGroup
variable [AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}
... | Mathlib/Algebra/ModEq.lean | 262 | 263 | theorem modEq_sub_iff_add_modEq' : a ≡ b - c [PMOD p] ↔ c + a ≡ b [PMOD p] := by |
simp [ModEq, sub_sub]
| 1 | 2.718282 | 0 | 0 | 3 | 72 |
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.modeq from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
namespace AddCommGroup
variable {α : Type*}
section AddCommGroup
variable [AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}
... | Mathlib/Algebra/ModEq.lean | 311 | 312 | theorem modEq_iff_int_modEq {a b z : ℤ} : a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z] := by |
simp [ModEq, dvd_iff_exists_eq_mul_left, Int.modEq_iff_dvd]
| 1 | 2.718282 | 0 | 0 | 3 | 72 |
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 79 | 79 | theorem compl_mem_ae_iff {s : Set α} : sᶜ ∈ ae μ ↔ μ s = 0 := by | simp only [mem_ae_iff, compl_compl]
| 1 | 2.718282 | 0 | 0 | 3 | 73 |
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 107 | 109 | theorem all_ae_of {ι : Sort*} {p : α → ι → Prop} (hp : ∀ᵐ a ∂μ, ∀ i, p a i) (i : ι) :
∀ᵐ a ∂μ, p a i := by |
filter_upwards [hp] with a ha using ha i
| 1 | 2.718282 | 0 | 0 | 3 | 73 |
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 148 | 151 | theorem ae_le_set : s ≤ᵐ[μ] t ↔ μ (s \ t) = 0 :=
calc
s ≤ᵐ[μ] t ↔ ∀ᵐ x ∂μ, x ∈ s → x ∈ t := Iff.rfl
_ ↔ μ (s \ t) = 0 := by | simp [ae_iff]; rfl
| 1 | 2.718282 | 0 | 0 | 3 | 73 |
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cas... | Mathlib/Data/Int/Cast/Basic.lean | 74 | 76 | theorem cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by |
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
| 1 | 2.718282 | 0 | 0 | 2 | 74 |
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cas... | Mathlib/Data/Int/Cast/Basic.lean | 79 | 80 | theorem cast_one : ((1 : ℤ) : R) = 1 := by |
erw [cast_natCast, Nat.cast_one]
| 1 | 2.718282 | 0 | 0 | 2 | 74 |
def SatisfiesM {m : Type u → Type v} [Functor m] (p : α → Prop) (x : m α) : Prop :=
∃ x' : m {a // p a}, Subtype.val <$> x' = x
@[simp] theorem SatisfiesM_Id_eq : SatisfiesM (m := Id) p x ↔ p x :=
⟨fun ⟨y, eq⟩ => eq ▸ y.2, fun h => ⟨⟨_, h⟩, rfl⟩⟩
@[simp] theorem SatisfiesM_Option_eq : SatisfiesM (m := Option... | .lake/packages/batteries/Batteries/Classes/SatisfiesM.lean | 165 | 166 | theorem SatisfiesM_StateRefT_eq [Monad m] :
SatisfiesM (m := StateRefT' ω σ m) p x ↔ ∀ s, SatisfiesM p (x s) := by | simp
| 1 | 2.718282 | 0 | 0 | 1 | 75 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import data.int.absolute_value from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
variable {R S : Type*} [Ring R] [Linea... | Mathlib/Data/Int/AbsoluteValue.lean | 28 | 29 | theorem AbsoluteValue.map_units_int (abv : AbsoluteValue ℤ S) (x : ℤˣ) : abv x = 1 := by |
rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
| 1 | 2.718282 | 0 | 0 | 3 | 76 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import data.int.absolute_value from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
variable {R S : Type*} [Ring R] [Linea... | Mathlib/Data/Int/AbsoluteValue.lean | 33 | 34 | theorem AbsoluteValue.map_units_intCast [Nontrivial R] (abv : AbsoluteValue R S) (x : ℤˣ) :
abv ((x : ℤ) : R) = 1 := by | rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
| 1 | 2.718282 | 0 | 0 | 3 | 76 |
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