statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
is_empty_right_moves_add (x y : pgame.{u})
[is_empty x.right_moves] [is_empty y.right_moves] : is_empty (x + y).right_moves | begin
unfreezingI { cases x, cases y },
apply is_empty_sum.2 ⟨_, _⟩,
assumption'
end | instance | pgame.is_empty_right_moves_add | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_zero_relabelling : Π (x : pgame.{u}), x + 0 ≡r x | | ⟨xl, xr, xL, xR⟩ :=
begin
refine ⟨equiv.sum_empty xl pempty, equiv.sum_empty xr pempty, _, _⟩;
rintro (⟨i⟩|⟨⟨⟩⟩);
apply add_zero_relabelling
end | def | pgame.add_zero_relabelling | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.sum_empty",
"pempty"
] | `x + 0` has exactly the same moves as `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_zero_equiv (x : pgame.{u}) : x + 0 ≈ x | (add_zero_relabelling x).equiv | lemma | pgame.add_zero_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv"
] | `x + 0` is equivalent to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_add_relabelling : Π (x : pgame.{u}), 0 + x ≡r x | | ⟨xl, xr, xL, xR⟩ :=
begin
refine ⟨equiv.empty_sum pempty xl, equiv.empty_sum pempty xr, _, _⟩;
rintro (⟨⟨⟩⟩|⟨i⟩);
apply zero_add_relabelling
end | def | pgame.zero_add_relabelling | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.empty_sum",
"pempty"
] | `0 + x` has exactly the same moves as `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_add_equiv (x : pgame.{u}) : 0 + x ≈ x | (zero_add_relabelling x).equiv | lemma | pgame.zero_add_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv"
] | `0 + x` is equivalent to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_moves_add : ∀ (x y : pgame.{u}),
(x + y).left_moves = (x.left_moves ⊕ y.left_moves) | | ⟨_, _, _, _⟩ ⟨_, _, _, _⟩ := rfl | theorem | pgame.left_moves_add | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_moves_add : ∀ (x y : pgame.{u}),
(x + y).right_moves = (x.right_moves ⊕ y.right_moves) | | ⟨_, _, _, _⟩ ⟨_, _, _, _⟩ := rfl | theorem | pgame.right_moves_add | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_left_moves_add {x y : pgame} :
x.left_moves ⊕ y.left_moves ≃ (x + y).left_moves | equiv.cast (left_moves_add x y).symm | def | pgame.to_left_moves_add | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.cast",
"pgame"
] | Converts a left move for `x` or `y` into a left move for `x + y` and vice versa.
Even though these types are the same (not definitionally so), this is the preferred way to convert
between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_right_moves_add {x y : pgame} :
x.right_moves ⊕ y.right_moves ≃ (x + y).right_moves | equiv.cast (right_moves_add x y).symm | def | pgame.to_right_moves_add | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.cast",
"pgame"
] | Converts a right move for `x` or `y` into a right move for `x + y` and vice versa.
Even though these types are the same (not definitionally so), this is the preferred way to convert
between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_add_move_left_inl {xl xr yl yr} {xL xR yL yR} {i} :
(mk xl xr xL xR + mk yl yr yL yR).move_left (sum.inl i) =
(mk xl xr xL xR).move_left i + (mk yl yr yL yR) | rfl | lemma | pgame.mk_add_move_left_inl | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_move_left_inl {x : pgame} (y : pgame) (i) :
(x + y).move_left (to_left_moves_add (sum.inl i)) = x.move_left i + y | by { cases x, cases y, refl } | lemma | pgame.add_move_left_inl | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_add_move_right_inl {xl xr yl yr} {xL xR yL yR} {i} :
(mk xl xr xL xR + mk yl yr yL yR).move_right (sum.inl i) =
(mk xl xr xL xR).move_right i + (mk yl yr yL yR) | rfl | lemma | pgame.mk_add_move_right_inl | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_move_right_inl {x : pgame} (y : pgame) (i) :
(x + y).move_right (to_right_moves_add (sum.inl i)) = x.move_right i + y | by { cases x, cases y, refl } | lemma | pgame.add_move_right_inl | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_add_move_left_inr {xl xr yl yr} {xL xR yL yR} {i} :
(mk xl xr xL xR + mk yl yr yL yR).move_left (sum.inr i) =
(mk xl xr xL xR) + (mk yl yr yL yR).move_left i | rfl | lemma | pgame.mk_add_move_left_inr | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_move_left_inr (x : pgame) {y : pgame} (i) :
(x + y).move_left (to_left_moves_add (sum.inr i)) = x + y.move_left i | by { cases x, cases y, refl } | lemma | pgame.add_move_left_inr | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_add_move_right_inr {xl xr yl yr} {xL xR yL yR} {i} :
(mk xl xr xL xR + mk yl yr yL yR).move_right (sum.inr i) =
(mk xl xr xL xR) + (mk yl yr yL yR).move_right i | rfl | lemma | pgame.mk_add_move_right_inr | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_move_right_inr (x : pgame) {y : pgame} (i) :
(x + y).move_right (to_right_moves_add (sum.inr i)) = x + y.move_right i | by { cases x, cases y, refl } | lemma | pgame.add_move_right_inr | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_moves_add_cases {x y : pgame} (k) {P : (x + y).left_moves → Prop}
(hl : ∀ i, P $ to_left_moves_add (sum.inl i))
(hr : ∀ i, P $ to_left_moves_add (sum.inr i)) : P k | begin
rw ←to_left_moves_add.apply_symm_apply k,
cases to_left_moves_add.symm k with i i,
{ exact hl i },
{ exact hr i }
end | lemma | pgame.left_moves_add_cases | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_moves_add_cases {x y : pgame} (k) {P : (x + y).right_moves → Prop}
(hl : ∀ j, P $ to_right_moves_add (sum.inl j))
(hr : ∀ j, P $ to_right_moves_add (sum.inr j)) : P k | begin
rw ←to_right_moves_add.apply_symm_apply k,
cases to_right_moves_add.symm k with i i,
{ exact hl i },
{ exact hr i }
end | lemma | pgame.right_moves_add_cases | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_empty_nat_right_moves : ∀ n : ℕ, is_empty (right_moves n) | | 0 := pempty.is_empty
| (n + 1) := begin
haveI := is_empty_nat_right_moves n,
rw [pgame.nat_succ, right_moves_add],
apply_instance
end | instance | pgame.is_empty_nat_right_moves | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"is_empty",
"pgame.nat_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabelling.add_congr : ∀ {w x y z : pgame.{u}}, w ≡r x → y ≡r z → w + y ≡r x + z | | ⟨wl, wr, wL, wR⟩ ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ ⟨zl, zr, zL, zR⟩
⟨L₁, R₁, hL₁, hR₁⟩ ⟨L₂, R₂, hL₂, hR₂⟩ :=
begin
let Hwx : ⟨wl, wr, wL, wR⟩ ≡r ⟨xl, xr, xL, xR⟩ := ⟨L₁, R₁, hL₁, hR₁⟩,
let Hyz : ⟨yl, yr, yL, yR⟩ ≡r ⟨zl, zr, zL, zR⟩ := ⟨L₂, R₂, hL₂, hR₂⟩,
refine ⟨equiv.sum_congr L₁ L₂, equiv.sum_congr R₁ R₂, _... | def | pgame.relabelling.add_congr | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.sum_congr"
] | If `w` has the same moves as `x` and `y` has the same moves as `z`,
then `w + y` has the same moves as `x + z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_zero (x : pgame) : x - 0 = x + 0 | show x + -0 = x + 0, by rw neg_zero | theorem | pgame.sub_zero | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabelling.sub_congr {w x y z : pgame} (h₁ : w ≡r x) (h₂ : y ≡r z) : w - y ≡r x - z | h₁.add_congr h₂.neg_congr | def | pgame.relabelling.sub_congr | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | If `w` has the same moves as `x` and `y` has the same moves as `z`,
then `w - y` has the same moves as `x - z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_add_relabelling : Π (x y : pgame), -(x + y) ≡r -x + -y | | ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ :=
begin
refine ⟨equiv.refl _, equiv.refl _, _, _⟩,
all_goals {
exact λ j, sum.cases_on j
(λ j, neg_add_relabelling _ _)
(λ j, neg_add_relabelling ⟨xl, xr, xL, xR⟩ _) }
end
using_well_founded { dec_tac := pgame_wf_tac } | def | pgame.neg_add_relabelling | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.refl",
"pgame"
] | `-(x + y)` has exactly the same moves as `-x + -y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_add_le {x y : pgame} : -(x + y) ≤ -x + -y | (neg_add_relabelling x y).le | theorem | pgame.neg_add_le | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_relabelling : Π (x y : pgame.{u}), x + y ≡r y + x | | (mk xl xr xL xR) (mk yl yr yL yR) :=
begin
refine ⟨equiv.sum_comm _ _, equiv.sum_comm _ _, _, _⟩;
rintros (_|_);
{ dsimp [left_moves_add, right_moves_add], apply add_comm_relabelling }
end
using_well_founded { dec_tac := pgame_wf_tac } | def | pgame.add_comm_relabelling | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.sum_comm"
] | `x + y` has exactly the same moves as `y + x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_comm_le {x y : pgame} : x + y ≤ y + x | (add_comm_relabelling x y).le | theorem | pgame.add_comm_le | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_equiv {x y : pgame} : x + y ≈ y + x | (add_comm_relabelling x y).equiv | theorem | pgame.add_comm_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_assoc_relabelling : Π (x y z : pgame.{u}), x + y + z ≡r x + (y + z) | | ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ ⟨zl, zr, zL, zR⟩ :=
begin
refine ⟨equiv.sum_assoc _ _ _, equiv.sum_assoc _ _ _, _, _⟩,
all_goals
{ rintro (⟨i|i⟩|i) <|> rintro (j|⟨j|j⟩),
{ apply add_assoc_relabelling },
{ apply add_assoc_relabelling ⟨xl, xr, xL, xR⟩ },
{ apply add_assoc_relabelling ⟨xl, xr, xL, xR... | def | pgame.add_assoc_relabelling | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.sum_assoc"
] | `(x + y) + z` has exactly the same moves as `x + (y + z)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_assoc_equiv {x y z : pgame} : (x + y) + z ≈ x + (y + z) | (add_assoc_relabelling x y z).equiv | theorem | pgame.add_assoc_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_neg_le_zero : ∀ (x : pgame), -x + x ≤ 0 | | ⟨xl, xr, xL, xR⟩ :=
le_zero.2 $ λ i, begin
cases i,
{ -- If Left played in -x, Right responds with the same move in x.
refine ⟨@to_right_moves_add _ ⟨_, _, _, _⟩ (sum.inr i), _⟩,
convert @add_left_neg_le_zero (xR i),
apply add_move_right_inr },
{ -- If Left in x, Right responds with the same move in... | theorem | pgame.add_left_neg_le_zero | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_add_left_neg (x : pgame) : 0 ≤ -x + x | begin
rw [←neg_le_neg_iff, neg_zero],
exact neg_add_le.trans (add_left_neg_le_zero _)
end | theorem | pgame.zero_le_add_left_neg | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_neg_equiv (x : pgame) : -x + x ≈ 0 | ⟨add_left_neg_le_zero x, zero_le_add_left_neg x⟩ | theorem | pgame.add_left_neg_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_right_neg_le_zero (x : pgame) : x + -x ≤ 0 | add_comm_le.trans (add_left_neg_le_zero x) | theorem | pgame.add_right_neg_le_zero | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_add_right_neg (x : pgame) : 0 ≤ x + -x | (zero_le_add_left_neg x).trans add_comm_le | theorem | pgame.zero_le_add_right_neg | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_right_neg_equiv (x : pgame) : x + -x ≈ 0 | ⟨add_right_neg_le_zero x, zero_le_add_right_neg x⟩ | theorem | pgame.add_right_neg_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self_equiv : ∀ x, x - x ≈ 0 | add_right_neg_equiv | theorem | pgame.sub_self_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_add_right' : ∀ {x y z : pgame} (h : x ≤ y), x + z ≤ y + z | | (mk xl xr xL xR) (mk yl yr yL yR) (mk zl zr zL zR) :=
λ h, begin
refine le_def.2 ⟨λ i, _, λ i, _⟩;
cases i,
{ rw le_def at h,
cases h,
rcases h_left i with ⟨i', ih⟩ | ⟨j, jh⟩,
{ exact or.inl ⟨to_left_moves_add (sum.inl i'), add_le_add_right' ih⟩ },
{ refine or.inr ⟨to_right_moves_add (sum.inl j)... | lemma | pgame.add_le_add_right' | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"ih",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_add_le : covariant_class pgame pgame (swap (+)) (≤) | ⟨λ x y z, add_le_add_right'⟩ | instance | pgame.covariant_class_swap_add_le | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"covariant_class",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_add_le : covariant_class pgame pgame (+) (≤) | ⟨λ x y z h, (add_comm_le.trans (add_le_add_right h x)).trans add_comm_le⟩ | instance | pgame.covariant_class_add_le | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"covariant_class",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lf_add_right {y z : pgame} (h : y ⧏ z) (x) : y + x ⧏ z + x | suffices z + x ≤ y + x → z ≤ y, by { rw ←pgame.not_le at ⊢ h, exact mt this h }, λ w,
calc z ≤ z + 0 : (add_zero_relabelling _).symm.le
... ≤ z + (x + -x) : add_le_add_left (zero_le_add_right_neg x) _
... ≤ z + x + -x : (add_assoc_relabelling _ _ _).symm.le
... ≤ y + x + -x : add_le_add_righ... | theorem | pgame.add_lf_add_right | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lf_add_left {y z : pgame} (h : y ⧏ z) (x) : x + y ⧏ x + z | by { rw lf_congr add_comm_equiv add_comm_equiv, apply add_lf_add_right h } | theorem | pgame.add_lf_add_left | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_add_lt : covariant_class pgame pgame (swap (+)) (<) | ⟨λ x y z h, ⟨add_le_add_right h.1 x, add_lf_add_right h.2 x⟩⟩ | instance | pgame.covariant_class_swap_add_lt | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"covariant_class",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_add_lt : covariant_class pgame pgame (+) (<) | ⟨λ x y z h, ⟨add_le_add_left h.1 x, add_lf_add_left h.2 x⟩⟩ | instance | pgame.covariant_class_add_lt | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"covariant_class",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lf_add_of_lf_of_le {w x y z : pgame} (hwx : w ⧏ x) (hyz : y ≤ z) : w + y ⧏ x + z | lf_of_lf_of_le (add_lf_add_right hwx y) (add_le_add_left hyz x) | theorem | pgame.add_lf_add_of_lf_of_le | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lf_add_of_le_of_lf {w x y z : pgame} (hwx : w ≤ x) (hyz : y ⧏ z) : w + y ⧏ x + z | lf_of_le_of_lf (add_le_add_right hwx y) (add_lf_add_left hyz x) | theorem | pgame.add_lf_add_of_le_of_lf | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_congr {w x y z : pgame} (h₁ : w ≈ x) (h₂ : y ≈ z) : w + y ≈ x + z | ⟨(add_le_add_left h₂.1 w).trans (add_le_add_right h₁.1 z),
(add_le_add_left h₂.2 x).trans (add_le_add_right h₁.2 y)⟩ | theorem | pgame.add_congr | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_congr_left {x y z : pgame} (h : x ≈ y) : x + z ≈ y + z | add_congr h equiv_rfl | theorem | pgame.add_congr_left | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_congr_right {x y z : pgame} : y ≈ z → x + y ≈ x + z | add_congr equiv_rfl | theorem | pgame.add_congr_right | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_congr {w x y z : pgame} (h₁ : w ≈ x) (h₂ : y ≈ z) : w - y ≈ x - z | add_congr h₁ (neg_equiv_neg_iff.2 h₂) | theorem | pgame.sub_congr | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_congr_left {x y z : pgame} (h : x ≈ y) : x - z ≈ y - z | sub_congr h equiv_rfl | theorem | pgame.sub_congr_left | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_congr_right {x y z : pgame} : y ≈ z → x - y ≈ x - z | sub_congr equiv_rfl | theorem | pgame.sub_congr_right | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_sub_nonneg {x y : pgame} : x ≤ y ↔ 0 ≤ y - x | ⟨λ h, (zero_le_add_right_neg x).trans (add_le_add_right h _),
λ h,
calc x ≤ 0 + x : (zero_add_relabelling x).symm.le
... ≤ y - x + x : add_le_add_right h _
... ≤ y + (-x + x) : (add_assoc_relabelling _ _ _).le
... ≤ y + 0 : add_le_add_left (add_left_neg_le_zero x) _
... ≤ y : (add_zero_relabellin... | theorem | pgame.le_iff_sub_nonneg | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf_iff_sub_zero_lf {x y : pgame} : x ⧏ y ↔ 0 ⧏ y - x | ⟨λ h, (zero_le_add_right_neg x).trans_lf (add_lf_add_right h _),
λ h,
calc x ≤ 0 + x : (zero_add_relabelling x).symm.le
... ⧏ y - x + x : add_lf_add_right h _
... ≤ y + (-x + x) : (add_assoc_relabelling _ _ _).le
... ≤ y + 0 : add_le_add_left (add_left_neg_le_zero x) _
... ≤ y : (add_zero_relabel... | theorem | pgame.lf_iff_sub_zero_lf | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_iff_sub_pos {x y : pgame} : x < y ↔ 0 < y - x | ⟨λ h, lt_of_le_of_lt (zero_le_add_right_neg x) (add_lt_add_right h _),
λ h,
calc x ≤ 0 + x : (zero_add_relabelling x).symm.le
... < y - x + x : add_lt_add_right h _
... ≤ y + (-x + x) : (add_assoc_relabelling _ _ _).le
... ≤ y + 0 : add_le_add_left (add_left_neg_le_zero x) _
... ≤ y : (add_zero_r... | theorem | pgame.lt_iff_sub_pos | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star : pgame.{u} | ⟨punit, punit, λ _, 0, λ _, 0⟩ | def | pgame.star | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | The pre-game `star`, which is fuzzy with zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_left_moves : star.left_moves = punit | rfl | theorem | pgame.star_left_moves | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_right_moves : star.right_moves = punit | rfl | theorem | pgame.star_right_moves | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_move_left (x) : star.move_left x = 0 | rfl | theorem | pgame.star_move_left | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_move_right (x) : star.move_right x = 0 | rfl | theorem | pgame.star_move_right | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_star_left_moves : unique star.left_moves | punit.unique | instance | pgame.unique_star_left_moves | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"punit.unique",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_star_right_moves : unique star.right_moves | punit.unique | instance | pgame.unique_star_right_moves | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"punit.unique",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_fuzzy_zero : star ‖ 0 | ⟨by { rw lf_zero, use default, rintros ⟨⟩ }, by { rw zero_lf, use default, rintros ⟨⟩ }⟩ | theorem | pgame.star_fuzzy_zero | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_star : -star = star | by simp [star] | theorem | pgame.neg_star | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_one : (0 : pgame) < 1 | lt_of_le_of_lf (zero_le_of_is_empty_right_moves 1) (zero_lf_le.2 ⟨default, le_rfl⟩) | theorem | pgame.zero_lt_one | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lf_one : (0 : pgame) ⧏ 1 | pgame.zero_lt_one.lf | theorem | pgame.zero_lf_one | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short : pgame.{u} → Type (u+1)
| mk : Π {α β : Type u} {L : α → pgame.{u}} {R : β → pgame.{u}}
(sL : ∀ i : α, short (L i)) (sR : ∀ j : β, short (R j))
[fintype α] [fintype β],
short ⟨α, β, L, R⟩ | inductive | pgame.short | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"fintype"
] | A short game is a game with a finite set of moves at every turn. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_short : Π (x : pgame), subsingleton (short x) | | (mk xl xr xL xR) :=
⟨λ a b, begin
cases a, cases b,
congr,
{ funext,
apply @subsingleton.elim _ (subsingleton_short (xL x)) },
{ funext,
apply @subsingleton.elim _ (subsingleton_short (xR x)) },
end⟩
using_well_founded { dec_tac := pgame_wf_tac } | instance | pgame.subsingleton_short | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short.mk' {x : pgame} [fintype x.left_moves] [fintype x.right_moves]
(sL : ∀ i : x.left_moves, short (x.move_left i))
(sR : ∀ j : x.right_moves, short (x.move_right j)) :
short x | by unfreezingI { cases x, dsimp at * }; exact short.mk sL sR | def | pgame.short.mk' | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"fintype",
"pgame"
] | A synonym for `short.mk` that specifies the pgame in an implicit argument. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype_left {α β : Type u} {L : α → pgame.{u}} {R : β → pgame.{u}} [S : short ⟨α, β, L, R⟩] :
fintype α | by { casesI S with _ _ _ _ _ _ F _, exact F } | def | pgame.fintype_left | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"fintype"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype_left_moves (x : pgame) [S : short x] : fintype (x.left_moves) | by { casesI x, dsimp, apply_instance } | instance | pgame.fintype_left_moves | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"fintype",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype_right {α β : Type u} {L : α → pgame.{u}} {R : β → pgame.{u}} [S : short ⟨α, β, L, R⟩] :
fintype β | by { casesI S with _ _ _ _ _ _ _ F, exact F } | def | pgame.fintype_right | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"fintype"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype_right_moves (x : pgame) [S : short x] : fintype (x.right_moves) | by { casesI x, dsimp, apply_instance } | instance | pgame.fintype_right_moves | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"fintype",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_short (x : pgame) [S : short x] (i : x.left_moves) : short (x.move_left i) | by { casesI S with _ _ _ _ L _ _ _, apply L } | instance | pgame.move_left_short | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_short' {xl xr} (xL xR) [S : short (mk xl xr xL xR)] (i : xl) : short (xL i) | by { casesI S with _ _ _ _ L _ _ _, apply L } | def | pgame.move_left_short' | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
move_right_short (x : pgame) [S : short x] (j : x.right_moves) : short (x.move_right j) | by { casesI S with _ _ _ _ _ R _ _, apply R } | instance | pgame.move_right_short | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_right_short' {xl xr} (xL xR) [S : short (mk xl xr xL xR)] (j : xr) : short (xR j) | by { casesI S with _ _ _ _ _ R _ _, apply R } | def | pgame.move_right_short' | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
short_birthday : ∀ (x : pgame.{u}) [short x], x.birthday < ordinal.omega | | ⟨xl, xr, xL, xR⟩ hs :=
begin
haveI := hs,
unfreezingI { rcases hs with ⟨sL, sR⟩ },
rw [birthday, max_lt_iff],
split, all_goals
{ rw ←cardinal.ord_aleph_0,
refine cardinal.lsub_lt_ord_of_is_regular.{u u} cardinal.is_regular_aleph_0
(cardinal.lt_aleph_0_of_finite _) (λ i, _),
rw cardinal.ord_ale... | theorem | pgame.short_birthday | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"cardinal.is_regular_aleph_0",
"cardinal.lt_aleph_0_of_finite",
"cardinal.ord_aleph_0",
"max_lt_iff",
"ordinal.omega"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short.of_is_empty {l r xL xR} [is_empty l] [is_empty r] : short (mk l r xL xR) | short.mk is_empty_elim is_empty_elim | def | pgame.short.of_is_empty | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"is_empty",
"is_empty_elim"
] | This leads to infinite loops if made into an instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
short_0 : short 0 | short.of_is_empty | instance | pgame.short_0 | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_1 : short 1 | short.mk (λ i, begin cases i, apply_instance, end) (λ j, by cases j) | instance | pgame.short_1 | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_short : list pgame.{u} → Type (u+1)
| nil : list_short []
| cons : Π (hd : pgame.{u}) [short hd] (tl : list pgame.{u}) [list_short tl], list_short (hd :: tl) | inductive | pgame.list_short | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | Evidence that every `pgame` in a list is `short`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_short_nth_le : Π (L : list pgame.{u}) [list_short L] (i : fin (list.length L)),
short (list.nth_le L i i.is_lt) | | [] _ n := begin exfalso, rcases n with ⟨_, ⟨⟩⟩, end
| (hd :: tl) (@list_short.cons _ S _ _) ⟨0, _⟩ := S
| (hd :: tl) (@list_short.cons _ _ _ S) ⟨n+1, h⟩ :=
@list_short_nth_le tl S ⟨n, (add_lt_add_iff_right 1).mp h⟩ | instance | pgame.list_short_nth_le | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_of_lists : Π (L R : list pgame) [list_short L] [list_short R],
short (pgame.of_lists L R) | | L R _ _ := by { resetI, apply short.mk,
{ intros, apply_instance },
{ intros, apply pgame.list_short_nth_le /- where does the subtype.val come from? -/ } } | instance | pgame.short_of_lists | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"pgame",
"pgame.list_short_nth_le",
"pgame.of_lists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_of_relabelling : Π {x y : pgame.{u}} (R : relabelling x y) (S : short x), short y | | x y ⟨L, R, rL, rR⟩ S :=
begin
resetI,
haveI := fintype.of_equiv _ L,
haveI := fintype.of_equiv _ R,
exact short.mk'
(λ i, by { rw ←(L.right_inv i), apply short_of_relabelling (rL (L.symm i)) infer_instance, })
(λ j, by simpa using short_of_relabelling (rR (R.symm j)) infer_instance)
end | def | pgame.short_of_relabelling | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"fintype.of_equiv"
] | If `x` is a short game, and `y` is a relabelling of `x`, then `y` is also short. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
short_neg : Π (x : pgame.{u}) [short x], short (-x) | | (mk xl xr xL xR) _ :=
by { resetI, exact short.mk (λ i, short_neg _) (λ i, short_neg _) }
using_well_founded { dec_tac := pgame_wf_tac } | instance | pgame.short_neg | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_add : Π (x y : pgame.{u}) [short x] [short y], short (x + y) | | (mk xl xr xL xR) (mk yl yr yL yR) _ _ :=
begin
resetI,
apply short.mk, all_goals
{ rintro ⟨i⟩,
{ apply short_add },
{ change short (mk xl xr xL xR + _), apply short_add } }
end
using_well_founded { dec_tac := pgame_wf_tac } | instance | pgame.short_add | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_nat : Π n : ℕ, short n | | 0 := pgame.short_0
| (n+1) := @pgame.short_add _ _ (short_nat n) pgame.short_1 | instance | pgame.short_nat | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"pgame.short_0",
"pgame.short_1",
"pgame.short_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_bit0 (x : pgame.{u}) [short x] : short (bit0 x) | by { dsimp [bit0], apply_instance } | instance | pgame.short_bit0 | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_bit1 (x : pgame.{u}) [short x] : short (bit1 x) | by { dsimp [bit1], apply_instance } | instance | pgame.short_bit1 | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_lf_decidable : Π (x y : pgame.{u}) [short x] [short y],
decidable (x ≤ y) × decidable (x ⧏ y) | | (mk xl xr xL xR) (mk yl yr yL yR) shortx shorty :=
begin
resetI,
split,
{ refine @decidable_of_iff' _ _ mk_le_mk (id _),
apply @and.decidable _ _ _ _,
{ apply @fintype.decidable_forall_fintype xl _ _ (by apply_instance),
intro i,
apply (@le_lf_decidable _ _ _ _).2; apply_instance, },
{ a... | def | pgame.le_lf_decidable | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [
"decidable_of_iff'",
"fintype.decidable_exists_fintype",
"fintype.decidable_forall_fintype"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_decidable (x y : pgame.{u}) [short x] [short y] : decidable (x ≤ y) | (le_lf_decidable x y).1 | instance | pgame.le_decidable | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf_decidable (x y : pgame.{u}) [short x] [short y] : decidable (x ⧏ y) | (le_lf_decidable x y).2 | instance | pgame.lf_decidable | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_decidable (x y : pgame.{u}) [short x] [short y] : decidable (x < y) | and.decidable | instance | pgame.lt_decidable | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_decidable (x y : pgame.{u}) [short x] [short y] : decidable (x ≈ y) | and.decidable | instance | pgame.equiv_decidable | set_theory.game | src/set_theory/game/short.lean | [
"data.fintype.basic",
"set_theory.cardinal.cofinality",
"set_theory.game.birthday"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
state (S : Type u) | (turn_bound : S → ℕ)
(L : S → finset S)
(R : S → finset S)
(left_bound : ∀ {s t : S} (m : t ∈ L s), turn_bound t < turn_bound s)
(right_bound : ∀ {s t : S} (m : t ∈ R s), turn_bound t < turn_bound s) | class | pgame.state | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [
"finset"
] | `pgame_state S` describes how to interpret `s : S` as a state of a combinatorial game.
Use `pgame.of_state s` or `game.of_state s` to construct the game.
`pgame_state.L : S → finset S` and `pgame_state.R : S → finset S` describe the states reachable
by a move by Left or Right. `pgame_state.turn_bound : S → ℕ` gives an... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
turn_bound_ne_zero_of_left_move {s t : S} (m : t ∈ L s) : turn_bound s ≠ 0 | begin
intro h,
have t := state.left_bound m,
rw h at t,
exact nat.not_succ_le_zero _ t,
end | lemma | pgame.turn_bound_ne_zero_of_left_move | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
turn_bound_ne_zero_of_right_move {s t : S} (m : t ∈ R s) : turn_bound s ≠ 0 | begin
intro h,
have t := state.right_bound m,
rw h at t,
exact nat.not_succ_le_zero _ t,
end | lemma | pgame.turn_bound_ne_zero_of_right_move | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
turn_bound_of_left {s t : S} (m : t ∈ L s) (n : ℕ) (h : turn_bound s ≤ n + 1) :
turn_bound t ≤ n | nat.le_of_lt_succ (nat.lt_of_lt_of_le (left_bound m) h) | lemma | pgame.turn_bound_of_left | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
turn_bound_of_right {s t : S} (m : t ∈ R s) (n : ℕ) (h : turn_bound s ≤ n + 1) :
turn_bound t ≤ n | nat.le_of_lt_succ (nat.lt_of_lt_of_le (right_bound m) h) | lemma | pgame.turn_bound_of_right | set_theory.game | src/set_theory/game/state.lean | [
"set_theory.game.short"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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