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chore(SetTheory/Game/Ordinal): make Ordinal.toGame an order embedding #19417

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chore(SetTheory/Game/Ordinal): make Ordinal.toGame an order embedding #19417

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058da98
Update Ordinal.lean
vihdzp Nov 23, 2024
8d70aa0
fix
vihdzp Nov 23, 2024
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2 changes: 1 addition & 1 deletion Mathlib/SetTheory/Game/Birthday.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -207,7 +207,7 @@ theorem birthday_ordinalToGame (o : Ordinal) : birthday o.toGame = o := by
apply birthday_quot_le_pGameBirthday
· let ⟨x, hx₁, hx₂⟩ := birthday_eq_pGameBirthday o.toGame
rw [← hx₂, ← toPGame_le_iff]
rw [← PGame.equiv_iff_game_eq] at hx₁
rw [← mk_toPGame, ← PGame.equiv_iff_game_eq] at hx₁
exact hx₁.2.trans (PGame.le_birthday x)

@[simp, norm_cast]
Expand Down
37 changes: 19 additions & 18 deletions Mathlib/SetTheory/Game/Ordinal.lean
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Expand Up @@ -155,15 +155,18 @@ noncomputable def toPGameEmbedding : Ordinal.{u} ↪o PGame.{u} where
map_rel_iff' := @toPGame_le_iff

/-- Converts an ordinal into the corresponding game. -/
noncomputable abbrev toGame (o : Ordinal) : Game := ⟦o.toPGame⟧

/-- The order embedding version of `toGame`. -/
@[simps]
noncomputable def toGameEmbedding : Ordinal.{u} ↪o Game.{u} where
toFun := Ordinal.toGame
noncomputable def toGame : Ordinal.{u} ↪o Game.{u} where
toFun o := ⟦o.toPGame⟧
inj' a b := by simpa [AntisymmRel] using le_antisymm
map_rel_iff' := toPGame_le_iff

@[simp]
theorem mk_toPGame (o : Ordinal) : ⟦o.toPGame⟧ = o.toGame :=
rfl

@[deprecated toGame (since := "2024-11-23")]
alias toGameEmbedding := toGame

@[simp]
theorem toGame_zero : toGame 0 = 0 :=
game_eq toPGame_zero
Expand All @@ -173,22 +176,20 @@ theorem toGame_one : toGame 1 = 1 :=
game_eq toPGame_one

theorem toGame_injective : Function.Injective toGame :=
toGameEmbedding.injective
toGame.injective

@[simp]
theorem toGame_lf_iff {a b : Ordinal} : Game.LF (toGame a) (toGame b) ↔ a < b :=
theorem toGame_lf_iff {a b : Ordinal} : Game.LF a.toGame b.toGame ↔ a < b :=
toPGame_lf_iff

@[simp]
theorem toGame_le_iff {a b : Ordinal} : toGame a ≤ toGame b ↔ a ≤ b :=
theorem toGame_le_iff {a b : Ordinal} : a.toGame ≤ b.toGame ↔ a ≤ b :=
toPGame_le_iff

@[simp]
theorem toGame_lt_iff {a b : Ordinal} : toGame a < toGame b ↔ a < b :=
theorem toGame_lt_iff {a b : Ordinal} : a.toGame < b.toGame ↔ a < b :=
toPGame_lt_iff

theorem toGame_eq_iff {a b : Ordinal} : toGame a = toGame b ↔ a = b :=
toGameEmbedding.inj
theorem toGame_eq_iff {a b : Ordinal} : a.toGame = b.toGame ↔ a = b :=
toGame.inj

/-- The natural addition of ordinals corresponds to their sum as games. -/
theorem toPGame_nadd (a b : Ordinal) : (a ♯ b).toPGame ≈ a.toPGame + b.toPGame := by
Expand Down Expand Up @@ -219,7 +220,7 @@ theorem toPGame_nmul (a b : Ordinal) : (a ⨳ b).toPGame ≈ a.toPGame * b.toPGa
refine ⟨le_of_forall_lf (fun i => ?_) isEmptyElim, le_of_forall_lf (fun i => ?_) isEmptyElim⟩
· rw [toPGame_moveLeft']
rcases lt_nmul_iff.1 (toLeftMovesToPGame_symm_lt i) with ⟨c, hc, d, hd, h⟩
rw [← toPGame_le_iff, le_iff_game_le, ← toGame, ← toGame, toGame_nadd _ _, toGame_nadd _ _,
rw [← toPGame_le_iff, le_iff_game_le, mk_toPGame, mk_toPGame, toGame_nadd _ _, toGame_nadd _ _,
← le_sub_iff_add_le] at h
refine lf_of_le_of_lf h <| (lf_congr_left ?_).1 <| moveLeft_lf <| toLeftMovesMul <| Sum.inl
⟨toLeftMovesToPGame ⟨c, hc⟩, toLeftMovesToPGame ⟨d, hd⟩⟩
Expand All @@ -232,15 +233,15 @@ theorem toPGame_nmul (a b : Ordinal) : (a ⨳ b).toPGame ≈ a.toPGame * b.toPGa
rw [mul_moveLeft_inl, toPGame_moveLeft', toPGame_moveLeft', lf_iff_game_lf,
quot_sub, quot_add, ← Game.not_le, le_sub_iff_add_le]
repeat rw [← game_eq (toPGame_nmul _ _)]
repeat rw [← toGame_nadd]
simp_rw [mk_toPGame, ← toGame_nadd]
apply toPGame_lf (nmul_nadd_lt _ _) <;>
exact toLeftMovesToPGame_symm_lt _
termination_by (a, b)

theorem toGame_nmul (a b : Ordinal) : (a ⨳ b).toGame = ⟦a.toPGame * b.toPGame⟧ :=
game_eq (toPGame_nmul a b)

@[simp, norm_cast]
@[simp]
Comment on lines -243 to +244
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Why this change?

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"badly shaped simp lemma, can't start with coe"

theorem toGame_natCast : ∀ n : ℕ, toGame n = n
| 0 => Quot.sound (zeroToPGameRelabelling).equiv
| n + 1 => by
Expand All @@ -249,6 +250,6 @@ theorem toGame_natCast : ∀ n : ℕ, toGame n = n
rfl

theorem toPGame_natCast (n : ℕ) : toPGame n ≈ n := by
rw [PGame.equiv_iff_game_eq, ← toGame, toGame_natCast, quot_natCast]
rw [PGame.equiv_iff_game_eq, mk_toPGame, toGame_natCast, quot_natCast]

end Ordinal