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 |
|---|---|---|---|---|---|---|
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
⊢ {a} = Iio (succ a) ∩ Ioi (pred 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... | suffices h_singleton_eq_inter' : {a} = Iic a ∩ Ici 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
have h_singleton_eq_inter : {a} = I... | 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
h_singleton_eq_inter' : {a} = Iic a ∩ Ici a
⊢ {a} = Iio (succ a) ∩ Ioi (pred 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 [h_singleton_eq_inter', ← Ioi_pred, ← Iio_succ] | 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
have h_singleton_eq_inter : {a} = I... | 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 |
case h_singleton_eq_inter'
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
⊢ {a} = Iic a ∩ Ici 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 [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
have h_singleton_eq_inter : {a} = I... | 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
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a)
⊢ 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... | rw [h_singleton_eq_inter] | 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
have h_singleton_eq_inter : {a} = I... | 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
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a)
⊢ IsOpen (Iio (succ a) ∩ Ioi (pred 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... | apply @IsOpen.inter _ _ _ (generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio 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
have h_singleton_eq_inter : {a} = I... | 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 |
case h₁
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
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a)
⊢ IsOpen (Iio (succ 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 isOpen_generateFrom_of_mem ⟨succ a, Or.inr rfl⟩ | 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
have h_singleton_eq_inter : {a} = I... | 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 |
case h₂
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
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a)
⊢ IsOpen (Ioi (pred 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 isOpen_generateFrom_of_mem ⟨pred a, Or.inl rfl⟩ | 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
have h_singleton_eq_inter : {a} = I... | 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
⊢ DiscreteTopology α ↔ OrderTopology α | /-
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' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩ | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α := by
| Mathlib.Topology.Instances.Discrete.60_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
case refine'_1
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
h : DiscreteTopology α
⊢ inst✝⁵ = 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... | rw [h.eq_bot] | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩
· | Mathlib.Topology.Instances.Discrete.60_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
case refine'_1
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
h : DiscreteTopology 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... | exact bot_topologicalSpace_eq_generateFrom_of_pred_succOrder | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩
· rw [h.eq_bot]
| Mathlib.Topology.Instances.Discrete.60_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
case refine'_2
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
h : OrderTopology α
⊢ inst✝⁵ = ⊥ | /-
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 [h.topology_eq_generate_intervals] | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩
· rw [h.eq_bot]
exact bot_topologicalSpace_eq_generateFrom_of_pred_succOrder
· | Mathlib.Topology.Instances.Discrete.60_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
case refine'_2
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
h : OrderTopology 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... | exact bot_topologicalSpace_eq_generateFrom_of_pred_succOrder.symm | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩
· rw [h.eq_bot]
exact bot_topologicalSpace_eq_generateFrom_of_pred_succOrder
· rw [h.topolog... | Mathlib.Topology.Instances.Discrete.60_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
[SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder 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 LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
| Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder 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} = Iic a ∩ Ici a := by rw [inter_comm, Ici_inter_Iic, Icc_self a] | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder 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.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
⊢ {a} = Iic a ∩ Ici 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 [inter_comm, Ici_inter_Iic, Icc_self a] | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iic a ∩ Ici a
⊢ 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... | by_cases ha_top : IsTop a | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iic a ∩ Ici a
ha_top : IsTop a
⊢ 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... | rw [ha_top.Iic_eq, inter_comm, inter_univ] at h_singleton_eq_inter | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Ici a
ha_top : IsTop a
⊢ 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... | by_cases ha_bot : IsBot a | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Ici a
ha_top : IsTop a
ha_bot : IsBot a
⊢ 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... | rw [ha_bot.Ici_eq] at h_singleton_eq_inter | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = univ
ha_top : IsTop a
ha_bot : IsBot a
⊢ 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... | rw [h_singleton_eq_inter] | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = univ
ha_top : IsTop a
ha_bot : IsBot a
⊢ IsOpen 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... | apply @isOpen_univ _ (generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }) | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Ici a
ha_top : IsTop a
ha_bot : ¬IsBot a
⊢ 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... | rw [isBot_iff_isMin] at ha_bot | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Ici a
ha_top : IsTop a
ha_bot : ¬IsMin a
⊢ 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... | rw [← Ioi_pred_of_not_isMin ha_bot] at h_singleton_eq_inter | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Ioi (pred a)
ha_top : IsTop a
ha_bot : ¬IsMin a
⊢ 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... | rw [h_singleton_eq_inter] | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Ioi (pred a)
ha_top : IsTop a
ha_bot : ¬IsMin a
⊢ IsOpen (Ioi (pred 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 isOpen_generateFrom_of_mem ⟨pred a, Or.inl rfl⟩ | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iic a ∩ Ici a
ha_top : ¬IsTop a
⊢ 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... | rw [isTop_iff_isMax] at ha_top | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iic a ∩ Ici a
ha_top : ¬IsMax a
⊢ 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... | rw [← Iio_succ_of_not_isMax ha_top] at h_singleton_eq_inter | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ici a
ha_top : ¬IsMax a
⊢ 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... | by_cases ha_bot : IsBot a | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ici a
ha_top : ¬IsMax a
ha_bot : IsBot a
⊢ 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... | rw [ha_bot.Ici_eq, inter_univ] at h_singleton_eq_inter | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a)
ha_top : ¬IsMax a
ha_bot : IsBot a
⊢ 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... | rw [h_singleton_eq_inter] | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a)
ha_top : ¬IsMax a
ha_bot : IsBot a
⊢ IsOpen (Iio (succ 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 isOpen_generateFrom_of_mem ⟨succ a, Or.inr rfl⟩ | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ici a
ha_top : ¬IsMax a
ha_bot : ¬IsBot a
⊢ 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... | rw [isBot_iff_isMin] at ha_bot | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ici a
ha_top : ¬IsMax a
ha_bot : ¬IsMin a
⊢ 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... | rw [← Ioi_pred_of_not_isMin ha_bot] at h_singleton_eq_inter | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a)
ha_top : ¬IsMax a
ha_bot : ¬IsMin a
⊢ 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... | rw [h_singleton_eq_inter] | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a)
ha_top : ¬IsMax a
ha_bot : ¬IsMin a
⊢ IsOpen (Iio (succ a) ∩ Ioi (pred 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... | apply @IsOpen.inter _ _ _ (generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a }) | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg.h₁
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a)
ha_top : ¬IsMax a
ha_bot : ¬IsMin a
⊢ IsOpen (Iio (succ 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 isOpen_generateFrom_of_mem ⟨succ a, Or.inr rfl⟩ | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
case neg.h₂
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
h_singleton_eq_inter : {a} = Iio (succ a) ∩ Ioi (pred a)
ha_top : ¬IsMax a
ha_bot : ¬IsMin a
⊢ IsOpen (Ioi (pred 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 isOpen_generateFrom_of_mem ⟨pred a, Or.inl rfl⟩ | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
refine' (eq_bot_of_singletons_open fun a => _).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_in... | Mathlib.Topology.Instances.Discrete.74_0.ZrbbXQYldP9CdIh | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
⊢ DiscreteTopology α ↔ OrderTopology α | /-
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' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩ | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by
| Mathlib.Topology.Instances.Discrete.104_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
case refine'_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
h : DiscreteTopology α
⊢ inst✝³ = 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... | rw [h.eq_bot] | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩
· | Mathlib.Topology.Instances.Discrete.104_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
case refine'_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
h : DiscreteTopology 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... | exact LinearOrder.bot_topologicalSpace_eq_generateFrom | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩
· rw [h.eq_bot]
| Mathlib.Topology.Instances.Discrete.104_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
case refine'_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
h : OrderTopology α
⊢ inst✝³ = ⊥ | /-
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 [h.topology_eq_generate_intervals] | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩
· rw [h.eq_bot]
exact LinearOrder.bot_topologicalSpace_eq_generateFrom
· | Mathlib.Topology.Instances.Discrete.104_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
case refine'_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
h : OrderTopology 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... | exact LinearOrder.bot_topologicalSpace_eq_generateFrom.symm | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by
refine' ⟨fun h => ⟨_⟩, fun h => ⟨_⟩⟩
· rw [h.eq_bot]
exact LinearOrder.bot_topologicalSpace_eq_generateFrom
· rw [h.topology_eq_generate_intervals]
| Mathlib.Topology.Instances.Discrete.104_0.ZrbbXQYldP9CdIh | theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
[SuccOrder α] : DiscreteTopology α ↔ OrderTopology α | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
⊢ MetrizableSpace α | /-
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... | obtain rfl := DiscreteTopology.eq_bot (α := α) | instance (priority := 100) DiscreteTopology.metrizableSpace [DiscreteTopology α] :
MetrizableSpace α := by
| Mathlib.Topology.Instances.Discrete.118_0.ZrbbXQYldP9CdIh | instance (priority | Mathlib_Topology_Instances_Discrete |
α : Type u_1
inst✝ : DiscreteTopology α
⊢ MetrizableSpace α | /-
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 @UniformSpace.metrizableSpace α ⊥ (isCountablyGenerated_principal _) _ | instance (priority := 100) DiscreteTopology.metrizableSpace [DiscreteTopology α] :
MetrizableSpace α := by
obtain rfl := DiscreteTopology.eq_bot (α := α)
| Mathlib.Topology.Instances.Discrete.118_0.ZrbbXQYldP9CdIh | instance (priority | Mathlib_Topology_Instances_Discrete |
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n : ℕ
⊢ (coeff R n) (derivativeFun f) = (coeff R (n + 1)) f * (↑n + 1) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [derivativeFun, coeff_mk] | theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
| Mathlib.RingTheory.PowerSeries.Derivative.40_0.PcMDeT7Qkkrj2IE | theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
f : R[X]
⊢ derivativeFun ↑f = ↑(derivative f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | ext | theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
| Mathlib.RingTheory.PowerSeries.Derivative.44_0.PcMDeT7Qkkrj2IE | theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
f : R[X]
n✝ : ℕ
⊢ (coeff R n✝) (derivativeFun ↑f) = (coeff R n✝) ↑(derivative f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative] | theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
ext
| Mathlib.RingTheory.PowerSeries.Derivative.44_0.PcMDeT7Qkkrj2IE | theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
f g : R⟦X⟧
⊢ derivativeFun (f + g) = derivativeFun f + derivativeFun g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | ext | theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
| Mathlib.RingTheory.PowerSeries.Derivative.48_0.PcMDeT7Qkkrj2IE | theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
f g : R⟦X⟧
n✝ : ℕ
⊢ (coeff R n✝) (derivativeFun (f + g)) = (coeff R n✝) (derivativeFun f + derivativeFun g) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul] | theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
ext
| Mathlib.RingTheory.PowerSeries.Derivative.48_0.PcMDeT7Qkkrj2IE | theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
r : R
⊢ derivativeFun ((C R) r) = 0 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | ext n | theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by
| Mathlib.RingTheory.PowerSeries.Derivative.54_0.PcMDeT7Qkkrj2IE | theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
r : R
n : ℕ
⊢ (coeff R n) (derivativeFun ((C R) r)) = (coeff R n) 0 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_derivativeFun, coeff_succ_C, zero_mul, map_zero] | theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by
ext n
| Mathlib.RingTheory.PowerSeries.Derivative.54_0.PcMDeT7Qkkrj2IE | theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n : ℕ
⊢ trunc n (derivativeFun f) = derivative (trunc (n + 1) f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | ext d | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
| Mathlib.RingTheory.PowerSeries.Derivative.58_0.PcMDeT7Qkkrj2IE | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) | Mathlib_RingTheory_PowerSeries_Derivative |
case a
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
⊢ Polynomial.coeff (trunc n (derivativeFun f)) d = Polynomial.coeff (derivative (trunc (n + 1) f)) d | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_trunc] | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
| Mathlib.RingTheory.PowerSeries.Derivative.58_0.PcMDeT7Qkkrj2IE | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) | Mathlib_RingTheory_PowerSeries_Derivative |
case a
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
⊢ (if d < n then (coeff R d) (derivativeFun f) else 0) = Polynomial.coeff (derivative (trunc (n + 1) f)) d | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | split_ifs with h | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
| Mathlib.RingTheory.PowerSeries.Derivative.58_0.PcMDeT7Qkkrj2IE | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) | Mathlib_RingTheory_PowerSeries_Derivative |
case pos
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
h : d < n
⊢ (coeff R d) (derivativeFun f) = Polynomial.coeff (derivative (trunc (n + 1) f)) d | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | have : d + 1 < n + 1 := succ_lt_succ_iff.2 h | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· | Mathlib.RingTheory.PowerSeries.Derivative.58_0.PcMDeT7Qkkrj2IE | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) | Mathlib_RingTheory_PowerSeries_Derivative |
case pos
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
h : d < n
this : d + 1 < n + 1
⊢ (coeff R d) (derivativeFun f) = Polynomial.coeff (derivative (trunc (n + 1) f)) d | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this] | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
| Mathlib.RingTheory.PowerSeries.Derivative.58_0.PcMDeT7Qkkrj2IE | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) | Mathlib_RingTheory_PowerSeries_Derivative |
case neg
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
h : ¬d < n
⊢ 0 = Polynomial.coeff (derivative (trunc (n + 1) f)) d | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | have : ¬d + 1 < n + 1 := by rwa [succ_lt_succ_iff] | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· | Mathlib.RingTheory.PowerSeries.Derivative.58_0.PcMDeT7Qkkrj2IE | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
h : ¬d < n
⊢ ¬d + 1 < n + 1 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rwa [succ_lt_succ_iff] | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by | Mathlib.RingTheory.PowerSeries.Derivative.58_0.PcMDeT7Qkkrj2IE | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) | Mathlib_RingTheory_PowerSeries_Derivative |
case neg
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
h : ¬d < n
this : ¬d + 1 < n + 1
⊢ 0 = Polynomial.coeff (derivative (trunc (n + 1) f)) d | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_derivative, coeff_trunc, if_neg this, zero_mul] | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by rwa [s... | Mathlib.RingTheory.PowerSeries.Derivative.58_0.PcMDeT7Qkkrj2IE | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
f g : R[X]
⊢ derivativeFun (↑f * ↑g) = ↑f * ↑(derivative g) + ↑g * ↑(derivative f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [← coe_mul, derivativeFun_coe, derivative_mul,
add_comm, mul_comm _ g, ← coe_mul, ← coe_mul, Polynomial.coe_add] | private theorem derivativeFun_coe_mul_coe (f g : R[X]) : derivativeFun (f * g : R⟦X⟧) =
f * derivative g + g * derivative f := by
| Mathlib.RingTheory.PowerSeries.Derivative.69_0.PcMDeT7Qkkrj2IE | private theorem derivativeFun_coe_mul_coe (f g : R[X]) : derivativeFun (f * g : R⟦X⟧) =
f * derivative g + g * derivative f | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
f g : R⟦X⟧
⊢ derivativeFun (f * g) = f • derivativeFun g + g • derivativeFun f | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | ext n | /-- **Leibniz rule for formal power series**.-/
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun := by
| Mathlib.RingTheory.PowerSeries.Derivative.74_0.PcMDeT7Qkkrj2IE | /-- **Leibniz rule for formal power series**.-/
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
f g : R⟦X⟧
n : ℕ
⊢ (coeff R n) (derivativeFun (f * g)) = (coeff R n) (f • derivativeFun g + g • derivativeFun f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | have h₁ : n < n + 1 := lt_succ_self n | /-- **Leibniz rule for formal power series**.-/
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun := by
ext n
| Mathlib.RingTheory.PowerSeries.Derivative.74_0.PcMDeT7Qkkrj2IE | /-- **Leibniz rule for formal power series**.-/
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
f g : R⟦X⟧
n : ℕ
h₁ : n < n + 1
⊢ (coeff R n) (derivativeFun (f * g)) = (coeff R n) (f • derivativeFun g + g • derivativeFun f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | have h₂ : n < n + 1 + 1 := Nat.lt_add_right _ h₁ | /-- **Leibniz rule for formal power series**.-/
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun := by
ext n
have h₁ : n < n + 1 := lt_succ_self n
| Mathlib.RingTheory.PowerSeries.Derivative.74_0.PcMDeT7Qkkrj2IE | /-- **Leibniz rule for formal power series**.-/
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
f g : R⟦X⟧
n : ℕ
h₁ : n < n + 1
h₂ : n < n + 1 + 1
⊢ (coeff R n) (derivativeFun (f * g)) = (coeff R n) (f • derivativeFun g + g • derivativeFun f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_derivativeFun, map_add, coeff_mul_eq_coeff_trunc_mul_trunc _ _ (lt_succ_self _),
smul_eq_mul, smul_eq_mul, coeff_mul_eq_coeff_trunc_mul_trunc₂ g f.derivativeFun h₂ h₁,
coeff_mul_eq_coeff_trunc_mul_trunc₂ f g.derivativeFun h₂ h₁, trunc_derivativeFun,
trunc_derivativeFun, ← map_add, ← derivativeFun_... | /-- **Leibniz rule for formal power series**.-/
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun := by
ext n
have h₁ : n < n + 1 := lt_succ_self n
have h₂ : n < n + 1 + 1 := Nat.lt_add_right _ h₁
| Mathlib.RingTheory.PowerSeries.Derivative.74_0.PcMDeT7Qkkrj2IE | /-- **Leibniz rule for formal power series**.-/
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
⊢ derivativeFun 1 = 0 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [← map_one (C R), derivativeFun_C (1 : R)] | theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 := by
| Mathlib.RingTheory.PowerSeries.Derivative.85_0.PcMDeT7Qkkrj2IE | theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
r : R
f : R⟦X⟧
⊢ derivativeFun (r • f) = r • derivativeFun f | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [smul_eq_C_mul, smul_eq_C_mul, derivativeFun_mul, derivativeFun_C, smul_zero, add_zero,
smul_eq_mul] | theorem derivativeFun_smul (r : R) (f : R⟦X⟧) : derivativeFun (r • f) = r • derivativeFun f := by
| Mathlib.RingTheory.PowerSeries.Derivative.88_0.PcMDeT7Qkkrj2IE | theorem derivativeFun_smul (r : R) (f : R⟦X⟧) : derivativeFun (r • f) = r • derivativeFun f | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
⊢ (d⁄dX R) X = 1 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | ext | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 := by
| Mathlib.RingTheory.PowerSeries.Derivative.112_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
n✝ : ℕ
⊢ (coeff R n✝) ((d⁄dX R) X) = (coeff R n✝) 1 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_derivative, coeff_one, coeff_X, boole_mul] | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 := by
ext
| Mathlib.RingTheory.PowerSeries.Derivative.112_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
n✝ : ℕ
⊢ (if n✝ + 1 = 1 then ↑n✝ + 1 else 0) = if n✝ = 0 then 1 else 0 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | simp_rw [add_left_eq_self] | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 := by
ext
rw [coeff_derivative, coeff_one, coeff_X, boole_mul]
| Mathlib.RingTheory.PowerSeries.Derivative.112_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommSemiring R
n✝ : ℕ
⊢ (if n✝ = 0 then ↑n✝ + 1 else 0) = if n✝ = 0 then 1 else 0 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | split_ifs with h | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 := by
ext
rw [coeff_derivative, coeff_one, coeff_X, boole_mul]
simp_rw [add_left_eq_self]
| Mathlib.RingTheory.PowerSeries.Derivative.112_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 | Mathlib_RingTheory_PowerSeries_Derivative |
case pos
R : Type u_1
inst✝ : CommSemiring R
n✝ : ℕ
h : n✝ = 0
⊢ ↑n✝ + 1 = 1 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [h, cast_zero, zero_add] | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 := by
ext
rw [coeff_derivative, coeff_one, coeff_X, boole_mul]
simp_rw [add_left_eq_self]
split_ifs with h
· | Mathlib.RingTheory.PowerSeries.Derivative.112_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 | Mathlib_RingTheory_PowerSeries_Derivative |
case neg
R : Type u_1
inst✝ : CommSemiring R
n✝ : ℕ
h : ¬n✝ = 0
⊢ 0 = 0 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rfl | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 := by
ext
rw [coeff_derivative, coeff_one, coeff_X, boole_mul]
simp_rw [add_left_eq_self]
split_ifs with h
· rw [h, cast_zero, zero_add]
· | Mathlib.RingTheory.PowerSeries.Derivative.112_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n : ℕ
⊢ trunc (n - 1) ((d⁄dX R) f) = Polynomial.derivative (trunc n f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | cases n with
| zero =>
simp
| succ n =>
rw [succ_sub_one, trunc_derivative] | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by
| Mathlib.RingTheory.PowerSeries.Derivative.124_0.PcMDeT7Qkkrj2IE | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n : ℕ
⊢ trunc (n - 1) ((d⁄dX R) f) = Polynomial.derivative (trunc n f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | cases n with
| zero =>
simp
| succ n =>
rw [succ_sub_one, trunc_derivative] | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by
| Mathlib.RingTheory.PowerSeries.Derivative.124_0.PcMDeT7Qkkrj2IE | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) | Mathlib_RingTheory_PowerSeries_Derivative |
case zero
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
⊢ trunc (Nat.zero - 1) ((d⁄dX R) f) = Polynomial.derivative (trunc Nat.zero f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | | zero =>
simp | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by
cases n with
| Mathlib.RingTheory.PowerSeries.Derivative.124_0.PcMDeT7Qkkrj2IE | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) | Mathlib_RingTheory_PowerSeries_Derivative |
case zero
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
⊢ trunc (Nat.zero - 1) ((d⁄dX R) f) = Polynomial.derivative (trunc Nat.zero f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | simp | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by
cases n with
| zero =>
| Mathlib.RingTheory.PowerSeries.Derivative.124_0.PcMDeT7Qkkrj2IE | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) | Mathlib_RingTheory_PowerSeries_Derivative |
case succ
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n : ℕ
⊢ trunc (succ n - 1) ((d⁄dX R) f) = Polynomial.derivative (trunc (succ n) f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | | succ n =>
rw [succ_sub_one, trunc_derivative] | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by
cases n with
| zero =>
simp
| Mathlib.RingTheory.PowerSeries.Derivative.124_0.PcMDeT7Qkkrj2IE | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) | Mathlib_RingTheory_PowerSeries_Derivative |
case succ
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n : ℕ
⊢ trunc (succ n - 1) ((d⁄dX R) f) = Polynomial.derivative (trunc (succ n) f) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [succ_sub_one, trunc_derivative] | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by
cases n with
| zero =>
simp
| succ n =>
| Mathlib.RingTheory.PowerSeries.Derivative.124_0.PcMDeT7Qkkrj2IE | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
⊢ f = g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | ext n | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
| Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
n : ℕ
⊢ (coeff R n) f = (coeff R n) g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | cases n with
| zero =>
rw [coeff_zero_eq_constantCoeff, hc]
| succ n =>
have equ : coeff R n (d⁄dX R f) = coeff R n (d⁄dX R g) := by rw [hD]
rwa [coeff_derivative, coeff_derivative, ← cast_succ, mul_comm, ← nsmul_eq_mul,
mul_comm, ← nsmul_eq_mul, smul_right_inj n.succ_ne_zero] at equ | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
ext n
| Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
n : ℕ
⊢ (coeff R n) f = (coeff R n) g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | cases n with
| zero =>
rw [coeff_zero_eq_constantCoeff, hc]
| succ n =>
have equ : coeff R n (d⁄dX R f) = coeff R n (d⁄dX R g) := by rw [hD]
rwa [coeff_derivative, coeff_derivative, ← cast_succ, mul_comm, ← nsmul_eq_mul,
mul_comm, ← nsmul_eq_mul, smul_right_inj n.succ_ne_zero] at equ | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
ext n
| Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
case h.zero
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
⊢ (coeff R Nat.zero) f = (coeff R Nat.zero) g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | | zero =>
rw [coeff_zero_eq_constantCoeff, hc] | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
ext n
cases n with
| Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
case h.zero
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
⊢ (coeff R Nat.zero) f = (coeff R Nat.zero) g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [coeff_zero_eq_constantCoeff, hc] | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
ext n
cases n with
| zero =>
| Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
case h.succ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
n : ℕ
⊢ (coeff R (succ n)) f = (coeff R (succ n)) g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | | succ n =>
have equ : coeff R n (d⁄dX R f) = coeff R n (d⁄dX R g) := by rw [hD]
rwa [coeff_derivative, coeff_derivative, ← cast_succ, mul_comm, ← nsmul_eq_mul,
mul_comm, ← nsmul_eq_mul, smul_right_inj n.succ_ne_zero] at equ | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
ext n
cases n with
| zero =>
rw [coeff_zero_eq_constantCoeff, h... | Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
case h.succ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
n : ℕ
⊢ (coeff R (succ n)) f = (coeff R (succ n)) g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | have equ : coeff R n (d⁄dX R f) = coeff R n (d⁄dX R g) := by rw [hD] | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
ext n
cases n with
| zero =>
rw [coeff_zero_eq_constantCoeff, h... | Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
n : ℕ
⊢ (coeff R n) ((d⁄dX R) f) = (coeff R n) ((d⁄dX R) g) | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [hD] | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
ext n
cases n with
| zero =>
rw [coeff_zero_eq_constantCoeff, h... | Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
case h.succ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroSMulDivisors ℕ R
f g : R⟦X⟧
hD : (d⁄dX R) f = (d⁄dX R) g
hc : (constantCoeff R) f = (constantCoeff R) g
n : ℕ
equ : (coeff R n) ((d⁄dX R) f) = (coeff R n) ((d⁄dX R) g)
⊢ (coeff R (succ n)) f = (coeff R (succ n)) g | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rwa [coeff_derivative, coeff_derivative, ← cast_succ, mul_comm, ← nsmul_eq_mul,
mul_comm, ← nsmul_eq_mul, smul_right_inj n.succ_ne_zero] at equ | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g := by
ext n
cases n with
| zero =>
rw [coeff_zero_eq_constantCoeff, h... | Mathlib.RingTheory.PowerSeries.Derivative.138_0.PcMDeT7Qkkrj2IE | /-- If `f` and `g` have the same constant term and derivative, then they are equal.-/
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
(hc : constantCoeff R f = constantCoeff R g) : f = g | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : CommRing R
f : R⟦X⟧ˣ
⊢ (d⁄dX R) ↑f⁻¹ = -↑f⁻¹ ^ 2 * (d⁄dX R) ↑f | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | apply Derivation.leibniz_of_mul_eq_one | @[simp] theorem derivative_inv {R} [CommRing R] (f : R⟦X⟧ˣ) :
d⁄dX R ↑f⁻¹ = -(↑f⁻¹ : R⟦X⟧) ^ 2 * d⁄dX R f := by
| Mathlib.RingTheory.PowerSeries.Derivative.150_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_inv {R} [CommRing R] (f : R⟦X⟧ˣ) :
d⁄dX R ↑f⁻¹ = -(↑f⁻¹ : R⟦X⟧) ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
case h
R : Type u_1
inst✝ : CommRing R
f : R⟦X⟧ˣ
⊢ ↑f⁻¹ * ↑f = 1 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | simp | @[simp] theorem derivative_inv {R} [CommRing R] (f : R⟦X⟧ˣ) :
d⁄dX R ↑f⁻¹ = -(↑f⁻¹ : R⟦X⟧) ^ 2 * d⁄dX R f := by
apply Derivation.leibniz_of_mul_eq_one
| Mathlib.RingTheory.PowerSeries.Derivative.150_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_inv {R} [CommRing R] (f : R⟦X⟧ˣ) :
d⁄dX R ↑f⁻¹ = -(↑f⁻¹ : R⟦X⟧) ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝¹ : CommRing R
f : R⟦X⟧
inst✝ : Invertible f
⊢ (d⁄dX R) ⅟f = -⅟f ^ 2 * (d⁄dX R) f | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [Derivation.leibniz_invOf, smul_eq_mul] | @[simp] theorem derivative_invOf {R} [CommRing R] (f : R⟦X⟧) [Invertible f] :
d⁄dX R ⅟f = - ⅟f ^ 2 * d⁄dX R f := by
| Mathlib.RingTheory.PowerSeries.Derivative.155_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_invOf {R} [CommRing R] (f : R⟦X⟧) [Invertible f] :
d⁄dX R ⅟f = - ⅟f ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : Field R
f : R⟦X⟧
⊢ (d⁄dX R) f⁻¹ = -f⁻¹ ^ 2 * (d⁄dX R) f | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | by_cases h : constantCoeff R f = 0 | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f := by
| Mathlib.RingTheory.PowerSeries.Derivative.163_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
case pos
R : Type u_1
inst✝ : Field R
f : R⟦X⟧
h : (constantCoeff R) f = 0
⊢ (d⁄dX R) f⁻¹ = -f⁻¹ ^ 2 * (d⁄dX R) f | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | suffices f⁻¹ = 0 by
rw [this, pow_two, zero_mul, neg_zero, zero_mul, map_zero] | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f := by
by_cases h : constantCoeff R f = 0
· | Mathlib.RingTheory.PowerSeries.Derivative.163_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝ : Field R
f : R⟦X⟧
h : (constantCoeff R) f = 0
this : f⁻¹ = 0
⊢ (d⁄dX R) f⁻¹ = -f⁻¹ ^ 2 * (d⁄dX R) f | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rw [this, pow_two, zero_mul, neg_zero, zero_mul, map_zero] | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f := by
by_cases h : constantCoeff R f = 0
· suffices f⁻¹ = 0 by
| Mathlib.RingTheory.PowerSeries.Derivative.163_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
case pos
R : Type u_1
inst✝ : Field R
f : R⟦X⟧
h : (constantCoeff R) f = 0
⊢ f⁻¹ = 0 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | rwa [MvPowerSeries.inv_eq_zero] | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f := by
by_cases h : constantCoeff R f = 0
· suffices f⁻¹ = 0 by
rw [this, pow_two, zero_mul, neg_zero, zero_mul, map_zero]
| Mathlib.RingTheory.PowerSeries.Derivative.163_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
case neg
R : Type u_1
inst✝ : Field R
f : R⟦X⟧
h : ¬(constantCoeff R) f = 0
⊢ (d⁄dX R) f⁻¹ = -f⁻¹ ^ 2 * (d⁄dX R) f | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | apply Derivation.leibniz_of_mul_eq_one | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f := by
by_cases h : constantCoeff R f = 0
· suffices f⁻¹ = 0 by
rw [this, pow_two, zero_mul, neg_zero, zero_mul, map_zero]
rwa [MvPowerSeries.inv_eq_zero]
| Mathlib.RingTheory.PowerSeries.Derivative.163_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
case neg.h
R : Type u_1
inst✝ : Field R
f : R⟦X⟧
h : ¬(constantCoeff R) f = 0
⊢ f⁻¹ * f = 1 | /-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this file we define an operation `derivative` (formal ... | exact PowerSeries.inv_mul_cancel (h := h) | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f := by
by_cases h : constantCoeff R f = 0
· suffices f⁻¹ = 0 by
rw [this, pow_two, zero_mul, neg_zero, zero_mul, map_zero]
rwa [MvPowerSeries.inv_eq_zero]
apply Derivation.leibniz_of_mul_eq_one
| Mathlib.RingTheory.PowerSeries.Derivative.163_0.PcMDeT7Qkkrj2IE | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f | Mathlib_RingTheory_PowerSeries_Derivative |
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1✝ D2✝ : Derivation R A A
a✝ : A
D1 D2 : Derivation R A A
a b : A
⊢ ⁅↑D1, ↑D2⁆ (a * b) = a • ⁅↑D1, ↑D2⁆ b + b • ⁅↑D1, ↑D2⁆ a | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-communit... | simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply] | /-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
| Mathlib.RingTheory.Derivation.Lie.32_0.lftH2oDc0WOWKWo | /-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) | Mathlib_RingTheory_Derivation_Lie |
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1✝ D2✝ : Derivation R A A
a✝ : A
D1 D2 : Derivation R A A
a b : A
⊢ a * D1 (D2 b) + D2 b * D1 a + (b * D1 (D2 a) + D2 a * D1 b) -
(a * D2 (D1 b) + D1 b * D2 a + (b * D2 (D1 a) + D1 a * D2 b)) =
a * (D1 (D2 b) - D2 (D1 b))... | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-communit... | ring | /-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, Li... | Mathlib.RingTheory.Derivation.Lie.32_0.lftH2oDc0WOWKWo | /-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) | Mathlib_RingTheory_Derivation_Lie |
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a : A
d e f : Derivation R A A
⊢ ⁅d + e, f⁆ = ⁅d, f⁆ + ⁅e, f⁆ | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-communit... | ext a | instance : LieRing (Derivation R A A) where
add_lie d e f := by | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d e f : Derivation R A A
a : A
⊢ ⁅d + e, f⁆ a = (⁅d, f⁆ + ⁅e, f⁆) a | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-communit... | simp only [commutator_apply, add_apply, map_add] | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.