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 |
|---|---|---|---|---|---|---|
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ p x✝ : P
⊢ x✝ ∈ ↑⊤ → x✝ ∈ ↑(perpBisector p p) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | simp [mem_perpBisector_iff_inner_eq_inner] | @[simp] theorem perpBisector_self (p : P) : perpBisector p p = ⊤ :=
top_unique <| fun _ ↦ by | Mathlib.Geometry.Euclidean.PerpBisector.110_0.WKtplj3xHYGfYbJ | @[simp] theorem perpBisector_self (p : P) : perpBisector p p = ⊤ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ perpBisector p₁ p₂ = ⊤ ↔ p₁ = p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | refine ⟨fun h ↦ ?_, fun h ↦ h ▸ perpBisector_self _⟩ | @[simp] theorem perpBisector_eq_top : perpBisector p₁ p₂ = ⊤ ↔ p₁ = p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.113_0.WKtplj3xHYGfYbJ | @[simp] theorem perpBisector_eq_top : perpBisector p₁ p₂ = ⊤ ↔ p₁ = p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
h : perpBisector p₁ p₂ = ⊤
⊢ p₁ = p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [← left_mem_perpBisector, h] | @[simp] theorem perpBisector_eq_top : perpBisector p₁ p₂ = ⊤ ↔ p₁ = p₂ := by
refine ⟨fun h ↦ ?_, fun h ↦ h ▸ perpBisector_self _⟩
| Mathlib.Geometry.Euclidean.PerpBisector.113_0.WKtplj3xHYGfYbJ | @[simp] theorem perpBisector_eq_top : perpBisector p₁ p₂ = ⊤ ↔ p₁ = p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
h : perpBisector p₁ p₂ = ⊤
⊢ p₁ ∈ ⊤ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | trivial | @[simp] theorem perpBisector_eq_top : perpBisector p₁ p₂ = ⊤ ↔ p₁ = p₂ := by
refine ⟨fun h ↦ ?_, fun h ↦ h ▸ perpBisector_self _⟩
rw [← left_mem_perpBisector, h]
| Mathlib.Geometry.Euclidean.PerpBisector.113_0.WKtplj3xHYGfYbJ | @[simp] theorem perpBisector_eq_top : perpBisector p₁ p₂ = ⊤ ↔ p₁ = p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ perpBisector p₁ p₂ ≠ ⊥ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [← nonempty_iff_ne_bot] | @[simp] theorem perpBisector_ne_bot : perpBisector p₁ p₂ ≠ ⊥ := by
| Mathlib.Geometry.Euclidean.PerpBisector.118_0.WKtplj3xHYGfYbJ | @[simp] theorem perpBisector_ne_bot : perpBisector p₁ p₂ ≠ ⊥ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c c₁ c₂ p₁ p₂ : P
⊢ Set.Nonempty ↑(perpBisector p₁ p₂) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | exact perpBisector_nonempty | @[simp] theorem perpBisector_ne_bot : perpBisector p₁ p₂ ≠ ⊥ := by
rw [← nonempty_iff_ne_bot]; | Mathlib.Geometry.Euclidean.PerpBisector.118_0.WKtplj3xHYGfYbJ | @[simp] theorem perpBisector_ne_bot : perpBisector p₁ p₂ ≠ ⊥ | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c₁ c₂ p₁ p₂ : P
hc₁ : dist p₁ c₁ = dist p₂ c₁
hc₂ : dist p₁ c₂ = dist p₂ c₂
⊢ inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0 | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rw [← Submodule.mem_orthogonal_singleton_iff_inner_left, ← direction_perpBisector] | /-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same applies to `c₂`. Then the
vector between `c₁` and `c₂` is orthogonal to that between `p₁` and `p₂`. (In two dimensions, this
says that the diagonals of a kite are orthogonal.) -/
theorem inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : ... | Mathlib.Geometry.Euclidean.PerpBisector.127_0.WKtplj3xHYGfYbJ | /-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same applies to `c₂`. Then the
vector between `c₁` and `c₂` is orthogonal to that between `p₁` and `p₂`. (In two dimensions, this
says that the diagonals of a kite are orthogonal.) -/
theorem inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : ... | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c₁ c₂ p₁ p₂ : P
hc₁ : dist p₁ c₁ = dist p₂ c₁
hc₂ : dist p₁ c₂ = dist p₂ c₂
⊢ c₂ -ᵥ c₁ ∈ direction (perpBisector p₁ p₂) | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | apply vsub_mem_direction | /-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same applies to `c₂`. Then the
vector between `c₁` and `c₂` is orthogonal to that between `p₁` and `p₂`. (In two dimensions, this
says that the diagonals of a kite are orthogonal.) -/
theorem inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : ... | Mathlib.Geometry.Euclidean.PerpBisector.127_0.WKtplj3xHYGfYbJ | /-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same applies to `c₂`. Then the
vector between `c₁` and `c₂` is orthogonal to that between `p₁` and `p₂`. (In two dimensions, this
says that the diagonals of a kite are orthogonal.) -/
theorem inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : ... | Mathlib_Geometry_Euclidean_PerpBisector |
case hp1
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c₁ c₂ p₁ p₂ : P
hc₁ : dist p₁ c₁ = dist p₂ c₁
hc₂ : dist p₁ c₂ = dist p₂ c₂
⊢ c₂ ∈ perpBisector p₁ p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rwa [mem_perpBisector_iff_dist_eq'] | /-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same applies to `c₂`. Then the
vector between `c₁` and `c₂` is orthogonal to that between `p₁` and `p₂`. (In two dimensions, this
says that the diagonals of a kite are orthogonal.) -/
theorem inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : ... | Mathlib.Geometry.Euclidean.PerpBisector.127_0.WKtplj3xHYGfYbJ | /-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same applies to `c₂`. Then the
vector between `c₁` and `c₂` is orthogonal to that between `p₁` and `p₂`. (In two dimensions, this
says that the diagonals of a kite are orthogonal.) -/
theorem inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : ... | Mathlib_Geometry_Euclidean_PerpBisector |
case hp2
V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c₁ c₂ p₁ p₂ : P
hc₁ : dist p₁ c₁ = dist p₂ c₁
hc₂ : dist p₁ c₂ = dist p₂ c₂
⊢ c₁ ∈ perpBisector p₁ p₂ | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | rwa [mem_perpBisector_iff_dist_eq'] | /-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same applies to `c₂`. Then the
vector between `c₁` and `c₂` is orthogonal to that between `p₁` and `p₂`. (In two dimensions, this
says that the diagonals of a kite are orthogonal.) -/
theorem inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : ... | Mathlib.Geometry.Euclidean.PerpBisector.127_0.WKtplj3xHYGfYbJ | /-- Suppose that `c₁` is equidistant from `p₁` and `p₂`, and the same applies to `c₂`. Then the
vector between `c₁` and `c₂` is orthogonal to that between `p₁` and `p₂`. (In two dimensions, this
says that the diagonals of a kite are orthogonal.) -/
theorem inner_vsub_vsub_of_dist_eq_of_dist_eq {c₁ c₂ p₁ p₂ : P} (hc₁ : ... | Mathlib_Geometry_Euclidean_PerpBisector |
V : Type u_4
P : Type u_1
V' : Type u_3
P' : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : NormedAddCommGroup V'
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : MetricSpace P'
inst✝ : NormedAddTorsor V' P'
f : P → P'
h : Isometry f
p₁ p₂ : P
... | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | ext x | theorem Isometry.preimage_perpBisector {f : P → P'} (h : Isometry f) (p₁ p₂ : P) :
f ⁻¹' (perpBisector (f p₁) (f p₂)) = perpBisector p₁ p₂ := by
| Mathlib.Geometry.Euclidean.PerpBisector.141_0.WKtplj3xHYGfYbJ | theorem Isometry.preimage_perpBisector {f : P → P'} (h : Isometry f) (p₁ p₂ : P) :
f ⁻¹' (perpBisector (f p₁) (f p₂)) = perpBisector p₁ p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
case h
V : Type u_4
P : Type u_1
V' : Type u_3
P' : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : NormedAddCommGroup V'
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : MetricSpace P'
inst✝ : NormedAddTorsor V' P'
f : P → P'
h : Isometry f
p₁ ... | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6f... | simp [mem_perpBisector_iff_dist_eq, h.dist_eq] | theorem Isometry.preimage_perpBisector {f : P → P'} (h : Isometry f) (p₁ p₂ : P) :
f ⁻¹' (perpBisector (f p₁) (f p₂)) = perpBisector p₁ p₂ := by
ext x; | Mathlib.Geometry.Euclidean.PerpBisector.141_0.WKtplj3xHYGfYbJ | theorem Isometry.preimage_perpBisector {f : P → P'} (h : Isometry f) (p₁ p₂ : P) :
f ⁻¹' (perpBisector (f p₁) (f p₂)) = perpBisector p₁ p₂ | Mathlib_Geometry_Euclidean_PerpBisector |
I : Type u
f : I → Type v
x y : (i : I) → f i
i✝ : I
α : Type u_1
inst✝² : (i : I) → SMul α (f i)
inst✝¹ : ∀ (i : I), Nonempty (f i)
i : I
inst✝ : FaithfulSMul α (f i)
m₁✝ m₂✝ : α
h : ∀ (a : (i : I) → f i), m₁✝ • a = m₂✝ • a
a : f i
⊢ m₁✝ • a = m₂✝ • a | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | classical
have :=
congr_fun (h <| Function.update (fun j => Classical.choice (‹∀ i, Nonempty (f i)› j)) i a)
i
simpa using this | /-- If `f i` has a faithful scalar action for a given `i`, then so does `Π i, f i`. This is
not an instance as `i` cannot be inferred. -/
@[to_additive
"If `f i` has a faithful additive action for a given `i`, then
so does `Π i, f i`. This is not an instance as `i` cannot be inferred"]
theorem faithfulSMul_at {α : ... | Mathlib.GroupTheory.GroupAction.Pi.102_0.o1qTP9EuP25B013 | /-- If `f i` has a faithful scalar action for a given `i`, then so does `Π i, f i`. This is
not an instance as `i` cannot be inferred. -/
@[to_additive
"If `f i` has a faithful additive action for a given `i`, then
so does `Π i, f i`. This is not an instance as `i` cannot be inferred"]
theorem faithfulSMul_at {α : ... | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i✝ : I
α : Type u_1
inst✝² : (i : I) → SMul α (f i)
inst✝¹ : ∀ (i : I), Nonempty (f i)
i : I
inst✝ : FaithfulSMul α (f i)
m₁✝ m₂✝ : α
h : ∀ (a : (i : I) → f i), m₁✝ • a = m₂✝ • a
a : f i
⊢ m₁✝ • a = m₂✝ • a | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | have :=
congr_fun (h <| Function.update (fun j => Classical.choice (‹∀ i, Nonempty (f i)› j)) i a)
i | /-- If `f i` has a faithful scalar action for a given `i`, then so does `Π i, f i`. This is
not an instance as `i` cannot be inferred. -/
@[to_additive
"If `f i` has a faithful additive action for a given `i`, then
so does `Π i, f i`. This is not an instance as `i` cannot be inferred"]
theorem faithfulSMul_at {α : ... | Mathlib.GroupTheory.GroupAction.Pi.102_0.o1qTP9EuP25B013 | /-- If `f i` has a faithful scalar action for a given `i`, then so does `Π i, f i`. This is
not an instance as `i` cannot be inferred. -/
@[to_additive
"If `f i` has a faithful additive action for a given `i`, then
so does `Π i, f i`. This is not an instance as `i` cannot be inferred"]
theorem faithfulSMul_at {α : ... | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i✝ : I
α : Type u_1
inst✝² : (i : I) → SMul α (f i)
inst✝¹ : ∀ (i : I), Nonempty (f i)
i : I
inst✝ : FaithfulSMul α (f i)
m₁✝ m₂✝ : α
h : ∀ (a : (i : I) → f i), m₁✝ • a = m₂✝ • a
a : f i
this :
(m₁✝ • Function.update (fun j => Classical.choice (_ : Nonempty (f j))) i a) i... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | simpa using this | /-- If `f i` has a faithful scalar action for a given `i`, then so does `Π i, f i`. This is
not an instance as `i` cannot be inferred. -/
@[to_additive
"If `f i` has a faithful additive action for a given `i`, then
so does `Π i, f i`. This is not an instance as `i` cannot be inferred"]
theorem faithfulSMul_at {α : ... | Mathlib.GroupTheory.GroupAction.Pi.102_0.o1qTP9EuP25B013 | /-- If `f i` has a faithful scalar action for a given `i`, then so does `Π i, f i`. This is
not an instance as `i` cannot be inferred. -/
@[to_additive
"If `f i` has a faithful additive action for a given `i`, then
so does `Π i, f i`. This is not an instance as `i` cannot be inferred"]
theorem faithfulSMul_at {α : ... | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → Zero (g i)
inst✝ : (i : I) → SMulZeroClass (f i) (g i)
⊢ ∀ (a : (i : I) → f i), a • 0 = 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | intros | instance smulZeroClass' {g : I → Type*} {n : ∀ i, Zero <| g i} [∀ i, SMulZeroClass (f i) (g i)] :
@SMulZeroClass (∀ i, f i) (∀ i : I, g i) (@Pi.instZero I g n) where
smul_zero := by | Mathlib.GroupTheory.GroupAction.Pi.151_0.o1qTP9EuP25B013 | instance smulZeroClass' {g : I → Type*} {n : ∀ i, Zero <| g i} [∀ i, SMulZeroClass (f i) (g i)] :
@SMulZeroClass (∀ i, f i) (∀ i : I, g i) (@Pi.instZero I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → Zero (g i)
inst✝ : (i : I) → SMulZeroClass (f i) (g i)
a✝ : (i : I) → f i
⊢ a✝ • 0 = 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | ext x | instance smulZeroClass' {g : I → Type*} {n : ∀ i, Zero <| g i} [∀ i, SMulZeroClass (f i) (g i)] :
@SMulZeroClass (∀ i, f i) (∀ i : I, g i) (@Pi.instZero I g n) where
smul_zero := by intros; | Mathlib.GroupTheory.GroupAction.Pi.151_0.o1qTP9EuP25B013 | instance smulZeroClass' {g : I → Type*} {n : ∀ i, Zero <| g i} [∀ i, SMulZeroClass (f i) (g i)] :
@SMulZeroClass (∀ i, f i) (∀ i : I, g i) (@Pi.instZero I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
case h
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → Zero (g i)
inst✝ : (i : I) → SMulZeroClass (f i) (g i)
a✝ : (i : I) → f i
x : I
⊢ (a✝ • 0) x = OfNat.ofNat 0 x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | exact smul_zero _ | instance smulZeroClass' {g : I → Type*} {n : ∀ i, Zero <| g i} [∀ i, SMulZeroClass (f i) (g i)] :
@SMulZeroClass (∀ i, f i) (∀ i : I, g i) (@Pi.instZero I g n) where
smul_zero := by intros; ext x; | Mathlib.GroupTheory.GroupAction.Pi.151_0.o1qTP9EuP25B013 | instance smulZeroClass' {g : I → Type*} {n : ∀ i, Zero <| g i} [∀ i, SMulZeroClass (f i) (g i)] :
@SMulZeroClass (∀ i, f i) (∀ i : I, g i) (@Pi.instZero I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → AddZeroClass (g i)
inst✝ : (i : I) → DistribSMul (f i) (g i)
⊢ ∀ (a : (i : I) → f i), a • 0 = 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | intros | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero := by | Mathlib.GroupTheory.GroupAction.Pi.162_0.o1qTP9EuP25B013 | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → AddZeroClass (g i)
inst✝ : (i : I) → DistribSMul (f i) (g i)
a✝ : (i : I) → f i
⊢ a✝ • 0 = 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | ext x | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero := by intros; | Mathlib.GroupTheory.GroupAction.Pi.162_0.o1qTP9EuP25B013 | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
case h
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → AddZeroClass (g i)
inst✝ : (i : I) → DistribSMul (f i) (g i)
a✝ : (i : I) → f i
x : I
⊢ (a✝ • 0) x = OfNat.ofNat 0 x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | exact smul_zero _ | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero := by intros; ext x; | Mathlib.GroupTheory.GroupAction.Pi.162_0.o1qTP9EuP25B013 | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → AddZeroClass (g i)
inst✝ : (i : I) → DistribSMul (f i) (g i)
⊢ ∀ (a : (i : I) → f i) (x y : (i : I) → g i), a • (x + y) = a • x + a • y | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | intros | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero := by intros; ext x; exact smul_zero _
smul_add := by | Mathlib.GroupTheory.GroupAction.Pi.162_0.o1qTP9EuP25B013 | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → AddZeroClass (g i)
inst✝ : (i : I) → DistribSMul (f i) (g i)
a✝ : (i : I) → f i
x✝ y✝ : (i : I) → g i
⊢ a✝ • (x✝ + y✝) = a✝ • x✝ + a✝ • y✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | ext x | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero := by intros; ext x; exact smul_zero _
smul_add := by intros; | Mathlib.GroupTheory.GroupAction.Pi.162_0.o1qTP9EuP25B013 | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
case h
I : Type u
f : I → Type v
x✝¹ y : (i : I) → f i
i : I
g : I → Type u_1
n : (i : I) → AddZeroClass (g i)
inst✝ : (i : I) → DistribSMul (f i) (g i)
a✝ : (i : I) → f i
x✝ y✝ : (i : I) → g i
x : I
⊢ (a✝ • (x✝ + y✝)) x = (a✝ • x✝ + a✝ • y✝) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | exact smul_add _ _ _ | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero := by intros; ext x; exact smul_zero _
smul_add := by intros; ext x; | Mathlib.GroupTheory.GroupAction.Pi.162_0.o1qTP9EuP25B013 | instance distribSMul' {g : I → Type*} {n : ∀ i, AddZeroClass <| g i}
[∀ i, DistribSMul (f i) (g i)] :
@DistribSMul (∀ i, f i) (∀ i : I, g i) (@Pi.addZeroClass I g n) where
smul_zero | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
m : (i : I) → Monoid (f i)
n : (i : I) → Monoid (g i)
inst✝ : (i : I) → MulDistribMulAction (f i) (g i)
⊢ ∀ (r : (i : I) → f i) (x y : (i : I) → g i), r • (x * y) = r • x * r • y | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | intros | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul := by
| Mathlib.GroupTheory.GroupAction.Pi.207_0.o1qTP9EuP25B013 | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
m : (i : I) → Monoid (f i)
n : (i : I) → Monoid (g i)
inst✝ : (i : I) → MulDistribMulAction (f i) (g i)
r✝ : (i : I) → f i
x✝ y✝ : (i : I) → g i
⊢ r✝ • (x✝ * y✝) = r✝ • x✝ * r✝ • y✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | ext x | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul := by
intros
| Mathlib.GroupTheory.GroupAction.Pi.207_0.o1qTP9EuP25B013 | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul | Mathlib_GroupTheory_GroupAction_Pi |
case h
I : Type u
f : I → Type v
x✝¹ y : (i : I) → f i
i : I
g : I → Type u_1
m : (i : I) → Monoid (f i)
n : (i : I) → Monoid (g i)
inst✝ : (i : I) → MulDistribMulAction (f i) (g i)
r✝ : (i : I) → f i
x✝ y✝ : (i : I) → g i
x : I
⊢ (r✝ • (x✝ * y✝)) x = (r✝ • x✝ * r✝ • y✝) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | apply smul_mul' | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul := by
intros
ext x
| Mathlib.GroupTheory.GroupAction.Pi.207_0.o1qTP9EuP25B013 | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
m : (i : I) → Monoid (f i)
n : (i : I) → Monoid (g i)
inst✝ : (i : I) → MulDistribMulAction (f i) (g i)
⊢ ∀ (r : (i : I) → f i), r • 1 = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | intros | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul := by
intros
ext x
apply smul_mul'
smul_one := by
| Mathlib.GroupTheory.GroupAction.Pi.207_0.o1qTP9EuP25B013 | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f : I → Type v
x y : (i : I) → f i
i : I
g : I → Type u_1
m : (i : I) → Monoid (f i)
n : (i : I) → Monoid (g i)
inst✝ : (i : I) → MulDistribMulAction (f i) (g i)
r✝ : (i : I) → f i
⊢ r✝ • 1 = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | ext x | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul := by
intros
ext x
apply smul_mul'
smul_one := by
intros... | Mathlib.GroupTheory.GroupAction.Pi.207_0.o1qTP9EuP25B013 | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul | Mathlib_GroupTheory_GroupAction_Pi |
case h
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i : I
g : I → Type u_1
m : (i : I) → Monoid (f i)
n : (i : I) → Monoid (g i)
inst✝ : (i : I) → MulDistribMulAction (f i) (g i)
r✝ : (i : I) → f i
x : I
⊢ (r✝ • 1) x = OfNat.ofNat 1 x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | apply smul_one | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul := by
intros
ext x
apply smul_mul'
smul_one := by
intros... | Mathlib.GroupTheory.GroupAction.Pi.207_0.o1qTP9EuP25B013 | instance mulDistribMulAction' {g : I → Type*} {m : ∀ i, Monoid (f i)} {n : ∀ i, Monoid <| g i}
[∀ i, MulDistribMulAction (f i) (g i)] :
@MulDistribMulAction (∀ i, f i) (∀ i : I, g i) (@Pi.monoid I f m) (@Pi.monoid I g n) where
smul_mul | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f✝ : I → Type v
x✝ y : (i : I) → f✝ i
i : I
R : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : SMul R γ
r : R
f : α → β
g : α → γ
e : β → γ
x : β
⊢ extend f (r • g) (r • e) x = (r • extend f g e) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | haveI : Decidable (∃ a : α, f a = x) := Classical.propDecidable _ | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e :=
funext fun x => by
-- Porting note: Lean4 is unable to automatically call `Classical.propDecidable`
| Mathlib.GroupTheory.GroupAction.Pi.267_0.o1qTP9EuP25B013 | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f✝ : I → Type v
x✝ y : (i : I) → f✝ i
i : I
R : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : SMul R γ
r : R
f : α → β
g : α → γ
e : β → γ
x : β
this : Decidable (∃ a, f a = x)
⊢ extend f (r • g) (r • e) x = (r • extend f g e) x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | rw [extend_def, Pi.smul_apply, Pi.smul_apply, extend_def] | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e :=
funext fun x => by
-- Porting note: Lean4 is unable to automatically call `Classical.propDecidable`
haveI : Decidable (∃ a : α... | Mathlib.GroupTheory.GroupAction.Pi.267_0.o1qTP9EuP25B013 | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e | Mathlib_GroupTheory_GroupAction_Pi |
I : Type u
f✝ : I → Type v
x✝ y : (i : I) → f✝ i
i : I
R : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : SMul R γ
r : R
f : α → β
g : α → γ
e : β → γ
x : β
this : Decidable (∃ a, f a = x)
⊢ (if h : ∃ a, f a = x then (r • g) (Classical.choose h) else r • e x) =
r • if h : ∃ a, f a = x then g (Classical.cho... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | split_ifs | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e :=
funext fun x => by
-- Porting note: Lean4 is unable to automatically call `Classical.propDecidable`
haveI : Decidable (∃ a : α... | Mathlib.GroupTheory.GroupAction.Pi.267_0.o1qTP9EuP25B013 | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e | Mathlib_GroupTheory_GroupAction_Pi |
case pos
I : Type u
f✝ : I → Type v
x✝ y : (i : I) → f✝ i
i : I
R : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : SMul R γ
r : R
f : α → β
g : α → γ
e : β → γ
x : β
this : Decidable (∃ a, f a = x)
h✝ : ∃ a, f a = x
⊢ (r • g) (Classical.choose h✝) = r • g (Classical.choose h✝) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | rfl | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e :=
funext fun x => by
-- Porting note: Lean4 is unable to automatically call `Classical.propDecidable`
haveI : Decidable (∃ a : α... | Mathlib.GroupTheory.GroupAction.Pi.267_0.o1qTP9EuP25B013 | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e | Mathlib_GroupTheory_GroupAction_Pi |
case neg
I : Type u
f✝ : I → Type v
x✝ y : (i : I) → f✝ i
i : I
R : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : SMul R γ
r : R
f : α → β
g : α → γ
e : β → γ
x : β
this : Decidable (∃ a, f a = x)
h✝ : ¬∃ a, f a = x
⊢ r • e x = r • e x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.Group.Pi
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.group_action.pi from "leanprover-community/mathlib"@... | rfl | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e :=
funext fun x => by
-- Porting note: Lean4 is unable to automatically call `Classical.propDecidable`
haveI : Decidable (∃ a : α... | Mathlib.GroupTheory.GroupAction.Pi.267_0.o1qTP9EuP25B013 | @[to_additive]
theorem Function.extend_smul {R α β γ : Type*} [SMul R γ] (r : R) (f : α → β) (g : α → γ)
(e : β → γ) : Function.extend f (r • g) (r • e) = r • Function.extend f g e | Mathlib_GroupTheory_GroupAction_Pi |
α : Type u_1
s : Set α
a : α
⊢ op a ∈ Set.op s ↔ a ∈ s | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | rfl | @[simp 1100]
theorem op_mem_op {s : Set α} {a : α} : op a ∈ s.op ↔ a ∈ s := by | Mathlib.Data.Set.Opposite.38_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem op_mem_op {s : Set α} {a : α} : op a ∈ s.op ↔ a ∈ s | Mathlib_Data_Set_Opposite |
α : Type u_1
s : Set αᵒᵖ
a : αᵒᵖ
⊢ a.unop ∈ Set.unop s ↔ a ∈ s | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | rfl | @[simp 1100]
theorem unop_mem_unop {s : Set αᵒᵖ} {a : αᵒᵖ} : unop a ∈ s.unop ↔ a ∈ s := by | Mathlib.Data.Set.Opposite.47_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem unop_mem_unop {s : Set αᵒᵖ} {a : αᵒᵖ} : unop a ∈ s.unop ↔ a ∈ s | Mathlib_Data_Set_Opposite |
α : Type u_1
x : α
⊢ Set.op {x} = {op x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | ext | @[simp]
theorem singleton_op (x : α) : ({x} : Set α).op = {op x} := by
| Mathlib.Data.Set.Opposite.75_0.2CXAU37XQBxMDqP | @[simp]
theorem singleton_op (x : α) : ({x} : Set α).op = {op x} | Mathlib_Data_Set_Opposite |
case h
α : Type u_1
x : α
x✝ : αᵒᵖ
⊢ x✝ ∈ Set.op {x} ↔ x✝ ∈ {op x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | constructor | @[simp]
theorem singleton_op (x : α) : ({x} : Set α).op = {op x} := by
ext
| Mathlib.Data.Set.Opposite.75_0.2CXAU37XQBxMDqP | @[simp]
theorem singleton_op (x : α) : ({x} : Set α).op = {op x} | Mathlib_Data_Set_Opposite |
case h.mp
α : Type u_1
x : α
x✝ : αᵒᵖ
⊢ x✝ ∈ Set.op {x} → x✝ ∈ {op x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | apply unop_injective | @[simp]
theorem singleton_op (x : α) : ({x} : Set α).op = {op x} := by
ext
constructor
· | Mathlib.Data.Set.Opposite.75_0.2CXAU37XQBxMDqP | @[simp]
theorem singleton_op (x : α) : ({x} : Set α).op = {op x} | Mathlib_Data_Set_Opposite |
case h.mpr
α : Type u_1
x : α
x✝ : αᵒᵖ
⊢ x✝ ∈ {op x} → x✝ ∈ Set.op {x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | apply op_injective | @[simp]
theorem singleton_op (x : α) : ({x} : Set α).op = {op x} := by
ext
constructor
· apply unop_injective
· | Mathlib.Data.Set.Opposite.75_0.2CXAU37XQBxMDqP | @[simp]
theorem singleton_op (x : α) : ({x} : Set α).op = {op x} | Mathlib_Data_Set_Opposite |
α : Type u_1
x : αᵒᵖ
⊢ Set.unop {x} = {x.unop} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | ext | @[simp]
theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} := by
| Mathlib.Data.Set.Opposite.83_0.2CXAU37XQBxMDqP | @[simp]
theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} | Mathlib_Data_Set_Opposite |
case h
α : Type u_1
x : αᵒᵖ
x✝ : α
⊢ x✝ ∈ Set.unop {x} ↔ x✝ ∈ {x.unop} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | constructor | @[simp]
theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} := by
ext
| Mathlib.Data.Set.Opposite.83_0.2CXAU37XQBxMDqP | @[simp]
theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} | Mathlib_Data_Set_Opposite |
case h.mp
α : Type u_1
x : αᵒᵖ
x✝ : α
⊢ x✝ ∈ Set.unop {x} → x✝ ∈ {x.unop} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | apply op_injective | @[simp]
theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} := by
ext
constructor
· | Mathlib.Data.Set.Opposite.83_0.2CXAU37XQBxMDqP | @[simp]
theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} | Mathlib_Data_Set_Opposite |
case h.mpr
α : Type u_1
x : αᵒᵖ
x✝ : α
⊢ x✝ ∈ {x.unop} → x✝ ∈ Set.unop {x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | apply unop_injective | @[simp]
theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} := by
ext
constructor
· apply op_injective
· | Mathlib.Data.Set.Opposite.83_0.2CXAU37XQBxMDqP | @[simp]
theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} | Mathlib_Data_Set_Opposite |
α : Type u_1
x : α
⊢ Set.unop {op x} = {x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | ext | @[simp 1100]
theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} := by
| Mathlib.Data.Set.Opposite.91_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} | Mathlib_Data_Set_Opposite |
case h
α : Type u_1
x x✝ : α
⊢ x✝ ∈ Set.unop {op x} ↔ x✝ ∈ {x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | constructor | @[simp 1100]
theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} := by
ext
| Mathlib.Data.Set.Opposite.91_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} | Mathlib_Data_Set_Opposite |
case h.mp
α : Type u_1
x x✝ : α
⊢ x✝ ∈ Set.unop {op x} → x✝ ∈ {x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | apply op_injective | @[simp 1100]
theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} := by
ext
constructor
· | Mathlib.Data.Set.Opposite.91_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} | Mathlib_Data_Set_Opposite |
case h.mpr
α : Type u_1
x x✝ : α
⊢ x✝ ∈ {x} → x✝ ∈ Set.unop {op x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | apply unop_injective | @[simp 1100]
theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} := by
ext
constructor
· apply op_injective
· | Mathlib.Data.Set.Opposite.91_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} | Mathlib_Data_Set_Opposite |
α : Type u_1
x : αᵒᵖ
⊢ Set.op {x.unop} = {x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | ext | @[simp 1100]
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} := by
| Mathlib.Data.Set.Opposite.99_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} | Mathlib_Data_Set_Opposite |
case h
α : Type u_1
x x✝ : αᵒᵖ
⊢ x✝ ∈ Set.op {x.unop} ↔ x✝ ∈ {x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | constructor | @[simp 1100]
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} := by
ext
| Mathlib.Data.Set.Opposite.99_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} | Mathlib_Data_Set_Opposite |
case h.mp
α : Type u_1
x x✝ : αᵒᵖ
⊢ x✝ ∈ Set.op {x.unop} → x✝ ∈ {x} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | apply unop_injective | @[simp 1100]
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} := by
ext
constructor
· | Mathlib.Data.Set.Opposite.99_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} | Mathlib_Data_Set_Opposite |
case h.mpr
α : Type u_1
x x✝ : αᵒᵖ
⊢ x✝ ∈ {x} → x✝ ∈ Set.op {x.unop} | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Image
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0... | apply op_injective | @[simp 1100]
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} := by
ext
constructor
· apply unop_injective
· | Mathlib.Data.Set.Opposite.99_0.2CXAU37XQBxMDqP | @[simp 1100]
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} | Mathlib_Data_Set_Opposite |
n : ℕ
⊢ bodd (succ n) = !bodd n | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp only [bodd, boddDiv2] | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) := by
| Mathlib.Init.Data.Nat.Bitwise.67_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
⊢ (match boddDiv2 n with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).fst =
!(boddDiv2 n).fst | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | let ⟨b,m⟩ := boddDiv2 n | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) := by
simp only [bodd, boddDiv2]
| Mathlib.Init.Data.Nat.Bitwise.67_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
b : Bool
m : ℕ
⊢ (match (b, m) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).fst =
!(b, m).fst | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases b | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) := by
simp only [bodd, boddDiv2]
let ⟨b,m⟩ := boddDiv2 n
| Mathlib.Init.Data.Nat.Bitwise.67_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case false
n m : ℕ
⊢ (match (false, m) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).fst =
!(false, m).fst | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) := by
simp only [bodd, boddDiv2]
let ⟨b,m⟩ := boddDiv2 n
cases b <;> | Mathlib.Init.Data.Nat.Bitwise.67_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case true
n m : ℕ
⊢ (match (true, m) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).fst =
!(true, m).fst | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) := by
simp only [bodd, boddDiv2]
let ⟨b,m⟩ := boddDiv2 n
cases b <;> | Mathlib.Init.Data.Nat.Bitwise.67_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_succ (n : ℕ) : bodd (succ n) = not (bodd n) | Mathlib_Init_Data_Nat_Bitwise |
m n : ℕ
⊢ bodd (m + n) = bxor (bodd m) (bodd n) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | induction n | @[simp]
theorem bodd_add (m n : ℕ) : bodd (m + n) = bxor (bodd m) (bodd n) := by
| Mathlib.Init.Data.Nat.Bitwise.74_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_add (m n : ℕ) : bodd (m + n) = bxor (bodd m) (bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case zero
m : ℕ
⊢ bodd (m + zero) = bxor (bodd m) (bodd zero) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp_all [add_succ, Bool.xor_not] | @[simp]
theorem bodd_add (m n : ℕ) : bodd (m + n) = bxor (bodd m) (bodd n) := by
induction n <;> | Mathlib.Init.Data.Nat.Bitwise.74_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_add (m n : ℕ) : bodd (m + n) = bxor (bodd m) (bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ
m n✝ : ℕ
n_ih✝ : bodd (m + n✝) = bxor (bodd m) (bodd n✝)
⊢ bodd (m + succ n✝) = bxor (bodd m) (bodd (succ n✝)) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp_all [add_succ, Bool.xor_not] | @[simp]
theorem bodd_add (m n : ℕ) : bodd (m + n) = bxor (bodd m) (bodd n) := by
induction n <;> | Mathlib.Init.Data.Nat.Bitwise.74_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_add (m n : ℕ) : bodd (m + n) = bxor (bodd m) (bodd n) | Mathlib_Init_Data_Nat_Bitwise |
m n : ℕ
⊢ bodd (m * n) = (bodd m && bodd n) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | induction' n with n IH | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
| Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case zero
m : ℕ
⊢ bodd (m * zero) = (bodd m && bodd zero) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· | Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ
m n : ℕ
IH : bodd (m * n) = (bodd m && bodd n)
⊢ bodd (m * succ n) = (bodd m && bodd (succ n)) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp [mul_succ, IH] | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· simp
· | Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ
m n : ℕ
IH : bodd (m * n) = (bodd m && bodd n)
⊢ bxor (bodd m && bodd n) (bodd m) = (bodd m && !bodd n) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases bodd m | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· simp
· simp [mul_succ, IH]
| Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ.false
m n : ℕ
IH : bodd (m * n) = (bodd m && bodd n)
⊢ bxor (false && bodd n) false = (false && !bodd n) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases bodd n | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· simp
· simp [mul_succ, IH]
cases bodd m <;> | Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ.true
m n : ℕ
IH : bodd (m * n) = (bodd m && bodd n)
⊢ bxor (true && bodd n) true = (true && !bodd n) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases bodd n | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· simp
· simp [mul_succ, IH]
cases bodd m <;> | Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ.false.false
m n : ℕ
IH : bodd (m * n) = (bodd m && bodd n)
⊢ bxor (false && false) false = (false && !false) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· simp
· simp [mul_succ, IH]
cases bodd m <;> cases bodd n <;> | Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ.false.true
m n : ℕ
IH : bodd (m * n) = (bodd m && bodd n)
⊢ bxor (false && true) false = (false && !true) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· simp
· simp [mul_succ, IH]
cases bodd m <;> cases bodd n <;> | Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ.true.false
m n : ℕ
IH : bodd (m * n) = (bodd m && bodd n)
⊢ bxor (true && false) true = (true && !false) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· simp
· simp [mul_succ, IH]
cases bodd m <;> cases bodd n <;> | Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
case succ.true.true
m n : ℕ
IH : bodd (m * n) = (bodd m && bodd n)
⊢ bxor (true && true) true = (true && !true) | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) := by
induction' n with n IH
· simp
· simp [mul_succ, IH]
cases bodd m <;> cases bodd n <;> | Mathlib.Init.Data.Nat.Bitwise.79_0.OFUBkIQvV236FCW | @[simp]
theorem bodd_mul (m n : ℕ) : bodd (m * n) = (bodd m && bodd n) | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
⊢ n % 2 = bif bodd n then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | have := congr_arg bodd (mod_add_div n 2) | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
| Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2 + 2 * (n / 2)) = bodd n
⊢ n % 2 = bif bodd n then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
| Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2 + 2 * (n / 2)) = bodd n
⊢ n % 2 = bif bodd n then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2) = bodd n
⊢ n % 2 = bif bodd n then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | have _ : ∀ b, and false b = false := by
intro b
cases b <;> rfl | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
| Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2) = bodd n
⊢ ∀ (b : Bool), (false && b) = false | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | intro b | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
| Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2) = bodd n
b : Bool
⊢ (false && b) = false | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases b | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
| Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
case false
n : ℕ
this : bodd (n % 2) = bodd n
⊢ (false && false) = false | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
case true
n : ℕ
this : bodd (n % 2) = bodd n
⊢ (false && true) = false | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2) = bodd n
x✝ : ∀ (b : Bool), (false && b) = false
⊢ n % 2 = bif bodd n then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | have _ : ∀ b, bxor b false = b := by
intro b
cases b <;> rfl | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2) = bodd n
x✝ : ∀ (b : Bool), (false && b) = false
⊢ ∀ (b : Bool), bxor b false = b | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | intro b | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2) = bodd n
x✝ : ∀ (b : Bool), (false && b) = false
b : Bool
⊢ bxor b false = b | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases b | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
case false
n : ℕ
this : bodd (n % 2) = bodd n
x✝ : ∀ (b : Bool), (false && b) = false
⊢ bxor false false = false | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
case true
n : ℕ
this : bodd (n % 2) = bodd n
x✝ : ∀ (b : Bool), (false && b) = false
⊢ bxor true false = true | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2) = bodd n
x✝¹ : ∀ (b : Bool), (false && b) = false
x✝ : ∀ (b : Bool), bxor b false = b
⊢ n % 2 = bif bodd n then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rw [← this] | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
this : bodd (n % 2) = bodd n
x✝¹ : ∀ (b : Bool), (false && b) = false
x✝ : ∀ (b : Bool), bxor b false = b
⊢ n % 2 = bif bodd (n % 2) then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases' mod_two_eq_zero_or_one n with h h | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
case inl
n : ℕ
this : bodd (n % 2) = bodd n
x✝¹ : ∀ (b : Bool), (false && b) = false
x✝ : ∀ (b : Bool), bxor b false = b
h : n % 2 = 0
⊢ n % 2 = bif bodd (n % 2) then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rw [h] | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
case inr
n : ℕ
this : bodd (n % 2) = bodd n
x✝¹ : ∀ (b : Bool), (false && b) = false
x✝ : ∀ (b : Bool), bxor b false = b
h : n % 2 = 1
⊢ n % 2 = bif bodd (n % 2) then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rw [h] | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
case inl
n : ℕ
this : bodd (n % 2) = bodd n
x✝¹ : ∀ (b : Bool), (false && b) = false
x✝ : ∀ (b : Bool), bxor b false = b
h : n % 2 = 0
⊢ 0 = bif bodd 0 then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
case inr
n : ℕ
this : bodd (n % 2) = bodd n
x✝¹ : ∀ (b : Bool), (false && b) = false
x✝ : ∀ (b : Bool), bxor b false = b
h : n % 2 = 1
⊢ 1 = bif bodd 1 then 1 else 0 | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | rfl | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this
says simp only [bodd_add, bodd_mul, bodd_succ, not, bodd_zero, Bool.false_and,
Bool.xor_false] at this
have _ : ∀ b, and false b = false := by
intro b
cases b <;> rf... | Mathlib.Init.Data.Nat.Bitwise.87_0.OFUBkIQvV236FCW | theorem mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
⊢ div2 (succ n) = bif bodd n then succ (div2 n) else div2 n | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp only [bodd, boddDiv2, div2] | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
⊢ (match boddDiv2 n with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).snd =
bif (boddDiv2 n).fst then succ (boddDiv2 n).snd else (boddDiv2 n).snd | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases' boddDiv2 n with fst snd | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
simp only [bodd, boddDiv2, div2]
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
case mk
n : ℕ
fst : Bool
snd : ℕ
⊢ (match (fst, snd) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).snd =
bif (fst, snd).fst then succ (fst, snd).snd else (fst, snd).snd | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | cases fst | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
simp only [bodd, boddDiv2, div2]
cases' boddDiv2 n with fst snd
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
case mk.false
n snd : ℕ
⊢ (match (false, snd) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).snd =
bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd
case mk.true
n snd : ℕ
⊢ (match (true, snd) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).... | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | case mk.false =>
simp | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
simp only [bodd, boddDiv2, div2]
cases' boddDiv2 n with fst snd
cases fst
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
n snd : ℕ
⊢ (match (false, snd) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).snd =
bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | case mk.false =>
simp | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
simp only [bodd, boddDiv2, div2]
cases' boddDiv2 n with fst snd
cases fst
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
n snd : ℕ
⊢ (match (false, snd) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).snd =
bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
simp only [bodd, boddDiv2, div2]
cases' boddDiv2 n with fst snd
cases fst
case mk.false =>
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
case mk.true
n snd : ℕ
⊢ (match (true, snd) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).snd =
bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | case mk.true =>
simp | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
simp only [bodd, boddDiv2, div2]
cases' boddDiv2 n with fst snd
cases fst
case mk.false =>
simp
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
n snd : ℕ
⊢ (match (true, snd) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).snd =
bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | case mk.true =>
simp | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
simp only [bodd, boddDiv2, div2]
cases' boddDiv2 n with fst snd
cases fst
case mk.false =>
simp
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
n snd : ℕ
⊢ (match (true, snd) with
| (false, m) => (true, m)
| (true, m) => (false, succ m)).snd =
bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
simp only [bodd, boddDiv2, div2]
cases' boddDiv2 n with fst snd
cases fst
case mk.false =>
simp
case mk.true =>
| Mathlib.Init.Data.Nat.Bitwise.115_0.OFUBkIQvV236FCW | @[simp]
theorem div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) | Mathlib_Init_Data_Nat_Bitwise |
n : ℕ
⊢ (bif bodd (succ n) then 1 else 0) + 2 * div2 (succ n) = succ n | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Lemmas
import Init.WFTactics
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Init.ZeroOne
impor... | simp only [bodd_succ, Bool.cond_not, div2_succ, Nat.mul_comm] | theorem bodd_add_div2 : ∀ n, cond (bodd n) 1 0 + 2 * div2 n = n
| 0 => rfl
| succ n => by
| Mathlib.Init.Data.Nat.Bitwise.128_0.OFUBkIQvV236FCW | theorem bodd_add_div2 : ∀ n, cond (bodd n) 1 0 + 2 * div2 n = n
| 0 => rfl
| succ n => by
simp only [bodd_succ, Bool.cond_not, div2_succ, Nat.mul_comm]
refine' Eq.trans _ (congr_arg succ (bodd_add_div2 n))
cases bodd n <;> simp [cond, not]
· rw [Nat.add_comm, Nat.add_succ]
· rw [succ_mul, Nat.ad... | Mathlib_Init_Data_Nat_Bitwise |
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