state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
s✝ t : Set α
s : Set β
h : ∀ (x : α), f x ∉ s
⊢ ∀ x ∈ s, ∀ (x_1 : α), ¬f x_1 = x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | intro y hy x hx | theorem preimage_eq_empty_iff {s : Set β} : f ⁻¹' s = ∅ ↔ Disjoint s (range f) :=
⟨fun h => by
simp only [eq_empty_iff_forall_not_mem, disjoint_iff_inter_eq_empty, not_exists, mem_inter_iff,
not_and, mem_range, mem_preimage] at h ⊢
| Mathlib.Data.Set.Image.1653_0.IJFiTzmYGOCpPSd | theorem preimage_eq_empty_iff {s : Set β} : f ⁻¹' s = ∅ ↔ Disjoint s (range f) | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
s✝ t : Set α
s : Set β
h : ∀ (x : α), f x ∉ s
y : β
hy : y ∈ s
x : α
hx : f x = y
⊢ False | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← hx] at hy | theorem preimage_eq_empty_iff {s : Set β} : f ⁻¹' s = ∅ ↔ Disjoint s (range f) :=
⟨fun h => by
simp only [eq_empty_iff_forall_not_mem, disjoint_iff_inter_eq_empty, not_exists, mem_inter_iff,
not_and, mem_range, mem_preimage] at h ⊢
intro y hy x hx
| Mathlib.Data.Set.Image.1653_0.IJFiTzmYGOCpPSd | theorem preimage_eq_empty_iff {s : Set β} : f ⁻¹' s = ∅ ↔ Disjoint s (range f) | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
s✝ t : Set α
s : Set β
h : ∀ (x : α), f x ∉ s
y : β
x : α
hy : f x ∈ s
hx : f x = y
⊢ False | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact h x hy | theorem preimage_eq_empty_iff {s : Set β} : f ⁻¹' s = ∅ ↔ Disjoint s (range f) :=
⟨fun h => by
simp only [eq_empty_iff_forall_not_mem, disjoint_iff_inter_eq_empty, not_exists, mem_inter_iff,
not_and, mem_range, mem_preimage] at h ⊢
intro y hy x hx
rw [← hx] at hy
| Mathlib.Data.Set.Image.1653_0.IJFiTzmYGOCpPSd | theorem preimage_eq_empty_iff {s : Set β} : f ⁻¹' s = ∅ ↔ Disjoint s (range f) | Mathlib_Data_Set_Image |
α : Type u_1
β : α → Type u_2
i j : α
s : Set (β i)
h : i ≠ j
⊢ Sigma.mk j ⁻¹' (Sigma.mk i '' s) = ∅ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp [image, h] | lemma sigma_mk_preimage_image' (h : i ≠ j) : Sigma.mk j ⁻¹' (Sigma.mk i '' s) = ∅ := by
| Mathlib.Data.Set.Image.1671_0.IJFiTzmYGOCpPSd | lemma sigma_mk_preimage_image' (h : i ≠ j) : Sigma.mk j ⁻¹' (Sigma.mk i '' s) = ∅ | Mathlib_Data_Set_Image |
α : Type u_1
β : α → Type u_2
i j : α
s : Set (β i)
⊢ Sigma.mk i ⁻¹' (Sigma.mk i '' s) = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp [image] | lemma sigma_mk_preimage_image_eq_self : Sigma.mk i ⁻¹' (Sigma.mk i '' s) = s := by
| Mathlib.Data.Set.Image.1674_0.IJFiTzmYGOCpPSd | lemma sigma_mk_preimage_image_eq_self : Sigma.mk i ⁻¹' (Sigma.mk i '' s) = s | Mathlib_Data_Set_Image |
o : Ordinal.{u_1}
⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1);
nim o =
mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim] | theorem nim_def (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
nim o =
PGame.mk o.out.α o.out.α (fun o₂ => nim (Ordinal.typein (· < ·) o₂)) fun o₂ =>
nim (Ordinal.typein (· < ·) o₂) := by
| Mathlib.SetTheory.Game.Nim.59_0.mmFMhRYSjViKjcP | theorem nim_def (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1);
(mk (Quotient.out o).α (Quotient.out o).α
(fun o₂ =>
let_fun x := (_ : typein (fun x x_1 => x < x_1) o₂ < o);
nim (typein (Quotient.out o).r o₂))
fun o₂ =>
let_fun x := (_ : typein (fun x x... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rfl | theorem nim_def (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
nim o =
PGame.mk o.out.α o.out.α (fun o₂ => nim (Ordinal.typein (· < ·) o₂)) fun o₂ =>
nim (Ordinal.typein (· < ·) o₂) := by
rw [nim]; | Mathlib.SetTheory.Game.Nim.59_0.mmFMhRYSjViKjcP | theorem nim_def (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ LeftMoves (nim o) = (Quotient.out o).α | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | theorem leftMoves_nim (o : Ordinal) : (nim o).LeftMoves = o.out.α := by | Mathlib.SetTheory.Game.Nim.67_0.mmFMhRYSjViKjcP | theorem leftMoves_nim (o : Ordinal) : (nim o).LeftMoves = o.out.α | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ LeftMoves
(mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)) =
(Quotient.out o).α | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rfl | theorem leftMoves_nim (o : Ordinal) : (nim o).LeftMoves = o.out.α := by rw [nim_def]; | Mathlib.SetTheory.Game.Nim.67_0.mmFMhRYSjViKjcP | theorem leftMoves_nim (o : Ordinal) : (nim o).LeftMoves = o.out.α | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ RightMoves (nim o) = (Quotient.out o).α | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | theorem rightMoves_nim (o : Ordinal) : (nim o).RightMoves = o.out.α := by | Mathlib.SetTheory.Game.Nim.70_0.mmFMhRYSjViKjcP | theorem rightMoves_nim (o : Ordinal) : (nim o).RightMoves = o.out.α | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ RightMoves
(mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)) =
(Quotient.out o).α | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rfl | theorem rightMoves_nim (o : Ordinal) : (nim o).RightMoves = o.out.α := by rw [nim_def]; | Mathlib.SetTheory.Game.Nim.70_0.mmFMhRYSjViKjcP | theorem rightMoves_nim (o : Ordinal) : (nim o).RightMoves = o.out.α | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1);
HEq (moveLeft (nim o)) fun i => nim (typein (fun x x_1 => x < x_1) i) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
HEq (nim o).moveLeft fun i : o.out.α => nim (typein (· < ·) i) := by | Mathlib.SetTheory.Game.Nim.73_0.mmFMhRYSjViKjcP | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1);
HEq
(moveLeft
(mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)))
fun i => nim (typein (fun x x_1 => x < x_1)... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rfl | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
HEq (nim o).moveLeft fun i : o.out.α => nim (typein (· < ·) i) := by rw [nim_def]; | Mathlib.SetTheory.Game.Nim.73_0.mmFMhRYSjViKjcP | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1);
HEq (moveRight (nim o)) fun i => nim (typein (fun x x_1 => x < x_1) i) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | theorem moveRight_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
HEq (nim o).moveRight fun i : o.out.α => nim (typein (· < ·) i) := by | Mathlib.SetTheory.Game.Nim.78_0.mmFMhRYSjViKjcP | theorem moveRight_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ let_fun this := (_ : IsWellOrder (Quotient.out o).α fun x x_1 => x < x_1);
HEq
(moveRight
(mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)))
fun i => nim (typein (fun x x_1 => x < x_1... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rfl | theorem moveRight_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
HEq (nim o).moveRight fun i : o.out.α => nim (typein (· < ·) i) := by rw [nim_def]; | Mathlib.SetTheory.Game.Nim.78_0.mmFMhRYSjViKjcP | theorem moveRight_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
i : ↑(Set.Iio o)
⊢ moveLeft (nim o) (toLeftMovesNim i) = nim ↑i | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp | theorem moveLeft_nim {o : Ordinal} (i) : (nim o).moveLeft (toLeftMovesNim i) = nim i := by | Mathlib.SetTheory.Game.Nim.111_0.mmFMhRYSjViKjcP | theorem moveLeft_nim {o : Ordinal} (i) : (nim o).moveLeft (toLeftMovesNim i) = nim i | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
i : ↑(Set.Iio o)
⊢ moveRight (nim o) (toRightMovesNim i) = nim ↑i | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp | theorem moveRight_nim {o : Ordinal} (i) : (nim o).moveRight (toRightMovesNim i) = nim i := by | Mathlib.SetTheory.Game.Nim.119_0.mmFMhRYSjViKjcP | theorem moveRight_nim {o : Ordinal} (i) : (nim o).moveRight (toRightMovesNim i) = nim i | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{?u.16604}
P : LeftMoves (nim o) → Sort u_1
i : LeftMoves (nim o)
H : (a : Ordinal.{?u.16604}) → (H : a < o) → P (toLeftMovesNim { val := a, property := H })
⊢ P i | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [← toLeftMovesNim.apply_symm_apply i] | /-- A recursion principle for left moves of a nim game. -/
@[elab_as_elim]
def leftMovesNimRecOn {o : Ordinal} {P : (nim o).LeftMoves → Sort*} (i : (nim o).LeftMoves)
(H : ∀ a (H : a < o), P <| toLeftMovesNim ⟨a, H⟩) : P i := by
| Mathlib.SetTheory.Game.Nim.122_0.mmFMhRYSjViKjcP | /-- A recursion principle for left moves of a nim game. -/
@[elab_as_elim]
def leftMovesNimRecOn {o : Ordinal} {P : (nim o).LeftMoves → Sort*} (i : (nim o).LeftMoves)
(H : ∀ a (H : a < o), P <| toLeftMovesNim ⟨a, H⟩) : P i | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{?u.16604}
P : LeftMoves (nim o) → Sort u_1
i : LeftMoves (nim o)
H : (a : Ordinal.{?u.16604}) → (H : a < o) → P (toLeftMovesNim { val := a, property := H })
⊢ P (toLeftMovesNim (toLeftMovesNim.symm i)) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | apply H | /-- A recursion principle for left moves of a nim game. -/
@[elab_as_elim]
def leftMovesNimRecOn {o : Ordinal} {P : (nim o).LeftMoves → Sort*} (i : (nim o).LeftMoves)
(H : ∀ a (H : a < o), P <| toLeftMovesNim ⟨a, H⟩) : P i := by
rw [← toLeftMovesNim.apply_symm_apply i]; | Mathlib.SetTheory.Game.Nim.122_0.mmFMhRYSjViKjcP | /-- A recursion principle for left moves of a nim game. -/
@[elab_as_elim]
def leftMovesNimRecOn {o : Ordinal} {P : (nim o).LeftMoves → Sort*} (i : (nim o).LeftMoves)
(H : ∀ a (H : a < o), P <| toLeftMovesNim ⟨a, H⟩) : P i | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{?u.17019}
P : RightMoves (nim o) → Sort u_1
i : RightMoves (nim o)
H : (a : Ordinal.{?u.17019}) → (H : a < o) → P (toRightMovesNim { val := a, property := H })
⊢ P i | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [← toRightMovesNim.apply_symm_apply i] | /-- A recursion principle for right moves of a nim game. -/
@[elab_as_elim]
def rightMovesNimRecOn {o : Ordinal} {P : (nim o).RightMoves → Sort*} (i : (nim o).RightMoves)
(H : ∀ a (H : a < o), P <| toRightMovesNim ⟨a, H⟩) : P i := by
| Mathlib.SetTheory.Game.Nim.129_0.mmFMhRYSjViKjcP | /-- A recursion principle for right moves of a nim game. -/
@[elab_as_elim]
def rightMovesNimRecOn {o : Ordinal} {P : (nim o).RightMoves → Sort*} (i : (nim o).RightMoves)
(H : ∀ a (H : a < o), P <| toRightMovesNim ⟨a, H⟩) : P i | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{?u.17019}
P : RightMoves (nim o) → Sort u_1
i : RightMoves (nim o)
H : (a : Ordinal.{?u.17019}) → (H : a < o) → P (toRightMovesNim { val := a, property := H })
⊢ P (toRightMovesNim (toRightMovesNim.symm i)) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | apply H | /-- A recursion principle for right moves of a nim game. -/
@[elab_as_elim]
def rightMovesNimRecOn {o : Ordinal} {P : (nim o).RightMoves → Sort*} (i : (nim o).RightMoves)
(H : ∀ a (H : a < o), P <| toRightMovesNim ⟨a, H⟩) : P i := by
rw [← toRightMovesNim.apply_symm_apply i]; | Mathlib.SetTheory.Game.Nim.129_0.mmFMhRYSjViKjcP | /-- A recursion principle for right moves of a nim game. -/
@[elab_as_elim]
def rightMovesNimRecOn {o : Ordinal} {P : (nim o).RightMoves → Sort*} (i : (nim o).RightMoves)
(H : ∀ a (H : a < o), P <| toRightMovesNim ⟨a, H⟩) : P i | Mathlib_SetTheory_Game_Nim |
⊢ IsEmpty (LeftMoves (nim 0)) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | instance isEmpty_nim_zero_leftMoves : IsEmpty (nim 0).LeftMoves := by
| Mathlib.SetTheory.Game.Nim.136_0.mmFMhRYSjViKjcP | instance isEmpty_nim_zero_leftMoves : IsEmpty (nim 0).LeftMoves | Mathlib_SetTheory_Game_Nim |
⊢ IsEmpty
(LeftMoves
(mk (Quotient.out 0).α (Quotient.out 0).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact Ordinal.isEmpty_out_zero | instance isEmpty_nim_zero_leftMoves : IsEmpty (nim 0).LeftMoves := by
rw [nim_def]
| Mathlib.SetTheory.Game.Nim.136_0.mmFMhRYSjViKjcP | instance isEmpty_nim_zero_leftMoves : IsEmpty (nim 0).LeftMoves | Mathlib_SetTheory_Game_Nim |
⊢ IsEmpty (RightMoves (nim 0)) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | instance isEmpty_nim_zero_rightMoves : IsEmpty (nim 0).RightMoves := by
| Mathlib.SetTheory.Game.Nim.141_0.mmFMhRYSjViKjcP | instance isEmpty_nim_zero_rightMoves : IsEmpty (nim 0).RightMoves | Mathlib_SetTheory_Game_Nim |
⊢ IsEmpty
(RightMoves
(mk (Quotient.out 0).α (Quotient.out 0).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact Ordinal.isEmpty_out_zero | instance isEmpty_nim_zero_rightMoves : IsEmpty (nim 0).RightMoves := by
rw [nim_def]
| Mathlib.SetTheory.Game.Nim.141_0.mmFMhRYSjViKjcP | instance isEmpty_nim_zero_rightMoves : IsEmpty (nim 0).RightMoves | Mathlib_SetTheory_Game_Nim |
i : LeftMoves (nim 1)
⊢ toLeftMovesNim.symm i = { val := 0, property := (_ : 0 ∈ Set.Iio 1) } | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp [eq_iff_true_of_subsingleton] | @[simp]
theorem toLeftMovesNim_one_symm (i) :
(@toLeftMovesNim 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ := by
| Mathlib.SetTheory.Game.Nim.175_0.mmFMhRYSjViKjcP | @[simp]
theorem toLeftMovesNim_one_symm (i) :
(@toLeftMovesNim 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ | Mathlib_SetTheory_Game_Nim |
i : RightMoves (nim 1)
⊢ toRightMovesNim.symm i = { val := 0, property := (_ : 0 ∈ Set.Iio 1) } | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp [eq_iff_true_of_subsingleton] | @[simp]
theorem toRightMovesNim_one_symm (i) :
(@toRightMovesNim 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ := by
| Mathlib.SetTheory.Game.Nim.181_0.mmFMhRYSjViKjcP | @[simp]
theorem toRightMovesNim_one_symm (i) :
(@toRightMovesNim 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ | Mathlib_SetTheory_Game_Nim |
x : LeftMoves (nim 1)
⊢ moveLeft (nim 1) x = nim 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp | theorem nim_one_moveLeft (x) : (nim 1).moveLeft x = nim 0 := by | Mathlib.SetTheory.Game.Nim.187_0.mmFMhRYSjViKjcP | theorem nim_one_moveLeft (x) : (nim 1).moveLeft x = nim 0 | Mathlib_SetTheory_Game_Nim |
x : RightMoves (nim 1)
⊢ moveRight (nim 1) x = nim 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp | theorem nim_one_moveRight (x) : (nim 1).moveRight x = nim 0 := by | Mathlib.SetTheory.Game.Nim.190_0.mmFMhRYSjViKjcP | theorem nim_one_moveRight (x) : (nim 1).moveRight x = nim 0 | Mathlib_SetTheory_Game_Nim |
⊢ nim 1 ≡r star | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
| Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
⊢ (mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)) ≡r
star | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | refine' ⟨_, _, fun i => _, fun j => _⟩ | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
| Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_1
⊢ LeftMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)) ≃
LeftMoves star
case refine'_2
⊢ RightMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x ... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | any_goals dsimp; apply Equiv.equivOfUnique | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
| Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_1
⊢ LeftMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)) ≃
LeftMoves star | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | dsimp | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_1
⊢ (Quotient.out 1).α ≃ PUnit.{?u.23865 + 1} | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | apply Equiv.equivOfUnique | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_2
⊢ RightMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)) ≃
RightMoves star | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | dsimp | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_2
⊢ (Quotient.out 1).α ≃ PUnit.{?u.23865 + 1} | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | apply Equiv.equivOfUnique | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_3
i :
LeftMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ moveLeft
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typ... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | dsimp | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_3
i :
LeftMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ nim (typein (fun x x_1 => x < x_1) i) ≡r 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | apply Equiv.equivOfUnique | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_4
j :
RightMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ moveRight
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (t... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | dsimp | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_4
j :
RightMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ nim (typein (fun x x_1 => x < x_1) j) ≡r 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | apply Equiv.equivOfUnique | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_3
i :
LeftMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ moveLeft
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typ... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | all_goals simp; exact nimZeroRelabelling | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; apply Equiv.equivOfUnique
| Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_3
i :
LeftMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ moveLeft
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typ... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; apply Equiv.equivOfUnique
all_goals | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_3
i :
LeftMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ nim 0 ≡r 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact nimZeroRelabelling | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; apply Equiv.equivOfUnique
all_goals simp; | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_4
j :
RightMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ moveRight
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (t... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; apply Equiv.equivOfUnique
all_goals | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
case refine'_4
j :
RightMoves
(mk (Quotient.out 1).α (Quotient.out 1).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
⊢ nim 0 ≡r 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact nimZeroRelabelling | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star := by
rw [nim_def]
refine' ⟨_, _, fun i => _, fun j => _⟩
any_goals dsimp; apply Equiv.equivOfUnique
all_goals simp; | Mathlib.SetTheory.Game.Nim.193_0.mmFMhRYSjViKjcP | /-- `nim 1` has exactly the same moves as `star`. -/
def nimOneRelabelling : nim 1 ≡r star | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ birthday (nim o) = o | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | induction' o using Ordinal.induction with o IH | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o := by
| Mathlib.SetTheory.Game.Nim.205_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{u_1}
IH : ∀ k < o, birthday (nim k) = k
⊢ birthday (nim o) = o | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def, birthday_def] | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o := by
induction' o using Ordinal.induction with o IH
| Mathlib.SetTheory.Game.Nim.205_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{u_1}
IH : ∀ k < o, birthday (nim k) = k
⊢ max
(lsub fun i =>
birthday
(moveLeft
(mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
i))
(lsub... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | dsimp | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o := by
induction' o using Ordinal.induction with o IH
rw [nim_def, birthday_def]
| Mathlib.SetTheory.Game.Nim.205_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{u_1}
IH : ∀ k < o, birthday (nim k) = k
⊢ max (lsub fun i => birthday (nim (typein (fun x x_1 => x < x_1) i)))
(lsub fun i => birthday (nim (typein (fun x x_1 => x < x_1) i))) =
o | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [max_eq_right le_rfl] | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o := by
induction' o using Ordinal.induction with o IH
rw [nim_def, birthday_def]
dsimp
| Mathlib.SetTheory.Game.Nim.205_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{u_1}
IH : ∀ k < o, birthday (nim k) = k
⊢ (lsub fun i => birthday (nim (typein (fun x x_1 => x < x_1) i))) = o | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | convert lsub_typein o with i | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o := by
induction' o using Ordinal.induction with o IH
rw [nim_def, birthday_def]
dsimp
rw [max_eq_right le_rfl]
| Mathlib.SetTheory.Game.Nim.205_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o | Mathlib_SetTheory_Game_Nim |
case h.e'_2.h.e'_2.h
o : Ordinal.{u_1}
IH : ∀ k < o, birthday (nim k) = k
i : (Quotient.out o).α
⊢ birthday (nim (typein (fun x x_1 => x < x_1) i)) = typein (fun x x_1 => x < x_1) i | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact IH _ (typein_lt_self i) | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o := by
induction' o using Ordinal.induction with o IH
rw [nim_def, birthday_def]
dsimp
rw [max_eq_right le_rfl]
convert lsub_typein o with i
| Mathlib.SetTheory.Game.Nim.205_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_birthday (o : Ordinal) : (nim o).birthday = o | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
⊢ -nim o = nim o | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | induction' o using Ordinal.induction with o IH | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o := by
| Mathlib.SetTheory.Game.Nim.215_0.mmFMhRYSjViKjcP | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ -nim o = nim o | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o := by
induction' o using Ordinal.induction with o IH
| Mathlib.SetTheory.Game.Nim.215_0.mmFMhRYSjViKjcP | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (-mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂)) =
mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | dsimp | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o := by
induction' o using Ordinal.induction with o IH
rw [nim_def]; | Mathlib.SetTheory.Game.Nim.215_0.mmFMhRYSjViKjcP | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (mk (Quotient.out o).α (Quotient.out o).α (fun j => -nim (typein (fun x x_1 => x < x_1) j)) fun i =>
-nim (typein (fun x x_1 => x < x_1) i)) =
mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | congr | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o := by
induction' o using Ordinal.induction with o IH
rw [nim_def]; dsimp; | Mathlib.SetTheory.Game.Nim.215_0.mmFMhRYSjViKjcP | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o | Mathlib_SetTheory_Game_Nim |
case h.e_a
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (fun j => -nim (typein (fun x x_1 => x < x_1) j)) = fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | funext i | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o := by
induction' o using Ordinal.induction with o IH
rw [nim_def]; dsimp; congr <;> | Mathlib.SetTheory.Game.Nim.215_0.mmFMhRYSjViKjcP | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o | Mathlib_SetTheory_Game_Nim |
case h.e_a
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (fun i => -nim (typein (fun x x_1 => x < x_1) i)) = fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | funext i | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o := by
induction' o using Ordinal.induction with o IH
rw [nim_def]; dsimp; congr <;> | Mathlib.SetTheory.Game.Nim.215_0.mmFMhRYSjViKjcP | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o | Mathlib_SetTheory_Game_Nim |
case h.e_a.h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
i : (Quotient.out o).α
⊢ -nim (typein (fun x x_1 => x < x_1) i) = nim (typein (fun x x_1 => x < x_1) i) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact IH _ (Ordinal.typein_lt_self i) | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o := by
induction' o using Ordinal.induction with o IH
rw [nim_def]; dsimp; congr <;> funext i <;> | Mathlib.SetTheory.Game.Nim.215_0.mmFMhRYSjViKjcP | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o | Mathlib_SetTheory_Game_Nim |
case h.e_a.h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
i : (Quotient.out o).α
⊢ -nim (typein (fun x x_1 => x < x_1) i) = nim (typein (fun x x_1 => x < x_1) i) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact IH _ (Ordinal.typein_lt_self i) | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o := by
induction' o using Ordinal.induction with o IH
rw [nim_def]; dsimp; congr <;> funext i <;> | Mathlib.SetTheory.Game.Nim.215_0.mmFMhRYSjViKjcP | @[simp]
theorem neg_nim (o : Ordinal) : -nim o = nim o | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{?u.34699}
⊢ Impartial (nim o) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | induction' o using Ordinal.induction with o IH | instance nim_impartial (o : Ordinal) : Impartial (nim o) := by
| Mathlib.SetTheory.Game.Nim.221_0.mmFMhRYSjViKjcP | instance nim_impartial (o : Ordinal) : Impartial (nim o) | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{?u.34717}
IH : ∀ k < o, Impartial (nim k)
⊢ Impartial (nim o) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [impartial_def, neg_nim] | instance nim_impartial (o : Ordinal) : Impartial (nim o) := by
induction' o using Ordinal.induction with o IH
| Mathlib.SetTheory.Game.Nim.221_0.mmFMhRYSjViKjcP | instance nim_impartial (o : Ordinal) : Impartial (nim o) | Mathlib_SetTheory_Game_Nim |
case h
o : Ordinal.{?u.34717}
IH : ∀ k < o, Impartial (nim k)
⊢ nim o ≈ nim o ∧
(∀ (i : LeftMoves (nim o)), Impartial (moveLeft (nim o) i)) ∧
∀ (j : RightMoves (nim o)), Impartial (moveRight (nim o) j) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | refine' ⟨equiv_rfl, fun i => _, fun i => _⟩ | instance nim_impartial (o : Ordinal) : Impartial (nim o) := by
induction' o using Ordinal.induction with o IH
rw [impartial_def, neg_nim]
| Mathlib.SetTheory.Game.Nim.221_0.mmFMhRYSjViKjcP | instance nim_impartial (o : Ordinal) : Impartial (nim o) | Mathlib_SetTheory_Game_Nim |
case h.refine'_1
o : Ordinal.{?u.34717}
IH : ∀ k < o, Impartial (nim k)
i : LeftMoves (nim o)
⊢ Impartial (moveLeft (nim o) i) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simpa using IH _ (typein_lt_self _) | instance nim_impartial (o : Ordinal) : Impartial (nim o) := by
induction' o using Ordinal.induction with o IH
rw [impartial_def, neg_nim]
refine' ⟨equiv_rfl, fun i => _, fun i => _⟩ <;> | Mathlib.SetTheory.Game.Nim.221_0.mmFMhRYSjViKjcP | instance nim_impartial (o : Ordinal) : Impartial (nim o) | Mathlib_SetTheory_Game_Nim |
case h.refine'_2
o : Ordinal.{?u.35327}
IH : ∀ k < o, Impartial (nim k)
i : RightMoves (nim o)
⊢ Impartial (moveRight (nim o) i) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simpa using IH _ (typein_lt_self _) | instance nim_impartial (o : Ordinal) : Impartial (nim o) := by
induction' o using Ordinal.induction with o IH
rw [impartial_def, neg_nim]
refine' ⟨equiv_rfl, fun i => _, fun i => _⟩ <;> | Mathlib.SetTheory.Game.Nim.221_0.mmFMhRYSjViKjcP | instance nim_impartial (o : Ordinal) : Impartial (nim o) | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ nim o ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [Impartial.fuzzy_zero_iff_lf, nim_def, lf_zero_le] | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 := by
| Mathlib.SetTheory.Game.Nim.227_0.mmFMhRYSjViKjcP | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ ∃ j,
moveRight
(mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
j ≤
0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [← Ordinal.pos_iff_ne_zero] at ho | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 := by
rw [Impartial.fuzzy_zero_iff_lf, nim_def, lf_zero_le]
| Mathlib.SetTheory.Game.Nim.227_0.mmFMhRYSjViKjcP | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
ho : 0 < o
⊢ ∃ j,
moveRight
(mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
j ≤
0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact ⟨(Ordinal.principalSegOut ho).top, by simp⟩ | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 := by
rw [Impartial.fuzzy_zero_iff_lf, nim_def, lf_zero_le]
rw [← Ordinal.pos_iff_ne_zero] at ho
| Mathlib.SetTheory.Game.Nim.227_0.mmFMhRYSjViKjcP | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 | Mathlib_SetTheory_Game_Nim |
o : Ordinal.{u_1}
ho : 0 < o
⊢ moveRight
(mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
nim (typein (fun x x_1 => x < x_1) o₂))
(principalSegOut ho).top ≤
0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simp | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 := by
rw [Impartial.fuzzy_zero_iff_lf, nim_def, lf_zero_le]
rw [← Ordinal.pos_iff_ne_zero] at ho
exact ⟨(Ordinal.principalSegOut ho).top, by | Mathlib.SetTheory.Game.Nim.227_0.mmFMhRYSjViKjcP | theorem nim_fuzzy_zero_of_ne_zero {o : Ordinal} (ho : o ≠ 0) : nim o ‖ 0 | Mathlib_SetTheory_Game_Nim |
o₁ o₂ : Ordinal.{u_1}
⊢ nim o₁ + nim o₂ ≈ 0 ↔ o₁ = o₂ | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | constructor | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
| Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
case mp
o₁ o₂ : Ordinal.{u_1}
⊢ nim o₁ + nim o₂ ≈ 0 → o₁ = o₂ | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _ | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· | Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
case mp
o₁ o₂ : Ordinal.{u_1}
hne : o₁ ≠ o₂
⊢ nim o₁ + nim o₂ ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | wlog h : o₁ < o₂ | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _
| Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
case mp.inr
o₁ o₂ : Ordinal.{u_1}
hne : o₁ ≠ o₂
this : ∀ (o₁ o₂ : Ordinal.{u_1}), o₁ ≠ o₂ → o₁ < o₂ → nim o₁ + nim o₂ ‖ 0
h : ¬o₁ < o₂
⊢ nim o₁ + nim o₂ ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact (fuzzy_congr_left add_comm_equiv).1 (this _ _ hne.symm (hne.lt_or_lt.resolve_left h)) | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _
wlog h : o₁ < o₂
· | Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
o₁ o₂ : Ordinal.{u_1}
hne : o₁ ≠ o₂
h : o₁ < o₂
⊢ nim o₁ + nim o₂ ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [Impartial.fuzzy_zero_iff_gf, zero_lf_le, nim_def o₂] | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _
wlog h : o₁ < o₂
· exact (fuzzy_congr_left add_comm_equiv).1 (this _ _ hne.symm (hne.lt_or_lt.resolve_l... | Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
o₁ o₂ : Ordinal.{u_1}
hne : o₁ ≠ o₂
h : o₁ < o₂
⊢ ∃ i,
0 ≤
moveLeft
(nim o₁ +
mk (Quotient.out o₂).α (Quotient.out o₂).α (fun o₂_1 => nim (typein (fun x x_1 => x < x_1) o₂_1)) fun o₂_1 =>
nim (typein (fun x x_1 => x < x_1) o₂_1))
i | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | refine' ⟨toLeftMovesAdd (Sum.inr _), _⟩ | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _
wlog h : o₁ < o₂
· exact (fuzzy_congr_left add_comm_equiv).1 (this _ _ hne.symm (hne.lt_or_lt.resolve_l... | Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
case refine'_1
o₁ o₂ : Ordinal.{u_1}
hne : o₁ ≠ o₂
h : o₁ < o₂
⊢ LeftMoves
(mk (Quotient.out o₂).α (Quotient.out o₂).α (fun o₂_1 => nim (typein (fun x x_1 => x < x_1) o₂_1)) fun o₂_1 =>
nim (typein (fun x x_1 => x < x_1) o₂_1)) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact (Ordinal.principalSegOut h).top | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _
wlog h : o₁ < o₂
· exact (fuzzy_congr_left add_comm_equiv).1 (this _ _ hne.symm (hne.lt_or_lt.resolve_l... | Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
case refine'_2
o₁ o₂ : Ordinal.{u_1}
hne : o₁ ≠ o₂
h : o₁ < o₂
⊢ 0 ≤
moveLeft
(nim o₁ +
mk (Quotient.out o₂).α (Quotient.out o₂).α (fun o₂_1 => nim (typein (fun x x_1 => x < x_1) o₂_1)) fun o₂_1 =>
nim (typein (fun x x_1 => x < x_1) o₂_1))
(toLeftMovesAdd (Sum.inr (principalSegOut h).t... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | simpa only [Ordinal.typein_top, Ordinal.type_lt, PGame.add_moveLeft_inr, PGame.moveLeft_mk]
using (Impartial.add_self (nim o₁)).2 | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _
wlog h : o₁ < o₂
· exact (fuzzy_congr_left add_comm_equiv).1 (this _ _ hne.symm (hne.lt_or_lt.resolve_l... | Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
case mpr
o₁ o₂ : Ordinal.{u_1}
⊢ o₁ = o₂ → nim o₁ + nim o₂ ≈ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rintro rfl | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _
wlog h : o₁ < o₂
· exact (fuzzy_congr_left add_comm_equiv).1 (this _ _ hne.symm (hne.lt_or_lt.resolve_l... | Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
case mpr
o₁ : Ordinal.{u_1}
⊢ nim o₁ + nim o₁ ≈ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact Impartial.add_self (nim o₁) | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ := by
constructor
· refine' not_imp_not.1 fun hne : _ ≠ _ => (Impartial.not_equiv_zero_iff (nim o₁ + nim o₂)).2 _
wlog h : o₁ < o₂
· exact (fuzzy_congr_left add_comm_equiv).1 (this _ _ hne.symm (hne.lt_or_lt.resolve_l... | Mathlib.SetTheory.Game.Nim.233_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_equiv_zero_iff (o₁ o₂ : Ordinal) : (nim o₁ + nim o₂ ≈ 0) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
o₁ o₂ : Ordinal.{u_1}
⊢ nim o₁ + nim o₂ ‖ 0 ↔ o₁ ≠ o₂ | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [iff_not_comm, Impartial.not_fuzzy_zero_iff, nim_add_equiv_zero_iff] | @[simp]
theorem nim_add_fuzzy_zero_iff {o₁ o₂ : Ordinal} : nim o₁ + nim o₂ ‖ 0 ↔ o₁ ≠ o₂ := by
| Mathlib.SetTheory.Game.Nim.249_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_add_fuzzy_zero_iff {o₁ o₂ : Ordinal} : nim o₁ + nim o₂ ‖ 0 ↔ o₁ ≠ o₂ | Mathlib_SetTheory_Game_Nim |
o₁ o₂ : Ordinal.{u_1}
⊢ nim o₁ ≈ nim o₂ ↔ o₁ = o₂ | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [Impartial.equiv_iff_add_equiv_zero, nim_add_equiv_zero_iff] | @[simp]
theorem nim_equiv_iff_eq {o₁ o₂ : Ordinal} : (nim o₁ ≈ nim o₂) ↔ o₁ = o₂ := by
| Mathlib.SetTheory.Game.Nim.254_0.mmFMhRYSjViKjcP | @[simp]
theorem nim_equiv_iff_eq {o₁ o₂ : Ordinal} : (nim o₁ ≈ nim o₂) ↔ o₁ = o₂ | Mathlib_SetTheory_Game_Nim |
x✝ : PGame
G : PGame := x✝
i : LeftMoves G
⊢ (invImage (fun a => a) instWellFoundedRelationPGame).1 (moveLeft G i) x✝ | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | pgame_wf_tac | /-- The Grundy value of an impartial game, the ordinal which corresponds to the game of nim that the
game is equivalent to -/
noncomputable def grundyValue : ∀ _ : PGame.{u}, Ordinal.{u}
| G => Ordinal.mex.{u, u} fun i => grundyValue (G.moveLeft i)
termination_by grundyValue G => G
decreasing_by | Mathlib.SetTheory.Game.Nim.259_0.mmFMhRYSjViKjcP | /-- The Grundy value of an impartial game, the ordinal which corresponds to the game of nim that the
game is equivalent to -/
noncomputable def grundyValue : ∀ _ : PGame.{u}, Ordinal.{u}
| G => Ordinal.mex.{u, u} fun i => grundyValue (G.moveLeft i)
termination_by grundyValue G => G
decreasing_by pgame_wf_tac | Mathlib_SetTheory_Game_Nim |
G : PGame
⊢ grundyValue G = mex fun i => grundyValue (moveLeft G i) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [grundyValue] | theorem grundyValue_eq_mex_left (G : PGame) :
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveLeft i) := by | Mathlib.SetTheory.Game.Nim.267_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_mex_left (G : PGame) :
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveLeft i) | Mathlib_SetTheory_Game_Nim |
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
⊢ x✝ ≈ nim (grundyValue x✝) | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [Impartial.equiv_iff_add_equiv_zero, ← Impartial.forall_leftMoves_fuzzy_iff_equiv_zero] | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
| Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
⊢ ∀ (i : LeftMoves (x✝ + nim (grundyValue x✝))), moveLeft (x✝ + nim (grundyValue x✝)) i ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | intro i | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
⊢ moveLeft (x✝ + nim (grundyValue x✝)) i ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | apply leftMoves_add_cases i | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
⊢ ∀ (i : LeftMoves x✝), moveLeft (x✝ + nim (grundyValue x✝)) (toLeftMovesAdd (Sum.inl i)) ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | intro i₁ | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₁ : LeftMoves x✝
⊢ moveLeft (x✝ + nim (grundyValue x✝)) (toLeftMovesAdd (Sum.inl i₁)) ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [add_moveLeft_inl] | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₁ : LeftMoves x✝
⊢ moveLeft x✝ i₁ + nim (grundyValue x✝) ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | apply
(fuzzy_congr_left (add_congr_left (Equiv.symm (equiv_nim_grundyValue (G.moveLeft i₁))))).1 | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₁ : LeftMoves x✝
⊢ nim (grundyValue (moveLeft G i₁)) + nim (grundyValue x✝) ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_add_fuzzy_zero_iff] | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₁ : LeftMoves x✝
⊢ grundyValue (moveLeft G i₁) ≠ grundyValue x✝ | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | intro heq | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₁ : LeftMoves x✝
heq : grundyValue (moveLeft G i₁) = grundyValue x✝
⊢ False | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [eq_comm, grundyValue_eq_mex_left G] at heq | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₁ : LeftMoves x✝
heq : (mex fun i => grundyValue (moveLeft G i)) = grundyValue (moveLeft G i₁)
⊢ False | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | have h := Ordinal.ne_mex.{u, u} (fun i ↦ grundyValue (moveLeft G i)) | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₁ : LeftMoves x✝
heq : (mex fun i => grundyValue (moveLeft G i)) = grundyValue (moveLeft G i₁)
h : ∀ (i : LeftMoves G), grundyValue (moveLeft G i) ≠ mex fun i => grundyValue (moveLeft G i)
⊢ False | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [heq] at h | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hl
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₁ : LeftMoves x✝
heq : (mex fun i => grundyValue (moveLeft G i)) = grundyValue (moveLeft G i₁)
h : ∀ (i : LeftMoves G), grundyValue (moveLeft G i) ≠ grundyValue (moveLeft G i₁)
⊢ False | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | exact (h i₁).irrefl | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hr
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
⊢ ∀ (i : LeftMoves (nim (grundyValue x✝))), moveLeft (x✝ + nim (grundyValue x✝)) (toLeftMovesAdd (Sum.inr i)) ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | intro i₂ | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hr
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₂ : LeftMoves (nim (grundyValue x✝))
⊢ moveLeft (x✝ + nim (grundyValue x✝)) (toLeftMovesAdd (Sum.inr i₂)) ‖ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [add_moveLeft_inr, ← Impartial.exists_left_move_equiv_iff_fuzzy_zero] | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hr
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₂ : LeftMoves (nim (grundyValue x✝))
⊢ ∃ i, moveLeft (x✝ + moveLeft (nim (grundyValue x✝)) i₂) i ≈ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | revert i₂ | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hr
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
⊢ ∀ (i₂ : LeftMoves (nim (grundyValue x✝))), ∃ i, moveLeft (x✝ + moveLeft (nim (grundyValue x✝)) i₂) i ≈ 0 | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | rw [nim_def] | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hr
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
⊢ ∀
(i₂ :
LeftMoves
(mk (Quotient.out (grundyValue x✝)).α (Quotient.out (grundyValue x✝)).α
(fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂))),... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | intro i₂ | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
case hr
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₂ :
LeftMoves
(mk (Quotient.out (grundyValue x✝)).α (Quotient.out (grundyValue x✝)).α
(fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂))
⊢ ∃ i,
moveLeft
... | /-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | have h' :
∃ i : G.LeftMoves,
grundyValue (G.moveLeft i) = Ordinal.typein (Quotient.out (grundyValue G)).r i₂ := by
revert i₂
rw [grundyValue_eq_mex_left]
intro i₂
have hnotin : _ ∉ _ := fun hin =>
(le_not_le_of_lt (Ordinal.typein_lt_self i₂)).2 (csInf_le' hin)... | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib.SetTheory.Game.Nim.271_0.mmFMhRYSjViKjcP | /-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
nim, namely the game of nim corresponding to the games Grundy value -/
theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
| G => by
rw [Impartial.equiv_iff_add_equiv_zero, ← Impa... | Mathlib_SetTheory_Game_Nim |
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