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 |
|---|---|---|---|---|---|---|
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, grundyValue (moveLeft G... | /-
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 |
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... | /-
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_eq_mex_left] | /-- 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✝))
⊢ ∀
(i₂ :
LeftMoves
(mk (Quotient.out (mex fun i => grundyValue (moveLeft x✝ i))).α
(Quotient.out (mex fun i => grundyValue (moveLeft x✝ i))).α (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 |
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₂ :
LeftMoves
(mk (Quotient.out (mex fun i => grundyValue (moveLeft x✝ i))).α
(Quotient.out (mex fun i => grundyValue (moveLeft x✝ i))).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂))
fun o₂ => nim (typein ... | /-
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 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 |
x✝ : PGame
inst✝ : Impartial x✝
G : PGame := x✝
i : LeftMoves (x✝ + nim (grundyValue x✝))
i₂ :
LeftMoves
(mk (Quotient.out (mex fun i => grundyValue (moveLeft x✝ i))).α
(Quotient.out (mex fun i => grundyValue (moveLeft x✝ i))).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂))
fun o₂ => nim (typein ... | /-
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 hnotin | /-- 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))
h' : ∃ i, grundyValue ... | /-
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... | cases' h' with i hi | /-- 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.intro
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 : LeftMoves G... | /-
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... | use toLeftMovesAdd (Sum.inl 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 h
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 : LeftMoves G
hi : g... | /-
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, moveLeft_mk] | /-- 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 h
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 : LeftMoves G
hi : g... | /-
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.trans (add_congr_left (equiv_nim_grundyValue (G.moveLeft 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 h
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 : LeftMoves G
hi : g... | /-
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 [hi] using Impartial.add_self (nim (grundyValue (G.moveLeft 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✝
a✝ :
∀ (y : (x : PGame) ×' Impartial x),
(invImage (fun a => PSigma.casesOn a fun G snd => G) instWellFoundedRelationPGame).1 y { fst := x✝, snd := inst✝ } →
y.1 ≈ nim (grundyValue y.1)
G : PGame := x✝
i₁ : LeftMoves x✝
⊢ (invImage (fun a => PSigma.casesOn a fun G snd => 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... | pgame_wf_tac | /-- 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✝
a✝ :
∀ (y : (x : PGame) ×' Impartial x),
(invImage (fun a => PSigma.casesOn a fun G snd => G) instWellFoundedRelationPGame).1 y { fst := x✝, snd := inst✝ } →
y.1 ≈ nim (grundyValue y.1)
G : PGame := x✝
i₂ :
LeftMoves
(mk (Quotient.out (grundyValue x✝)).α (Quotient.out (... | /-
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 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 |
G : PGame
inst✝ : Impartial G
o : Ordinal.{u_1}
⊢ grundyValue G = o → G ≈ 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... | rintro rfl | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) :=
⟨by | Mathlib.SetTheory.Game.Nim.312_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) | Mathlib_SetTheory_Game_Nim |
G : PGame
inst✝ : Impartial G
⊢ G ≈ nim (grundyValue G) | /-
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 equiv_nim_grundyValue G | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) :=
⟨by rintro rfl; | Mathlib.SetTheory.Game.Nim.312_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) | Mathlib_SetTheory_Game_Nim |
G : PGame
inst✝ : Impartial G
o : Ordinal.{u_1}
⊢ G ≈ nim o → grundyValue G = 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... | intro h | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) :=
⟨by rintro rfl; exact equiv_nim_grundyValue G,
by | Mathlib.SetTheory.Game.Nim.312_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) | Mathlib_SetTheory_Game_Nim |
G : PGame
inst✝ : Impartial G
o : Ordinal.{u_1}
h : G ≈ nim o
⊢ grundyValue G = 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_equiv_iff_eq] | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) :=
⟨by rintro rfl; exact equiv_nim_grundyValue G,
by intro h; | Mathlib.SetTheory.Game.Nim.312_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) | Mathlib_SetTheory_Game_Nim |
G : PGame
inst✝ : Impartial G
o : Ordinal.{u_1}
h : G ≈ nim o
⊢ nim (grundyValue G) ≈ 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... | exact Equiv.trans (Equiv.symm (equiv_nim_grundyValue G)) h | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) :=
⟨by rintro rfl; exact equiv_nim_grundyValue G,
by intro h; rw [← nim_equiv_iff_eq]; | Mathlib.SetTheory.Game.Nim.312_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
grundyValue G = o ↔ (G ≈ nim o) | Mathlib_SetTheory_Game_Nim |
G : PGame
inst✝ : Impartial G
⊢ grundyValue G = 0 ↔ G ≈ 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 [← grundyValue_eq_iff_equiv, grundyValue_zero] | theorem grundyValue_iff_equiv_zero (G : PGame) [G.Impartial] : grundyValue G = 0 ↔ (G ≈ 0) := by
| Mathlib.SetTheory.Game.Nim.333_0.mmFMhRYSjViKjcP | theorem grundyValue_iff_equiv_zero (G : PGame) [G.Impartial] : grundyValue G = 0 ↔ (G ≈ 0) | Mathlib_SetTheory_Game_Nim |
G : PGame
inst✝ : Impartial G
⊢ grundyValue (-G) = grundyValue G | /-
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_eq_iff_equiv_nim, neg_equiv_iff, neg_nim, ← grundyValue_eq_iff_equiv_nim] | @[simp]
theorem grundyValue_neg (G : PGame) [G.Impartial] : grundyValue (-G) = grundyValue G := by
| Mathlib.SetTheory.Game.Nim.342_0.mmFMhRYSjViKjcP | @[simp]
theorem grundyValue_neg (G : PGame) [G.Impartial] : grundyValue (-G) = grundyValue G | Mathlib_SetTheory_Game_Nim |
l r : Type u
L : l → PGame
R : r → PGame
x✝ : Impartial (mk l r L R)
⊢ grundyValue (mk l r L R) = mex fun i => grundyValue (moveRight (mk l r L R) 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_neg, grundyValue_eq_mex_left] | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
| Mathlib.SetTheory.Game.Nim.347_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
congr
ext i
haveI : (R i).Impartial | Mathlib_SetTheory_Game_Nim |
l r : Type u
L : l → PGame
R : r → PGame
x✝ : Impartial (mk l r L R)
⊢ (mex fun i => grundyValue (moveLeft (-mk l r L R) i)) = mex fun i => grundyValue (moveRight (mk l r L R) 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... | congr | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
| Mathlib.SetTheory.Game.Nim.347_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
congr
ext i
haveI : (R i).Impartial | Mathlib_SetTheory_Game_Nim |
case e_f
l r : Type u
L : l → PGame
R : r → PGame
x✝ : Impartial (mk l r L R)
⊢ (fun i => grundyValue (moveLeft (-mk l r L R) i)) = fun i => grundyValue (moveRight (mk l r L R) 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... | ext i | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
congr
| Mathlib.SetTheory.Game.Nim.347_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
congr
ext i
haveI : (R i).Impartial | Mathlib_SetTheory_Game_Nim |
case e_f.h
l r : Type u
L : l → PGame
R : r → PGame
x✝ : Impartial (mk l r L R)
i : LeftMoves (-mk l r L R)
⊢ grundyValue (moveLeft (-mk l r L R) i) = grundyValue (moveRight (mk l r L R) 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... | haveI : (R i).Impartial := @Impartial.moveRight_impartial ⟨l, r, L, R⟩ _ i | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
congr
ext i
| Mathlib.SetTheory.Game.Nim.347_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
congr
ext i
haveI : (R i).Impartial | Mathlib_SetTheory_Game_Nim |
case e_f.h
l r : Type u
L : l → PGame
R : r → PGame
x✝ : Impartial (mk l r L R)
i : LeftMoves (-mk l r L R)
this : Impartial (R i)
⊢ grundyValue (moveLeft (-mk l r L R) i) = grundyValue (moveRight (mk l r L R) 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 grundyValue_neg | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
congr
ext i
haveI : (R i).Impartial := @Impartial.moveRight_impartial ⟨l, r, L, R⟩ ... | Mathlib.SetTheory.Game.Nim.347_0.mmFMhRYSjViKjcP | theorem grundyValue_eq_mex_right :
∀ (G : PGame) [G.Impartial],
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
| ⟨l, r, L, R⟩, _ => by
rw [← grundyValue_neg, grundyValue_eq_mex_left]
congr
ext i
haveI : (R i).Impartial | Mathlib_SetTheory_Game_Nim |
n m : ℕ
⊢ grundyValue (nim ↑n + nim ↑m) = ↑(n ^^^ m) | /-
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' n using Nat.strong_induction_on with n hn generalizing m | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
| Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
⊢ grundyValue (nim ↑n + nim ↑m) = ↑(n ^^^ m) | /-
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' m using Nat.strong_induction_on with m hm | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
⊢ grundyValue (nim ↑n + nim ↑m) = ↑(n ^^^ m) | /-
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_eq_mex_left] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
⊢ (mex fun i => grundyValue (moveLeft (nim ↑n + nim ↑m) i)) = ↑(n ^^^ m) | /-
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 (Ordinal.mex_le_of_ne.{u, u} fun i => ?_).antisymm
(Ordinal.le_mex_of_forall fun ou hu => ?_) | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
⊢ grundyValue (moveLeft (nim ↑n + nim ↑m) i) ≠ ↑(n ^^^ m) | /-
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 Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
⊢ ∀ (i : LeftMoves (nim ↑n)), grundyValue (moveLeft (nim ↑n + nim ↑m) (toLeftMovesAdd (Sum.inl i))) ≠ ↑(n ^^^ m) | /-
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 a => leftMovesNimRecOn a fun ok hk => _ | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
ok : Ordinal.{u}
hk : ok < ↑n
⊢ grundyValue (moveLeft (nim ↑n + nim ↑m) (toLeftMovesAdd (Sum.in... | /-
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... | obtain ⟨k, rfl⟩ := Ordinal.lt_omega.1 (hk.trans (Ordinal.nat_lt_omega _)) | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
k : ℕ
hk : ↑k < ↑n
⊢ grundyValue (moveLeft (nim ↑n + nim ↑m) (toLeftMovesAdd (Sum.inl (to... | /-
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 only [add_moveLeft_inl, add_moveLeft_inr, moveLeft_nim', Equiv.symm_apply_apply] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
k : ℕ
hk : ↑k < ↑n
⊢ grundyValue (nim ↑k + nim ↑m) ≠ ↑(n ^^^ m) | /-
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 [nat_cast_lt] at hk | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
k : ℕ
hk : k < n
⊢ grundyValue (nim ↑k + nim ↑m) ≠ ↑(n ^^^ m) | /-
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... | first
| rw [hn _ hk]
| rw [hm _ hk] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
k : ℕ
hk : k < n
⊢ grundyValue (nim ↑k + nim ↑m) ≠ ↑(n ^^^ m) | /-
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 [hn _ hk] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
k : ℕ
hk : k < n
⊢ ↑(k ^^^ m) ≠ ↑(n ^^^ m) | /-
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 h => hk.ne _ | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
k : ℕ
hk : k < n
h : ↑(k ^^^ m) = ↑(n ^^^ m)
⊢ k = n | /-
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.nat_cast_inj] at h | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
k : ℕ
hk : k < n
h : k ^^^ m = n ^^^ m
⊢ k = n | /-
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... | first
| rwa [Nat.xor_left_inj] at h
| rwa [Nat.xor_right_inj] at h | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hl.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑n)
k : ℕ
hk : k < n
h : k ^^^ m = n ^^^ m
⊢ k = n | /-
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... | rwa [Nat.xor_left_inj] at h | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
⊢ ∀ (i : LeftMoves (nim ↑m)), grundyValue (moveLeft (nim ↑n + nim ↑m) (toLeftMovesAdd (Sum.inr i))) ≠ ↑(n ^^^ m) | /-
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 a => leftMovesNimRecOn a fun ok hk => _ | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
ok : Ordinal.{u}
hk : ok < ↑m
⊢ grundyValue (moveLeft (nim ↑n + nim ↑m) (toLeftMovesAdd (Sum.in... | /-
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... | obtain ⟨k, rfl⟩ := Ordinal.lt_omega.1 (hk.trans (Ordinal.nat_lt_omega _)) | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : ↑k < ↑m
⊢ grundyValue (moveLeft (nim ↑n + nim ↑m) (toLeftMovesAdd (Sum.inr (to... | /-
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 only [add_moveLeft_inl, add_moveLeft_inr, moveLeft_nim', Equiv.symm_apply_apply] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : ↑k < ↑m
⊢ grundyValue (nim ↑n + nim ↑k) ≠ ↑(n ^^^ m) | /-
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 [nat_cast_lt] at hk | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : k < m
⊢ grundyValue (nim ↑n + nim ↑k) ≠ ↑(n ^^^ m) | /-
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... | first
| rw [hn _ hk]
| rw [hm _ hk] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : k < m
⊢ grundyValue (nim ↑n + nim ↑k) ≠ ↑(n ^^^ m) | /-
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 [hn _ hk] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : k < m
⊢ grundyValue (nim ↑n + nim ↑k) ≠ ↑(n ^^^ m) | /-
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 [hm _ hk] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : k < m
⊢ ↑(n ^^^ k) ≠ ↑(n ^^^ m) | /-
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 h => hk.ne _ | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : k < m
h : ↑(n ^^^ k) = ↑(n ^^^ m)
⊢ k = m | /-
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.nat_cast_inj] at h | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : k < m
h : n ^^^ k = n ^^^ m
⊢ k = m | /-
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... | first
| rwa [Nat.xor_left_inj] at h
| rwa [Nat.xor_right_inj] at h | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : k < m
h : n ^^^ k = n ^^^ m
⊢ k = m | /-
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... | rwa [Nat.xor_left_inj] at h | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_1.hr.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
i : LeftMoves (nim ↑n + nim ↑m)
a : LeftMoves (nim ↑m)
k : ℕ
hk : k < m
h : n ^^^ k = n ^^^ m
⊢ k = m | /-
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... | rwa [Nat.xor_right_inj] at h | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_2
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
ou : Ordinal.{u}
hu : ou < ↑(n ^^^ m)
⊢ ∃ i, grundyValue (moveLeft (nim ↑n + nim ↑m) i) = ou | /-
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... | obtain ⟨u, rfl⟩ := Ordinal.lt_omega.1 (hu.trans (Ordinal.nat_lt_omega _)) | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_2.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
u : ℕ
hu : ↑u < ↑(n ^^^ m)
⊢ ∃ i, grundyValue (moveLeft (nim ↑n + nim ↑m) i) = ↑u | /-
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... | replace hu := Ordinal.nat_cast_lt.1 hu | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_2.intro
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
u : ℕ
hu : u < n ^^^ m
⊢ ∃ i, grundyValue (moveLeft (nim ↑n + nim ↑m) i) = ↑u | /-
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... | cases' Nat.lt_xor_cases hu with h h | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_2.intro.inl
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
u : ℕ
hu : u < n ^^^ m
h : u ^^^ m < n
⊢ ∃ i, grundyValue (moveLeft (nim ↑n + nim ↑m) i) = ↑u | /-
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.inl <| toLeftMovesNim ⟨_, Ordinal.nat_cast_lt.2 h⟩), _⟩ | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_2.intro.inl
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
u : ℕ
hu : u < n ^^^ m
h : u ^^^ m < n
⊢ grundyValue
(moveLeft (nim ↑n + nim ↑m)
(toLeftMovesAdd (Sum.inl (toLeftMovesNim { val :... | /-
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 [Nat.lxor_cancel_right, hn _ h] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_2.intro.inr
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
u : ℕ
hu : u < n ^^^ m
h : u ^^^ n < m
⊢ ∃ i, grundyValue (moveLeft (nim ↑n + nim ↑m) i) = ↑u | /-
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 <| toLeftMovesNim ⟨_, Ordinal.nat_cast_lt.2 h⟩), _⟩ | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_2.intro.inr
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
u : ℕ
hu : u < n ^^^ m
h : u ^^^ n < m
⊢ grundyValue
(moveLeft (nim ↑n + nim ↑m)
(toLeftMovesAdd (Sum.inr (toLeftMovesNim { val :... | /-
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 : n ^^^ (u ^^^ n) = u | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case this
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
u : ℕ
hu : u < n ^^^ m
h : u ^^^ n < m
⊢ n ^^^ (u ^^^ n) = u
case h.h.refine_2.intro.inr
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^... | /-
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 [Nat.xor_comm u, Nat.xor_cancel_left] | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.h.refine_2.intro.inr
n : ℕ
hn : ∀ m < n, ∀ (m_1 : ℕ), grundyValue (nim ↑m + nim ↑m_1) = ↑(m ^^^ m_1)
m : ℕ
hm : ∀ m_1 < m, grundyValue (nim ↑n + nim ↑m_1) = ↑(n ^^^ m_1)
u : ℕ
hu : u < n ^^^ m
h : u ^^^ n < m
this : n ^^^ (u ^^^ n) = u
⊢ grundyValue
(moveLeft (nim ↑n + nim ↑m)
(toLeftMovesAdd (Sum.... | /-
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 [hm _ h] using this | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-- We do strong induction on both variables.
induction' n using Nat.strong_induction_on with n hn ge... | Mathlib.SetTheory.Game.Nim.360_0.mmFMhRYSjViKjcP | /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
xor. -/
@[simp]
theorem grundyValue_nim_add_nim (n m : ℕ) :
grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
n m : ℕ
⊢ nim ↑n + nim ↑m ≈ nim ↑(n ^^^ m) | /-
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_eq_iff_equiv_nim, grundyValue_nim_add_nim] | theorem nim_add_nim_equiv {n m : ℕ} : nim n + nim m ≈ nim (n ^^^ m) := by
| Mathlib.SetTheory.Game.Nim.402_0.mmFMhRYSjViKjcP | theorem nim_add_nim_equiv {n m : ℕ} : nim n + nim m ≈ nim (n ^^^ m) | Mathlib_SetTheory_Game_Nim |
G H : PGame
inst✝¹ : Impartial G
inst✝ : Impartial H
n m : ℕ
hG : grundyValue G = ↑n
hH : grundyValue H = ↑m
⊢ grundyValue (G + H) = ↑(n ^^^ m) | /-
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_grundyValue (n ^^^ m), grundyValue_eq_iff_equiv] | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m := by
| Mathlib.SetTheory.Game.Nim.406_0.mmFMhRYSjViKjcP | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
G H : PGame
inst✝¹ : Impartial G
inst✝ : Impartial H
n m : ℕ
hG : grundyValue G = ↑n
hH : grundyValue H = ↑m
⊢ G + H ≈ nim ↑(n ^^^ m) | /-
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.trans _ nim_add_nim_equiv | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m := by
rw [← nim_grundyValue (n ^^^ m), grundyValue_eq_iff_equiv]
| Mathlib.SetTheory.Game.Nim.406_0.mmFMhRYSjViKjcP | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
G H : PGame
inst✝¹ : Impartial G
inst✝ : Impartial H
n m : ℕ
hG : grundyValue G = ↑n
hH : grundyValue H = ↑m
⊢ G + H ≈ nim ↑n + nim ↑m | /-
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 add_congr (equiv_nim_grundyValue G) (equiv_nim_grundyValue H) | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m := by
rw [← nim_grundyValue (n ^^^ m), grundyValue_eq_iff_equiv]
refine' Equiv.trans _ nim_add_nim_equiv
| Mathlib.SetTheory.Game.Nim.406_0.mmFMhRYSjViKjcP | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.e'_4.h.e'_5.h.e'_1
G H : PGame
inst✝¹ : Impartial G
inst✝ : Impartial H
n m : ℕ
hG : grundyValue G = ↑n
hH : grundyValue H = ↑m
⊢ ↑n = grundyValue G | /-
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 only [hG, hH] | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m := by
rw [← nim_grundyValue (n ^^^ m), grundyValue_eq_iff_equiv]
refine' Equiv.trans _ nim_add_nim_equiv
convert add_congr (equiv_nim_grundyValue G) (eq... | Mathlib.SetTheory.Game.Nim.406_0.mmFMhRYSjViKjcP | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
case h.e'_4.h.e'_6.h.e'_1
G H : PGame
inst✝¹ : Impartial G
inst✝ : Impartial H
n m : ℕ
hG : grundyValue G = ↑n
hH : grundyValue H = ↑m
⊢ ↑m = grundyValue H | /-
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 only [hG, hH] | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m := by
rw [← nim_grundyValue (n ^^^ m), grundyValue_eq_iff_equiv]
refine' Equiv.trans _ nim_add_nim_equiv
convert add_congr (equiv_nim_grundyValue G) (eq... | Mathlib.SetTheory.Game.Nim.406_0.mmFMhRYSjViKjcP | theorem grundyValue_add (G H : PGame) [G.Impartial] [H.Impartial] {n m : ℕ} (hG : grundyValue G = n)
(hH : grundyValue H = m) : grundyValue (G + H) = n ^^^ m | Mathlib_SetTheory_Game_Nim |
β : Type v
f : β → Type v
P : Type v
s : (b : β) → P ⟶ f b
b : β
x : P
⊢ Pi.π f b (Pi.lift s x) = s b x | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | simp | /-- A restatement of `Types.Limit.lift_π_apply` that uses `Pi.π` and `Pi.lift`,
with specialized universes. -/
theorem pi_lift_π_apply' {β : Type v} (f : β → Type v) {P : Type v}
(s : ∀ b, P ⟶ f b) (b : β) (x : P) :
(Pi.π f b : (piObj f) → f b) (@Pi.lift β _ _ f _ P s x) = s b x := by
| Mathlib.CategoryTheory.Limits.Shapes.Types.63_0.ctQAUYXLRXnvMGw | /-- A restatement of `Types.Limit.lift_π_apply` that uses `Pi.π` and `Pi.lift`,
with specialized universes. -/
theorem pi_lift_π_apply' {β : Type v} (f : β → Type v) {P : Type v}
(s : ∀ b, P ⟶ f b) (b : β) (x : P) :
(Pi.π f b : (piObj f) → f b) (@Pi.lift β _ _ f _ P s x) = s b x | Mathlib_CategoryTheory_Limits_Shapes_Types |
β : Type v
f g : β → Type v
α : (j : β) → f j ⟶ g j
b : β
x : ∏ fun b => f b
⊢ Pi.π g b (Pi.map α x) = α b (Pi.π f b x) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | simp | /-- A restatement of `Types.Limit.map_π_apply` that uses `Pi.π` and `Pi.map`,
with specialized universes. -/
theorem pi_map_π_apply' {β : Type v} {f g : β → Type v} (α : ∀ j, f j ⟶ g j) (b : β) (x) :
(Pi.π g b : ∏ g → g b) (Pi.map α x) = α b ((Pi.π f b : ∏ f → f b) x) := by
| Mathlib.CategoryTheory.Limits.Shapes.Types.79_0.ctQAUYXLRXnvMGw | /-- A restatement of `Types.Limit.map_π_apply` that uses `Pi.π` and `Pi.map`,
with specialized universes. -/
theorem pi_map_π_apply' {β : Type v} {f g : β → Type v} (α : ∀ j, f j ⟶ g j) (b : β) (x) :
(Pi.π g b : ∏ g → g b) (Pi.map α x) = α b ((Pi.π f b : ∏ f → f b) x) | Mathlib_CategoryTheory_Limits_Shapes_Types |
x✝ : Cone (Functor.empty (Type u))
⊢ ∀ (j : Discrete PEmpty.{?u.3735 + 1}),
(fun x x => PUnit.unit) x✝ ≫
{ pt := PUnit.{u + 1},
π :=
(Functor.uniqueFromEmpty
((Functor.const (Discrete PEmpty.{?u.3735 + 1})).obj PUnit.{u + 1})).hom }.π.app
j =... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rintro ⟨⟨⟩⟩ | /-- The category of types has `PUnit` as a terminal object. -/
def terminalLimitCone : Limits.LimitCone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cone :=
{ pt := PUnit
π := (Functor.uniqueFromEmpty _).hom }
isLimit :=
{ lift := fun _ _ => PUnit... | Mathlib.CategoryTheory.Limits.Shapes.Types.86_0.ctQAUYXLRXnvMGw | /-- The category of types has `PUnit` as a terminal object. -/
def terminalLimitCone : Limits.LimitCone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cone | Mathlib_CategoryTheory_Limits_Shapes_Types |
x✝² : Cone (Functor.empty (Type u))
x✝¹ :
x✝².pt ⟶
{ pt := PUnit.{u + 1},
π := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty.{?u.3735 + 1})).obj PUnit.{u + 1})).hom }.pt
x✝ :
∀ (j : Discrete PEmpty.{?u.3735 + 1}),
x✝¹ ≫
{ pt := PUnit.{u + 1},
π :=
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | funext | /-- The category of types has `PUnit` as a terminal object. -/
def terminalLimitCone : Limits.LimitCone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cone :=
{ pt := PUnit
π := (Functor.uniqueFromEmpty _).hom }
isLimit :=
{ lift := fun _ _ => PUnit... | Mathlib.CategoryTheory.Limits.Shapes.Types.86_0.ctQAUYXLRXnvMGw | /-- The category of types has `PUnit` as a terminal object. -/
def terminalLimitCone : Limits.LimitCone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cone | Mathlib_CategoryTheory_Limits_Shapes_Types |
case h
x✝³ : Cone (Functor.empty (Type u))
x✝² :
x✝³.pt ⟶
{ pt := PUnit.{u + 1},
π := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty.{?u.3735 + 1})).obj PUnit.{u + 1})).hom }.pt
x✝¹ :
∀ (j : Discrete PEmpty.{?u.3735 + 1}),
x✝² ≫
{ pt := PUnit.{u + 1},
π :=
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | apply Subsingleton.elim | /-- The category of types has `PUnit` as a terminal object. -/
def terminalLimitCone : Limits.LimitCone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cone :=
{ pt := PUnit
π := (Functor.uniqueFromEmpty _).hom }
isLimit :=
{ lift := fun _ _ => PUnit... | Mathlib.CategoryTheory.Limits.Shapes.Types.86_0.ctQAUYXLRXnvMGw | /-- The category of types has `PUnit` as a terminal object. -/
def terminalLimitCone : Limits.LimitCone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cone | Mathlib_CategoryTheory_Limits_Shapes_Types |
X : Type u
⊢ IsTerminal X ≃ (X ≅ PUnit.{u + 1}) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | calc
IsTerminal X ≃ Unique X := isTerminalEquivUnique _
_ ≃ (X ≃ PUnit.{u + 1}) := uniqueEquivEquivUnique _ _
_ ≃ (X ≅ PUnit) := equivEquivIso | /-- A type is terminal if and only if it is isomorphic to `PUnit`. -/
noncomputable def isTerminalEquivIsoPUnit (X : Type u) : IsTerminal X ≃ (X ≅ PUnit) := by
| Mathlib.CategoryTheory.Limits.Shapes.Types.127_0.ctQAUYXLRXnvMGw | /-- A type is terminal if and only if it is isomorphic to `PUnit`. -/
noncomputable def isTerminalEquivIsoPUnit (X : Type u) : IsTerminal X ≃ (X ≅ PUnit) | Mathlib_CategoryTheory_Limits_Shapes_Types |
x✝ : Cocone (Functor.empty (Type u))
⊢ { pt := PEmpty.{u + 1},
ι := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty.{?u.11885 + 1})).obj PEmpty.{u + 1})).inv }.pt ⟶
x✝.pt | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rintro ⟨⟩ | /-- The category of types has `PEmpty` as an initial object. -/
def initialColimitCocone : Limits.ColimitCocone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cocone :=
{ pt := PEmpty
ι := (Functor.uniqueFromEmpty _).inv }
isColimit :=
{ desc := fun... | Mathlib.CategoryTheory.Limits.Shapes.Types.134_0.ctQAUYXLRXnvMGw | /-- The category of types has `PEmpty` as an initial object. -/
def initialColimitCocone : Limits.ColimitCocone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cocone | Mathlib_CategoryTheory_Limits_Shapes_Types |
x✝ : Cocone (Functor.empty (Type u))
⊢ ∀ (j : Discrete PEmpty.{?u.11885 + 1}),
{ pt := PEmpty.{u + 1},
ι :=
(Functor.uniqueFromEmpty
((Functor.const (Discrete PEmpty.{?u.11885 + 1})).obj PEmpty.{u + 1})).inv }.ι.app
j ≫
(fun x a => PEmpty.cas... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rintro ⟨⟨⟩⟩ | /-- The category of types has `PEmpty` as an initial object. -/
def initialColimitCocone : Limits.ColimitCocone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cocone :=
{ pt := PEmpty
ι := (Functor.uniqueFromEmpty _).inv }
isColimit :=
{ desc := fun... | Mathlib.CategoryTheory.Limits.Shapes.Types.134_0.ctQAUYXLRXnvMGw | /-- The category of types has `PEmpty` as an initial object. -/
def initialColimitCocone : Limits.ColimitCocone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cocone | Mathlib_CategoryTheory_Limits_Shapes_Types |
x✝² : Cocone (Functor.empty (Type u))
x✝¹ :
{ pt := PEmpty.{u + 1},
ι := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty.{?u.11885 + 1})).obj PEmpty.{u + 1})).inv }.pt ⟶
x✝².pt
x✝ :
∀ (j : Discrete PEmpty.{?u.11885 + 1}),
{ pt := PEmpty.{u + 1},
ι :=
(Func... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | funext x | /-- The category of types has `PEmpty` as an initial object. -/
def initialColimitCocone : Limits.ColimitCocone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cocone :=
{ pt := PEmpty
ι := (Functor.uniqueFromEmpty _).inv }
isColimit :=
{ desc := fun... | Mathlib.CategoryTheory.Limits.Shapes.Types.134_0.ctQAUYXLRXnvMGw | /-- The category of types has `PEmpty` as an initial object. -/
def initialColimitCocone : Limits.ColimitCocone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cocone | Mathlib_CategoryTheory_Limits_Shapes_Types |
case h
x✝² : Cocone (Functor.empty (Type u))
x✝¹ :
{ pt := PEmpty.{u + 1},
ι := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty.{?u.11885 + 1})).obj PEmpty.{u + 1})).inv }.pt ⟶
x✝².pt
x✝ :
∀ (j : Discrete PEmpty.{?u.11885 + 1}),
{ pt := PEmpty.{u + 1},
ι :=
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | cases x | /-- The category of types has `PEmpty` as an initial object. -/
def initialColimitCocone : Limits.ColimitCocone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cocone :=
{ pt := PEmpty
ι := (Functor.uniqueFromEmpty _).inv }
isColimit :=
{ desc := fun... | Mathlib.CategoryTheory.Limits.Shapes.Types.134_0.ctQAUYXLRXnvMGw | /-- The category of types has `PEmpty` as an initial object. -/
def initialColimitCocone : Limits.ColimitCocone (Functor.empty (Type u)) where
-- porting note: tidy was able to fill the structure automatically
cocone | Mathlib_CategoryTheory_Limits_Shapes_Types |
⊢ binaryProductFunctor ≅ prod.functor | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | refine' NatIso.ofComponents (fun X => _) (fun _ => _) | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
| Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_1
X : Type u
⊢ binaryProductFunctor.obj X ≅ prod.functor.obj X | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | refine' NatIso.ofComponents (fun Y => _) (fun _ => _) | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_1.refine'_1
X Y : Type u
⊢ (binaryProductFunctor.obj X).obj Y ≅ (prod.functor.obj X).obj Y | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | exact ((limit.isLimit _).conePointUniqueUpToIso (binaryProductLimit X Y)).symm | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_1.refine'_2
X X✝ Y✝ : Type u
x✝ : X✝ ⟶ Y✝
⊢ (binaryProductFunctor.obj X).map x✝ ≫
((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) Y✝).hom =
((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) X... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | apply Limits.prod.hom_ext | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_1.refine'_2.h₁
X X✝ Y✝ : Type u
x✝ : X✝ ⟶ Y✝
⊢ ((binaryProductFunctor.obj X).map x✝ ≫
((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) Y✝).hom) ≫
prod.fst =
(((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryP... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | simp | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_1.refine'_2.h₂
X X✝ Y✝ : Type u
x✝ : X✝ ⟶ Y✝
⊢ ((binaryProductFunctor.obj X).map x✝ ≫
((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) Y✝).hom) ≫
prod.snd =
(((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryP... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | simp | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_1.refine'_2.h₁
X X✝ Y✝ : Type u
x✝ : X✝ ⟶ Y✝
⊢ (binaryProductFunctor.obj X).map x✝ ≫ _root_.Prod.fst = _root_.Prod.fst | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rfl | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_1.refine'_2.h₂
X X✝ Y✝ : Type u
x✝ : X✝ ⟶ Y✝
⊢ (binaryProductFunctor.obj X).map x✝ ≫ _root_.Prod.snd = _root_.Prod.snd ≫ x✝ | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rfl | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_2
X✝ Y✝ : Type u
x✝ : X✝ ⟶ Y✝
⊢ binaryProductFunctor.map x✝ ≫
((fun X =>
NatIso.ofComponents fun Y =>
(IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)
Y✝).hom =
((fun X =>
NatIso.ofComponents fun Y =>
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | ext : 2 | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_2.w.h
X✝ Y✝ : Type u
x✝¹ : X✝ ⟶ Y✝
x✝ : Type u
⊢ (binaryProductFunctor.map x✝¹ ≫
((fun X =>
NatIso.ofComponents fun Y =>
(IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)
Y✝).hom).app
x✝ =
(((fun X ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | apply Limits.prod.hom_ext | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_2.w.h.h₁
X✝ Y✝ : Type u
x✝¹ : X✝ ⟶ Y✝
x✝ : Type u
⊢ (binaryProductFunctor.map x✝¹ ≫
((fun X =>
NatIso.ofComponents fun Y =>
(IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)
Y✝).hom).app
x✝ ≫
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | simp | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_2.w.h.h₂
X✝ Y✝ : Type u
x✝¹ : X✝ ⟶ Y✝
x✝ : Type u
⊢ (binaryProductFunctor.map x✝¹ ≫
((fun X =>
NatIso.ofComponents fun Y =>
(IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)
Y✝).hom).app
x✝ ≫
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | simp | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_2.w.h.h₁
X✝ Y✝ : Type u
x✝¹ : X✝ ⟶ Y✝
x✝ : Type u
⊢ (binaryProductFunctor.map x✝¹).app x✝ ≫ _root_.Prod.fst = _root_.Prod.fst ≫ x✝¹ | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rfl | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case refine'_2.w.h.h₂
X✝ Y✝ : Type u
x✝¹ : X✝ ⟶ Y✝
x✝ : Type u
⊢ (binaryProductFunctor.map x✝¹).app x✝ ≫ _root_.Prod.snd = _root_.Prod.snd | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rfl | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) := by
refine' NatIso.ofComponents (fun X => _) (fun _ => _)
· refi... | Mathlib.CategoryTheory.Limits.Shapes.Types.236_0.ctQAUYXLRXnvMGw | /-- The product functor given by the instance `HasBinaryProducts (Type u)` is isomorphic to the
explicit binary product functor given by the product type.
-/
noncomputable def binaryProductIsoProd : binaryProductFunctor ≅ (prod.functor : Type u ⥤ _) | Mathlib_CategoryTheory_Limits_Shapes_Types |
X Y : Type u
c : BinaryCofan X Y
⊢ Nonempty (IsColimit c) ↔
Injective (BinaryCofan.inl c) ∧
Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | classical
constructor
· rintro ⟨h⟩
rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) ⟨WalkingPair.left⟩,
← show _ = c.inr from
h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) ⟨WalkingPair.right⟩]
dsimp [binaryCopro... | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
| Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
X Y : Type u
c : BinaryCofan X Y
⊢ Nonempty (IsColimit c) ↔
Injective (BinaryCofan.inl c) ∧
Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | constructor | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
| Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case mp
X Y : Type u
c : BinaryCofan X Y
⊢ Nonempty (IsColimit c) →
Injective (BinaryCofan.inl c) ∧
Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rintro ⟨h⟩ | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
constructor
· | Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case mp.intro
X Y : Type u
c : BinaryCofan X Y
h : IsColimit c
⊢ Injective (BinaryCofan.inl c) ∧
Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) ⟨WalkingPair.left⟩,
← show _ = c.inr from
h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) ⟨WalkingPair.right⟩] | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
constructor
· rintro ⟨h⟩
| Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case mp.intro
X Y : Type u
c : BinaryCofan X Y
h : IsColimit c
⊢ Injective
((binaryCoproductCocone X Y).ι.app { as := left } ≫
(IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧
Injective
((binaryCoproductCocone X Y).ι.app { as := right } ≫
(IsColimit.coconePo... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | dsimp [binaryCoproductCocone] | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
constructor
· rintro ⟨h⟩
rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (bi... | Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case mp.intro
X Y : Type u
c : BinaryCofan X Y
h : IsColimit c
⊢ Injective (Sum.inl ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧
Injective (Sum.inr ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) ∧
IsCompl (Set.range (Sum.inl ≫ (IsColimit.coconePointU... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | refine'
⟨(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp
Sum.inl_injective,
(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp
Sum.inr_injective, _⟩ | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
constructor
· rintro ⟨h⟩
rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (bi... | Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case mp.intro
X Y : Type u
c : BinaryCofan X Y
h : IsColimit c
⊢ IsCompl (Set.range (Sum.inl ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))
(Set.range (Sum.inr ≫ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | erw [Set.range_comp, ← eq_compl_iff_isCompl, Set.range_comp _ Sum.inr, ←
Set.image_compl_eq
(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.bijective] | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
constructor
· rintro ⟨h⟩
rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (bi... | Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case mp.intro
X Y : Type u
c : BinaryCofan X Y
h : IsColimit c
⊢ (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv '' Set.range Sum.inl =
⇑(IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).symm.toEquiv '' (Set.range Sum.inr)ᶜ | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | simp | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
constructor
· rintro ⟨h⟩
rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (bi... | Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case mpr
X Y : Type u
c : BinaryCofan X Y
⊢ Injective (BinaryCofan.inl c) ∧
Injective (BinaryCofan.inr c) ∧ IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) →
Nonempty (IsColimit c) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | rintro ⟨h₁, h₂, h₃⟩ | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
constructor
· rintro ⟨h⟩
rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (bi... | Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
case mpr.intro.intro
X Y : Type u
c : BinaryCofan X Y
h₁ : Injective (BinaryCofan.inl c)
h₂ : Injective (BinaryCofan.inr c)
h₃ : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))
⊢ Nonempty (IsColimit c) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | have : ∀ x, x ∈ Set.range c.inl ∨ x ∈ Set.range c.inr := by
rw [eq_compl_iff_isCompl.mpr h₃.symm]
exact fun _ => or_not | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
classical
constructor
· rintro ⟨h⟩
rw [← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (bi... | Mathlib.CategoryTheory.Limits.Shapes.Types.302_0.ctQAUYXLRXnvMGw | theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) | Mathlib_CategoryTheory_Limits_Shapes_Types |
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