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
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|---|---|---|---|---|---|---|---|---|---|---|
mul_assoc_equiv (x y z : pgame) : x * y * z ≈ x * (y * z) | quotient.exact $ quot_mul_assoc _ _ _ | theorem | pgame.mul_assoc_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `x * y * z` is equivalent to `x * (y * z).` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_ty (l r : Type u) : bool → Type u
| zero : inv_ty ff
| left₁ : r → inv_ty ff → inv_ty ff
| left₂ : l → inv_ty tt → inv_ty ff
| right₁ : l → inv_ty ff → inv_ty tt
| right₂ : r → inv_ty tt → inv_ty tt | inductive | pgame.inv_ty | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | Because the two halves of the definition of `inv` produce more elements
on each side, we have to define the two families inductively.
This is the indexing set for the function, and `inv_val` is the function part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_inv_ty (l r : Type u) [is_empty l] [is_empty r] : unique (inv_ty l r ff) | { uniq := by { rintro (a|a|a), refl, all_goals { exact is_empty_elim a } },
..inv_ty.inhabited l r } | instance | pgame.unique_inv_ty | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"is_empty",
"is_empty_elim",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_val {l r} (L : l → pgame) (R : r → pgame)
(IHl : l → pgame) (IHr : r → pgame) : ∀ {b}, inv_ty l r b → pgame | | _ inv_ty.zero := 0
| _ (inv_ty.left₁ i j) := (1 + (R i - mk l r L R) * inv_val j) * IHr i
| _ (inv_ty.left₂ i j) := (1 + (L i - mk l r L R) * inv_val j) * IHl i
| _ (inv_ty.right₁ i j) := (1 + (L i - mk l r L R) * inv_val j) * IHl i
| _ (inv_ty.right₂ i j) := (1 + (R i - mk l r L R) * inv_val j) * IHr i | def | pgame.inv_val | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | Because the two halves of the definition of `inv` produce more elements
of each side, we have to define the two families inductively.
This is the function part, defined by recursion on `inv_ty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_val_is_empty {l r : Type u} {b} (L R IHl IHr) (i : inv_ty l r b)
[is_empty l] [is_empty r] : inv_val L R IHl IHr i = 0 | begin
cases i with a _ a _ a _ a,
{ refl },
all_goals { exact is_empty_elim a }
end | theorem | pgame.inv_val_is_empty | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"is_empty",
"is_empty_elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv' : pgame → pgame | | ⟨l, r, L, R⟩ :=
let l' := {i // 0 < L i},
L' : l' → pgame := λ i, L i.1,
IHl' : l' → pgame := λ i, inv' (L i.1),
IHr := λ i, inv' (R i) in
⟨inv_ty l' r ff, inv_ty l' r tt,
inv_val L' R IHl' IHr, inv_val L' R IHl' IHr⟩ | def | pgame.inv' | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | The inverse of a positive surreal number `x = {L | R}` is
given by `x⁻¹ = {0,
(1 + (R - x) * x⁻¹L) * R, (1 + (L - x) * x⁻¹R) * L |
(1 + (L - x) * x⁻¹L) * L, (1 + (R - x) * x⁻¹R) * R}`.
Because the two halves `x⁻¹L, x⁻¹R` of `x⁻¹` are used in their own
definition, the sets and elements are inductively generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lf_inv' : ∀ (x : pgame), 0 ⧏ inv' x | | ⟨xl, xr, xL, xR⟩ := by { convert lf_mk _ _ inv_ty.zero, refl } | theorem | pgame.zero_lf_inv' | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv'_zero : inv' 0 ≡r 1 | begin
change mk _ _ _ _ ≡r 1,
refine ⟨_, _, λ i, _, is_empty.elim _⟩,
{ apply equiv.equiv_punit (inv_ty _ _ _),
apply_instance },
{ apply equiv.equiv_pempty (inv_ty _ _ _),
apply_instance },
{ simp },
{ dsimp,
apply_instance }
end | def | pgame.inv'_zero | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv.equiv_pempty",
"equiv.equiv_punit",
"is_empty.elim"
] | `inv' 0` has exactly the same moves as `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv'_zero_equiv : inv' 0 ≈ 1 | inv'_zero.equiv | theorem | pgame.inv'_zero_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv'_one : inv' 1 ≡r (1 : pgame.{u}) | begin
change relabelling (mk _ _ _ _) 1,
haveI : is_empty {i : punit.{u+1} // (0 : pgame.{u}) < 0},
{ rw lt_self_iff_false, apply_instance },
refine ⟨_, _, λ i, _, is_empty.elim _⟩; dsimp,
{ apply equiv.equiv_punit },
{ apply equiv.equiv_of_is_empty },
{ simp },
{ apply_instance }
end | def | pgame.inv'_one | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv.equiv_of_is_empty",
"equiv.equiv_punit",
"is_empty",
"is_empty.elim",
"lt_self_iff_false"
] | `inv' 1` has exactly the same moves as `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv'_one_equiv : inv' 1 ≈ 1 | inv'_one.equiv | theorem | pgame.inv'_one_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_equiv_zero {x : pgame} (h : x ≈ 0) : x⁻¹ = 0 | by { classical, exact if_pos h } | theorem | pgame.inv_eq_of_equiv_zero | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_zero : (0 : pgame)⁻¹ = 0 | inv_eq_of_equiv_zero (equiv_refl _) | theorem | pgame.inv_zero | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"inv_zero",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_pos {x : pgame} (h : 0 < x) : x⁻¹ = inv' x | by { classical, exact (if_neg h.lf.not_equiv').trans (if_pos h) } | theorem | pgame.inv_eq_of_pos | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_lf_zero {x : pgame} (h : x ⧏ 0) : x⁻¹ = -inv' (-x) | by { classical, exact (if_neg h.not_equiv).trans (if_neg h.not_gt) } | theorem | pgame.inv_eq_of_lf_zero | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_one : 1⁻¹ ≡r 1 | by { rw inv_eq_of_pos pgame.zero_lt_one, exact inv'_one } | def | pgame.inv_one | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"inv_one",
"pgame.zero_lt_one"
] | `1⁻¹` has exactly the same moves as `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_one_equiv : 1⁻¹ ≈ 1 | inv_one.equiv | theorem | pgame.inv_one_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday : pgame.{u} → ordinal.{u} | | ⟨xl, xr, xL, xR⟩ :=
max (lsub.{u u} $ λ i, birthday (xL i)) (lsub.{u u} $ λ i, birthday (xR i)) | def | pgame.birthday | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | The birthday of a pre-game is inductively defined as the least strict upper bound of the
birthdays of its left and right games. It may be thought as the "step" in which a certain game is
constructed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
birthday_def (x : pgame) : birthday x = max
(lsub.{u u} (λ i, birthday (x.move_left i)))
(lsub.{u u} (λ i, birthday (x.move_right i))) | by { cases x, rw birthday, refl } | theorem | pgame.birthday_def | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_move_left_lt {x : pgame} (i : x.left_moves) :
(x.move_left i).birthday < x.birthday | by { cases x, rw birthday, exact lt_max_of_lt_left (lt_lsub _ i) } | theorem | pgame.birthday_move_left_lt | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"lt_max_of_lt_left",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_move_right_lt {x : pgame} (i : x.right_moves) :
(x.move_right i).birthday < x.birthday | by { cases x, rw birthday, exact lt_max_of_lt_right (lt_lsub _ i) } | theorem | pgame.birthday_move_right_lt | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"lt_max_of_lt_right",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_birthday_iff {x : pgame} {o : ordinal} : o < x.birthday ↔
(∃ i : x.left_moves, o ≤ (x.move_left i).birthday) ∨
(∃ i : x.right_moves, o ≤ (x.move_right i).birthday) | begin
split,
{ rw birthday_def,
intro h,
cases lt_max_iff.1 h with h' h',
{ left,
rwa lt_lsub_iff at h' },
{ right,
rwa lt_lsub_iff at h' } },
{ rintro (⟨i, hi⟩ | ⟨i, hi⟩),
{ exact hi.trans_lt (birthday_move_left_lt i) },
{ exact hi.trans_lt (birthday_move_right_lt i) } }
end | theorem | pgame.lt_birthday_iff | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"ordinal",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabelling.birthday_congr : ∀ {x y : pgame.{u}}, x ≡r y → birthday x = birthday y | | ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ r := begin
unfold birthday,
congr' 1,
all_goals
{ apply lsub_eq_of_range_eq.{u u u},
ext i, split },
all_goals { rintro ⟨j, rfl⟩ },
{ exact ⟨_, (r.move_left j).birthday_congr.symm⟩ },
{ exact ⟨_, (r.move_left_symm j).birthday_congr⟩ },
{ exact ⟨_, (r.move_right j)... | theorem | pgame.relabelling.birthday_congr | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_eq_zero {x : pgame} :
birthday x = 0 ↔ is_empty x.left_moves ∧ is_empty x.right_moves | by rw [birthday_def, max_eq_zero, lsub_eq_zero_iff, lsub_eq_zero_iff] | theorem | pgame.birthday_eq_zero | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"is_empty",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_zero : birthday 0 = 0 | by simp [pempty.is_empty] | theorem | pgame.birthday_zero | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_one : birthday 1 = 1 | by { rw birthday_def, simp } | theorem | pgame.birthday_one | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_star : birthday star = 1 | by { rw birthday_def, simp } | theorem | pgame.birthday_star | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_birthday : ∀ x : pgame, (-x).birthday = x.birthday | | ⟨xl, xr, xL, xR⟩ := begin
rw [birthday_def, birthday_def, max_comm],
congr; funext; apply neg_birthday
end | theorem | pgame.neg_birthday | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pgame_birthday (o : ordinal) : o.to_pgame.birthday = o | begin
induction o using ordinal.induction with o IH,
rw [to_pgame_def, pgame.birthday],
simp only [lsub_empty, max_zero_right],
nth_rewrite 0 ←lsub_typein o,
congr' with x,
exact IH _ (typein_lt_self x)
end | theorem | pgame.to_pgame_birthday | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"ordinal",
"ordinal.induction",
"pgame.birthday"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_birthday : ∀ x : pgame, x ≤ x.birthday.to_pgame | | ⟨xl, _, xL, _⟩ :=
le_def.2 ⟨λ i, or.inl ⟨to_left_moves_to_pgame ⟨_, birthday_move_left_lt i⟩,
by simp [le_birthday (xL i)]⟩, is_empty_elim⟩ | theorem | pgame.le_birthday | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_birthday_le : -x.birthday.to_pgame ≤ x | by simpa only [neg_birthday, ←neg_le_iff] using le_birthday (-x) | theorem | pgame.neg_birthday_le | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_add : ∀ x y : pgame.{u}, (x + y).birthday = x.birthday ♯ y.birthday | | ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ := begin
rw [birthday_def, nadd_def],
simp only [birthday_add, lsub_sum, mk_add_move_left_inl, move_left_mk, mk_add_move_left_inr,
mk_add_move_right_inl, move_right_mk, mk_add_move_right_inr],
rw max_max_max_comm,
congr; apply le_antisymm,
any_goals
{ exact max_le_iff... | theorem | pgame.birthday_add | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"lt_max_of_lt_left",
"lt_max_of_lt_right",
"max_max_max_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_add_zero : (a + 0).birthday = a.birthday | by simp | theorem | pgame.birthday_add_zero | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_zero_add : (0 + a).birthday = a.birthday | by simp | theorem | pgame.birthday_zero_add | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_add_one : (a + 1).birthday = order.succ a.birthday | by simp | theorem | pgame.birthday_add_one | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"order.succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_one_add : (1 + a).birthday = order.succ a.birthday | by simp | theorem | pgame.birthday_one_add | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [
"order.succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_nat_cast : ∀ n : ℕ, birthday n = n | | 0 := birthday_zero
| (n + 1) := by simp [birthday_nat_cast] | theorem | pgame.birthday_nat_cast | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_add_nat (n : ℕ) : (a + n).birthday = a.birthday + n | by simp | theorem | pgame.birthday_add_nat | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_nat_add (n : ℕ) : (↑n + a).birthday = a.birthday + n | by simp | theorem | pgame.birthday_nat_add | set_theory.game | src/set_theory/game/birthday.lean | [
"set_theory.game.ordinal",
"set_theory.ordinal.natural_ops"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
shift_up : ℤ × ℤ ≃ ℤ × ℤ | (equiv.refl ℤ).prod_congr (equiv.add_right (1 : ℤ)) | def | pgame.domineering.shift_up | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"equiv.refl"
] | The equivalence `(x, y) ↦ (x, y+1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
shift_right : ℤ × ℤ ≃ ℤ × ℤ | (equiv.add_right (1 : ℤ)).prod_congr (equiv.refl ℤ) | def | pgame.domineering.shift_right | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"equiv.refl"
] | The equivalence `(x, y) ↦ (x+1, y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
board | finset (ℤ × ℤ) | def | pgame.domineering.board | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset"
] | A Domineering board is an arbitrary finite subset of `ℤ × ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left (b : board) : finset (ℤ × ℤ) | b ∩ b.map shift_up | def | pgame.domineering.left | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset"
] | Left can play anywhere that a square and the square below it are open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right (b : board) : finset (ℤ × ℤ) | b ∩ b.map shift_right | def | pgame.domineering.right | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset"
] | Right can play anywhere that a square and the square to the left are open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_left {b : board} (x : ℤ × ℤ) : x ∈ left b ↔ x ∈ b ∧ (x.1, x.2 - 1) ∈ b | finset.mem_inter.trans (and_congr iff.rfl finset.mem_map_equiv) | lemma | pgame.domineering.mem_left | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.mem_map_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_right {b : board} (x : ℤ × ℤ) : x ∈ right b ↔ x ∈ b ∧ (x.1 - 1, x.2) ∈ b | finset.mem_inter.trans (and_congr iff.rfl finset.mem_map_equiv) | lemma | pgame.domineering.mem_right | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.mem_map_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left (b : board) (m : ℤ × ℤ) : board | (b.erase m).erase (m.1, m.2 - 1) | def | pgame.domineering.move_left | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [] | After Left moves, two vertically adjacent squares are removed from the board. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
move_right (b : board) (m : ℤ × ℤ) : board | (b.erase m).erase (m.1 - 1, m.2) | def | pgame.domineering.move_right | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [] | After Left moves, two horizontally adjacent squares are removed from the board. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_pred_mem_erase_of_mem_right {b : board} {m : ℤ × ℤ} (h : m ∈ right b) :
(m.1 - 1, m.2) ∈ b.erase m | begin
rw mem_right at h,
apply finset.mem_erase_of_ne_of_mem _ h.2,
exact ne_of_apply_ne prod.fst (pred_ne_self m.1),
end | lemma | pgame.domineering.fst_pred_mem_erase_of_mem_right | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.mem_erase_of_ne_of_mem",
"ne_of_apply_ne",
"pred_ne_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_pred_mem_erase_of_mem_left {b : board} {m : ℤ × ℤ} (h : m ∈ left b) :
(m.1, m.2 - 1) ∈ b.erase m | begin
rw mem_left at h,
apply finset.mem_erase_of_ne_of_mem _ h.2,
exact ne_of_apply_ne prod.snd (pred_ne_self m.2),
end | lemma | pgame.domineering.snd_pred_mem_erase_of_mem_left | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.mem_erase_of_ne_of_mem",
"ne_of_apply_ne",
"pred_ne_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_of_mem_left {b : board} {m : ℤ × ℤ} (h : m ∈ left b) : 2 ≤ finset.card b | begin
have w₁ : m ∈ b := (finset.mem_inter.1 h).1,
have w₂ : (m.1, m.2 - 1) ∈ b.erase m := snd_pred_mem_erase_of_mem_left h,
have i₁ := finset.card_erase_lt_of_mem w₁,
have i₂ := nat.lt_of_le_of_lt (nat.zero_le _) (finset.card_erase_lt_of_mem w₂),
exact nat.lt_of_le_of_lt i₂ i₁,
end | lemma | pgame.domineering.card_of_mem_left | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.card",
"finset.card_erase_lt_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_of_mem_right {b : board} {m : ℤ × ℤ} (h : m ∈ right b) : 2 ≤ finset.card b | begin
have w₁ : m ∈ b := (finset.mem_inter.1 h).1,
have w₂ := fst_pred_mem_erase_of_mem_right h,
have i₁ := finset.card_erase_lt_of_mem w₁,
have i₂ := nat.lt_of_le_of_lt (nat.zero_le _) (finset.card_erase_lt_of_mem w₂),
exact nat.lt_of_le_of_lt i₂ i₁,
end | lemma | pgame.domineering.card_of_mem_right | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.card",
"finset.card_erase_lt_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_card {b : board} {m : ℤ × ℤ} (h : m ∈ left b) :
finset.card (move_left b m) + 2 = finset.card b | begin
dsimp [move_left],
rw finset.card_erase_of_mem (snd_pred_mem_erase_of_mem_left h),
rw finset.card_erase_of_mem (finset.mem_of_mem_inter_left h),
exact tsub_add_cancel_of_le (card_of_mem_left h),
end | lemma | pgame.domineering.move_left_card | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.card",
"finset.card_erase_of_mem",
"finset.mem_of_mem_inter_left",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_right_card {b : board} {m : ℤ × ℤ} (h : m ∈ right b) :
finset.card (move_right b m) + 2 = finset.card b | begin
dsimp [move_right],
rw finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h),
rw finset.card_erase_of_mem (finset.mem_of_mem_inter_left h),
exact tsub_add_cancel_of_le (card_of_mem_right h),
end | lemma | pgame.domineering.move_right_card | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.card",
"finset.card_erase_of_mem",
"finset.mem_of_mem_inter_left",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_smaller {b : board} {m : ℤ × ℤ} (h : m ∈ left b) :
finset.card (move_left b m) / 2 < finset.card b / 2 | by simp [←move_left_card h, lt_add_one] | lemma | pgame.domineering.move_left_smaller | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.card",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_right_smaller {b : board} {m : ℤ × ℤ} (h : m ∈ right b) :
finset.card (move_right b m) / 2 < finset.card b / 2 | by simp [←move_right_card h, lt_add_one] | lemma | pgame.domineering.move_right_smaller | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.card",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
state : state board | { turn_bound := λ s, s.card / 2,
L := λ s, (left s).image (move_left s),
R := λ s, (right s).image (move_right s),
left_bound := λ s t m,
begin
simp only [finset.mem_image, prod.exists] at m,
rcases m with ⟨_, _, ⟨h, rfl⟩⟩,
exact move_left_smaller h
end,
right_bound := λ s t m,
begin
simp ... | instance | pgame.domineering.state | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"finset.mem_image"
] | The instance describing allowed moves on a Domineering board. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
domineering (b : domineering.board) : pgame | pgame.of_state b | def | pgame.domineering | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [
"pgame",
"pgame.of_state"
] | Construct a pre-game from a Domineering board. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
short_domineering (b : domineering.board) : short (domineering b) | by { dsimp [domineering], apply_instance } | instance | pgame.short_domineering | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [] | All games of Domineering are short, because each move removes two squares. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
domineering.one | domineering ([(0,0), (0,1)].to_finset) | def | pgame.domineering.one | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [] | The Domineering board with two squares arranged vertically, in which Left has the only move. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
domineering.L | domineering ([(0,2), (0,1), (0,0), (1,0)].to_finset) | def | pgame.domineering.L | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [] | The `L` shaped Domineering board, in which Left is exactly half a move ahead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
short_one : short domineering.one | by { dsimp [domineering.one], apply_instance } | instance | pgame.short_one | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
short_L : short domineering.L | by { dsimp [domineering.L], apply_instance } | instance | pgame.short_L | set_theory.game | src/set_theory/game/domineering.lean | [
"set_theory.game.state"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
impartial_aux : pgame → Prop | | G := G ≈ -G ∧ (∀ i, impartial_aux (G.move_left i)) ∧ ∀ j, impartial_aux (G.move_right j)
using_well_founded { dec_tac := pgame_wf_tac } | def | pgame.impartial_aux | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | The definition for a impartial game, defined using Conway induction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
impartial_aux_def {G : pgame} : G.impartial_aux ↔ G ≈ -G ∧
(∀ i, impartial_aux (G.move_left i)) ∧ ∀ j, impartial_aux (G.move_right j) | by rw impartial_aux | lemma | pgame.impartial_aux_def | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
impartial (G : pgame) : Prop | (out : impartial_aux G) | class | pgame.impartial | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | A typeclass on impartial games. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
impartial_iff_aux {G : pgame} : G.impartial ↔ G.impartial_aux | ⟨λ h, h.1, λ h, ⟨h⟩⟩ | lemma | pgame.impartial_iff_aux | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
impartial_def {G : pgame} : G.impartial ↔ G ≈ -G ∧
(∀ i, impartial (G.move_left i)) ∧ ∀ j, impartial (G.move_right j) | by simpa only [impartial_iff_aux] using impartial_aux_def | lemma | pgame.impartial_def | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
impartial_zero : impartial 0 | by { rw impartial_def, dsimp, simp } | instance | pgame.impartial.impartial_zero | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
impartial_star : impartial star | by { rw impartial_def, simpa using impartial.impartial_zero } | instance | pgame.impartial.impartial_star | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_equiv_self (G : pgame) [h : G.impartial] : G ≈ -G | (impartial_def.1 h).1 | lemma | pgame.impartial.neg_equiv_self | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_neg_equiv_self (G : pgame) [h : G.impartial] : -⟦G⟧ = ⟦G⟧ | quot.sound (neg_equiv_self G).symm | lemma | pgame.impartial.mk_neg_equiv_self | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_impartial {G : pgame} [h : G.impartial] (i : G.left_moves) :
(G.move_left i).impartial | (impartial_def.1 h).2.1 i | instance | pgame.impartial.move_left_impartial | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_right_impartial {G : pgame} [h : G.impartial] (j : G.right_moves) :
(G.move_right j).impartial | (impartial_def.1 h).2.2 j | instance | pgame.impartial.move_right_impartial | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
impartial_congr : ∀ {G H : pgame} (e : G ≡r H) [G.impartial], H.impartial | | G H := λ e, begin
introI h,
exact impartial_def.2
⟨e.symm.equiv.trans ((neg_equiv_self G).trans (neg_equiv_neg_iff.2 e.equiv)),
λ i, impartial_congr (e.move_left_symm i), λ j, impartial_congr (e.move_right_symm j)⟩
end
using_well_founded { dec_tac := pgame_wf_tac } | theorem | pgame.impartial.impartial_congr | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
impartial_add : ∀ (G H : pgame) [G.impartial] [H.impartial], (G + H).impartial | | G H :=
begin
introsI hG hH,
rw impartial_def,
refine ⟨(add_congr (neg_equiv_self _) (neg_equiv_self _)).trans
(neg_add_relabelling _ _).equiv.symm, λ k, _, λ k, _⟩,
{ apply left_moves_add_cases k,
all_goals
{ intro i, simp only [add_move_left_inl, add_move_left_inr],
apply impartial_add } },... | instance | pgame.impartial.impartial_add | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"equiv.symm",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
impartial_neg : ∀ (G : pgame) [G.impartial], (-G).impartial | | G :=
begin
introI hG,
rw impartial_def,
refine ⟨_, λ i, _, λ i, _⟩,
{ rw neg_neg,
exact (neg_equiv_self G).symm },
{ rw move_left_neg',
apply impartial_neg },
{ rw move_right_neg',
apply impartial_neg }
end
using_well_founded { dec_tac := pgame_wf_tac } | instance | pgame.impartial.impartial_neg | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonpos : ¬ 0 < G | λ h, begin
have h' := neg_lt_neg_iff.2 h,
rw [neg_zero, lt_congr_left (neg_equiv_self G).symm] at h',
exact (h.trans h').false
end | lemma | pgame.impartial.nonpos | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg : ¬ G < 0 | λ h, begin
have h' := neg_lt_neg_iff.2 h,
rw [neg_zero, lt_congr_right (neg_equiv_self G).symm] at h',
exact (h.trans h').false
end | lemma | pgame.impartial.nonneg | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_or_fuzzy_zero : G ≈ 0 ∨ G ‖ 0 | begin
rcases lt_or_equiv_or_gt_or_fuzzy G 0 with h | h | h | h,
{ exact ((nonneg G) h).elim },
{ exact or.inl h },
{ exact ((nonpos G) h).elim },
{ exact or.inr h }
end | lemma | pgame.impartial.equiv_or_fuzzy_zero | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | In an impartial game, either the first player always wins, or the second player always wins. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_equiv_zero_iff : ¬ G ≈ 0 ↔ G ‖ 0 | ⟨(equiv_or_fuzzy_zero G).resolve_left, fuzzy.not_equiv⟩ | lemma | pgame.impartial.not_equiv_zero_iff | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_fuzzy_zero_iff : ¬ G ‖ 0 ↔ G ≈ 0 | ⟨(equiv_or_fuzzy_zero G).resolve_right, equiv.not_fuzzy⟩ | lemma | pgame.impartial.not_fuzzy_zero_iff | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_self : G + G ≈ 0 | (add_congr_left (neg_equiv_self G)).trans (add_left_neg_equiv G) | lemma | pgame.impartial.add_self | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"add_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_add_self : ⟦G⟧ + ⟦G⟧ = 0 | quot.sound (add_self G) | lemma | pgame.impartial.mk_add_self | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"add_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_iff_add_equiv_zero (H : pgame) : H ≈ G ↔ H + G ≈ 0 | by { rw [equiv_iff_game_eq, equiv_iff_game_eq, ←@add_right_cancel_iff _ _ _ (-⟦G⟧)], simpa } | lemma | pgame.impartial.equiv_iff_add_equiv_zero | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | This lemma doesn't require `H` to be impartial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_iff_add_equiv_zero' (H : pgame) : G ≈ H ↔ G + H ≈ 0 | by { rw [equiv_iff_game_eq, equiv_iff_game_eq, ←@add_left_cancel_iff _ _ _ (-⟦G⟧), eq_comm], simpa } | lemma | pgame.impartial.equiv_iff_add_equiv_zero' | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | This lemma doesn't require `H` to be impartial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_zero_iff {G : pgame} [G.impartial] : G ≤ 0 ↔ 0 ≤ G | by rw [←zero_le_neg_iff, le_congr_right (neg_equiv_self G)] | lemma | pgame.impartial.le_zero_iff | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"le_zero_iff",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf_zero_iff {G : pgame} [G.impartial] : G ⧏ 0 ↔ 0 ⧏ G | by rw [←zero_lf_neg_iff, lf_congr_right (neg_equiv_self G)] | lemma | pgame.impartial.lf_zero_iff | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_zero_iff_le: G ≈ 0 ↔ G ≤ 0 | ⟨and.left, λ h, ⟨h, le_zero_iff.1 h⟩⟩ | lemma | pgame.impartial.equiv_zero_iff_le | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fuzzy_zero_iff_lf : G ‖ 0 ↔ G ⧏ 0 | ⟨and.left, λ h, ⟨h, lf_zero_iff.1 h⟩⟩ | lemma | pgame.impartial.fuzzy_zero_iff_lf | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_zero_iff_ge : G ≈ 0 ↔ 0 ≤ G | ⟨and.right, λ h, ⟨le_zero_iff.2 h, h⟩⟩ | lemma | pgame.impartial.equiv_zero_iff_ge | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fuzzy_zero_iff_gf : G ‖ 0 ↔ 0 ⧏ G | ⟨and.right, λ h, ⟨lf_zero_iff.2 h, h⟩⟩ | lemma | pgame.impartial.fuzzy_zero_iff_gf | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_left_moves_fuzzy_iff_equiv_zero : (∀ i, G.move_left i ‖ 0) ↔ G ≈ 0 | begin
refine ⟨λ hb, _, λ hp i, _⟩,
{ rw [equiv_zero_iff_le G, le_zero_lf],
exact λ i, (hb i).1 },
{ rw fuzzy_zero_iff_lf,
exact hp.1.move_left_lf i }
end | lemma | pgame.impartial.forall_left_moves_fuzzy_iff_equiv_zero | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_right_moves_fuzzy_iff_equiv_zero : (∀ j, G.move_right j ‖ 0) ↔ G ≈ 0 | begin
refine ⟨λ hb, _, λ hp i, _⟩,
{ rw [equiv_zero_iff_ge G, zero_le_lf],
exact λ i, (hb i).2 },
{ rw fuzzy_zero_iff_gf,
exact hp.2.lf_move_right i }
end | lemma | pgame.impartial.forall_right_moves_fuzzy_iff_equiv_zero | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_left_move_equiv_iff_fuzzy_zero : (∃ i, G.move_left i ≈ 0) ↔ G ‖ 0 | begin
refine ⟨λ ⟨i, hi⟩, (fuzzy_zero_iff_gf G).2 (lf_of_le_move_left hi.2), λ hn, _⟩,
rw [fuzzy_zero_iff_gf G, zero_lf_le] at hn,
cases hn with i hi,
exact ⟨i, (equiv_zero_iff_ge _).2 hi⟩
end | lemma | pgame.impartial.exists_left_move_equiv_iff_fuzzy_zero | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_right_move_equiv_iff_fuzzy_zero : (∃ j, G.move_right j ≈ 0) ↔ G ‖ 0 | begin
refine ⟨λ ⟨i, hi⟩, (fuzzy_zero_iff_lf G).2 (lf_of_move_right_le hi.1), λ hn, _⟩,
rw [fuzzy_zero_iff_lf G, lf_zero_le] at hn,
cases hn with i hi,
exact ⟨i, (equiv_zero_iff_le _).2 hi⟩
end | lemma | pgame.impartial.exists_right_move_equiv_iff_fuzzy_zero | set_theory.game | src/set_theory/game/impartial.lean | [
"set_theory.game.basic",
"tactic.nth_rewrite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nim_def (o : ordinal) : nim o = pgame.mk o.out.α o.out.α
(λ o₂, nim (ordinal.typein (<) o₂))
(λ o₂, nim (ordinal.typein (<) o₂)) | by { rw nim, refl } | lemma | pgame.nim_def | set_theory.game | src/set_theory/game/nim.lean | [
"data.nat.bitwise",
"set_theory.game.birthday",
"set_theory.game.impartial"
] | [
"ordinal",
"ordinal.typein"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_moves_nim (o : ordinal) : (nim o).left_moves = o.out.α | by { rw nim_def, refl } | lemma | pgame.left_moves_nim | set_theory.game | src/set_theory/game/nim.lean | [
"data.nat.bitwise",
"set_theory.game.birthday",
"set_theory.game.impartial"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_moves_nim (o : ordinal) : (nim o).right_moves = o.out.α | by { rw nim_def, refl } | lemma | pgame.right_moves_nim | set_theory.game | src/set_theory/game/nim.lean | [
"data.nat.bitwise",
"set_theory.game.birthday",
"set_theory.game.impartial"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_nim_heq (o : ordinal) : (nim o).move_left == λ i : o.out.α, nim (typein (<) i) | by { rw nim_def, refl } | lemma | pgame.move_left_nim_heq | set_theory.game | src/set_theory/game/nim.lean | [
"data.nat.bitwise",
"set_theory.game.birthday",
"set_theory.game.impartial"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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