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 |
|---|---|---|---|---|---|---|
𝕜✝ : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝⁴ : SeminormedAddCommGroup E✝
inst✝³ : NormedSpace 𝕜✝ E✝
𝕜 : Type u_3
E : Type u_4
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
h : x' ∈ polar 𝕜 (closedBall 0 r)
⊢ ‖x'‖ ≤ r⁻¹ | /-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | refine' ContinuousLinearMap.op_norm_le_of_ball hr (inv_nonneg.mpr hr.le) fun z _ => _ | /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ := by
refine' Subset.ant... | Mathlib.Analysis.NormedSpace.Dual.243_0.WirVfj6f5oiZZ2w | /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ | Mathlib_Analysis_NormedSpace_Dual |
𝕜✝ : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝⁴ : SeminormedAddCommGroup E✝
inst✝³ : NormedSpace 𝕜✝ E✝
𝕜 : Type u_3
E : Type u_4
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
h : x' ∈ polar 𝕜 (closedBall 0 r)
z : E
x✝ : z ∈ ball 0 r... | /-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | simpa only [one_div] using LinearMap.bound_of_ball_bound' hr 1 x'.toLinearMap h z | /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ := by
refine' Subset.ant... | Mathlib.Analysis.NormedSpace.Dual.243_0.WirVfj6f5oiZZ2w | /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
s_nhd : s ∈ 𝓝 0
⊢ IsBounded (polar 𝕜 s) | /-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | obtain ⟨a, ha⟩ : ∃ a : 𝕜, 1 < ‖a‖ := NormedField.exists_one_lt_norm 𝕜 | /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
IsBounded (polar 𝕜 s) := by
| Mathlib.Analysis.NormedSpace.Dual.254_0.WirVfj6f5oiZZ2w | /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
IsBounded (polar 𝕜 s) | Mathlib_Analysis_NormedSpace_Dual |
case intro
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
s_nhd : s ∈ 𝓝 0
a : 𝕜
ha : 1 < ‖a‖
⊢ IsBounded (polar 𝕜 s) | /-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | obtain ⟨r, r_pos, r_ball⟩ : ∃ r : ℝ, 0 < r ∧ ball 0 r ⊆ s := Metric.mem_nhds_iff.1 s_nhd | /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
IsBounded (polar 𝕜 s) := by
obtain ⟨a, ha⟩ : ∃ a : 𝕜, 1 < ‖a‖ := NormedField.ex... | Mathlib.Analysis.NormedSpace.Dual.254_0.WirVfj6f5oiZZ2w | /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
IsBounded (polar 𝕜 s) | Mathlib_Analysis_NormedSpace_Dual |
case intro.intro.intro
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
s_nhd : s ∈ 𝓝 0
a : 𝕜
ha : 1 < ‖a‖
r : ℝ
r_pos : 0 < r
r_ball : ball 0 r ⊆ s
⊢ IsBounded (polar 𝕜 s) | /-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | exact isBounded_closedBall.subset
(((dualPairing 𝕜 E).flip.polar_antitone r_ball).trans <|
polar_ball_subset_closedBall_div ha r_pos) | /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
IsBounded (polar 𝕜 s) := by
obtain ⟨a, ha⟩ : ∃ a : 𝕜, 1 < ‖a‖ := NormedField.ex... | Mathlib.Analysis.NormedSpace.Dual.254_0.WirVfj6f5oiZZ2w | /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
IsBounded (polar 𝕜 s) | Mathlib_Analysis_NormedSpace_Dual |
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
⊢ CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) ⇑toFinsupp | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rintro s t ⟨u, a, hr, he⟩ | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
| Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
hr : ∀ a' ∈ u, r a' a
he : s + {a} = t + u
⊢ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | replace hr := fun a' ↦ mt (hr a') | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
| Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
he : s + {a} = t + u
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
⊢ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> simp_rw [toFinsupp_apply]
· apply_fun count b at he
simpa only [count_add, count_singleton, if_neg h.2, add_zero, count_eq_zero.2 (hr b h.1)]
using he
· apply_fun count a at he
simp only [count_add, count_singleton_self, count_eq_zero.2 (hr _ (irrefl_of r a... | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
| Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
he : s + {a} = t + u
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
⊢ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | refine ⟨a, fun b h ↦ ?_, ?_⟩ | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
classical
| Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro.refine_1
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
he : s + {a} = t + u
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
b : α
h : (rᶜ ⊓ fun x x_1 => x ≠ x_1) b a
⊢ (toFinsupp s) b = (toFinsupp t) b | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | simp_rw [toFinsupp_apply] | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> | Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro.refine_2
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
he : s + {a} = t + u
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
⊢ (fun {i} x x_1 => x < x_1) ((toFinsupp s) a) ((toFinsupp t) a) | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | simp_rw [toFinsupp_apply] | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> | Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro.refine_1
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
he : s + {a} = t + u
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
b : α
h : (rᶜ ⊓ fun x x_1 => x ≠ x_1) b a
⊢ count b s = count b t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | apply_fun count b at he | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> simp_rw [toFinsupp_apply]
· | Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro.refine_1
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
b : α
h : (rᶜ ⊓ fun x x_1 => x ≠ x_1) b a
he : count b (s + {a}) = count b (t + u)
⊢ count b s = count b t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | simpa only [count_add, count_singleton, if_neg h.2, add_zero, count_eq_zero.2 (hr b h.1)]
using he | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> simp_rw [toFinsupp_apply]
· apply_fun count b at he
| Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro.refine_2
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
he : s + {a} = t + u
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
⊢ count a s < count a t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | apply_fun count a at he | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> simp_rw [toFinsupp_apply]
· apply_fun count b at he
simp... | Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro.refine_2
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
he : count a (s + {a}) = count a (t + u)
⊢ count a s < count a t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | simp only [count_add, count_singleton_self, count_eq_zero.2 (hr _ (irrefl_of r a)),
add_zero] at he | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> simp_rw [toFinsupp_apply]
· apply_fun count b at he
simp... | Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
case intro.intro.intro.refine_2
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t u : Multiset α
a : α
hr : ∀ (a' : α), ¬r a' a → a' ∉ u
he : count a s + 1 = count a t
⊢ count a s < count a t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | exact he ▸ Nat.lt_succ_self _ | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp := by
rintro s t ⟨u, a, hr, he⟩
replace hr := fun a' ↦ mt (hr a')
classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> simp_rw [toFinsupp_apply]
· apply_fun count b at he
simp... | Mathlib.Logic.Hydra.62_0.cWRHz2gehQLFc75 | theorem cutExpand_le_invImage_lex [DecidableEq α] [IsIrrefl α r] :
CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ (· ≠ ·)) (· < ·)) toFinsupp | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
x' x : α
h✝ : r x' x
a : α
h : a ∈ {x'}
⊢ r a x | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rwa [mem_singleton.1 h] | theorem cutExpand_singleton_singleton {x' x} (h : r x' x) : CutExpand r {x'} {x} :=
cutExpand_singleton fun a h ↦ by | Mathlib.Logic.Hydra.81_0.cWRHz2gehQLFc75 | theorem cutExpand_singleton_singleton {x' x} (h : r x' x) : CutExpand r {x'} {x} | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
t u s x✝¹ : Multiset α
x✝ : α
⊢ s + t + {x✝} = s + u + x✝¹ ↔ t + {x✝} = u + x✝¹ | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [add_assoc, add_assoc, add_left_cancel_iff] | theorem cutExpand_add_left {t u} (s) : CutExpand r (s + t) (s + u) ↔ CutExpand r t u :=
exists₂_congr fun _ _ ↦ and_congr Iff.rfl <| by | Mathlib.Logic.Hydra.85_0.cWRHz2gehQLFc75 | theorem cutExpand_add_left {t u} (s) : CutExpand r (s + t) (s + u) ↔ CutExpand r t u | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s' s : Multiset α
⊢ CutExpand r s' s ↔ ∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = erase s a + t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | simp_rw [CutExpand, add_singleton_eq_iff] | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t := by
| Mathlib.Logic.Hydra.89_0.cWRHz2gehQLFc75 | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s' s : Multiset α
⊢ (∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ s' = erase (s + t) a) ↔
∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = erase s a + t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | refine' exists₂_congr fun t a ↦ ⟨_, _⟩ | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t := by
simp_rw [CutExpand, add_singleton_eq_iff]
| Mathlib.Logic.Hydra.89_0.cWRHz2gehQLFc75 | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | Mathlib_Logic_Hydra |
case refine'_1
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s' s t : Multiset α
a : α
⊢ (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ s' = erase (s + t) a → (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = erase s a + t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rintro ⟨ht, ha, rfl⟩ | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t := by
simp_rw [CutExpand, add_singleton_eq_iff]
refine' exists₂_congr fun t a ↦ ⟨_, _⟩
· | Mathlib.Logic.Hydra.89_0.cWRHz2gehQLFc75 | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | Mathlib_Logic_Hydra |
case refine'_1.intro.intro
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t : Multiset α
a : α
ht : ∀ a' ∈ t, r a' a
ha : a ∈ s + t
⊢ (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ erase (s + t) a = erase s a + t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | obtain h | h := mem_add.1 ha | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t := by
simp_rw [CutExpand, add_singleton_eq_iff]
refine' exists₂_congr fun t a ↦ ⟨_, _⟩
· rintro ⟨ht, ha, rfl⟩
| Mathlib.Logic.Hydra.89_0.cWRHz2gehQLFc75 | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | Mathlib_Logic_Hydra |
case refine'_1.intro.intro.inl
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t : Multiset α
a : α
ht : ∀ a' ∈ t, r a' a
ha : a ∈ s + t
h : a ∈ s
⊢ (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ erase (s + t) a = erase s a + t
case refine'_1.intro.intro.inr
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq ... | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | exacts [⟨ht, h, erase_add_left_pos t h⟩, (@irrefl α r _ a (ht a h)).elim] | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t := by
simp_rw [CutExpand, add_singleton_eq_iff]
refine' exists₂_congr fun t a ↦ ⟨_, _⟩
· rintro ⟨ht, ha, rfl⟩
obtain h | h := mem_... | Mathlib.Logic.Hydra.89_0.cWRHz2gehQLFc75 | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | Mathlib_Logic_Hydra |
case refine'_2
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s' s t : Multiset α
a : α
⊢ (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = erase s a + t → (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ s' = erase (s + t) a | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rintro ⟨ht, h, rfl⟩ | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t := by
simp_rw [CutExpand, add_singleton_eq_iff]
refine' exists₂_congr fun t a ↦ ⟨_, _⟩
· rintro ⟨ht, ha, rfl⟩
obtain h | h := mem_... | Mathlib.Logic.Hydra.89_0.cWRHz2gehQLFc75 | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | Mathlib_Logic_Hydra |
case refine'_2.intro.intro
α : Type u_1
r : α → α → Prop
inst✝¹ : DecidableEq α
inst✝ : IsIrrefl α r
s t : Multiset α
a : α
ht : ∀ a' ∈ t, r a' a
h : a ∈ s
⊢ (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ erase s a + t = erase (s + t) a | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | exact ⟨ht, mem_add.2 (Or.inl h), (erase_add_left_pos t h).symm⟩ | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t := by
simp_rw [CutExpand, add_singleton_eq_iff]
refine' exists₂_congr fun t a ↦ ⟨_, _⟩
· rintro ⟨ht, ha, rfl⟩
obtain h | h := mem_... | Mathlib.Logic.Hydra.89_0.cWRHz2gehQLFc75 | theorem cutExpand_iff [DecidableEq α] [IsIrrefl α r] {s' s : Multiset α} :
CutExpand r s' s ↔
∃ (t : Multiset α) (a : α), (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
s : Multiset α
⊢ ¬CutExpand r s 0 | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | classical
rw [cutExpand_iff]
rintro ⟨_, _, _, ⟨⟩, _⟩ | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 := by
| Mathlib.Logic.Hydra.101_0.cWRHz2gehQLFc75 | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
s : Multiset α
⊢ ¬CutExpand r s 0 | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [cutExpand_iff] | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 := by
classical
| Mathlib.Logic.Hydra.101_0.cWRHz2gehQLFc75 | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
s : Multiset α
⊢ ¬∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ 0 ∧ s = erase 0 a + t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rintro ⟨_, _, _, ⟨⟩, _⟩ | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 := by
classical
rw [cutExpand_iff]
| Mathlib.Logic.Hydra.101_0.cWRHz2gehQLFc75 | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 | Mathlib_Logic_Hydra |
α : Type u_1
r✝ r : α → α → Prop
⊢ Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s => s.1 + s.2 | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rintro ⟨s₁, s₂⟩ s ⟨t, a, hr, he⟩ | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ s t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : s + {a} = (fun s => s.1 + s.2) (s₁, s₂) + t
⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ (fun s => s.1 + s.2) a' = s | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | dsimp at he ⊢ | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ s t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : s + {a} = s₁ + s₂ + t
⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = s | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | classical
obtain ⟨ha, rfl⟩ := add_singleton_eq_iff.1 he
rw [add_assoc, mem_add] at ha
obtain h | h := ha
· refine' ⟨(s₁.erase a + t, s₂), GameAdd.fst ⟨t, a, hr, _⟩, _⟩
· rw [add_comm, ← add_assoc, singleton_add, cons_erase h]
· rw [add_assoc s₁, erase_add_left_pos _ h, add_right_comm, add_assoc]
· ref... | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ s t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : s + {a} = s₁ + s₂ + t
⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = s | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | obtain ⟨ha, rfl⟩ := add_singleton_eq_iff.1 he | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro.intro
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
ha : a ∈ s₁ + s₂ + t
he : erase (s₁ + s₂ + t) a + {a} = s₁ + s₂ + t
⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = erase (s₁ + s₂ + t) a | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [add_assoc, mem_add] at ha | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro.intro
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
ha : a ∈ s₁ ∨ a ∈ s₂ + t
he : erase (s₁ + s₂ + t) a + {a} = s₁ + s₂ + t
⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = erase (s₁ + s₂ + t) a | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | obtain h | h := ha | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro.intro.inl
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : erase (s₁ + s₂ + t) a + {a} = s₁ + s₂ + t
h : a ∈ s₁
⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = erase (s₁ + s₂ + t) a | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | refine' ⟨(s₁.erase a + t, s₂), GameAdd.fst ⟨t, a, hr, _⟩, _⟩ | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro.intro.inl.refine'_1
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : erase (s₁ + s₂ + t) a + {a} = s₁ + s₂ + t
h : a ∈ s₁
⊢ erase s₁ a + t + {a} = s₁ + t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [add_comm, ← add_assoc, singleton_add, cons_erase h] | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro.intro.inl.refine'_2
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : erase (s₁ + s₂ + t) a + {a} = s₁ + s₂ + t
h : a ∈ s₁
⊢ (erase s₁ a + t, s₂).1 + (erase s₁ a + t, s₂).2 = erase (s₁ + s₂ + t) a | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [add_assoc s₁, erase_add_left_pos _ h, add_right_comm, add_assoc] | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro.intro.inr
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : erase (s₁ + s₂ + t) a + {a} = s₁ + s₂ + t
h : a ∈ s₂ + t
⊢ ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = erase (s₁ + s₂ + t) a | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | refine' ⟨(s₁, (s₂ + t).erase a), GameAdd.snd ⟨t, a, hr, _⟩, _⟩ | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro.intro.inr.refine'_1
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : erase (s₁ + s₂ + t) a + {a} = s₁ + s₂ + t
h : a ∈ s₂ + t
⊢ erase (s₂ + t) a + {a} = s₂ + t | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [add_comm, singleton_add, cons_erase h] | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
case mk.intro.intro.intro.intro.inr.refine'_2
α : Type u_1
r✝ r : α → α → Prop
s₁ s₂ t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
he : erase (s₁ + s₂ + t) a + {a} = s₁ + s₂ + t
h : a ∈ s₂ + t
⊢ (s₁, erase (s₂ + t) a).1 + (s₁, erase (s₂ + t) a).2 = erase (s₁ + s₂ + t) a | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [add_assoc, erase_add_right_pos _ h] | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 :... | Mathlib.Logic.Hydra.107_0.cWRHz2gehQLFc75 | /-- For any relation `r` on `α`, multiset addition `Multiset α × Multiset α → Multiset α` is a
fibration between the game sum of `CutExpand r` with itself and `CutExpand r` itself. -/
theorem cutExpand_fibration (r : α → α → Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
s : Multiset α
hs : ∀ a ∈ s, Acc (CutExpand r) {a}
⊢ Acc (CutExpand r) s | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | induction s using Multiset.induction | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
| Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
case empty
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
hs : ∀ a ∈ 0, Acc (CutExpand r) {a}
⊢ Acc (CutExpand r) 0
case cons
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝¹ : α
s✝ : Multiset α
a✝ : (∀ a ∈ s✝, Acc (CutExpand r) {a}) → Acc (CutExpand r) s✝
hs : ∀ a ∈ a✝¹ ::ₘ s✝, Acc (CutExpand r) {a}
⊢ Acc (C... | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | case empty => exact Acc.intro 0 fun s h ↦ (not_cutExpand_zero s h).elim | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
induction s using Multiset.induction
| Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
hs : ∀ a ∈ 0, Acc (CutExpand r) {a}
⊢ Acc (CutExpand r) 0 | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | case empty => exact Acc.intro 0 fun s h ↦ (not_cutExpand_zero s h).elim | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
induction s using Multiset.induction
| Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
hs : ∀ a ∈ 0, Acc (CutExpand r) {a}
⊢ Acc (CutExpand r) 0 | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | exact Acc.intro 0 fun s h ↦ (not_cutExpand_zero s h).elim | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
induction s using Multiset.induction
case empty => | Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
case cons
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝¹ : α
s✝ : Multiset α
a✝ : (∀ a ∈ s✝, Acc (CutExpand r) {a}) → Acc (CutExpand r) s✝
hs : ∀ a ∈ a✝¹ ::ₘ s✝, Acc (CutExpand r) {a}
⊢ Acc (CutExpand r) (a✝¹ ::ₘ s✝) | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | case cons a s ihs =>
rw [← s.singleton_add a]
rw [forall_mem_cons] at hs
exact (hs.1.prod_gameAdd <| ihs fun a ha ↦ hs.2 a ha).of_fibration _ (cutExpand_fibration r) | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
induction s using Multiset.induction
case empty => exact Acc.intro 0 fun s... | Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a : α
s : Multiset α
ihs : (∀ a ∈ s, Acc (CutExpand r) {a}) → Acc (CutExpand r) s
hs : ∀ a_1 ∈ a ::ₘ s, Acc (CutExpand r) {a_1}
⊢ Acc (CutExpand r) (a ::ₘ s) | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | case cons a s ihs =>
rw [← s.singleton_add a]
rw [forall_mem_cons] at hs
exact (hs.1.prod_gameAdd <| ihs fun a ha ↦ hs.2 a ha).of_fibration _ (cutExpand_fibration r) | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
induction s using Multiset.induction
case empty => exact Acc.intro 0 fun s... | Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a : α
s : Multiset α
ihs : (∀ a ∈ s, Acc (CutExpand r) {a}) → Acc (CutExpand r) s
hs : ∀ a_1 ∈ a ::ₘ s, Acc (CutExpand r) {a_1}
⊢ Acc (CutExpand r) (a ::ₘ s) | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [← s.singleton_add a] | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
induction s using Multiset.induction
case empty => exact Acc.intro 0 fun s... | Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a : α
s : Multiset α
ihs : (∀ a ∈ s, Acc (CutExpand r) {a}) → Acc (CutExpand r) s
hs : ∀ a_1 ∈ a ::ₘ s, Acc (CutExpand r) {a_1}
⊢ Acc (CutExpand r) ({a} + s) | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [forall_mem_cons] at hs | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
induction s using Multiset.induction
case empty => exact Acc.intro 0 fun s... | Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a : α
s : Multiset α
ihs : (∀ a ∈ s, Acc (CutExpand r) {a}) → Acc (CutExpand r) s
hs : Acc (CutExpand r) {a} ∧ ∀ x ∈ s, Acc (CutExpand r) {x}
⊢ Acc (CutExpand r) ({a} + s) | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | exact (hs.1.prod_gameAdd <| ihs fun a ha ↦ hs.2 a ha).of_fibration _ (cutExpand_fibration r) | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by
induction s using Multiset.induction
case empty => exact Acc.intro 0 fun s... | Mathlib.Logic.Hydra.124_0.cWRHz2gehQLFc75 | /-- A multiset is accessible under `CutExpand` if all its singleton subsets are,
assuming `r` is irreflexive. -/
theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | Mathlib_Logic_Hydra |
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a : α
hacc : Acc r a
⊢ Acc (CutExpand r) {a} | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | induction' hacc with a h ih | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by
| Mathlib.Logic.Hydra.136_0.cWRHz2gehQLFc75 | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} | Mathlib_Logic_Hydra |
case intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ a : α
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
⊢ Acc (CutExpand r) {a} | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | refine' Acc.intro _ fun s ↦ _ | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by
induction' hacc with a h ih
| Mathlib.Logic.Hydra.136_0.cWRHz2gehQLFc75 | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} | Mathlib_Logic_Hydra |
case intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ a : α
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
s : Multiset α
⊢ CutExpand r s {a} → Acc (CutExpand r) s | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | classical
simp only [cutExpand_iff, mem_singleton]
rintro ⟨t, a, hr, rfl, rfl⟩
refine' acc_of_singleton fun a' ↦ _
rw [erase_singleton, zero_add]
exact ih a' ∘ hr a' | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by
induction' hacc with a h ih
refine' Acc.intro _ fun s ↦ _
| Mathlib.Logic.Hydra.136_0.cWRHz2gehQLFc75 | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} | Mathlib_Logic_Hydra |
case intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ a : α
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
s : Multiset α
⊢ CutExpand r s {a} → Acc (CutExpand r) s | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | simp only [cutExpand_iff, mem_singleton] | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by
induction' hacc with a h ih
refine' Acc.intro _ fun s ↦ _
classical
| Mathlib.Logic.Hydra.136_0.cWRHz2gehQLFc75 | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} | Mathlib_Logic_Hydra |
case intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ a : α
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
s : Multiset α
⊢ (∃ t a_1, (∀ a' ∈ t, r a' a_1) ∧ a_1 = a ∧ s = erase {a} a_1 + t) → Acc (CutExpand r) s | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rintro ⟨t, a, hr, rfl, rfl⟩ | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by
induction' hacc with a h ih
refine' Acc.intro _ fun s ↦ _
classical
simp only [cutExpand_if... | Mathlib.Logic.Hydra.136_0.cWRHz2gehQLFc75 | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} | Mathlib_Logic_Hydra |
case intro.intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ : α
t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
⊢ Acc (CutExpand r) (erase {a} a + t) | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | refine' acc_of_singleton fun a' ↦ _ | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by
induction' hacc with a h ih
refine' Acc.intro _ fun s ↦ _
classical
simp only [cutExpand_if... | Mathlib.Logic.Hydra.136_0.cWRHz2gehQLFc75 | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} | Mathlib_Logic_Hydra |
case intro.intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ : α
t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
a' : α
⊢ a' ∈ erase {a} a + t → Acc (CutExpand r) {a'} | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | rw [erase_singleton, zero_add] | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by
induction' hacc with a h ih
refine' Acc.intro _ fun s ↦ _
classical
simp only [cutExpand_if... | Mathlib.Logic.Hydra.136_0.cWRHz2gehQLFc75 | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} | Mathlib_Logic_Hydra |
case intro.intro.intro.intro.intro
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a✝ : α
t : Multiset α
a : α
hr : ∀ a' ∈ t, r a' a
h : ∀ (y : α), r y a → Acc r y
ih : ∀ (y : α), r y a → Acc (CutExpand r) {y}
a' : α
⊢ a' ∈ t → Acc (CutExpand r) {a'} | /-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e6843... | exact ih a' ∘ hr a' | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} := by
induction' hacc with a h ih
refine' Acc.intro _ fun s ↦ _
classical
simp only [cutExpand_if... | Mathlib.Logic.Hydra.136_0.cWRHz2gehQLFc75 | /-- A singleton `{a}` is accessible under `CutExpand r` if `a` is accessible under `r`,
assuming `r` is irreflexive. -/
theorem _root_.Acc.cutExpand [IsIrrefl α r] {a : α} (hacc : Acc r a) : Acc (CutExpand r) {a} | Mathlib_Logic_Hydra |
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
⊢ (x + y) ^ 0 = x ^ 0 + ↑0 * x ^ (0 - 1) * y + 0 * y ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | simp | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by | Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
⊢ (x + y) ^ 1 = x ^ 1 + ↑1 * x ^ (1 - 1) * y + 0 * y ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | simp | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by | Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n✝ : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
n : ℕ
⊢ { k // (x + y) ^ (n + 2) = x ^ (n + 2) + ↑(n + 2) * x ^ (n + 2 - 1) * y + k * y ^ 2 } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | cases' (powAddExpansion x y (n + 1)) with z hz | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
| Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
case mk
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n✝ : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
n : ℕ
z : R
hz : (x + y) ^ (n + 1) = x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2
⊢ { k // (x + y) ^ (n + 2) = x ^ (n + 2) + ↑(n + 2) * x ^ (n + 2 - 1) * y + k * y ^ 2 } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | exists x * z + (n + 1) * x ^ n + z * y | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
case mk
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n✝ : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
n : ℕ
z : R
hz : (x + y) ^ (n + 1) = x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2
⊢ (x + y) ^ (n + 2) = x ^ (n + 2) + ↑(n + 2) * x ^ (n + 2 - 1) * y + (x * z + (↑n + 1) * ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | calc
(x + y) ^ (n + 2) = (x + y) * (x + y) ^ (n + 1) := by ring
_ = (x + y) * (x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2) := by rw [hz]
_ = x ^ (n + 2) + ↑(n + 2) * x ^ (n + 1) * y + (x * z + (n + 1) * x ^ n + z * y) * y ^ 2 := by
push_cast
ring! | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n✝ : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
n : ℕ
z : R
hz : (x + y) ^ (n + 1) = x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2
⊢ (x + y) ^ (n + 2) = (x + y) * (x + y) ^ (n + 1) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | ring | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n✝ : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
n : ℕ
z : R
hz : (x + y) ^ (n + 1) = x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2
⊢ (x + y) * (x + y) ^ (n + 1) = (x + y) * (x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | rw [hz] | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n✝ : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
n : ℕ
z : R
hz : (x + y) ^ (n + 1) = x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2
⊢ (x + y) * (x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2) =
x ^ (n + 2) + ↑(n + ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | push_cast | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
R✝ : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R✝
m n✝ : ℕ
R : Type u_1
inst✝ : CommSemiring R
x y : R
n : ℕ
z : R
hz : (x + y) ^ (n + 1) = x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2
⊢ (x + y) * (x ^ (n + 1) + (↑n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2) =
x ^ (n + 2) + (↑n + ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | ring! | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib.Data.Polynomial.Identities.37_0.o6IrpyrTENfZuiK | /-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
cases' (powAddExpansion ... | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a✝ b : R
m n : ℕ
inst✝ : CommRing R
x y : R
e : ℕ
a : R
⊢ { k // a * (x + y) ^ e = a * (x ^ e + ↑e * x ^ (e - 1) * y + k * y ^ 2) } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | exists (powAddExpansion x y e).val | private def polyBinomAux1 (x y : R) (e : ℕ) (a : R) :
{ k : R // a * (x + y) ^ e = a * (x ^ e + e * x ^ (e - 1) * y + k * y ^ 2) } := by
| Mathlib.Data.Polynomial.Identities.56_0.o6IrpyrTENfZuiK | private def polyBinomAux1 (x y : R) (e : ℕ) (a : R) :
{ k : R // a * (x + y) ^ e = a * (x ^ e + e * x ^ (e - 1) * y + k * y ^ 2) } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a✝ b : R
m n : ℕ
inst✝ : CommRing R
x y : R
e : ℕ
a : R
⊢ a * (x + y) ^ e = a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(powAddExpansion x y e) * y ^ 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | congr | private def polyBinomAux1 (x y : R) (e : ℕ) (a : R) :
{ k : R // a * (x + y) ^ e = a * (x ^ e + e * x ^ (e - 1) * y + k * y ^ 2) } := by
exists (powAddExpansion x y e).val
| Mathlib.Data.Polynomial.Identities.56_0.o6IrpyrTENfZuiK | private def polyBinomAux1 (x y : R) (e : ℕ) (a : R) :
{ k : R // a * (x + y) ^ e = a * (x ^ e + e * x ^ (e - 1) * y + k * y ^ 2) } | Mathlib_Data_Polynomial_Identities |
case e_a
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a✝ b : R
m n : ℕ
inst✝ : CommRing R
x y : R
e : ℕ
a : R
⊢ (x + y) ^ e = x ^ e + ↑e * x ^ (e - 1) * y + ↑(powAddExpansion x y e) * y ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | apply (powAddExpansion _ _ _).property | private def polyBinomAux1 (x y : R) (e : ℕ) (a : R) :
{ k : R // a * (x + y) ^ e = a * (x ^ e + e * x ^ (e - 1) * y + k * y ^ 2) } := by
exists (powAddExpansion x y e).val
congr
| Mathlib.Data.Polynomial.Identities.56_0.o6IrpyrTENfZuiK | private def polyBinomAux1 (x y : R) (e : ℕ) (a : R) :
{ k : R // a * (x + y) ^ e = a * (x ^ e + e * x ^ (e - 1) * y + k * y ^ 2) } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ eval (x + y) f = sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | unfold eval | private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) := by
| Mathlib.Data.Polynomial.Identities.62_0.o6IrpyrTENfZuiK | private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ eval₂ (RingHom.id R) (x + y) f =
sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | rw [eval₂_eq_sum] | private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) := by
unfold eval; | Mathlib.Data.Polynomial.Identities.62_0.o6IrpyrTENfZuiK | private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (sum f fun e a => (RingHom.id R) a * (x + y) ^ e) =
sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | congr with (n z) | private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) := by
unfold eval; rw [eval₂_eq_sum]; | Mathlib.Data.Polynomial.Identities.62_0.o6IrpyrTENfZuiK | private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) | Mathlib_Data_Polynomial_Identities |
case e_f.h.h
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n✝ : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
n : ℕ
z : R
⊢ (RingHom.id R) z * (x + y) ^ n = z * (x ^ n + ↑n * x ^ (n - 1) * y + ↑(Polynomial.polyBinomAux1 x y n z) * y ^ 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | apply (polyBinomAux1 x y _ _).property | private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) := by
unfold eval; rw [eval₂_eq_sum]; congr with (n z)
| Mathlib.Data.Polynomial.Identities.62_0.o6IrpyrTENfZuiK | private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ eval (x + y) f =
((sum f fun e a => a * x ^ e) + sum f fun e a => a * ↑e * x ^ (e - 1) * y) +
sum f fun e a => a * ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | rw [poly_binom_aux2] | private theorem poly_binom_aux3 (f : R[X]) (x y : R) :
f.eval (x + y) =
((f.sum fun e a => a * x ^ e) + f.sum fun e a => a * e * x ^ (e - 1) * y) +
f.sum fun e a => a * (polyBinomAux1 x y e a).val * y ^ 2 := by
| Mathlib.Data.Polynomial.Identities.68_0.o6IrpyrTENfZuiK | private theorem poly_binom_aux3 (f : R[X]) (x y : R) :
f.eval (x + y) =
((f.sum fun e a => a * x ^ e) + f.sum fun e a => a * e * x ^ (e - 1) * y) +
f.sum fun e a => a * (polyBinomAux1 x y e a).val * y ^ 2 | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (sum f fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)) =
((sum f fun e a => a * x ^ e) + sum f fun e a => a * ↑e * x ^ (e - 1) * y) +
sum f fun e... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | simp [left_distrib, sum_add, mul_assoc] | private theorem poly_binom_aux3 (f : R[X]) (x y : R) :
f.eval (x + y) =
((f.sum fun e a => a * x ^ e) + f.sum fun e a => a * e * x ^ (e - 1) * y) +
f.sum fun e a => a * (polyBinomAux1 x y e a).val * y ^ 2 := by
rw [poly_binom_aux2]
| Mathlib.Data.Polynomial.Identities.68_0.o6IrpyrTENfZuiK | private theorem poly_binom_aux3 (f : R[X]) (x y : R) :
f.eval (x + y) =
((f.sum fun e a => a * x ^ e) + f.sum fun e a => a * e * x ^ (e - 1) * y) +
f.sum fun e a => a * (polyBinomAux1 x y e a).val * y ^ 2 | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ { k // eval (x + y) f = eval x f + eval x (derivative f) * y + k * y ^ 2 } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | exists f.sum fun e a => a * (polyBinomAux1 x y e a).val | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } := by... | Mathlib.Data.Polynomial.Identities.75_0.o6IrpyrTENfZuiK | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ eval (x + y) f =
eval x f + eval x (derivative f) * y + (sum f fun e a => a * ↑(Polynomial.polyBinomAux1 x y e a)) * y ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | rw [poly_binom_aux3] | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } := by... | Mathlib.Data.Polynomial.Identities.75_0.o6IrpyrTENfZuiK | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (((sum f fun e a => a * x ^ e) + sum f fun e a => a * ↑e * x ^ (e - 1) * y) +
sum f fun e a => a * ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2) =
eval x f + eval x (derivative f) * y + (sum ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | congr | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } := by... | Mathlib.Data.Polynomial.Identities.75_0.o6IrpyrTENfZuiK | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } | Mathlib_Data_Polynomial_Identities |
case e_a.e_a
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (sum f fun e a => a * x ^ e) = eval x f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | rw [← eval_eq_sum] | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } := by... | Mathlib.Data.Polynomial.Identities.75_0.o6IrpyrTENfZuiK | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } | Mathlib_Data_Polynomial_Identities |
case e_a.e_a
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (sum f fun e a => a * ↑e * x ^ (e - 1) * y) = eval x (derivative f) * y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | rw [derivative_eval] | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } := by... | Mathlib.Data.Polynomial.Identities.75_0.o6IrpyrTENfZuiK | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } | Mathlib_Data_Polynomial_Identities |
case e_a.e_a
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (sum f fun e a => a * ↑e * x ^ (e - 1) * y) = (sum f fun n a => a * ↑n * x ^ (n - 1)) * y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | exact Finset.sum_mul.symm | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } := by... | Mathlib.Data.Polynomial.Identities.75_0.o6IrpyrTENfZuiK | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } | Mathlib_Data_Polynomial_Identities |
case e_a
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (sum f fun e a => a * ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2) =
(sum f fun e a => a * ↑(Polynomial.polyBinomAux1 x y e a)) * y ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | exact Finset.sum_mul.symm | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } := by... | Mathlib.Data.Polynomial.Identities.75_0.o6IrpyrTENfZuiK | /-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
x y : R
⊢ x ^ 0 - y ^ 0 = 0 * (x - y) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | simp | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by | Mathlib.Data.Polynomial.Identities.90_0.o6IrpyrTENfZuiK | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
exists z * x + y ^ (k + 1)
linear_combina... | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
x y : R
⊢ x ^ 1 - y ^ 1 = 1 * (x - y) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | simp | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by | Mathlib.Data.Polynomial.Identities.90_0.o6IrpyrTENfZuiK | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
exists z * x + y ^ (k + 1)
linear_combina... | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k✝ : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
x y : R
k : ℕ
⊢ { z // x ^ (k + 2) - y ^ (k + 2) = z * (x - y) } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | cases' @powSubPowFactor x y (k + 1) with z hz | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
| Mathlib.Data.Polynomial.Identities.90_0.o6IrpyrTENfZuiK | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
exists z * x + y ^ (k + 1)
linear_combina... | Mathlib_Data_Polynomial_Identities |
case mk
R : Type u
S : Type v
T : Type w
ι : Type x
k✝ : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
x y : R
k : ℕ
z : R
hz : x ^ (k + 1) - y ^ (k + 1) = z * (x - y)
⊢ { z // x ^ (k + 2) - y ^ (k + 2) = z * (x - y) } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | exists z * x + y ^ (k + 1) | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
| Mathlib.Data.Polynomial.Identities.90_0.o6IrpyrTENfZuiK | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
exists z * x + y ^ (k + 1)
linear_combina... | Mathlib_Data_Polynomial_Identities |
case mk
R : Type u
S : Type v
T : Type w
ι : Type x
k✝ : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
x y : R
k : ℕ
z : R
hz : x ^ (k + 1) - y ^ (k + 1) = z * (x - y)
⊢ x ^ (k + 2) - y ^ (k + 2) = (z * x + y ^ (k + 1)) * (x - y) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | linear_combination (norm := ring) x * hz | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
exists z * x + y ^ (k + 1)
| Mathlib.Data.Polynomial.Identities.90_0.o6IrpyrTENfZuiK | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
exists z * x + y ^ (k + 1)
linear_combina... | Mathlib_Data_Polynomial_Identities |
case a
R : Type u
S : Type v
T : Type w
ι : Type x
k✝ : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
x y : R
k : ℕ
z : R
hz : x ^ (k + 1) - y ^ (k + 1) = z * (x - y)
⊢ x ^ (k + 2) - y ^ (k + 2) - (z * x + y ^ (k + 1)) * (x - y) - (x * (x ^ (k + 1) - y ^ (k + 1)) - x * (z * (x - y))) =
0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | ring | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
exists z * x + y ^ (k + 1)
linear_combina... | Mathlib.Data.Polynomial.Identities.90_0.o6IrpyrTENfZuiK | /-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
cases' @powSubPowFactor x y (k + 1) with z hz
exists z * x + y ^ (k + 1)
linear_combina... | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ { z // eval x f - eval y f = z * (x - y) } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | refine' ⟨f.sum fun i r => r * (powSubPowFactor x y i).val, _⟩ | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } := by
| Mathlib.Data.Polynomial.Identities.101_0.o6IrpyrTENfZuiK | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ eval x f - eval y f = (sum f fun i r => r * ↑(powSubPowFactor x y i)) * (x - y) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | delta eval | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } := by
refine' ⟨f.sum fun i r => r * (powSubPowFactor x y i).val, _⟩
| Mathlib.Data.Polynomial.Identities.101_0.o6IrpyrTENfZuiK | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ eval₂ (RingHom.id R) x f - eval₂ (RingHom.id R) y f = (sum f fun i r => r * ↑(powSubPowFactor x y i)) * (x - y) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | rw [eval₂_eq_sum, eval₂_eq_sum] | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } := by
refine' ⟨f.sum fun i r => r * (powSubPowFactor x y i).val, _⟩
delta eval; | Mathlib.Data.Polynomial.Identities.101_0.o6IrpyrTENfZuiK | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ ((sum f fun e a => (RingHom.id R) a * x ^ e) - sum f fun e a => (RingHom.id R) a * y ^ e) =
(sum f fun i r => r * ↑(powSubPowFactor x y i)) * (x - y) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | simp only [sum, ← Finset.sum_sub_distrib, Finset.sum_mul] | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } := by
refine' ⟨f.sum fun i r => r * (powSubPowFactor x y i).val, _⟩
delta eval; rw [eval₂_eq_sum, eval₂_eq_sum];
| Mathlib.Data.Polynomial.Identities.101_0.o6IrpyrTENfZuiK | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (Finset.sum (support f) fun x_1 => (RingHom.id R) (coeff f x_1) * x ^ x_1 - (RingHom.id R) (coeff f x_1) * y ^ x_1) =
Finset.sum (support f) fun x_1 => coeff f x_1 * ↑(powSubPowFactor x y x_1) * (... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | dsimp | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } := by
refine' ⟨f.sum fun i r => r * (powSubPowFactor x y i).val, _⟩
delta eval; rw [eval₂_eq_sum, eval₂_eq_sum];
... | Mathlib.Data.Polynomial.Identities.101_0.o6IrpyrTENfZuiK | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } | Mathlib_Data_Polynomial_Identities |
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
⊢ (Finset.sum (support f) fun x_1 => coeff f x_1 * x ^ x_1 - coeff f x_1 * y ^ x_1) =
Finset.sum (support f) fun x_1 => coeff f x_1 * ↑(powSubPowFactor x y x_1) * (x - y) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | congr with i | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } := by
refine' ⟨f.sum fun i r => r * (powSubPowFactor x y i).val, _⟩
delta eval; rw [eval₂_eq_sum, eval₂_eq_sum];
... | Mathlib.Data.Polynomial.Identities.101_0.o6IrpyrTENfZuiK | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } | Mathlib_Data_Polynomial_Identities |
case e_f.h
R : Type u
S : Type v
T : Type w
ι : Type x
k : Type y
A : Type z
a b : R
m n : ℕ
inst✝ : CommRing R
f : R[X]
x y : R
i : ℕ
⊢ coeff f i * x ^ i - coeff f i * y ^ i = coeff f i * ↑(powSubPowFactor x y i) * (x - y) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Data.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
#align_impor... | rw [mul_assoc, ← (powSubPowFactor x y _).prop, mul_sub] | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } := by
refine' ⟨f.sum fun i r => r * (powSubPowFactor x y i).val, _⟩
delta eval; rw [eval₂_eq_sum, eval₂_eq_sum];
... | Mathlib.Data.Polynomial.Identities.101_0.o6IrpyrTENfZuiK | /-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } | Mathlib_Data_Polynomial_Identities |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Mul M
inst✝ : ContinuousMul M
a b : M
⊢ 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [← map₂_mul, ← map_uncurry_prod, ← nhds_prod_eq] | @[to_additive]
theorem le_nhds_mul (a b : M) : 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) := by
| Mathlib.Topology.Algebra.Monoid.139_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem le_nhds_mul (a b : M) : 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Mul M
inst✝ : ContinuousMul M
a b : M
⊢ map (Function.uncurry fun x x_1 => x * x_1) (𝓝 (a, b)) ≤ 𝓝 (a * b) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact continuous_mul.tendsto _ | @[to_additive]
theorem le_nhds_mul (a b : M) : 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) := by
rw [← map₂_mul, ← map_uncurry_prod, ← nhds_prod_eq]
| Mathlib.Topology.Algebra.Monoid.139_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem le_nhds_mul (a b : M) : 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Mul M
inst✝⁵ : ContinuousMul M
inst✝⁴ : TopologicalSpace N
inst✝³ : Monoid N
inst✝² : ContinuousMul N
inst✝¹ : T2Space N
f : ι → Nˣ
r₁ r₂ : N
l : Filter ι
inst✝ : NeBot l
h₁ : Tendsto (fun x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | symm | /-- Construct a unit from limits of units and their inverses. -/
@[to_additive (attr := simps)
"Construct an additive unit from limits of additive units and their negatives."]
def Filter.Tendsto.units [TopologicalSpace N] [Monoid N] [ContinuousMul N] [T2Space N]
{f : ι → Nˣ} {r₁ r₂ : N} {l : Filter ι} [l.NeBot] (... | Mathlib.Topology.Algebra.Monoid.197_0.3p9EZf9ZWFxWOAq | /-- Construct a unit from limits of units and their inverses. -/
@[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Mul M
inst✝⁵ : ContinuousMul M
inst✝⁴ : TopologicalSpace N
inst✝³ : Monoid N
inst✝² : ContinuousMul N
inst✝¹ : T2Space N
f : ι → Nˣ
r₁ r₂ : N
l : Filter ι
inst✝ : NeBot l
h₁ : Tendsto (fun x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa using h₁.mul h₂ | /-- Construct a unit from limits of units and their inverses. -/
@[to_additive (attr := simps)
"Construct an additive unit from limits of additive units and their negatives."]
def Filter.Tendsto.units [TopologicalSpace N] [Monoid N] [ContinuousMul N] [T2Space N]
{f : ι → Nˣ} {r₁ r₂ : N} {l : Filter ι} [l.NeBot] (... | Mathlib.Topology.Algebra.Monoid.197_0.3p9EZf9ZWFxWOAq | /-- Construct a unit from limits of units and their inverses. -/
@[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Mul M
inst✝⁵ : ContinuousMul M
inst✝⁴ : TopologicalSpace N
inst✝³ : Monoid N
inst✝² : ContinuousMul N
inst✝¹ : T2Space N
f : ι → Nˣ
r₁ r₂ : N
l : Filter ι
inst✝ : NeBot l
h₁ : Tendsto (fun x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | symm | /-- Construct a unit from limits of units and their inverses. -/
@[to_additive (attr := simps)
"Construct an additive unit from limits of additive units and their negatives."]
def Filter.Tendsto.units [TopologicalSpace N] [Monoid N] [ContinuousMul N] [T2Space N]
{f : ι → Nˣ} {r₁ r₂ : N} {l : Filter ι} [l.NeBot] (... | Mathlib.Topology.Algebra.Monoid.197_0.3p9EZf9ZWFxWOAq | /-- Construct a unit from limits of units and their inverses. -/
@[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
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