Short games

A combinatorial game is Short [Conway, ch.9][conway2001] if it has only finitely many positions. In particular, this means there is a finite set of moves at every point.

We prove that the order relations ≤ and <, and the equivalence relation ≈, are decidable on short games, although unfortunately in practice decide doesn't seem to be able to prove anything using these instances.

class inductive SetTheory.PGame.Short : PGame → Type (u + 1)

A short game is a game with a finite set of moves at every turn.

• mk {α β : Type u} {L : α → PGame} {R : β → PGame} : ((i : α) → (L i).Short) → ((j : β) → (R j).Short) → [Fintype α] → [Fintype β] → (PGame.mk α β L R).Short

instance SetTheory.PGame.subsingleton_short (x : PGame) : Subsingleton x.Short

@[irreducible]

theorem SetTheory.PGame.subsingleton_short_example (x : PGame) : Subsingleton x.Short

def SetTheory.PGame.Short.mk' {x : PGame} [Fintype x.LeftMoves] [Fintype x.RightMoves] (sL : (i : x.LeftMoves) → (x.moveLeft i).Short) (sR : (j : x.RightMoves) → (x.moveRight j).Short) : x.Short

A synonym for Short.mk that specifies the pgame in an implicit argument.

Equations

• SetTheory.PGame.Short.mk' sL sR = ⋯.mpr (SetTheory.PGame.Short.mk sL sR)

def SetTheory.PGame.fintypeLeft {α β : Type u} {L : α → PGame} {R : β → PGame} [S : (mk α β L R).Short] : Fintype α

Extracting the Fintype instance for the indexing type for Left's moves in a short game. This is an unindexed typeclass, so it can't be made a global instance.

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instance SetTheory.PGame.fintypeLeftMoves (x : PGame) [S : x.Short] : Fintype x.LeftMoves

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def SetTheory.PGame.fintypeRight {α β : Type u} {L : α → PGame} {R : β → PGame} [S : (mk α β L R).Short] : Fintype β

Extracting the Fintype instance for the indexing type for Right's moves in a short game. This is an unindexed typeclass, so it can't be made a global instance.

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instance SetTheory.PGame.fintypeRightMoves (x : PGame) [S : x.Short] : Fintype x.RightMoves

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instance SetTheory.PGame.moveLeftShort (x : PGame) [S : x.Short] (i : x.LeftMoves) : (x.moveLeft i).Short

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def SetTheory.PGame.moveLeftShort' {xl xr : Type u_1} (xL : xl → PGame) (xR : xr → PGame) [S : (mk xl xr xL xR).Short] (i : xl) : (xL i).Short

Extracting the Short instance for a move by Left. This would be a dangerous instance potentially introducing new metavariables in typeclass search, so we only make it an instance locally.

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instance SetTheory.PGame.moveRightShort (x : PGame) [S : x.Short] (j : x.RightMoves) : (x.moveRight j).Short

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def SetTheory.PGame.moveRightShort' {xl xr : Type u_1} (xL : xl → PGame) (xR : xr → PGame) [S : (mk xl xr xL xR).Short] (j : xr) : (xR j).Short

Extracting the Short instance for a move by Right. This would be a dangerous instance potentially introducing new metavariables in typeclass search, so we only make it an instance locally.

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theorem SetTheory.PGame.short_birthday (x : PGame) [x.Short] : x.birthday < Ordinal.omega0

def SetTheory.PGame.Short.ofIsEmpty {l r : Type u_1} {xL : l → PGame} {xR : r → PGame} [IsEmpty l] [IsEmpty r] : (PGame.mk l r xL xR).Short

This leads to infinite loops if made into an instance.

Equations

• SetTheory.PGame.Short.ofIsEmpty = SetTheory.PGame.Short.mk (fun (a : l) => isEmptyElim a) fun (a : r) => isEmptyElim a

instance SetTheory.PGame.short0 : Short 0

Equations

• SetTheory.PGame.short0 = SetTheory.PGame.Short.ofIsEmpty

instance SetTheory.PGame.short1 : Short 1

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class inductive SetTheory.PGame.ListShort : List PGame → Type (u + 1)

Evidence that every PGame in a list is Short.

• nil : ListShort []
• cons' {hd : PGame} {tl : List PGame} : hd.Short → ListShort tl → ListShort (hd :: tl)

instance SetTheory.PGame.ListShort.cons (hd : PGame) [short_hd : hd.Short] (tl : List PGame) [short_tl : ListShort tl] : ListShort (hd :: tl)

Equations

• SetTheory.PGame.ListShort.cons hd tl = SetTheory.PGame.ListShort.cons' short_hd short_tl

instance SetTheory.PGame.listShortGet (L : List PGame) [ListShort L] (i : ℕ) (h : i < L.length) : L[i].Short

Equations

• SetTheory.PGame.listShortGet (head :: tail) 0 x_4 = S
• SetTheory.PGame.listShortGet (head :: tl) n.succ h = SetTheory.PGame.listShortGet tl n ⋯

instance SetTheory.PGame.shortOfLists (L R : List PGame) [ListShort L] [ListShort R] : (ofLists L R).Short

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def SetTheory.PGame.shortOfRelabelling {x y : PGame} : x.Relabelling y → x.Short → y.Short

If x is a short game, and y is a relabelling of x, then y is also short.

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instance SetTheory.PGame.shortNeg (x : PGame) [x.Short] : (-x).Short

Equations

• (SetTheory.PGame.mk xl xr xL xR).shortNeg = SetTheory.PGame.Short.mk (fun (i : xr) => (xR i).shortNeg) fun (i : xl) => (xL i).shortNeg

@[irreducible]

instance SetTheory.PGame.shortAdd (x y : PGame) [x.Short] [y.Short] : (x + y).Short

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instance SetTheory.PGame.shortNat (n : ℕ) : (↑n).Short

Equations

• SetTheory.PGame.shortNat 0 = SetTheory.PGame.short0
• SetTheory.PGame.shortNat n.succ = n.unaryCast.shortAdd 1

instance SetTheory.PGame.shortOfNat (n : ℕ) [n.AtLeastTwo] : (OfNat.ofNat n).Short

Equations

• SetTheory.PGame.shortOfNat n = SetTheory.PGame.shortNat n

instance SetTheory.PGame.shortBit0 (x : PGame) [x.Short] : (x + x).Short

Equations

• x.shortBit0 = inferInstance

instance SetTheory.PGame.shortBit1 (x : PGame) [x.Short] : (x + x + 1).Short

Equations

• x.shortBit1 = (x + x).shortAdd 1

@[irreducible]

def SetTheory.PGame.leLFDecidable (x y : PGame) [x.Short] [y.Short] : Decidable (x ≤ y) × Decidable (x.LF y)

Auxiliary construction of decidability instances. We build Decidable (x ≤ y) and Decidable (x ⧏ y) in a simultaneous induction. Instances for the two projections separately are provided below.

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instance SetTheory.PGame.leDecidable (x y : PGame) [x.Short] [y.Short] : Decidable (x ≤ y)

Equations

• x.leDecidable y = (x.leLFDecidable y).1

instance SetTheory.PGame.lfDecidable (x y : PGame) [x.Short] [y.Short] : Decidable (x.LF y)

Equations

• x.lfDecidable y = (x.leLFDecidable y).2

instance SetTheory.PGame.ltDecidable (x y : PGame) [x.Short] [y.Short] : Decidable (x < y)

Equations

• x.ltDecidable y = inferInstanceAs (Decidable (x ≤ y ∧ x.LF y))

instance SetTheory.PGame.equivDecidable (x y : PGame) [x.Short] [y.Short] : Decidable (x ≈ y)

Equations

• x.equivDecidable y = inferInstanceAs (Decidable (x ≤ y ∧ y ≤ x))