Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 157 | 158 | theorem preimage_const_add_Ioc : (fun x => a + x) ⁻¹' Ioc b c = Ioc (b - a) (c - a) := by |
simp [← Ioi_inter_Iic]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 162 | 163 | theorem preimage_const_add_Ioo : (fun x => a + x) ⁻¹' Ioo b c = Ioo (b - a) (c - a) := by |
simp [← Ioi_inter_Iio]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 192 | 193 | theorem preimage_add_const_Icc : (fun x => x + a) ⁻¹' Icc b c = Icc (b - a) (c - a) := by |
simp [← Ici_inter_Iic]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 197 | 198 | theorem preimage_add_const_Ico : (fun x => x + a) ⁻¹' Ico b c = Ico (b - a) (c - a) := by |
simp [← Ici_inter_Iio]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 202 | 203 | theorem preimage_add_const_Ioc : (fun x => x + a) ⁻¹' Ioc b c = Ioc (b - a) (c - a) := by |
simp [← Ioi_inter_Iic]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 207 | 208 | theorem preimage_add_const_Ioo : (fun x => x + a) ⁻¹' Ioo b c = Ioo (b - a) (c - a) := by |
simp [← Ioi_inter_Iio]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 237 | 237 | theorem preimage_neg_Icc : -Icc a b = Icc (-b) (-a) := by | simp [← Ici_inter_Iic, inter_comm]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 241 | 242 | theorem preimage_neg_Ico : -Ico a b = Ioc (-b) (-a) := by |
simp [← Ici_inter_Iio, ← Ioi_inter_Iic, inter_comm]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 619 | 620 | theorem preimage_mul_const_Ioo (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Ioo a b = Ioo (a / c) (b / c) := by | simp [← Ioi_inter_Iio, h]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 624 | 625 | theorem preimage_mul_const_Ioc (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Ioc a b = Ioc (a / c) (b / c) := by | simp [← Ioi_inter_Iic, h]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 629 | 630 | theorem preimage_mul_const_Ico (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Ico a b = Ico (a / c) (b / c) := by | simp [← Ici_inter_Iio, h]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 634 | 635 | theorem preimage_mul_const_Icc (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Icc a b = Icc (a / c) (b / c) := by | simp [← Ici_inter_Iic, h]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 663 | 664 | theorem preimage_mul_const_Ioo_of_neg (a b : α) {c : α} (h : c < 0) :
(fun x => x * c) ⁻¹' Ioo a b = Ioo (b / c) (a / c) := by | simp [← Ioi_inter_Iio, h, inter_comm]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 668 | 670 | theorem preimage_mul_const_Ioc_of_neg (a b : α) {c : α} (h : c < 0) :
(fun x => x * c) ⁻¹' Ioc a b = Ico (b / c) (a / c) := by |
simp [← Ioi_inter_Iic, ← Ici_inter_Iio, h, inter_comm]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 674 | 676 | theorem preimage_mul_const_Ico_of_neg (a b : α) {c : α} (h : c < 0) :
(fun x => x * c) ⁻¹' Ico a b = Ioc (b / c) (a / c) := by |
simp [← Ici_inter_Iio, ← Ioi_inter_Iic, h, inter_comm]
| 957 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 680 | 681 | theorem preimage_mul_const_Icc_of_neg (a b : α) {c : α} (h : c < 0) :
(fun x => x * c) ⁻¹' Icc a b = Icc (b / c) (a / c) := by | simp [← Ici_inter_Iic, h, inter_comm]
| 957 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
| Mathlib/Data/Nat/Cast/Field.lean | 29 | 33 | theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_dvd : n ∣ m) (hn : (n : α) ≠ 0) :
((m / n : ℕ) : α) = m / n := by |
rcases n_dvd with ⟨k, rfl⟩
have : n ≠ 0 := by rintro rfl; simp at hn
rw [Nat.mul_div_cancel_left _ this.bot_lt, mul_comm n, cast_mul, mul_div_cancel_right₀ _ hn]
| 958 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 36 | 41 | theorem cast_div_div_div_cancel_right [DivisionSemiring α] [CharZero α] {m n d : ℕ}
(hn : d ∣ n) (hm : d ∣ m) :
(↑(m / d) : α) / (↑(n / d) : α) = (m : α) / n := by |
rcases eq_or_ne d 0 with (rfl | hd); · simp [Nat.zero_dvd.1 hm]
replace hd : (d : α) ≠ 0 := by norm_cast
rw [cast_div hm, cast_div hn, div_div_div_cancel_right _ hd] <;> exact hd
| 958 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 53 | 58 | theorem cast_div_le {m n : ℕ} : ((m / n : ℕ) : α) ≤ m / n := by |
cases n
· rw [cast_zero, div_zero, Nat.div_zero, cast_zero]
rw [le_div_iff, ← Nat.cast_mul, @Nat.cast_le]
· exact Nat.div_mul_le_self m _
· exact Nat.cast_pos.2 (Nat.succ_pos _)
| 958 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 65 | 67 | theorem one_div_pos_of_nat {n : ℕ} : 0 < 1 / ((n : α) + 1) := by |
rw [one_div]
exact inv_pos_of_nat
| 958 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 70 | 73 | theorem one_div_le_one_div {n m : ℕ} (h : n ≤ m) : 1 / ((m : α) + 1) ≤ 1 / ((n : α) + 1) := by |
refine one_div_le_one_div_of_le ?_ ?_
· exact Nat.cast_add_one_pos _
· simpa
| 958 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 76 | 79 | theorem one_div_lt_one_div {n m : ℕ} (h : n < m) : 1 / ((m : α) + 1) < 1 / ((n : α) + 1) := by |
refine one_div_lt_one_div_of_lt ?_ ?_
· exact Nat.cast_add_one_pos _
· simpa
| 958 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 89 | 92 | theorem image (hs : IsAntichain r s) (f : α → β) (h : ∀ ⦃a b⦄, r' (f a) (f b) → r a b) :
IsAntichain r' (f '' s) := by |
rintro _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ hbc hr
exact hs hb hc (ne_of_apply_ne _ hbc) (h hr)
| 959 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 120 | 124 | theorem image_relEmbedding (hs : IsAntichain r s) (φ : r ↪r r') : IsAntichain r' (φ '' s) := by |
intro b hb b' hb' h₁ h₂
rw [Set.mem_image] at hb hb'
obtain ⟨⟨a, has, rfl⟩, ⟨a', has', rfl⟩⟩ := hb, hb'
exact hs has has' (fun haa' => h₁ (by rw [haa'])) (φ.map_rel_iff.mp h₂)
| 959 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 313 | 317 | theorem image (hs : IsStrongAntichain r s) {f : α → β} (hf : Surjective f)
(h : ∀ a b, r' (f a) (f b) → r a b) : IsStrongAntichain r' (f '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ hab c
obtain ⟨c, rfl⟩ := hf c
exact (hs ha hb (ne_of_apply_ne _ hab) _).imp (mt <| h _ _) (mt <| h _ _)
| 959 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.Antichain
import Mathlib.Order.Interval.Finset.Nat
#align_import data.finset.slice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Finset Nat
variable {α : Type*} {ι : Sort*} {κ : ι → Sort*}
namespace Set
... | Mathlib/Data/Finset/Slice.lean | 64 | 66 | theorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r := by |
simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]
exact forall_swap
| 960 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.Antichain
import Mathlib.Order.Interval.Finset.Nat
#align_import data.finset.slice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Finset Nat
variable {α : Type*} {ι : Sort*} {κ : ι → Sort*}
namespace Set
... | Mathlib/Data/Finset/Slice.lean | 70 | 72 | theorem sized_iUnion₂ {f : ∀ i, κ i → Set (Finset α)} :
(⋃ (i) (j), f i j).Sized r ↔ ∀ i j, (f i j).Sized r := by |
simp only [Set.sized_iUnion]
| 960 |
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
#align_import data.set.intervals.ord_connected from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
open scoped Interval
open Set
open OrderDual (toDual ofDual)
namespace Set
... | Mathlib/Order/Interval/Set/OrdConnected.lean | 57 | 63 | theorem ordConnected_of_Ioo {α : Type*} [PartialOrder α] {s : Set α}
(hs : ∀ x ∈ s, ∀ y ∈ s, x < y → Ioo x y ⊆ s) : OrdConnected s := by |
rw [ordConnected_iff]
intro x hx y hy hxy
rcases eq_or_lt_of_le hxy with (rfl | hxy'); · simpa
rw [← Ioc_insert_left hxy, ← Ioo_insert_right hxy']
exact insert_subset_iff.2 ⟨hx, insert_subset_iff.2 ⟨hy, hs x hx y hy hxy'⟩⟩
| 961 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {α β : Type*}
section WeaklyCovers
section Preorder
variable [Preorder α] [Preorder β] {a ... | Mathlib/Order/Cover.lean | 96 | 97 | theorem not_wcovBy_iff (h : a ≤ b) : ¬a ⩿ b ↔ ∃ c, a < c ∧ c < b := by |
simp_rw [WCovBy, h, true_and_iff, not_forall, exists_prop, not_not]
| 962 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {α β : Type*}
section WeaklyCovers
section Preorder
variable [Preorder α] [Preorder β] {a ... | Mathlib/Order/Cover.lean | 122 | 126 | theorem WCovBy.image (f : α ↪o β) (hab : a ⩿ b) (h : (range f).OrdConnected) : f a ⩿ f b := by |
refine ⟨f.monotone hab.le, fun c ha hb => ?_⟩
obtain ⟨c, rfl⟩ := h.out (mem_range_self _) (mem_range_self _) ⟨ha.le, hb.le⟩
rw [f.lt_iff_lt] at ha hb
exact hab.2 ha hb
| 962 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {α β : Type*}
section WeaklyCovers
section LT
variable [LT α] {a b : α}
def CovBy (a b :... | Mathlib/Order/Cover.lean | 233 | 234 | theorem not_covBy_iff (h : a < b) : ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b := by |
simp_rw [CovBy, h, true_and_iff, not_forall, exists_prop, not_not]
| 962 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 73 | 76 | theorem mulSupport_eq_iff {f : α → M} {s : Set α} :
mulSupport f = s ↔ (∀ x, x ∈ s → f x ≠ 1) ∧ ∀ x, x ∉ s → f x = 1 := by |
simp (config := { contextual := true }) only [ext_iff, mem_mulSupport, ne_eq, iff_def,
not_imp_comm, and_comm, forall_and]
| 963 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 88 | 90 | theorem mulSupport_update_of_ne_one [DecidableEq α] (f : α → M) (x : α) {y : M} (hy : y ≠ 1) :
mulSupport (update f x y) = insert x (mulSupport f) := by |
ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]
| 963 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 93 | 95 | theorem mulSupport_update_one [DecidableEq α] (f : α → M) (x : α) :
mulSupport (update f x 1) = mulSupport f \ {x} := by |
ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]
| 963 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 98 | 100 | theorem mulSupport_update_eq_ite [DecidableEq α] [DecidableEq M] (f : α → M) (x : α) (y : M) :
mulSupport (update f x y) = if y = 1 then mulSupport f \ {x} else insert x (mulSupport f) := by |
rcases eq_or_ne y 1 with rfl | hy <;> simp [mulSupport_update_one, mulSupport_update_of_ne_one, *]
| 963 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 119 | 122 | theorem mulSupport_disjoint_iff {f : α → M} {s : Set α} :
Disjoint (mulSupport f) s ↔ EqOn f 1 s := by |
simp_rw [← subset_compl_iff_disjoint_right, mulSupport_subset_iff', not_mem_compl_iff, EqOn,
Pi.one_apply]
| 963 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 127 | 129 | theorem disjoint_mulSupport_iff {f : α → M} {s : Set α} :
Disjoint s (mulSupport f) ↔ EqOn f 1 s := by |
rw [disjoint_comm, mulSupport_disjoint_iff]
| 963 |
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Int.Cast.Field
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.int.char_zero from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Nat Set
variable {α β : Type*}
namespace Int
@[simp, norm_cast]
| Mathlib/Data/Int/CharZero.lean | 24 | 28 | theorem cast_div_charZero {k : Type*} [DivisionRing k] [CharZero k] {m n : ℤ} (n_dvd : n ∣ m) :
((m / n : ℤ) : k) = m / n := by |
rcases eq_or_ne n 0 with (rfl | hn)
· simp [Int.ediv_zero]
· exact cast_div n_dvd (cast_ne_zero.mpr hn)
| 964 |
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Int.Cast.Field
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.int.char_zero from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Nat Set
variable {α β : Type*}
namespace Int
@[simp, norm_cast]
theorem cast_div_charZe... | Mathlib/Data/Int/CharZero.lean | 33 | 35 | theorem cast_div_ofNat_charZero {k : Type*} [DivisionRing k] [CharZero k] {m n : ℕ}
(n_dvd : n ∣ m) : (((m : ℤ) / (n : ℤ) : ℤ) : k) = m / n := by |
rw [cast_div_charZero (Int.ofNat_dvd.mpr n_dvd), cast_natCast, cast_natCast]
| 964 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
namespace Nat
variable {R : Type*} [AddMonoidWithOne R] [Char... | Mathlib/Algebra/CharZero/Lemmas.lean | 39 | 42 | theorem cast_pow_eq_one {R : Type*} [Semiring R] [CharZero R] (q : ℕ) (n : ℕ) (hn : n ≠ 0) :
(q : R) ^ n = 1 ↔ q = 1 := by |
rw [← cast_pow, cast_eq_one]
exact pow_eq_one_iff hn
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
namespace Nat
variable {R : Type*} [AddMonoidWithOne R] [Char... | Mathlib/Algebra/CharZero/Lemmas.lean | 46 | 50 | theorem cast_div_charZero {k : Type*} [DivisionSemiring k] [CharZero k] {m n : ℕ} (n_dvd : n ∣ m) :
((m / n : ℕ) : k) = m / n := by |
rcases eq_or_ne n 0 with (rfl | hn)
· simp
· exact cast_div n_dvd (cast_ne_zero.2 hn)
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 88 | 89 | theorem add_self_eq_zero {a : R} : a + a = 0 ↔ a = 0 := by |
simp only [(two_mul a).symm, mul_eq_zero, two_ne_zero, false_or_iff]
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 100 | 102 | theorem zero_eq_bit0 {a : R} : 0 = bit0 a ↔ a = 0 := by |
rw [eq_comm]
exact bit0_eq_zero
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 127 | 129 | theorem nat_mul_inj {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) : n = 0 ∨ a = b := by |
rw [← sub_eq_zero, ← mul_sub, mul_eq_zero, sub_eq_zero] at h
exact mod_cast h
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 132 | 133 | theorem nat_mul_inj' {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) (w : n ≠ 0) : a = b := by |
simpa [w] using nat_mul_inj h
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 161 | 162 | theorem bit1_eq_one {a : R} : bit1 a = 1 ↔ a = 0 := by |
rw [show (1 : R) = bit1 0 by simp, bit1_eq_bit1]
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 166 | 168 | theorem one_eq_bit1 {a : R} : 1 = bit1 a ↔ a = 0 := by |
rw [eq_comm]
exact bit1_eq_one
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 182 | 182 | theorem add_halves' (a : R) : a / 2 + a / 2 = a := by | rw [← add_div, half_add_self]
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 185 | 185 | theorem sub_half (a : R) : a - a / 2 = a / 2 := by | rw [sub_eq_iff_eq_add, add_halves']
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 188 | 188 | theorem half_sub (a : R) : a / 2 - a = -(a / 2) := by | rw [← neg_sub, sub_half]
| 965 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 236 | 238 | theorem units_ne_neg_self (u : Rˣ) : u ≠ -u := by |
simp_rw [ne_eq, Units.ext_iff, Units.val_neg, eq_neg_iff_add_eq_zero, ← two_mul,
Units.mul_left_eq_zero, two_ne_zero, not_false_iff]
| 965 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 30 | 37 | theorem zpow_le_of_le (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n := by |
have ha₀ : 0 < a := one_pos.trans_le ha
lift n - m to ℕ using sub_nonneg.2 h with k hk
calc
a ^ m = a ^ m * 1 := (mul_one _).symm
_ ≤ a ^ m * a ^ k :=
mul_le_mul_of_nonneg_left (one_le_pow_of_one_le ha _) (zpow_nonneg ha₀.le _)
_ = a ^ n := by rw [← zpow_natCast, ← zpow_add₀ ha₀.ne', hk, add_su... | 966 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 97 | 100 | theorem zpow_injective (h₀ : 0 < a) (h₁ : a ≠ 1) : Injective (a ^ · : ℤ → α) := by |
rcases h₁.lt_or_lt with (H | H)
· exact (zpow_strictAnti h₀ H).injective
· exact (zpow_strictMono H).injective
| 966 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 114 | 116 | theorem zpow_le_max_iff_min_le {x : α} (hx : 1 < x) {a b c : ℤ} :
x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b)) ↔ min a b ≤ c := by |
simp_rw [le_max_iff, min_le_iff, zpow_le_iff_le hx, neg_le_neg_iff]
| 966 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 150 | 152 | theorem Even.zpow_pos_iff (hn : Even n) (h : n ≠ 0) : 0 < a ^ n ↔ a ≠ 0 := by |
obtain ⟨k, rfl⟩ := hn
rw [zpow_add' (by simp [em']), mul_self_pos, zpow_ne_zero_iff (by simpa using h)]
| 966 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 155 | 159 | theorem Odd.zpow_neg_iff (hn : Odd n) : a ^ n < 0 ↔ a < 0 := by |
refine ⟨lt_imp_lt_of_le_imp_le (zpow_nonneg · _), fun ha ↦ ?_⟩
obtain ⟨k, rfl⟩ := hn
rw [zpow_add_one₀ ha.ne]
exact mul_neg_of_pos_of_neg (Even.zpow_pos (even_two_mul _) ha.ne) ha
| 966 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 166 | 169 | theorem Odd.zpow_nonpos_iff (hn : Odd n) : a ^ n ≤ 0 ↔ a ≤ 0 := by |
rw [le_iff_lt_or_eq, le_iff_lt_or_eq, hn.zpow_neg_iff, zpow_eq_zero_iff]
rintro rfl
exact Int.odd_iff_not_even.1 hn even_zero
| 966 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 181 | 182 | theorem Even.zpow_abs {p : ℤ} (hp : Even p) (a : α) : |a| ^ p = a ^ p := by |
cases' abs_choice a with h h <;> simp only [h, hp.neg_zpow _]
| 966 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemmas
import Mathlib.Data.Set.Subsingleton
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Order.GaloisConnection
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith... | Mathlib/Algebra/Order/Floor.lean | 577 | 589 | theorem subsingleton_floorSemiring {α} [LinearOrderedSemiring α] :
Subsingleton (FloorSemiring α) := by |
refine ⟨fun H₁ H₂ => ?_⟩
have : H₁.ceil = H₂.ceil := funext fun a => (H₁.gc_ceil.l_unique H₂.gc_ceil) fun n => rfl
have : H₁.floor = H₂.floor := by
ext a
cases' lt_or_le a 0 with h h
· rw [H₁.floor_of_neg, H₂.floor_of_neg] <;> exact h
· refine eq_of_forall_le_iff fun n => ?_
rw [H₁.gc_floor... | 967 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import data.rat.floor from "leanprover-community/mathlib"@"e1bccd6e40ae78370f01659715d3c948716e3b7e"
open Int
namespace Rat
variable {α : Type*} [LinearOrderedField α] [FloorRi... | Mathlib/Data/Rat/Floor.lean | 56 | 66 | theorem floor_int_div_nat_eq_div {n : ℤ} {d : ℕ} : ⌊(↑n : ℚ) / (↑d : ℚ)⌋ = n / (↑d : ℤ) := by |
rw [Rat.floor_def]
obtain rfl | hd := @eq_zero_or_pos _ _ d
· simp
set q := (n : ℚ) / d with q_eq
obtain ⟨c, n_eq_c_mul_num, d_eq_c_mul_denom⟩ : ∃ c, n = c * q.num ∧ (d : ℤ) = c * q.den := by
rw [q_eq]
exact mod_cast @Rat.exists_eq_mul_div_num_and_eq_mul_div_den n d (mod_cast hd.ne')
rw [n_eq_c_mul... | 968 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import data.rat.floor from "leanprover-community/mathlib"@"e1bccd6e40ae78370f01659715d3c948716e3b7e"
open Int
namespace Rat
variable {α : Type*} [LinearOrderedField α] [FloorRi... | Mathlib/Data/Rat/Floor.lean | 75 | 76 | theorem ceil_cast (x : ℚ) : ⌈(x : α)⌉ = ⌈x⌉ := by |
rw [← neg_inj, ← floor_neg, ← floor_neg, ← Rat.cast_neg, Rat.floor_cast]
| 968 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import data.rat.floor from "leanprover-community/mathlib"@"e1bccd6e40ae78370f01659715d3c948716e3b7e"
open Int
namespace Rat
variable {α : Type*} [LinearOrderedField α] [FloorRi... | Mathlib/Data/Rat/Floor.lean | 80 | 82 | theorem round_cast (x : ℚ) : round (x : α) = round x := by |
have : ((x + 1 / 2 : ℚ) : α) = x + 1 / 2 := by simp
rw [round_eq, round_eq, ← this, floor_cast]
| 968 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import data.rat.floor from "leanprover-community/mathlib"@"e1bccd6e40ae78370f01659715d3c948716e3b7e"
open Int
namespace Rat
variable {α : Type*} [LinearOrderedField α] [FloorRi... | Mathlib/Data/Rat/Floor.lean | 86 | 87 | theorem cast_fract (x : ℚ) : (↑(fract x) : α) = fract (x : α) := by |
simp only [fract, cast_sub, cast_intCast, floor_cast]
| 968 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 64 | 81 | theorem existsUnique_zsmul_near_of_pos {a : α} (ha : 0 < a) (g : α) :
∃! k : ℤ, k • a ≤ g ∧ g < (k + 1) • a := by |
let s : Set ℤ := { n : ℤ | n • a ≤ g }
obtain ⟨k, hk : -g ≤ k • a⟩ := Archimedean.arch (-g) ha
have h_ne : s.Nonempty := ⟨-k, by simpa [s] using neg_le_neg hk⟩
obtain ⟨k, hk⟩ := Archimedean.arch g ha
have h_bdd : ∀ n ∈ s, n ≤ (k : ℤ) := by
intro n hn
apply (zsmul_le_zsmul_iff ha).mp
rw [← natCast... | 969 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 84 | 87 | theorem existsUnique_zsmul_near_of_pos' {a : α} (ha : 0 < a) (g : α) :
∃! k : ℤ, 0 ≤ g - k • a ∧ g - k • a < a := by |
simpa only [sub_nonneg, add_zsmul, one_zsmul, sub_lt_iff_lt_add'] using
existsUnique_zsmul_near_of_pos ha g
| 969 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 90 | 93 | theorem existsUnique_sub_zsmul_mem_Ico {a : α} (ha : 0 < a) (b c : α) :
∃! m : ℤ, b - m • a ∈ Set.Ico c (c + a) := by |
simpa only [mem_Ico, le_sub_iff_add_le, zero_add, add_comm c, sub_lt_iff_lt_add', add_assoc] using
existsUnique_zsmul_near_of_pos' ha (b - c)
| 969 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 120 | 122 | theorem exists_nat_ge [OrderedSemiring α] [Archimedean α] (x : α) : ∃ n : ℕ, x ≤ n := by |
nontriviality α
exact (Archimedean.arch x one_pos).imp fun n h => by rwa [← nsmul_one]
| 969 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 138 | 148 | theorem add_one_pow_unbounded_of_pos (x : α) (hy : 0 < y) : ∃ n : ℕ, x < (y + 1) ^ n :=
have : 0 ≤ 1 + y := add_nonneg zero_le_one hy.le
(Archimedean.arch x hy).imp fun n h ↦
calc
x ≤ n • y := h
_ = n * y := nsmul_eq_mul _ _
_ < 1 + n * y := lt_one_add _
_ ≤ (1 + y) ^ n :=
one_ad... | rw [add_comm]
| 969 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Set.Lattice
#align_import group_theory.archimedean from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set
variable {G : Type*} [LinearOrderedAddCommGroup G] [Archimedean G]
| Mathlib/GroupTheory/Archimedean.lean | 40 | 54 | theorem AddSubgroup.cyclic_of_min {H : AddSubgroup G} {a : G}
(ha : IsLeast { g : G | g ∈ H ∧ 0 < g } a) : H = AddSubgroup.closure {a} := by |
obtain ⟨⟨a_in, a_pos⟩, a_min⟩ := ha
refine le_antisymm ?_ (H.closure_le.mpr <| by simp [a_in])
intro g g_in
obtain ⟨k, ⟨nonneg, lt⟩, _⟩ := existsUnique_zsmul_near_of_pos' a_pos g
have h_zero : g - k • a = 0 := by
by_contra h
have h : a ≤ g - k • a := by
refine a_min ⟨?_, ?_⟩
· exact AddSu... | 970 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Set.Lattice
#align_import group_theory.archimedean from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set
variable {G : Type*} [LinearOrderedAddCommGroup G] [Archimedean G]
th... | Mathlib/GroupTheory/Archimedean.lean | 60 | 87 | theorem AddSubgroup.exists_isLeast_pos {H : AddSubgroup G} (hbot : H ≠ ⊥) {a : G} (h₀ : 0 < a)
(hd : Disjoint (H : Set G) (Ioo 0 a)) : ∃ b, IsLeast { g : G | g ∈ H ∧ 0 < g } b := by |
-- todo: move to a lemma?
have hex : ∀ g > 0, ∃ n : ℕ, g ∈ Ioc (n • a) ((n + 1) • a) := fun g hg => by
rcases existsUnique_add_zsmul_mem_Ico h₀ 0 (g - a) with ⟨m, ⟨hm, hm'⟩, -⟩
simp only [zero_add, sub_le_iff_le_add, sub_add_cancel, ← add_one_zsmul] at hm hm'
lift m to ℕ
· rw [← Int.lt_add_one_iff,... | 970 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Set.Lattice
#align_import group_theory.archimedean from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set
variable {G : Type*} [LinearOrderedAddCommGroup G] [Archimedean G]
th... | Mathlib/GroupTheory/Archimedean.lean | 91 | 95 | theorem AddSubgroup.cyclic_of_isolated_zero {H : AddSubgroup G} {a : G} (h₀ : 0 < a)
(hd : Disjoint (H : Set G) (Ioo 0 a)) : ∃ b, H = closure {b} := by |
rcases eq_or_ne H ⊥ with rfl | hbot
· exact ⟨0, closure_singleton_zero.symm⟩
· exact (exists_isLeast_pos hbot h₀ hd).imp fun _ => cyclic_of_min
| 970 |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 42 | 53 | theorem dense_of_not_isolated_zero (S : AddSubgroup G) (hS : ∀ ε > 0, ∃ g ∈ S, g ∈ Ioo 0 ε) :
Dense (S : Set G) := by |
cases subsingleton_or_nontrivial G
· refine fun x => _root_.subset_closure ?_
rw [Subsingleton.elim x 0]
exact zero_mem S
refine dense_of_exists_between fun a b hlt => ?_
rcases hS (b - a) (sub_pos.2 hlt) with ⟨g, hgS, hg0, hg⟩
rcases (existsUnique_add_zsmul_mem_Ioc hg0 0 a).exists with ⟨m, hm⟩
rw ... | 971 |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 58 | 62 | theorem dense_of_no_min (S : AddSubgroup G) (hbot : S ≠ ⊥)
(H : ¬∃ a : G, IsLeast { g : G | g ∈ S ∧ 0 < g } a) : Dense (S : Set G) := by |
refine S.dense_of_not_isolated_zero fun ε ε0 => ?_
contrapose! H
exact exists_isLeast_pos hbot ε0 (disjoint_left.2 H)
| 971 |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 67 | 71 | theorem dense_or_cyclic (S : AddSubgroup G) : Dense (S : Set G) ∨ ∃ a : G, S = closure {a} := by |
refine (em _).imp (dense_of_not_isolated_zero S) fun h => ?_
push_neg at h
rcases h with ⟨ε, ε0, hε⟩
exact cyclic_of_isolated_zero ε0 (disjoint_left.2 hε)
| 971 |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.CauSeq.Basic
#align_import data.real.cau_seq from "leanprover-community/mathlib"@"9116dd6709f303dcf781632e15fdef382b0fc579"
open Finset IsAbsoluteValue
namespace IsCauSeq
variable {α β : Type*} [LinearOrderedField... | Mathlib/Algebra/Order/CauSeq/BigOperators.lean | 57 | 141 | theorem _root_.cauchy_product (ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n))
(hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ, ∀ j ≥ i,
abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k -
∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε := by |
let ⟨P, hP⟩ := ha.bounded
let ⟨Q, hQ⟩ := hb.bounded
have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
let ⟨N, hN⟩ := hb.cauchy₂ hPε0
have hQε0 : 0 < ε / (4 * Q) :=
div_pos ε0 (mul_pos (show (0 : α) < 4 by norm... | 972 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 73 | 75 | theorem map_add_nsmul [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
(f : F) (x : G) (n : ℕ) : f (x + n • a) = f x + n • b := by |
simpa using (AddConstMapClass.semiconj f).iterate_right n x
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 78 | 79 | theorem map_add_nat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (x : G) (n : ℕ) : f (x + n) = f x + n • b := by | simp [← map_add_nsmul]
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 90 | 91 | theorem map_add_nat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (x : G) (n : ℕ) : f (x + n) = f x + n := by | simp
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 98 | 100 | theorem map_const [AddZeroClass G] [Add H] [AddConstMapClass F G H a b] (f : F) :
f a = f 0 + b := by |
simpa using map_add_const f 0
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 107 | 109 | theorem map_nsmul_const [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
(f : F) (n : ℕ) : f (n • a) = f 0 + n • b := by |
simpa using map_add_nsmul f 0 n
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 112 | 114 | theorem map_nat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (n : ℕ) : f n = f 0 + n • b := by |
simpa using map_add_nat' f 0 n
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 121 | 122 | theorem map_nat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (n : ℕ) : f n = f 0 + n := by | simp
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 129 | 131 | theorem map_const_add [AddCommSemigroup G] [Add H] [AddConstMapClass F G H a b]
(f : F) (x : G) : f (a + x) = f x + b := by |
rw [add_comm, map_add_const]
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 137 | 139 | theorem map_nsmul_add [AddCommMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
(f : F) (n : ℕ) (x : G) : f (n • a + x) = f x + n • b := by |
rw [add_comm, map_add_nsmul]
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 142 | 144 | theorem map_nat_add' [AddCommMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n • b := by |
simpa using map_nsmul_add f n x
| 973 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 151 | 152 | theorem map_nat_add [AddCommMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n := by | simp
| 973 |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
open Finset Int
namespace Int
variable (a b : ℤ) {r : ℤ} (hr : 0 < r)
lemma Ico_filter_dvd_eq : (Ico a b).filter (r ∣ ·) =
(Ico ⌈a / (r : ℚ)⌉ ⌈b... | Mathlib/Data/Int/CardIntervalMod.lean | 42 | 44 | theorem Ico_filter_dvd_card : ((Ico a b).filter (r ∣ ·)).card =
max (⌈b / (r : ℚ)⌉ - ⌈a / (r : ℚ)⌉) 0 := by |
rw [Ico_filter_dvd_eq _ _ hr, card_map, card_Ico, toNat_eq_max]
| 974 |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
open Finset Int
namespace Int
variable (a b : ℤ) {r : ℤ} (hr : 0 < r)
lemma Ico_filter_dvd_eq : (Ico a b).filter (r ∣ ·) =
(Ico ⌈a / (r : ℚ)⌉ ⌈b... | Mathlib/Data/Int/CardIntervalMod.lean | 47 | 49 | theorem Ioc_filter_dvd_card : ((Ioc a b).filter (r ∣ ·)).card =
max (⌊b / (r : ℚ)⌋ - ⌊a / (r : ℚ)⌋) 0 := by |
rw [Ioc_filter_dvd_eq _ _ hr, card_map, card_Ioc, toNat_eq_max]
| 974 |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
open Finset Int
namespace Int
variable (a b : ℤ) {r : ℤ} (hr : 0 < r)
lemma Ico_filter_dvd_eq : (Ico a b).filter (r ∣ ·) =
(Ico ⌈a / (r : ℚ)⌉ ⌈b... | Mathlib/Data/Int/CardIntervalMod.lean | 65 | 67 | theorem Ico_filter_modEq_card (v : ℤ) : ((Ico a b).filter (· ≡ v [ZMOD r])).card =
max (⌈(b - v) / (r : ℚ)⌉ - ⌈(a - v) / (r : ℚ)⌉) 0 := by |
simp [Ico_filter_modEq_eq, Ico_filter_dvd_eq, toNat_eq_max, hr]
| 974 |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
open Finset Int
namespace Int
variable (a b : ℤ) {r : ℤ} (hr : 0 < r)
lemma Ico_filter_dvd_eq : (Ico a b).filter (r ∣ ·) =
(Ico ⌈a / (r : ℚ)⌉ ⌈b... | Mathlib/Data/Int/CardIntervalMod.lean | 71 | 73 | theorem Ioc_filter_modEq_card (v : ℤ) : ((Ioc a b).filter (· ≡ v [ZMOD r])).card =
max (⌊(b - v) / (r : ℚ)⌋ - ⌊(a - v) / (r : ℚ)⌋) 0 := by |
simp [Ioc_filter_modEq_eq, Ioc_filter_dvd_eq, toNat_eq_max, hr]
| 974 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 66 | 70 | theorem log_of_right_le_one (b : ℕ) {r : R} (hr : r ≤ 1) : log b r = -Nat.clog b ⌈r⁻¹⌉₊ := by |
obtain rfl | hr := hr.eq_or_lt
· rw [log, if_pos hr, inv_one, Nat.ceil_one, Nat.floor_one, Nat.log_one_right, Nat.clog_one_right,
Int.ofNat_zero, neg_zero]
· exact if_neg hr.not_le
| 975 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 74 | 78 | theorem log_natCast (b : ℕ) (n : ℕ) : log b (n : R) = Nat.log b n := by |
cases n
· simp [log_of_right_le_one]
· rw [log_of_one_le_right, Nat.floor_natCast]
simp
| 975 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 87 | 90 | theorem log_of_left_le_one {b : ℕ} (hb : b ≤ 1) (r : R) : log b r = 0 := by |
rcases le_total 1 r with h | h
· rw [log_of_one_le_right _ h, Nat.log_of_left_le_one hb, Int.ofNat_zero]
· rw [log_of_right_le_one _ h, Nat.clog_of_left_le_one hb, Int.ofNat_zero, neg_zero]
| 975 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 93 | 96 | theorem log_of_right_le_zero (b : ℕ) {r : R} (hr : r ≤ 0) : log b r = 0 := by |
rw [log_of_right_le_one _ (hr.trans zero_le_one),
Nat.clog_of_right_le_one ((Nat.ceil_eq_zero.mpr <| inv_nonpos.2 hr).trans_le zero_le_one),
Int.ofNat_zero, neg_zero]
| 975 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 99 | 105 | theorem zpow_log_le_self {b : ℕ} {r : R} (hb : 1 < b) (hr : 0 < r) : (b : R) ^ log b r ≤ r := by |
rcases le_total 1 r with hr1 | hr1
· rw [log_of_one_le_right _ hr1]
rw [zpow_natCast, ← Nat.cast_pow, ← Nat.le_floor_iff hr.le]
exact Nat.pow_log_le_self b (Nat.floor_pos.mpr hr1).ne'
· rw [log_of_right_le_one _ hr1, zpow_neg, zpow_natCast, ← Nat.cast_pow]
exact inv_le_of_inv_le hr (Nat.ceil_le.1 <| ... | 975 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 108 | 124 | theorem lt_zpow_succ_log_self {b : ℕ} (hb : 1 < b) (r : R) : r < (b : R) ^ (log b r + 1) := by |
rcases le_or_lt r 0 with hr | hr
· rw [log_of_right_le_zero _ hr, zero_add, zpow_one]
exact hr.trans_lt (zero_lt_one.trans_le <| mod_cast hb.le)
rcases le_or_lt 1 r with hr1 | hr1
· rw [log_of_one_le_right _ hr1]
rw [Int.ofNat_add_one_out, zpow_natCast, ← Nat.cast_pow]
apply Nat.lt_of_floor_lt
... | 975 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 133 | 134 | theorem log_one_right (b : ℕ) : log b (1 : R) = 0 := by |
rw [log_of_one_le_right _ le_rfl, Nat.floor_one, Nat.log_one_right, Int.ofNat_zero]
| 975 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 138 | 145 | theorem log_zpow {b : ℕ} (hb : 1 < b) (z : ℤ) : log b ((b : R) ^ z : R) = z := by |
obtain ⟨n, rfl | rfl⟩ := Int.eq_nat_or_neg z
· rw [log_of_one_le_right _ (one_le_zpow_of_nonneg _ <| Int.natCast_nonneg _), zpow_natCast, ←
Nat.cast_pow, Nat.floor_natCast, Nat.log_pow hb]
exact mod_cast hb.le
· rw [log_of_right_le_one _ (zpow_le_one_of_nonpos _ <| neg_nonpos.mpr (Int.natCast_nonneg _)... | 975 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.ContinuedFractions.Basic
#align_import algebra.continued_fractions.computation.basic from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
-- Fix a carrier `K`.
variable (K : Type*)
structu... | Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean | 159 | 161 | theorem stream_isSeq (v : K) : (IntFractPair.stream v).IsSeq := by |
intro _ hyp
simp [IntFractPair.stream, hyp]
| 976 |
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 66 | 71 | theorem stream_eq_none_of_fr_eq_zero {ifp_n : IntFractPair K}
(stream_nth_eq : IntFractPair.stream v n = some ifp_n) (nth_fr_eq_zero : ifp_n.fr = 0) :
IntFractPair.stream v (n + 1) = none := by |
cases' ifp_n with _ fr
change fr = 0 at nth_fr_eq_zero
simp [IntFractPair.stream, stream_nth_eq, nth_fr_eq_zero]
| 977 |
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