state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case intro.intro.intro.refine'_1
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : Set.Countab... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact (diff_subset _ _).trans hts | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2 | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib_Topology_Order_Basic |
case intro.intro.intro.refine'_2
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : Set.Countab... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact htc.mono (diff_subset _ _) | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2 | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib_Topology_Order_Basic |
case intro.intro.intro.refine'_3
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : Set.Countab... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact htd.diff_finite ((subsingleton_isBot α).finite.union (subsingleton_isTop α).finite) | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2 | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib_Topology_Order_Basic |
case intro.intro.intro.refine'_4
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : Set.Countab... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [hx] | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2 | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib_Topology_Order_Basic |
case intro.intro.intro.refine'_5
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a b : α
s✝ : Set α
l : Filter β
f : α → β
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : Set.Countab... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [hx] | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib.Topology.Order.Basic.2643_0.Npdof1X5b8sCkZ2 | /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivi... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
a b : α
s : Set α
l : Filter β
f : α → β
inst✝¹ : SeparableSpace α
inst✝ : Nontrivial α
⊢ ∃ s, Set.Countable s ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∉ s) ∧ ∀ (x : α), IsTop x → x ∉ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa using dense_univ.exists_countable_dense_subset_no_bot_top | /-- If `α` is a nontrivial separable dense linear order, then there exists a
countable dense set `s : Set α` that contains neither top nor bottom elements of `α`.
For a dense set containing both bot and top elements, see
`exists_countable_dense_bot_top`. -/
theorem exists_countable_dense_no_bot_top [SeparableSpace α] [... | Mathlib.Topology.Order.Basic.2660_0.Npdof1X5b8sCkZ2 | /-- If `α` is a nontrivial separable dense linear order, then there exists a
countable dense set `s : Set α` that contains neither top nor bottom elements of `α`.
For a dense set containing both bot and top elements, see
`exists_countable_dense_bot_top`. -/
theorem exists_countable_dense_no_bot_top [SeparableSpace α] [... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : Continuous... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [iSup, Monotone.map_sSup_of_continuousAt' Cf Mf (range_nonempty g) bdd, ← range_comp, iSup] | /-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
(bdd : BddAbov... | Mathlib.Topology.Order.Basic.2686_0.Npdof1X5b8sCkZ2 | /-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
(bdd : BddAbov... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : Continuous... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | /-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
(bdd : BddAbov... | Mathlib.Topology.Order.Basic.2686_0.Npdof1X5b8sCkZ2 | /-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
(bdd : BddAbov... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : Continuous... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [iInf, Monotone.map_sInf_of_continuousAt' Cf Mf (range_nonempty g) bdd, ← range_comp, iInf] | /-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. -/
theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
(bdd : BddBelow (... | Mathlib.Topology.Order.Basic.2703_0.Npdof1X5b8sCkZ2 | /-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. -/
theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
(bdd : BddBelow (... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : Continuous... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | /-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. -/
theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
(bdd : BddBelow (... | Mathlib.Topology.Order.Basic.2703_0.Npdof1X5b8sCkZ2 | /-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. -/
theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
(bdd : BddBelow (... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : Continuous... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [iInf, Antitone.map_sInf_of_continuousAt' Cf Af (range_nonempty g) bdd, ← range_comp, iSup] | /-- An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed supremum of the composition. -/
theorem Antitone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
(bdd : BddBelow... | Mathlib.Topology.Order.Basic.2720_0.Npdof1X5b8sCkZ2 | /-- An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed supremum of the composition. -/
theorem Antitone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
(bdd : BddBelow... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : Continuous... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | /-- An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed supremum of the composition. -/
theorem Antitone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
(bdd : BddBelow... | Mathlib.Topology.Order.Basic.2720_0.Npdof1X5b8sCkZ2 | /-- An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed supremum of the composition. -/
theorem Antitone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
(bdd : BddBelow... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : Continuous... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [iSup, Antitone.map_sSup_of_continuousAt' Cf Af (range_nonempty g) bdd, ← range_comp, iInf] | /-- An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
indexed supremum to the indexed infimum of the composition. -/
theorem Antitone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
(bdd : BddAbov... | Mathlib.Topology.Order.Basic.2737_0.Npdof1X5b8sCkZ2 | /-- An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
indexed supremum to the indexed infimum of the composition. -/
theorem Antitone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
(bdd : BddAbov... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderClosedTopology β
inst✝¹ : Nonempty γ
ι : Sort u_1
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : Continuous... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | /-- An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
indexed supremum to the indexed infimum of the composition. -/
theorem Antitone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
(bdd : BddAbov... | Mathlib.Topology.Order.Basic.2737_0.Npdof1X5b8sCkZ2 | /-- An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
indexed supremum to the indexed infimum of the composition. -/
theorem Antitone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
(bdd : BddAbov... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁶ : CompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : CompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
f : α → β
s : Set α
Cf : ContinuousAt f (sSup s)
Mf : Monotone f
fbot : f ⊥ = ⊥
⊢ f (sSup s) =... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases s.eq_empty_or_nonempty with h | h | /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s)... | Mathlib.Topology.Order.Basic.2773_0.Npdof1X5b8sCkZ2 | /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s)... | Mathlib_Topology_Order_Basic |
case inl
α : Type u
β : Type v
γ : Type w
inst✝⁶ : CompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : CompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
f : α → β
s : Set α
Cf : ContinuousAt f (sSup s)
Mf : Monotone f
fbot : f ⊥ = ⊥
h : s... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [h, fbot] | /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s)... | Mathlib.Topology.Order.Basic.2773_0.Npdof1X5b8sCkZ2 | /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s)... | Mathlib_Topology_Order_Basic |
case inr
α : Type u
β : Type v
γ : Type w
inst✝⁶ : CompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : CompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
f : α → β
s : Set α
Cf : ContinuousAt f (sSup s)
Mf : Monotone f
fbot : f ⊥ = ⊥
h : S... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Mf.map_sSup_of_continuousAt' Cf h | /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s)... | Mathlib.Topology.Order.Basic.2773_0.Npdof1X5b8sCkZ2 | /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. -/
theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s)... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁶ : CompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : CompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
ι : Sort u_1
f : α → β
g : ι → α
Cf : ContinuousAt f (iSup g)
Mf : Monotone f
fbot : f ⊥ = ⊥
⊢... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [iSup, Mf.map_sSup_of_continuousAt Cf fbot, ← range_comp, iSup] | /-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fb... | Mathlib.Topology.Order.Basic.2782_0.Npdof1X5b8sCkZ2 | /-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fb... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁶ : CompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : CompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
ι : Sort u_1
f : α → β
g : ι → α
Cf : ContinuousAt f (iSup g)
Mf : Monotone f
fbot : f ⊥ = ⊥
⊢... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | /-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fb... | Mathlib.Topology.Order.Basic.2782_0.Npdof1X5b8sCkZ2 | /-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
theorem Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fb... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
f : α → β
s : Set α
Cf : ContinuousAt f (sSup s)
Mf : Monotone f
ne ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' ((isLUB_csSup (ne.image f) (Mf.map_bddAbove H)).unique _).symm | /-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f... | Mathlib.Topology.Order.Basic.2870_0.Npdof1X5b8sCkZ2 | /-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
f : α → β
s : Set α
Cf : ContinuousAt f (sSup s)
Mf : Monotone f
ne ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' (isLUB_csSup ne H).isLUB_of_tendsto (fun x _ y _ xy => Mf xy) ne _ | /-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f... | Mathlib.Topology.Order.Basic.2870_0.Npdof1X5b8sCkZ2 | /-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
f : α → β
s : Set α
Cf : ContinuousAt f (sSup s)
Mf : Monotone f
ne ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact Cf.mono_left inf_le_left | /-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f... | Mathlib.Topology.Order.Basic.2870_0.Npdof1X5b8sCkZ2 | /-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. -/
theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
(Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
f : α → β
g : γ → α
Cf : ContinuousAt f (⨆ i, g i)
Mf : Monotone f
H... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rw [iSup, Mf.map_csSup_of_continuousAt Cf (range_nonempty _) H, ← range_comp, iSup] | /-- If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the supremum of the composition. -/
theorem Monotone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
(Mf : Monotone f) (H : BddAbove (range g... | Mathlib.Topology.Order.Basic.2879_0.Npdof1X5b8sCkZ2 | /-- If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the supremum of the composition. -/
theorem Monotone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
(Mf : Monotone f) (H : BddAbove (range g... | Mathlib_Topology_Order_Basic |
α : Type u
β : Type v
γ : Type w
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
inst✝ : Nonempty γ
f : α → β
g : γ → α
Cf : ContinuousAt f (⨆ i, g i)
Mf : Monotone f
H... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rfl | /-- If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the supremum of the composition. -/
theorem Monotone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
(Mf : Monotone f) (H : BddAbove (range g... | Mathlib.Topology.Order.Basic.2879_0.Npdof1X5b8sCkZ2 | /-- If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the supremum of the composition. -/
theorem Monotone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
(Mf : Monotone f) (H : BddAbove (range g... | Mathlib_Topology_Order_Basic |
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | rcases eq_empty_or_nonempty (Iio x) with (h | h) | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib.Topology.Order.Basic.2932_0.Npdof1X5b8sCkZ2 | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib_Topology_Order_Basic |
case inl
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simp [h] | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib.Topology.Order.Basic.2932_0.Npdof1X5b8sCkZ2 | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib_Topology_Order_Basic |
case inr
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine' tendsto_order.2 ⟨fun l hl => _, fun m hm => _⟩ | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib.Topology.Order.Basic.2932_0.Npdof1X5b8sCkZ2 | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib_Topology_Order_Basic |
case inr.refine'_1
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : Li... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | obtain ⟨z, zx, lz⟩ : ∃ a : α, a < x ∧ l < f a := by
simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using
exists_lt_of_lt_csSup (nonempty_image_iff.2 h) hl | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib.Topology.Order.Basic.2932_0.Npdof1X5b8sCkZ2 | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib_Topology_Order_Basic |
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using
exists_lt_of_lt_csSup (nonempty_image_iff.2 h) hl | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib.Topology.Order.Basic.2932_0.Npdof1X5b8sCkZ2 | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib_Topology_Order_Basic |
case inr.refine'_1.intro.intro
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact mem_of_superset (Ioo_mem_nhdsWithin_Iio' zx) fun y hy => lz.trans_le (Mf hy.1.le) | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib.Topology.Order.Basic.2932_0.Npdof1X5b8sCkZ2 | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib_Topology_Order_Basic |
case inr.refine'_2
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : Li... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | refine mem_of_superset self_mem_nhdsWithin fun _ hy => lt_of_le_of_lt ?_ hm | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib.Topology.Order.Basic.2932_0.Npdof1X5b8sCkZ2 | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib_Topology_Order_Basic |
case inr.refine'_2
α✝ : Type u
β✝ : Type v
γ : Type w
inst✝¹² : ConditionallyCompleteLinearOrder α✝
inst✝¹¹ : TopologicalSpace α✝
inst✝¹⁰ : OrderTopology α✝
inst✝⁹ : ConditionallyCompleteLinearOrder β✝
inst✝⁸ : TopologicalSpace β✝
inst✝⁷ : OrderClosedTopology β✝
inst✝⁶ : Nonempty γ
α : Type u_1
β : Type u_2
inst✝⁵ : Li... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Intervals.Pi
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Order.Filter.Interval
import Mathlib.T... | exact le_csSup (Mf.map_bddAbove bddAbove_Iio) (mem_image_of_mem _ hy) | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib.Topology.Order.Basic.2932_0.Npdof1X5b8sCkZ2 | /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/
theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (... | Mathlib_Topology_Order_Basic |
p✝ n✝ k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ n / p < n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | apply Nat.div_lt_self | /--
Tail recursive function which returns the largest `k : ℕ` such that `p ^ k ∣ n` for any `p : ℕ`.
`padicValNat_eq_maxPowDiv` allows the code generator to use this definition for `padicValNat`
-/
def maxPowDiv (p n : ℕ) : ℕ :=
go 0 p n
where go (k p n : ℕ) : ℕ :=
if h : 1 < p ∧ 0 < n ∧ n % p = 0 then
have : n... | Mathlib.Data.Nat.MaxPowDiv.26_0.2jhZnOeTQwLUwDJ | /--
Tail recursive function which returns the largest `k : ℕ` such that `p ^ k ∣ n` for any `p : ℕ`.
`padicValNat_eq_maxPowDiv` allows the code generator to use this definition for `padicValNat`
-/
def maxPowDiv (p n : ℕ) : ℕ | Mathlib_Data_Nat_MaxPowDiv |
case hLtN
p✝ n✝ k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ 0 < n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | aesop | /--
Tail recursive function which returns the largest `k : ℕ` such that `p ^ k ∣ n` for any `p : ℕ`.
`padicValNat_eq_maxPowDiv` allows the code generator to use this definition for `padicValNat`
-/
def maxPowDiv (p n : ℕ) : ℕ :=
go 0 p n
where go (k p n : ℕ) : ℕ :=
if h : 1 < p ∧ 0 < n ∧ n % p = 0 then
have : n... | Mathlib.Data.Nat.MaxPowDiv.26_0.2jhZnOeTQwLUwDJ | /--
Tail recursive function which returns the largest `k : ℕ` such that `p ^ k ∣ n` for any `p : ℕ`.
`padicValNat_eq_maxPowDiv` allows the code generator to use this definition for `padicValNat`
-/
def maxPowDiv (p n : ℕ) : ℕ | Mathlib_Data_Nat_MaxPowDiv |
case hLtK
p✝ n✝ k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ 1 < p | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | aesop | /--
Tail recursive function which returns the largest `k : ℕ` such that `p ^ k ∣ n` for any `p : ℕ`.
`padicValNat_eq_maxPowDiv` allows the code generator to use this definition for `padicValNat`
-/
def maxPowDiv (p n : ℕ) : ℕ :=
go 0 p n
where go (k p n : ℕ) : ℕ :=
if h : 1 < p ∧ 0 < n ∧ n % p = 0 then
have : n... | Mathlib.Data.Nat.MaxPowDiv.26_0.2jhZnOeTQwLUwDJ | /--
Tail recursive function which returns the largest `k : ℕ` such that `p ^ k ∣ n` for any `p : ℕ`.
`padicValNat_eq_maxPowDiv` allows the code generator to use this definition for `padicValNat`
-/
def maxPowDiv (p n : ℕ) : ℕ | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
⊢ go k p n = if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1) p (n / p) else k | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | dsimp [go, go._unary] | theorem go_eq {k p n : ℕ} :
go k p n = if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k+1) p (n / p) else k := by
| Mathlib.Data.Nat.MaxPowDiv.45_0.2jhZnOeTQwLUwDJ | theorem go_eq {k p n : ℕ} :
go k p n = if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k+1) p (n / p) else k | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
⊢ WellFounded.fix go._unary.proof_1
(fun _x a =>
if h : 1 < _x.snd.fst ∧ 0 < _x.snd.snd ∧ _x.snd.snd % _x.snd.fst = 0 then
a { fst := _x.fst + 1, snd := { fst := _x.snd.fst, snd := _x.snd.snd / _x.snd.fst } }
(_ : _x.snd.snd / _x.snd.fst < _x.snd.snd)
else _x.fst)
... | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [WellFounded.fix_eq] | theorem go_eq {k p n : ℕ} :
go k p n = if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k+1) p (n / p) else k := by
dsimp [go, go._unary]
| Mathlib.Data.Nat.MaxPowDiv.45_0.2jhZnOeTQwLUwDJ | theorem go_eq {k p n : ℕ} :
go k p n = if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k+1) p (n / p) else k | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
⊢ (if h :
1 < { fst := k, snd := { fst := p, snd := n } }.snd.fst ∧
0 < { fst := k, snd := { fst := p, snd := n } }.snd.snd ∧
{ fst := k, snd := { fst := p, snd := n } }.snd.snd % { fst := k, snd := { fst := p, snd := n } }.snd.fst =
0 then
(fun y x =>
... | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp | theorem go_eq {k p n : ℕ} :
go k p n = if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k+1) p (n / p) else k := by
dsimp [go, go._unary]
rw [WellFounded.fix_eq]
| Mathlib.Data.Nat.MaxPowDiv.45_0.2jhZnOeTQwLUwDJ | theorem go_eq {k p n : ℕ} :
go k p n = if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k+1) p (n / p) else k | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
⊢ go (k + 1) p n = go k p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [go_eq] | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
| Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
⊢ (if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1 + 1) p (n / p) else k + 1) = go k p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | conv_rhs => rw [go_eq] | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
| Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
| go k p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [go_eq] | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
| go k p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [go_eq] | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
| go k p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [go_eq] | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
⊢ (if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1 + 1) p (n / p) else k + 1) =
(if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1) p (n / p) else k) + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0) | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
| Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case pos
k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ (if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1 + 1) p (n / p) else k + 1) =
(if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1) p (n / p) else k) + 1
case neg
k p n : ℕ
h : ¬(1 < p ∧ 0 < n ∧ n % p = 0)
⊢ (if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1 + 1) p (n / p) else k + ... | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | swap | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0); | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case neg
k p n : ℕ
h : ¬(1 < p ∧ 0 < n ∧ n % p = 0)
⊢ (if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1 + 1) p (n / p) else k + 1) =
(if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1) p (n / p) else k) + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp only [if_neg h] | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0); swap
· | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case pos
k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ (if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1 + 1) p (n / p) else k + 1) =
(if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1) p (n / p) else k) + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | have : n / p < n := by apply Nat.div_lt_self <;> aesop | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0); swap
· simp only [if_neg h]
· | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ n / p < n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | apply Nat.div_lt_self | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0); swap
· simp only [if_neg h]
· have : n / p < n := by | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case hLtN
k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ 0 < n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | aesop | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0); swap
· simp only [if_neg h]
· have : n / p < n := by apply Nat.div_lt_self <;> | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case hLtK
k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ 1 < p | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | aesop | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0); swap
· simp only [if_neg h]
· have : n / p < n := by apply Nat.div_lt_self <;> | Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case pos
k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
⊢ (if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1 + 1) p (n / p) else k + 1) =
(if 1 < p ∧ 0 < n ∧ n % p = 0 then go (k + 1) p (n / p) else k) + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp only [if_pos h] | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0); swap
· simp only [if_neg h]
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
| Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case pos
k p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
⊢ go (k + 1 + 1) p (n / p) = go (k + 1) p (n / p) + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | apply go_succ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by
rw [go_eq]
conv_rhs => rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0); swap
· simp only [if_neg h]
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
simp only [if_pos h]
| Mathlib.Data.Nat.MaxPowDiv.51_0.2jhZnOeTQwLUwDJ | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 | Mathlib_Data_Nat_MaxPowDiv |
n : ℕ
⊢ maxPowDiv 0 n = 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | dsimp [maxPowDiv] | @[simp]
theorem zero_base {n : ℕ} : maxPowDiv 0 n = 0 := by
| Mathlib.Data.Nat.MaxPowDiv.60_0.2jhZnOeTQwLUwDJ | @[simp]
theorem zero_base {n : ℕ} : maxPowDiv 0 n = 0 | Mathlib_Data_Nat_MaxPowDiv |
n : ℕ
⊢ go 0 0 n = 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [maxPowDiv.go_eq] | @[simp]
theorem zero_base {n : ℕ} : maxPowDiv 0 n = 0 := by
dsimp [maxPowDiv]
| Mathlib.Data.Nat.MaxPowDiv.60_0.2jhZnOeTQwLUwDJ | @[simp]
theorem zero_base {n : ℕ} : maxPowDiv 0 n = 0 | Mathlib_Data_Nat_MaxPowDiv |
n : ℕ
⊢ (if 1 < 0 ∧ 0 < n ∧ n % 0 = 0 then go (0 + 1) 0 (n / 0) else 0) = 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp | @[simp]
theorem zero_base {n : ℕ} : maxPowDiv 0 n = 0 := by
dsimp [maxPowDiv]
rw [maxPowDiv.go_eq]
| Mathlib.Data.Nat.MaxPowDiv.60_0.2jhZnOeTQwLUwDJ | @[simp]
theorem zero_base {n : ℕ} : maxPowDiv 0 n = 0 | Mathlib_Data_Nat_MaxPowDiv |
p : ℕ
⊢ maxPowDiv p 0 = 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | dsimp [maxPowDiv] | @[simp]
theorem zero {p : ℕ} : maxPowDiv p 0 = 0 := by
| Mathlib.Data.Nat.MaxPowDiv.66_0.2jhZnOeTQwLUwDJ | @[simp]
theorem zero {p : ℕ} : maxPowDiv p 0 = 0 | Mathlib_Data_Nat_MaxPowDiv |
p : ℕ
⊢ go 0 p 0 = 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [maxPowDiv.go_eq] | @[simp]
theorem zero {p : ℕ} : maxPowDiv p 0 = 0 := by
dsimp [maxPowDiv]
| Mathlib.Data.Nat.MaxPowDiv.66_0.2jhZnOeTQwLUwDJ | @[simp]
theorem zero {p : ℕ} : maxPowDiv p 0 = 0 | Mathlib_Data_Nat_MaxPowDiv |
p : ℕ
⊢ (if 1 < p ∧ 0 < 0 ∧ 0 % p = 0 then go (0 + 1) p (0 / p) else 0) = 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp | @[simp]
theorem zero {p : ℕ} : maxPowDiv p 0 = 0 := by
dsimp [maxPowDiv]
rw [maxPowDiv.go_eq]
| Mathlib.Data.Nat.MaxPowDiv.66_0.2jhZnOeTQwLUwDJ | @[simp]
theorem zero {p : ℕ} : maxPowDiv p 0 = 0 | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
hp : 1 < p
hn : 0 < n
⊢ maxPowDiv p (p * n) = maxPowDiv p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | have : 0 < p := lt_trans (b := 1) (by simp) hp | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by
| Mathlib.Data.Nat.MaxPowDiv.72_0.2jhZnOeTQwLUwDJ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
hp : 1 < p
hn : 0 < n
⊢ 0 < 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by
have : 0 < p := lt_trans (b := 1) (by | Mathlib.Data.Nat.MaxPowDiv.72_0.2jhZnOeTQwLUwDJ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
hp : 1 < p
hn : 0 < n
this : 0 < p
⊢ maxPowDiv p (p * n) = maxPowDiv p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | dsimp [maxPowDiv] | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by
have : 0 < p := lt_trans (b := 1) (by simp) hp
| Mathlib.Data.Nat.MaxPowDiv.72_0.2jhZnOeTQwLUwDJ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
hp : 1 < p
hn : 0 < n
this : 0 < p
⊢ go 0 p (p * n) = go 0 p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [maxPowDiv.go_eq, if_pos, mul_div_right _ this] | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by
have : 0 < p := lt_trans (b := 1) (by simp) hp
dsimp [maxPowDiv]
| Mathlib.Data.Nat.MaxPowDiv.72_0.2jhZnOeTQwLUwDJ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
hp : 1 < p
hn : 0 < n
this : 0 < p
⊢ go (0 + 1) p n = go 0 p n + 1 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | apply go_succ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by
have : 0 < p := lt_trans (b := 1) (by simp) hp
dsimp [maxPowDiv]
rw [maxPowDiv.go_eq, if_pos, mul_div_right _ this]
· | Mathlib.Data.Nat.MaxPowDiv.72_0.2jhZnOeTQwLUwDJ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case hc
p n : ℕ
hp : 1 < p
hn : 0 < n
this : 0 < p
⊢ 1 < p ∧ 0 < p * n ∧ p * n % p = 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | refine ⟨hp, ?_, by simp⟩ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by
have : 0 < p := lt_trans (b := 1) (by simp) hp
dsimp [maxPowDiv]
rw [maxPowDiv.go_eq, if_pos, mul_div_right _ this]
· apply go_succ
· | Mathlib.Data.Nat.MaxPowDiv.72_0.2jhZnOeTQwLUwDJ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
hp : 1 < p
hn : 0 < n
this : 0 < p
⊢ p * n % p = 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by
have : 0 < p := lt_trans (b := 1) (by simp) hp
dsimp [maxPowDiv]
rw [maxPowDiv.go_eq, if_pos, mul_div_right _ this]
· apply go_succ
· refine ⟨hp, ?_, by | Mathlib.Data.Nat.MaxPowDiv.72_0.2jhZnOeTQwLUwDJ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 | Mathlib_Data_Nat_MaxPowDiv |
case hc
p n : ℕ
hp : 1 < p
hn : 0 < n
this : 0 < p
⊢ 0 < p * n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | apply Nat.mul_pos this hn | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by
have : 0 < p := lt_trans (b := 1) (by simp) hp
dsimp [maxPowDiv]
rw [maxPowDiv.go_eq, if_pos, mul_div_right _ this]
· apply go_succ
· refine ⟨hp, ?_, by simp⟩
| Mathlib.Data.Nat.MaxPowDiv.72_0.2jhZnOeTQwLUwDJ | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 | Mathlib_Data_Nat_MaxPowDiv |
p n exp : ℕ
hp : 1 < p
hn : 0 < n
⊢ maxPowDiv p (p ^ exp * n) = maxPowDiv p n + exp | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | match exp with
| 0 => simp
| e + 1 =>
rw [pow_succ, mul_assoc, mul_comm, mul_assoc, base_mul_eq_succ hp, mul_comm, base_pow_mul hp hn]
· ac_rfl
· apply Nat.mul_pos hn <| pow_pos (pos_of_gt hp) e | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp := by
| Mathlib.Data.Nat.MaxPowDiv.81_0.2jhZnOeTQwLUwDJ | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp | Mathlib_Data_Nat_MaxPowDiv |
p n exp : ℕ
hp : 1 < p
hn : 0 < n
⊢ maxPowDiv p (p ^ 0 * n) = maxPowDiv p n + 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp := by
match exp with
| 0 => | Mathlib.Data.Nat.MaxPowDiv.81_0.2jhZnOeTQwLUwDJ | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp | Mathlib_Data_Nat_MaxPowDiv |
p n exp : ℕ
hp : 1 < p
hn : 0 < n
e : ℕ
⊢ maxPowDiv p (p ^ (e + 1) * n) = maxPowDiv p n + (e + 1) | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [pow_succ, mul_assoc, mul_comm, mul_assoc, base_mul_eq_succ hp, mul_comm, base_pow_mul hp hn] | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp := by
match exp with
| 0 => simp
| e + 1 =>
| Mathlib.Data.Nat.MaxPowDiv.81_0.2jhZnOeTQwLUwDJ | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp | Mathlib_Data_Nat_MaxPowDiv |
p n exp : ℕ
hp : 1 < p
hn : 0 < n
e : ℕ
⊢ maxPowDiv p n + e + 1 = maxPowDiv p n + (e + 1) | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | ac_rfl | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp := by
match exp with
| 0 => simp
| e + 1 =>
rw [pow_succ, mul_assoc, mul_comm, mul_assoc, base_mul_eq_succ hp, mul_comm, base_pow_mul hp hn]
· | Mathlib.Data.Nat.MaxPowDiv.81_0.2jhZnOeTQwLUwDJ | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp | Mathlib_Data_Nat_MaxPowDiv |
p n exp : ℕ
hp : 1 < p
hn : 0 < n
e : ℕ
⊢ 0 < n * p ^ e | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | apply Nat.mul_pos hn <| pow_pos (pos_of_gt hp) e | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp := by
match exp with
| 0 => simp
| e + 1 =>
rw [pow_succ, mul_assoc, mul_comm, mul_assoc, base_mul_eq_succ hp, mul_comm, base_pow_mul hp hn]
· ac_rfl
· | Mathlib.Data.Nat.MaxPowDiv.81_0.2jhZnOeTQwLUwDJ | theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p ^ exp * n) = p.maxPowDiv n + exp | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
⊢ p ^ maxPowDiv p n ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | dsimp [maxPowDiv] | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
| Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
⊢ p ^ go 0 p n ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [go_eq] | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
| Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
⊢ (p ^ if 1 < p ∧ 0 < n ∧ n % p = 0 then go (0 + 1) p (n / p) else 0) ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0) | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
| Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case pos
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ (p ^ if 1 < p ∧ 0 < n ∧ n % p = 0 then go (0 + 1) p (n / p) else 0) ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | have : n / p < n := by apply Nat.div_lt_self <;> aesop | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ n / p < n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | apply Nat.div_lt_self | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case hLtN
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ 0 < n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | aesop | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case hLtK
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ 1 < p | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | aesop | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case pos
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
⊢ (p ^ if 1 < p ∧ 0 < n ∧ n % p = 0 then go (0 + 1) p (n / p) else 0) ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [if_pos h] | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
| Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case pos
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
⊢ p ^ go (0 + 1) p (n / p) ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | have ⟨c,hc⟩ := pow_dvd p (n / p) | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
| Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case pos
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
c : ℕ
hc : n / p = p ^ maxPowDiv p (n / p) * c
⊢ p ^ go (0 + 1) p (n / p) ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [go_succ, pow_succ] | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
| Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case pos
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
c : ℕ
hc : n / p = p ^ maxPowDiv p (n / p) * c
⊢ p ^ go 0 p (n / p) * p ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | nth_rw 2 [← mod_add_div' n p] | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
rw [go_succ, pow_succ]
| Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case pos
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
c : ℕ
hc : n / p = p ^ maxPowDiv p (n / p) * c
⊢ p ^ go 0 p (n / p) * p ∣ n % p + n / p * p | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [h.right.right, zero_add] | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
rw [go_succ, pow_succ]
nth_rw 2 [← mod_add_div' n p]
| Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case pos
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
c : ℕ
hc : n / p = p ^ maxPowDiv p (n / p) * c
⊢ p ^ go 0 p (n / p) * p ∣ n / p * p | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | exact ⟨c,by nth_rw 1 [hc]; ac_rfl⟩ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
rw [go_succ, pow_succ]
nth_rw 2 [← mod_add_div' n p]
rw [h.r... | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
c : ℕ
hc : n / p = p ^ maxPowDiv p (n / p) * c
⊢ n / p * p = p ^ go 0 p (n / p) * p * c | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | nth_rw 1 [hc] | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
rw [go_succ, pow_succ]
nth_rw 2 [← mod_add_div' n p]
rw [h.r... | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
this : n / p < n
c : ℕ
hc : n / p = p ^ maxPowDiv p (n / p) * c
⊢ p ^ maxPowDiv p (n / p) * c * p = p ^ go 0 p (n / p) * p * c | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | ac_rfl | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
rw [go_succ, pow_succ]
nth_rw 2 [← mod_add_div' n p]
rw [h.r... | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case neg
p n : ℕ
h : ¬(1 < p ∧ 0 < n ∧ n % p = 0)
⊢ (p ^ if 1 < p ∧ 0 < n ∧ n % p = 0 then go (0 + 1) p (n / p) else 0) ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [if_neg h] | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
rw [go_succ, pow_succ]
nth_rw 2 [← mod_add_div' n p]
rw [h.r... | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
case neg
p n : ℕ
h : ¬(1 < p ∧ 0 < n ∧ n % p = 0)
⊢ p ^ 0 ∣ n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by
dsimp [maxPowDiv]
rw [go_eq]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
rw [go_succ, pow_succ]
nth_rw 2 [← mod_add_div' n p]
rw [h.r... | Mathlib.Data.Nat.MaxPowDiv.90_0.2jhZnOeTQwLUwDJ | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | Mathlib_Data_Nat_MaxPowDiv |
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
⊢ pow ≤ maxPowDiv p n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | have ⟨c, hc⟩ := h | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n := by
| Mathlib.Data.Nat.MaxPowDiv.104_0.2jhZnOeTQwLUwDJ | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n | Mathlib_Data_Nat_MaxPowDiv |
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
⊢ pow ≤ maxPowDiv p n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | have : 0 < c := by
apply Nat.pos_of_ne_zero
intro h'
rw [h',mul_zero] at hc
exact not_eq_zero_of_lt hn hc | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n := by
have ⟨c, hc⟩ := h
| Mathlib.Data.Nat.MaxPowDiv.104_0.2jhZnOeTQwLUwDJ | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n | Mathlib_Data_Nat_MaxPowDiv |
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
⊢ 0 < c | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | apply Nat.pos_of_ne_zero | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n := by
have ⟨c, hc⟩ := h
have : 0 < c := by
| Mathlib.Data.Nat.MaxPowDiv.104_0.2jhZnOeTQwLUwDJ | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n | Mathlib_Data_Nat_MaxPowDiv |
case a
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
⊢ c ≠ 0 | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | intro h' | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n := by
have ⟨c, hc⟩ := h
have : 0 < c := by
apply Nat.pos_of_ne_zero
| Mathlib.Data.Nat.MaxPowDiv.104_0.2jhZnOeTQwLUwDJ | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n | Mathlib_Data_Nat_MaxPowDiv |
case a
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
h' : c = 0
⊢ False | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | rw [h',mul_zero] at hc | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n := by
have ⟨c, hc⟩ := h
have : 0 < c := by
apply Nat.pos_of_ne_zero
intro h'
| Mathlib.Data.Nat.MaxPowDiv.104_0.2jhZnOeTQwLUwDJ | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n | Mathlib_Data_Nat_MaxPowDiv |
case a
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = 0
h' : c = 0
⊢ False | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | exact not_eq_zero_of_lt hn hc | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n := by
have ⟨c, hc⟩ := h
have : 0 < c := by
apply Nat.pos_of_ne_zero
intro h'
rw [h',mul_zero] at hc
| Mathlib.Data.Nat.MaxPowDiv.104_0.2jhZnOeTQwLUwDJ | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n | Mathlib_Data_Nat_MaxPowDiv |
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
this : 0 < c
⊢ pow ≤ maxPowDiv p n | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Data.Nat.Pow
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural number dividing another
Here we introduce `p.maxPow... | simp [hc, base_pow_mul hp this] | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n := by
have ⟨c, hc⟩ := h
have : 0 < c := by
apply Nat.pos_of_ne_zero
intro h'
rw [h',mul_zero] at hc
exact not_eq_zero_of_lt hn hc
| Mathlib.Data.Nat.MaxPowDiv.104_0.2jhZnOeTQwLUwDJ | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n | Mathlib_Data_Nat_MaxPowDiv |
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
⊢ ∀ (a : α), IsCountablyGenerated (nhds a) | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "lea... | rw [nhds_discrete] | instance (priority := 100) DiscreteTopology.firstCountableTopology [DiscreteTopology α] :
FirstCountableTopology α where
nhds_generated_countable := by | Mathlib.Topology.Instances.Discrete.31_0.ZrbbXQYldP9CdIh | instance (priority | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
⊢ ∀ (a : α), IsCountablyGenerated (pure a) | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "lea... | exact isCountablyGenerated_pure | instance (priority := 100) DiscreteTopology.firstCountableTopology [DiscreteTopology α] :
FirstCountableTopology α where
nhds_generated_countable := by rw [nhds_discrete]; | Mathlib.Topology.Instances.Discrete.31_0.ZrbbXQYldP9CdIh | instance (priority | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝¹ : TopologicalSpace α
hd : DiscreteTopology α
inst✝ : Encodable α
i : α
⊢ instTopologicalSpaceSubtype = generateFrom {univ} | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "lea... | simp only [eq_iff_true_of_subsingleton] | instance (priority := 100) DiscreteTopology.secondCountableTopology_of_encodable
[hd : DiscreteTopology α] [Encodable α] : SecondCountableTopology α :=
haveI : ∀ i : α, SecondCountableTopology (↥({i} : Set α)) := fun i =>
{ is_open_generated_countable :=
⟨{univ}, countable_singleton _, by | Mathlib.Topology.Instances.Discrete.36_0.ZrbbXQYldP9CdIh | instance (priority | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : PartialOrder α
inst✝³ : PredOrder α
inst✝² : SuccOrder α
inst✝¹ : NoMinOrder α
inst✝ : NoMaxOrder α
⊢ ⊥ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "lea... | refine' (eq_bot_of_singletons_open fun a => _).symm | theorem bot_topologicalSpace_eq_generateFrom_of_pred_succOrder [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] :
(⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
| Mathlib.Topology.Instances.Discrete.45_0.ZrbbXQYldP9CdIh | theorem bot_topologicalSpace_eq_generateFrom_of_pred_succOrder [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] :
(⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : PartialOrder α
inst✝³ : PredOrder α
inst✝² : SuccOrder α
inst✝¹ : NoMinOrder α
inst✝ : NoMaxOrder α
a : α
⊢ IsOpen {a} | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "lea... | have h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a) := by
suffices h_singleton_eq_inter' : {a} = Iic a ∩ Ici a
· rw [h_singleton_eq_inter', ← Ioi_pred, ← Iio_succ]
rw [inter_comm, Ici_inter_Iic, Icc_self a] | theorem bot_topologicalSpace_eq_generateFrom_of_pred_succOrder [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] :
(⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
| Mathlib.Topology.Instances.Discrete.45_0.ZrbbXQYldP9CdIh | theorem bot_topologicalSpace_eq_generateFrom_of_pred_succOrder [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] :
(⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
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