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.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 138 | 139 | theorem lift_smul (r : R) (x : ℍ[R,c₁,c₂]) : q.lift (r • x) = r • q.lift x := by |
simp [lift, mul_smul, ← Algebra.smul_def]
| 1 | 2.718282 | 0 | 0.4 | 10 | 394 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
... | Mathlib/Order/OrdContinuous.lean | 98 | 99 | theorem le_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x ≤ f y ↔ x ≤ y := by |
simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]
| 1 | 2.718282 | 0 | 0.4 | 5 | 395 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
... | Mathlib/Order/OrdContinuous.lean | 102 | 103 | theorem lt_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x < f y ↔ x < y := by |
simp only [lt_iff_le_not_le, hf.le_iff h]
| 1 | 2.718282 | 0 | 0.4 | 5 | 395 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
... | Mathlib/Order/OrdContinuous.lean | 131 | 132 | theorem map_sSup (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = ⨆ x ∈ s, f x := by |
rw [hf.map_sSup', sSup_image]
| 1 | 2.718282 | 0 | 0.4 | 5 | 395 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
... | Mathlib/Order/OrdContinuous.lean | 135 | 137 | theorem map_iSup (hf : LeftOrdContinuous f) (g : ι → α) : f (⨆ i, g i) = ⨆ i, f (g i) := by |
simp only [iSup, hf.map_sSup', ← range_comp]
rfl
| 2 | 7.389056 | 1 | 0.4 | 5 | 395 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
... | Mathlib/Order/OrdContinuous.lean | 151 | 154 | theorem map_ciSup (hf : LeftOrdContinuous f) {g : ι → α} (hg : BddAbove (range g)) :
f (⨆ i, g i) = ⨆ i, f (g i) := by |
simp only [iSup, hf.map_csSup (range_nonempty _) hg, ← range_comp]
rfl
| 2 | 7.389056 | 1 | 0.4 | 5 | 395 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 95 | 97 | theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by | simp [hfp.2]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 126 | 127 | theorem toLp_congr {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) (hfg : f =ᵐ[μ] g) :
hf.toLp f = hg.toLp g := by | simp [toLp, hfg]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 131 | 132 | theorem toLp_eq_toLp_iff {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
hf.toLp f = hg.toLp g ↔ f =ᵐ[μ] g := by | simp [toLp]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 163 | 167 | theorem ext {f g : Lp E p μ} (h : f =ᵐ[μ] g) : f = g := by |
cases f
cases g
simp only [Subtype.mk_eq_mk]
exact AEEqFun.ext h
| 4 | 54.59815 | 2 | 0.4 | 5 | 396 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 177 | 178 | theorem mem_Lp_iff_memℒp {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ Memℒp f p μ := by |
simp [mem_Lp_iff_snorm_lt_top, Memℒp, f.stronglyMeasurable.aestronglyMeasurable]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 73 | 77 | theorem toDistrib_injective : Function.Injective (@toDistrib R) := by |
intro _ _ h
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
| 4 | 54.59815 | 2 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 90 | 92 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 133 | 135 | theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 195 | 201 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
intro _ _ h
-- Use above extensionality lemma to prove injectivity by showing that `h_add` and `h_mul` hold.
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
| 5 | 148.413159 | 2 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 405 | 407 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 427 | 429 | theorem toNonUnitalSemiring_injective :
Function.Injective (@toNonUnitalSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 451 | 453 | theorem toNonUnitalNonAssocRing_injective :
Function.Injective (@toNonUnitalNonAssocRing R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 473 | 475 | theorem toNonUnitalRing_injective :
Function.Injective (@toNonUnitalRing R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 497 | 499 | theorem toSemiring_injective :
Function.Injective (@toSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 519 | 520 | theorem toRing_injective : Function.Injective (@toRing R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 80 | 81 | theorem exp_diagonal (v : m → 𝔸) : exp 𝕂 (diagonal v) = diagonal (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, diagonal_pow, ← diagonal_smul, ← diagonal_tsum]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 84 | 86 | theorem exp_blockDiagonal (v : m → Matrix n n 𝔸) :
exp 𝕂 (blockDiagonal v) = blockDiagonal (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, ← blockDiagonal_pow, ← blockDiagonal_smul, ← blockDiagonal_tsum]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 89 | 91 | theorem exp_blockDiagonal' (v : ∀ i, Matrix (n' i) (n' i) 𝔸) :
exp 𝕂 (blockDiagonal' v) = blockDiagonal' (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, ← blockDiagonal'_pow, ← blockDiagonal'_smul, ← blockDiagonal'_tsum]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 111 | 112 | theorem exp_transpose (A : Matrix m m 𝔸) : exp 𝕂 Aᵀ = (exp 𝕂 A)ᵀ := by |
simp_rw [exp_eq_tsum, transpose_tsum, transpose_smul, transpose_pow]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 182 | 187 | theorem exp_neg (A : Matrix m m 𝔸) : exp 𝕂 (-A) = (exp 𝕂 A)⁻¹ := by |
rw [nonsing_inv_eq_ring_inverse]
letI : SeminormedRing (Matrix m m 𝔸) := Matrix.linftyOpSemiNormedRing
letI : NormedRing (Matrix m m 𝔸) := Matrix.linftyOpNormedRing
letI : NormedAlgebra 𝕂 (Matrix m m 𝔸) := Matrix.linftyOpNormedAlgebra
exact (Ring.inverse_exp _ A).symm
| 5 | 148.413159 | 2 | 0.4 | 5 | 398 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 57 | 59 | theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by |
rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero]
exact Real.rpow_eq_zero_iff_of_nonneg x.2
| 2 | 7.389056 | 1 | 0.4 | 5 | 399 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 97 | 97 | theorem rpow_neg_one (x : ℝ≥0) : x ^ (-1 : ℝ) = x⁻¹ := by | simp [rpow_neg]
| 1 | 2.718282 | 0 | 0.4 | 5 | 399 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 108 | 109 | theorem rpow_inv_rpow_self {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ (1 / y) = x := by |
field_simp [← rpow_mul]
| 1 | 2.718282 | 0 | 0.4 | 5 | 399 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 112 | 113 | theorem rpow_self_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ (1 / y)) ^ y = x := by |
field_simp [← rpow_mul]
| 1 | 2.718282 | 0 | 0.4 | 5 | 399 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 124 | 127 | theorem sqrt_eq_rpow (x : ℝ≥0) : sqrt x = x ^ (1 / (2 : ℝ)) := by |
refine NNReal.eq ?_
push_cast
exact Real.sqrt_eq_rpow x.1
| 3 | 20.085537 | 1 | 0.4 | 5 | 399 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 71 | 73 | theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by | simpa using mul_le_mul_left' bc a⁻¹
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 82 | 84 | theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by | simpa using mul_le_mul_right' bc a⁻¹
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 98 | 100 | theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by |
rw [← mul_le_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 106 | 108 | theorem Left.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by |
rw [← mul_le_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 113 | 115 | theorem le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c := by |
rw [← mul_le_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 120 | 121 | theorem inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c := by |
rw [← mul_le_mul_iff_left b, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 138 | 139 | theorem le_inv_mul_iff_le : 1 ≤ b⁻¹ * a ↔ b ≤ a := by |
rw [← mul_le_mul_iff_left b, mul_one, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 158 | 159 | theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by |
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 165 | 166 | theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by |
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 171 | 173 | theorem lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c := by |
rw [← mul_lt_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 178 | 179 | theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c := by |
rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 196 | 197 | theorem lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < a := by |
rw [← mul_lt_mul_iff_left b, mul_one, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 215 | 217 | theorem Right.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by |
rw [← mul_le_mul_iff_right a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 223 | 225 | theorem Right.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by |
rw [← mul_le_mul_iff_right a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 280 | 281 | theorem Right.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by |
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 287 | 288 | theorem Right.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by |
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 305 | 306 | theorem mul_inv_lt_iff_lt_mul : a * b⁻¹ < c ↔ a < c * b := by |
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 318 | 319 | theorem inv_mul_lt_one_iff_lt : a * b⁻¹ < 1 ↔ a < b := by |
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right, one_mul]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 343 | 345 | theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := by |
rw [← mul_le_mul_iff_left a, ← mul_le_mul_iff_right b]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 353 | 355 | theorem mul_inv_le_inv_mul_iff : a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b := by |
rw [← mul_le_mul_iff_left d, ← mul_le_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 382 | 384 | theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a := by |
rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 389 | 389 | theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < a := by | rw [← inv_lt_inv_iff, inv_inv]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 394 | 394 | theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ := by | rw [← inv_lt_inv_iff, inv_inv]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 411 | 413 | theorem mul_inv_lt_inv_mul_iff : a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b := by |
rw [← mul_lt_mul_iff_left d, ← mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 875 | 876 | theorem div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b := by |
simpa only [div_eq_mul_inv] using mul_lt_mul_iff_right _
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 33 | 35 | theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by |
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 59 | 59 | theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by | simp [← Ici_inter_Iic]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 63 | 63 | theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by | simp [← Ici_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 67 | 67 | theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by | simp [← Ioi_inter_Iic]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 71 | 71 | theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by | simp [← Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 75 | 76 | theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by |
rw [← range_coe, preimage_range]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 80 | 81 | theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by |
simp [← Ici_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 85 | 86 | theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by |
simp [← Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 89 | 90 | theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 93 | 94 | theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 97 | 99 | theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by |
rw [← preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 102 | 104 | theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) := by |
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 107 | 110 | theorem image_coe_Icc : (some : α → WithTop α) '' Icc a b = Icc (a : WithTop α) b := by |
rw [← preimage_coe_Icc, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left
(Subset.trans Icc_subset_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)]
| 3 | 20.085537 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 113 | 115 | theorem image_coe_Ico : (some : α → WithTop α) '' Ico a b = Ico (a : WithTop α) b := by |
rw [← preimage_coe_Ico, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Subset.trans Ico_subset_Iio_self <| Iio_subset_Iio le_top)]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 118 | 121 | theorem image_coe_Ioc : (some : α → WithTop α) '' Ioc a b = Ioc (a : WithTop α) b := by |
rw [← preimage_coe_Ioc, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left
(Subset.trans Ioc_subset_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)]
| 3 | 20.085537 | 1 | 0.4 | 15 | 401 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 87 | 88 | theorem snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ENNReal.toReal p) μ := by | simp [snorm, hp_ne_zero, hp_ne_top]
| 1 | 2.718282 | 0 | 0.4 | 5 | 402 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 91 | 93 | theorem snorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) := by |
rw [snorm_eq_snorm' hp_ne_zero hp_ne_top, snorm']
| 1 | 2.718282 | 0 | 0.4 | 5 | 402 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 96 | 98 | theorem snorm_one_eq_lintegral_nnnorm {f : α → F} : snorm f 1 μ = ∫⁻ x, ‖f x‖₊ ∂μ := by |
simp_rw [snorm_eq_lintegral_rpow_nnnorm one_ne_zero ENNReal.coe_ne_top, ENNReal.one_toReal,
one_div_one, ENNReal.rpow_one]
| 2 | 7.389056 | 1 | 0.4 | 5 | 402 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 102 | 102 | theorem snorm_exponent_top {f : α → F} : snorm f ∞ μ = snormEssSup f μ := by | simp [snorm]
| 1 | 2.718282 | 0 | 0.4 | 5 | 402 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 117 | 120 | theorem lintegral_rpow_nnnorm_eq_rpow_snorm' {f : α → F} (hq0_lt : 0 < q) :
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) = snorm' f q μ ^ q := by |
rw [snorm', ← ENNReal.rpow_mul, one_div, inv_mul_cancel, ENNReal.rpow_one]
exact (ne_of_lt hq0_lt).symm
| 2 | 7.389056 | 1 | 0.4 | 5 | 402 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 50 | 57 | theorem continuousAt_angle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∠ y.1 y.2.1 y.2.2) x := by |
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp
((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd... | 6 | 403.428793 | 2 | 0.4 | 5 | 403 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 61 | 64 | theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
(f : P →ᵃⁱ[ℝ] P₂) (p₁ p₂ p₃ : P) : ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃ := by |
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 113 | 114 | theorem angle_sub_const (v₁ v₂ v₃ : V) (v : V) : ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃ := by |
simpa only [vsub_eq_sub] using angle_vsub_const v₁ v₂ v₃ v
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 119 | 120 | theorem angle_const_sub (v : V) (v₁ v₂ v₃ : V) : ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃ := by |
simpa only [vsub_eq_sub] using angle_const_vsub v v₁ v₂ v₃
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 125 | 126 | theorem angle_neg (v₁ v₂ v₃ : V) : ∠ (-v₁) (-v₂) (-v₃) = ∠ v₁ v₂ v₃ := by |
simpa only [zero_sub] using angle_const_sub 0 v₁ v₂ v₃
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 75 | 76 | theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by |
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 89 | 91 | theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by |
simp only [HAdd.hAdd, Add.add, RatFunc.add]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 104 | 106 | theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by |
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 117 | 118 | theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by | simp only [Neg.neg, RatFunc.neg]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 131 | 132 | theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by |
simp only [One.one, OfNat.ofNat, RatFunc.one]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 145 | 147 | theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by |
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 164 | 166 | theorem ofFractionRing_div (p q : FractionRing K[X]) :
ofFractionRing (p / q) = ofFractionRing p / ofFractionRing q := by |
simp only [Div.div, HDiv.hDiv, RatFunc.div]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 177 | 179 | theorem ofFractionRing_inv (p : FractionRing K[X]) :
ofFractionRing p⁻¹ = (ofFractionRing p)⁻¹ := by |
simp only [Inv.inv, RatFunc.inv]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 209 | 211 | theorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :
ofFractionRing (c • p) = c • ofFractionRing p := by |
simp only [SMul.smul, HSMul.hSMul, RatFunc.smul]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 214 | 217 | theorem toFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : RatFunc K) :
toFractionRing (c • p) = c • toFractionRing p := by |
cases p
rw [← ofFractionRing_smul]
| 2 | 7.389056 | 1 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 220 | 225 | theorem smul_eq_C_smul (x : RatFunc K) (r : K) : r • x = Polynomial.C r • x := by |
cases' x with x
-- Porting note: had to specify the induction principle manually
induction x using Localization.induction_on
rw [← ofFractionRing_smul, ← ofFractionRing_smul, Localization.smul_mk,
Localization.smul_mk, smul_eq_mul, Polynomial.smul_eq_C_mul]
| 5 | 148.413159 | 2 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 235 | 239 | theorem mk_smul (c : R) (p q : K[X]) : RatFunc.mk (c • p) q = c • RatFunc.mk p q := by |
by_cases hq : q = 0
· rw [hq, mk_zero, mk_zero, ← ofFractionRing_smul, smul_zero]
· rw [mk_eq_localization_mk _ hq, mk_eq_localization_mk _ hq, ← Localization.smul_mk, ←
ofFractionRing_smul]
| 4 | 54.59815 | 2 | 0.416667 | 12 | 404 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 58 | 59 | theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by |
simp only [chain_cons, Chain.nil, and_true_iff]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 62 | 65 | theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by |
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
| 2 | 7.389056 | 1 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 69 | 71 | theorem chain_append_cons_cons {a b c : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: c :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ R b c ∧ Chain R c l₂ := by |
rw [chain_split, chain_cons]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 82 | 83 | theorem chain_append_singleton_iff_forall₂ :
Chain R a (l ++ [b]) ↔ Forall₂ R (a :: l) (l ++ [b]) := by | simp [chain_iff_forall₂]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 86 | 88 | theorem chain_map (f : β → α) {b : β} {l : List β} :
Chain R (f b) (map f l) ↔ Chain (fun a b : β => R (f a) (f b)) b l := by |
induction l generalizing b <;> simp only [map, Chain.nil, chain_cons, *]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 101 | 106 | theorem chain_pmap_of_chain {S : β → β → Prop} {p : α → Prop} {f : ∀ a, p a → β}
(H : ∀ a b ha hb, R a b → S (f a ha) (f b hb)) {a : α} {l : List α} (hl₁ : Chain R a l)
(ha : p a) (hl₂ : ∀ a ∈ l, p a) : Chain S (f a ha) (List.pmap f l hl₂) := by |
induction' l with lh lt l_ih generalizing a
· simp
· simp [H _ _ _ _ (rel_of_chain_cons hl₁), l_ih (chain_of_chain_cons hl₁)]
| 3 | 20.085537 | 1 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 109 | 114 | theorem chain_of_chain_pmap {S : β → β → Prop} {p : α → Prop} (f : ∀ a, p a → β) {l : List α}
(hl₁ : ∀ a ∈ l, p a) {a : α} (ha : p a) (hl₂ : Chain S (f a ha) (List.pmap f l hl₁))
(H : ∀ a b ha hb, S (f a ha) (f b hb) → R a b) : Chain R a l := by |
induction' l with lh lt l_ih generalizing a
· simp
· simp [H _ _ _ _ (rel_of_chain_cons hl₂), l_ih _ _ (chain_of_chain_cons hl₂)]
| 3 | 20.085537 | 1 | 0.428571 | 7 | 405 |
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