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
value |
|---|---|---|---|---|---|---|---|---|---|---|
lt_congr_imp {x₁ y₁ x₂ y₂} (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) (h : x₁ < y₁) : x₂ < y₂ | hx.2.trans_lt (h.trans_le hy.1) | theorem | pgame.lt_congr_imp | 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 | |
lt_congr {x₁ y₁ x₂ y₂} (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ < y₁ ↔ x₂ < y₂ | ⟨lt_congr_imp hx hy, lt_congr_imp hx.symm hy.symm⟩ | theorem | pgame.lt_congr | 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 | |
lt_congr_left {x₁ x₂ y} (hx : x₁ ≈ x₂) : x₁ < y ↔ x₂ < y | lt_congr hx equiv_rfl | theorem | pgame.lt_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_congr_right {x y₁ y₂} (hy : y₁ ≈ y₂) : x < y₁ ↔ x < y₂ | lt_congr equiv_rfl hy | theorem | pgame.lt_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_or_equiv_of_le {x y : pgame} (h : x ≤ y) : x < y ∨ x ≈ y | and_or_distrib_left.mp ⟨h, (em $ y ≤ x).swap.imp_left pgame.not_le.1⟩ | theorem | pgame.lt_or_equiv_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"
] | [
"em",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf_or_equiv_or_gf (x y : pgame) : x ⧏ y ∨ x ≈ y ∨ y ⧏ x | begin
by_cases h : x ⧏ y,
{ exact or.inl h },
{ right,
cases (lt_or_equiv_of_le (pgame.not_lf.1 h)) with h' h',
{ exact or.inr h'.lf },
{ exact or.inl h'.symm } }
end | theorem | pgame.lf_or_equiv_or_gf | 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 | |
equiv_congr_left {y₁ y₂} : y₁ ≈ y₂ ↔ ∀ x₁, x₁ ≈ y₁ ↔ x₁ ≈ y₂ | ⟨λ h x₁, ⟨λ h', h'.trans h, λ h', h'.trans h.symm⟩,
λ h, (h y₁).1 $ equiv_rfl⟩ | theorem | pgame.equiv_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_congr_right {x₁ x₂} : x₁ ≈ x₂ ↔ ∀ y₁, x₁ ≈ y₁ ↔ x₂ ≈ y₁ | ⟨λ h y₁, ⟨λ h', h.symm.trans h', λ h', h.trans h'⟩,
λ h, (h x₂).2 $ equiv_rfl⟩ | theorem | pgame.equiv_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_mk_equiv {x y : pgame}
(L : x.left_moves ≃ y.left_moves) (R : x.right_moves ≃ y.right_moves)
(hl : ∀ i, x.move_left i ≈ y.move_left (L i))
(hr : ∀ j, x.move_right j ≈ y.move_right (R j)) : x ≈ y | begin
fsplit; rw le_def,
{ exact ⟨λ i, or.inl ⟨_, (hl i).1⟩, λ j, or.inr ⟨_, by simpa using (hr (R.symm j)).1⟩⟩ },
{ exact ⟨λ i, or.inl ⟨_, by simpa using (hl (L.symm i)).2⟩, λ j, or.inr ⟨_, (hr j).2⟩⟩ }
end | theorem | pgame.equiv_of_mk_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 | |
fuzzy (x y : pgame) : Prop | x ⧏ y ∧ y ⧏ x | def | pgame.fuzzy | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | The fuzzy, confused, or incomparable relation on pre-games.
If `x ‖ 0`, then the first player can always win `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fuzzy.swap {x y : pgame} : x ‖ y → y ‖ x | and.swap | theorem | pgame.fuzzy.swap | 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 | |
fuzzy.swap_iff {x y : pgame} : x ‖ y ↔ y ‖ x | ⟨fuzzy.swap, fuzzy.swap⟩ | theorem | pgame.fuzzy.swap_iff | 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 | |
fuzzy_irrefl (x : pgame) : ¬ x ‖ x | λ h, lf_irrefl x h.1 | theorem | pgame.fuzzy_irrefl | 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_lt_or_fuzzy {x y : pgame} : x ⧏ y ↔ x < y ∨ x ‖ y | by { simp only [lt_iff_le_and_lf, fuzzy, ←pgame.not_le], tauto! } | theorem | pgame.lf_iff_lt_or_fuzzy | 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_of_fuzzy {x y : pgame} (h : x ‖ y) : x ⧏ y | lf_iff_lt_or_fuzzy.2 (or.inr h) | theorem | pgame.lf_of_fuzzy | 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_or_fuzzy_of_lf {x y : pgame} : x ⧏ y → x < y ∨ x ‖ y | lf_iff_lt_or_fuzzy.1 | theorem | pgame.lt_or_fuzzy_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 | |
fuzzy.not_equiv {x y : pgame} (h : x ‖ y) : ¬ x ≈ y | λ h', h'.1.not_gf h.2 | theorem | pgame.fuzzy.not_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 | |
fuzzy.not_equiv' {x y : pgame} (h : x ‖ y) : ¬ y ≈ x | λ h', h'.2.not_gf h.2 | theorem | pgame.fuzzy.not_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 | |
not_fuzzy_of_le {x y : pgame} (h : x ≤ y) : ¬ x ‖ y | λ h', h'.2.not_ge h | theorem | pgame.not_fuzzy_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 | |
not_fuzzy_of_ge {x y : pgame} (h : y ≤ x) : ¬ x ‖ y | λ h', h'.1.not_ge h | theorem | pgame.not_fuzzy_of_ge | 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 | |
equiv.not_fuzzy {x y : pgame} (h : x ≈ y) : ¬ x ‖ y | not_fuzzy_of_le h.1 | theorem | pgame.equiv.not_fuzzy | 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 | |
equiv.not_fuzzy' {x y : pgame} (h : x ≈ y) : ¬ y ‖ x | not_fuzzy_of_le h.2 | theorem | pgame.equiv.not_fuzzy' | 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 | |
fuzzy_congr {x₁ y₁ x₂ y₂ : pgame} (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ ‖ y₁ ↔ x₂ ‖ y₂ | show _ ∧ _ ↔ _ ∧ _, by rw [lf_congr hx hy, lf_congr hy hx] | theorem | pgame.fuzzy_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 | |
fuzzy_congr_imp {x₁ y₁ x₂ y₂ : pgame} (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ ‖ y₁ → x₂ ‖ y₂ | (fuzzy_congr hx hy).1 | theorem | pgame.fuzzy_congr_imp | 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 | |
fuzzy_congr_left {x₁ x₂ y} (hx : x₁ ≈ x₂) : x₁ ‖ y ↔ x₂ ‖ y | fuzzy_congr hx equiv_rfl | theorem | pgame.fuzzy_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fuzzy_congr_right {x y₁ y₂} (hy : y₁ ≈ y₂) : x ‖ y₁ ↔ x ‖ y₂ | fuzzy_congr equiv_rfl hy | theorem | pgame.fuzzy_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fuzzy_of_fuzzy_of_equiv {x y z} (h₁ : x ‖ y) (h₂ : y ≈ z) : x ‖ z | (fuzzy_congr_right h₂).1 h₁ | theorem | pgame.fuzzy_of_fuzzy_of_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 | |
fuzzy_of_equiv_of_fuzzy {x y z} (h₁ : x ≈ y) (h₂ : y ‖ z) : x ‖ z | (fuzzy_congr_left h₁).2 h₂ | theorem | pgame.fuzzy_of_equiv_of_fuzzy | 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 | |
lt_or_equiv_or_gt_or_fuzzy (x y : pgame) : x < y ∨ x ≈ y ∨ y < x ∨ x ‖ y | begin
cases le_or_gf x y with h₁ h₁;
cases le_or_gf y x with h₂ h₂,
{ right, left, exact ⟨h₁, h₂⟩ },
{ left, exact ⟨h₁, h₂⟩ },
{ right, right, left, exact ⟨h₂, h₁⟩ },
{ right, right, right, exact ⟨h₂, h₁⟩ }
end | theorem | pgame.lt_or_equiv_or_gt_or_fuzzy | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | Exactly one of the following is true (although we don't prove this here). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_or_equiv_or_gf (x y : pgame) : x < y ∨ x ≈ y ∨ y ⧏ x | begin
rw [lf_iff_lt_or_fuzzy, fuzzy.swap_iff],
exact lt_or_equiv_or_gt_or_fuzzy x y
end | theorem | pgame.lt_or_equiv_or_gf | 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 : pgame.{u} → pgame.{u} → Type (u+1)
| mk : Π {x y : pgame} (L : x.left_moves ≃ y.left_moves) (R : x.right_moves ≃ y.right_moves),
(∀ i, relabelling (x.move_left i) (y.move_left (L i))) →
(∀ j, relabelling (x.move_right j) (y.move_right (R j))) →
relabelling x y | inductive | pgame.relabelling | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | `relabelling x y` says that `x` and `y` are really the same game, just dressed up differently.
Specifically, there is a bijection between the moves for Left in `x` and in `y`, and similarly
for Right, and under these bijections we inductively have `relabelling`s for the consequent games. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (L : y.left_moves ≃ x.left_moves) (R : y.right_moves ≃ x.right_moves)
(hL : ∀ i, x.move_left (L i) ≡r y.move_left i)
(hR : ∀ j, x.move_right (R j) ≡r y.move_right j) : x ≡r y | ⟨L.symm, R.symm, λ i, by simpa using hL (L.symm i), λ j, by simpa using hR (R.symm j)⟩ | def | pgame.relabelling.mk' | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"mk'"
] | A constructor for relabellings swapping the equivalences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_moves_equiv : Π (r : x ≡r y), x.left_moves ≃ y.left_moves | | ⟨L, R, hL, hR⟩ := L | def | pgame.relabelling.left_moves_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | The equivalence between left moves of `x` and `y` given by the relabelling. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_left_moves_equiv {x y L R hL hR} :
(@relabelling.mk x y L R hL hR).left_moves_equiv = L | rfl | theorem | pgame.relabelling.mk_left_moves_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 | |
mk'_left_moves_equiv {x y L R hL hR} :
(@relabelling.mk' x y L R hL hR).left_moves_equiv = L.symm | rfl | theorem | pgame.relabelling.mk'_left_moves_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 | |
right_moves_equiv : Π (r : x ≡r y), x.right_moves ≃ y.right_moves | | ⟨L, R, hL, hR⟩ := R | def | pgame.relabelling.right_moves_equiv | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | The equivalence between right moves of `x` and `y` given by the relabelling. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_right_moves_equiv {x y L R hL hR} :
(@relabelling.mk x y L R hL hR).right_moves_equiv = R | rfl | theorem | pgame.relabelling.mk_right_moves_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 | |
mk'_right_moves_equiv {x y L R hL hR} :
(@relabelling.mk' x y L R hL hR).right_moves_equiv = R.symm | rfl | theorem | pgame.relabelling.mk'_right_moves_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 | |
move_left : ∀ (r : x ≡r y) (i : x.left_moves),
x.move_left i ≡r y.move_left (r.left_moves_equiv i) | | ⟨L, R, hL, hR⟩ := hL | def | pgame.relabelling.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"
] | [] | A left move of `x` is a relabelling of a left move of `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
move_left_symm : ∀ (r : x ≡r y) (i : y.left_moves),
x.move_left (r.left_moves_equiv.symm i) ≡r y.move_left i | | ⟨L, R, hL, hR⟩ i := by simpa using hL (L.symm i) | def | pgame.relabelling.move_left_symm | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | A left move of `y` is a relabelling of a left move of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
move_right : ∀ (r : x ≡r y) (i : x.right_moves),
x.move_right i ≡r y.move_right (r.right_moves_equiv i) | | ⟨L, R, hL, hR⟩ := hR | def | pgame.relabelling.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"
] | [] | A right move of `x` is a relabelling of a right move of `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
move_right_symm : ∀ (r : x ≡r y) (i : y.right_moves),
x.move_right (r.right_moves_equiv.symm i) ≡r y.move_right i | | ⟨L, R, hL, hR⟩ i := by simpa using hR (R.symm i) | def | pgame.relabelling.move_right_symm | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | A right move of `y` is a relabelling of a right move of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl : Π (x : pgame), x ≡r x | | x := ⟨equiv.refl _, equiv.refl _, λ i, refl _, λ j, refl _⟩
using_well_founded { dec_tac := pgame_wf_tac } | def | pgame.relabelling.refl | 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"
] | The identity relabelling. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm : Π {x y : pgame}, x ≡r y → y ≡r x | | x y ⟨L, R, hL, hR⟩ := mk' L R (λ i, (hL i).symm) (λ j, (hR j).symm) | def | pgame.relabelling.symm | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"mk'",
"pgame"
] | Flip a relabelling. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le : ∀ {x y : pgame} (r : x ≡r y), x ≤ y | | x y r := le_def.2 ⟨λ i, or.inl ⟨_, (r.move_left i).le⟩, λ j, or.inr ⟨_, (r.move_right_symm j).le⟩⟩
using_well_founded { dec_tac := pgame_wf_tac } | theorem | pgame.relabelling.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 | |
ge {x y : pgame} (r : x ≡r y) : y ≤ x | r.symm.le | theorem | pgame.relabelling.ge | 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 | |
equiv (r : x ≡r y) : x ≈ y | ⟨r.le, r.ge⟩ | theorem | pgame.relabelling.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"
] | A relabelling lets us prove equivalence of games. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans : Π {x y z : pgame}, x ≡r y → y ≡r z → x ≡r z | | x y z ⟨L₁, R₁, hL₁, hR₁⟩ ⟨L₂, R₂, hL₂, hR₂⟩ :=
⟨L₁.trans L₂, R₁.trans R₂, λ i, (hL₁ i).trans (hL₂ _), λ j, (hR₁ j).trans (hR₂ _)⟩ | def | pgame.relabelling.trans | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | Transitivity of relabelling. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_empty (x : pgame) [is_empty x.left_moves] [is_empty x.right_moves] : x ≡r 0 | ⟨equiv.equiv_pempty _, equiv.equiv_of_is_empty _ _, is_empty_elim, is_empty_elim⟩ | def | pgame.relabelling.is_empty | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.equiv_of_is_empty",
"is_empty",
"is_empty_elim",
"pgame"
] | Any game without left or right moves is a relabelling of 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv.is_empty (x : pgame) [is_empty x.left_moves] [is_empty x.right_moves] : x ≈ 0 | (relabelling.is_empty x).equiv | theorem | pgame.equiv.is_empty | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv",
"equiv.is_empty",
"is_empty",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabel {x : pgame} {xl' xr'} (el : xl' ≃ x.left_moves) (er : xr' ≃ x.right_moves) : pgame | ⟨xl', xr', x.move_left ∘ el, x.move_right ∘ er⟩ | def | pgame.relabel | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | Replace the types indexing the next moves for Left and Right by equivalent types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
relabel_move_left' {x : pgame} {xl' xr'}
(el : xl' ≃ x.left_moves) (er : xr' ≃ x.right_moves) (i : xl') :
move_left (relabel el er) i = x.move_left (el i) | rfl | lemma | pgame.relabel_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"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabel_move_left {x : pgame} {xl' xr'}
(el : xl' ≃ x.left_moves) (er : xr' ≃ x.right_moves) (i : x.left_moves) :
move_left (relabel el er) (el.symm i) = x.move_left i | by simp | lemma | pgame.relabel_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"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabel_move_right' {x : pgame} {xl' xr'}
(el : xl' ≃ x.left_moves) (er : xr' ≃ x.right_moves) (j : xr') :
move_right (relabel el er) j = x.move_right (er j) | rfl | lemma | pgame.relabel_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"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabel_move_right {x : pgame} {xl' xr'}
(el : xl' ≃ x.left_moves) (er : xr' ≃ x.right_moves) (j : x.right_moves) :
move_right (relabel el er) (er.symm j) = x.move_right j | by simp | lemma | pgame.relabel_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"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabel_relabelling {x : pgame} {xl' xr'} (el : xl' ≃ x.left_moves) (er : xr' ≃ x.right_moves) :
x ≡r relabel el er | relabelling.mk' el er (λ i, by simp) (λ j, by simp) | def | pgame.relabel_relabelling | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | The game obtained by relabelling the next moves is a relabelling of the original game. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg : pgame → pgame | | ⟨l, r, L, R⟩ := ⟨r, l, λ i, neg (R i), λ i, neg (L i)⟩ | def | pgame.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"
] | The negation of `{L | R}` is `{-R | -L}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_def {xl xr xL xR} : -(mk xl xr xL xR) = mk xr xl (λ j, -(xR j)) (λ i, -(xL i)) | rfl | lemma | pgame.neg_def | 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_of_lists (L R : list pgame) :
-of_lists L R = of_lists (R.map (λ x, -x)) (L.map (λ x, -x)) | begin
simp only [of_lists, neg_def, list.length_map, list.nth_le_map', eq_self_iff_true, true_and],
split, all_goals
{ apply hfunext,
{ simp },
{ intros a a' ha,
congr' 2,
have : ∀ {m n} (h₁ : m = n) {b : ulift (fin m)} {c : ulift (fin n)} (h₂ : b == c),
(b.down : ℕ) = ↑c.down,
{... | lemma | pgame.neg_of_lists | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"list.nth_le_map'",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_option_neg {x y : pgame} : is_option x (-y) ↔ is_option (-x) y | begin
rw [is_option_iff, is_option_iff, or_comm],
cases y, apply or_congr;
{ apply exists_congr, intro, rw neg_eq_iff_eq_neg, refl },
end | theorem | pgame.is_option_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 | |
is_option_neg_neg {x y : pgame} : is_option (-x) (-y) ↔ is_option x y | by rw [is_option_neg, neg_neg] | theorem | pgame.is_option_neg_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 | |
left_moves_neg : ∀ x : pgame, (-x).left_moves = x.right_moves | | ⟨_, _, _, _⟩ := rfl | theorem | pgame.left_moves_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 | |
right_moves_neg : ∀ x : pgame, (-x).right_moves = x.left_moves | | ⟨_, _, _, _⟩ := rfl | theorem | pgame.right_moves_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 | |
to_left_moves_neg {x : pgame} : x.right_moves ≃ (-x).left_moves | equiv.cast (left_moves_neg x).symm | def | pgame.to_left_moves_neg | 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"
] | Turns a right move for `x` into a left move for `-x` 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_neg {x : pgame} : x.left_moves ≃ (-x).right_moves | equiv.cast (right_moves_neg x).symm | def | pgame.to_right_moves_neg | 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"
] | Turns a left move for `x` into a right move for `-x` 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 |
move_left_neg {x : pgame} (i) :
(-x).move_left (to_left_moves_neg i) = -x.move_right i | by { cases x, refl } | lemma | pgame.move_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 | |
move_left_neg' {x : pgame} (i) :
(-x).move_left i = -x.move_right (to_left_moves_neg.symm i) | by { cases x, refl } | lemma | pgame.move_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 | |
move_right_neg {x : pgame} (i) :
(-x).move_right (to_right_moves_neg i) = -(x.move_left i) | by { cases x, refl } | lemma | pgame.move_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 | |
move_right_neg' {x : pgame} (i) :
(-x).move_right i = -x.move_left (to_right_moves_neg.symm i) | by { cases x, refl } | lemma | pgame.move_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 | |
move_left_neg_symm {x : pgame} (i) :
x.move_left (to_right_moves_neg.symm i) = -(-x).move_right i | by simp | lemma | pgame.move_left_neg_symm | 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 | |
move_left_neg_symm' {x : pgame} (i) :
x.move_left i = -(-x).move_right (to_right_moves_neg i) | by simp | lemma | pgame.move_left_neg_symm' | 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 | |
move_right_neg_symm {x : pgame} (i) :
x.move_right (to_left_moves_neg.symm i) = -(-x).move_left i | by simp | lemma | pgame.move_right_neg_symm | 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 | |
move_right_neg_symm' {x : pgame} (i) :
x.move_right i = -(-x).move_left (to_left_moves_neg i) | by simp | lemma | pgame.move_right_neg_symm' | 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.neg_congr : ∀ {x y : pgame}, x ≡r y → -x ≡r -y | | ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ ⟨L, R, hL, hR⟩ :=
⟨R, L, λ j, (hR j).neg_congr, λ i, (hL i).neg_congr⟩ | def | pgame.relabelling.neg_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 `x` has the same moves as `y`, then `-x` has the sames moves as `-y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_le_lf_neg_iff :
Π {x y : pgame.{u}}, (-y ≤ -x ↔ x ≤ y) ∧ (-y ⧏ -x ↔ x ⧏ y) | | (mk xl xr xL xR) (mk yl yr yL yR) :=
begin
simp_rw [neg_def, mk_le_mk, mk_lf_mk, ← neg_def],
split,
{ rw and_comm, apply and_congr; exact forall_congr (λ _, neg_le_lf_neg_iff.2) },
{ rw or_comm, apply or_congr; exact exists_congr (λ _, neg_le_lf_neg_iff.1) },
end
using_well_founded { dec_tac := pgame_wf_tac } | theorem | pgame.neg_le_lf_neg_iff | 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_le_neg_iff {x y : pgame} : -y ≤ -x ↔ x ≤ y | neg_le_lf_neg_iff.1 | theorem | pgame.neg_le_neg_iff | 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 | |
neg_lf_neg_iff {x y : pgame} : -y ⧏ -x ↔ x ⧏ y | neg_le_lf_neg_iff.2 | theorem | pgame.neg_lf_neg_iff | 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 | |
neg_lt_neg_iff {x y : pgame} : -y < -x ↔ x < y | by rw [lt_iff_le_and_lf, lt_iff_le_and_lf, neg_le_neg_iff, neg_lf_neg_iff] | theorem | pgame.neg_lt_neg_iff | 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 | |
neg_equiv_neg_iff {x y : pgame} : -x ≈ -y ↔ x ≈ y | by rw [equiv, equiv, neg_le_neg_iff, neg_le_neg_iff, and.comm] | theorem | pgame.neg_equiv_neg_iff | 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 | |
neg_fuzzy_neg_iff {x y : pgame} : -x ‖ -y ↔ x ‖ y | by rw [fuzzy, fuzzy, neg_lf_neg_iff, neg_lf_neg_iff, and.comm] | theorem | pgame.neg_fuzzy_neg_iff | 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 | |
neg_le_iff {x y : pgame} : -y ≤ x ↔ -x ≤ y | by rw [←neg_neg x, neg_le_neg_iff, neg_neg] | theorem | pgame.neg_le_iff | 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 | |
neg_lf_iff {x y : pgame} : -y ⧏ x ↔ -x ⧏ y | by rw [←neg_neg x, neg_lf_neg_iff, neg_neg] | theorem | pgame.neg_lf_iff | 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 | |
neg_lt_iff {x y : pgame} : -y < x ↔ -x < y | by rw [←neg_neg x, neg_lt_neg_iff, neg_neg] | theorem | pgame.neg_lt_iff | 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 | |
neg_equiv_iff {x y : pgame} : -x ≈ y ↔ x ≈ -y | by rw [←neg_neg y, neg_equiv_neg_iff, neg_neg] | theorem | pgame.neg_equiv_iff | 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 | |
neg_fuzzy_iff {x y : pgame} : -x ‖ y ↔ x ‖ -y | by rw [←neg_neg y, neg_fuzzy_neg_iff, neg_neg] | theorem | pgame.neg_fuzzy_iff | 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_neg_iff {x y : pgame} : y ≤ -x ↔ x ≤ -y | by rw [←neg_neg x, neg_le_neg_iff, neg_neg] | theorem | pgame.le_neg_iff | 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_neg_iff {x y : pgame} : y ⧏ -x ↔ x ⧏ -y | by rw [←neg_neg x, neg_lf_neg_iff, neg_neg] | theorem | pgame.lf_neg_iff | 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_neg_iff {x y : pgame} : y < -x ↔ x < -y | by rw [←neg_neg x, neg_lt_neg_iff, neg_neg] | theorem | pgame.lt_neg_iff | 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 | |
neg_le_zero_iff {x : pgame} : -x ≤ 0 ↔ 0 ≤ x | by rw [neg_le_iff, neg_zero] | theorem | pgame.neg_le_zero_iff | 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_neg_iff {x : pgame} : 0 ≤ -x ↔ x ≤ 0 | by rw [le_neg_iff, neg_zero] | theorem | pgame.zero_le_neg_iff | 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 | |
neg_lf_zero_iff {x : pgame} : -x ⧏ 0 ↔ 0 ⧏ x | by rw [neg_lf_iff, neg_zero] | theorem | pgame.neg_lf_zero_iff | 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_lf_neg_iff {x : pgame} : 0 ⧏ -x ↔ x ⧏ 0 | by rw [lf_neg_iff, neg_zero] | theorem | pgame.zero_lf_neg_iff | 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 | |
neg_lt_zero_iff {x : pgame} : -x < 0 ↔ 0 < x | by rw [neg_lt_iff, neg_zero] | theorem | pgame.neg_lt_zero_iff | 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_lt_neg_iff {x : pgame} : 0 < -x ↔ x < 0 | by rw [lt_neg_iff, neg_zero] | theorem | pgame.zero_lt_neg_iff | 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 | |
neg_equiv_zero_iff {x : pgame} : -x ≈ 0 ↔ x ≈ 0 | by rw [neg_equiv_iff, neg_zero] | theorem | pgame.neg_equiv_zero_iff | 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 | |
neg_fuzzy_zero_iff {x : pgame} : -x ‖ 0 ↔ x ‖ 0 | by rw [neg_fuzzy_iff, neg_zero] | theorem | pgame.neg_fuzzy_zero_iff | 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_equiv_neg_iff {x : pgame} : 0 ≈ -x ↔ 0 ≈ x | by rw [←neg_equiv_iff, neg_zero] | theorem | pgame.zero_equiv_neg_iff | 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_fuzzy_neg_iff {x : pgame} : 0 ‖ -x ↔ 0 ‖ x | by rw [←neg_fuzzy_iff, neg_zero] | theorem | pgame.zero_fuzzy_neg_iff | 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 | |
nat_succ (n : ℕ) : ((n + 1 : ℕ) : pgame) = n + 1 | rfl | theorem | pgame.nat_succ | 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_left_moves_add (x y : pgame.{u})
[is_empty x.left_moves] [is_empty y.left_moves] : is_empty (x + y).left_moves | begin
unfreezingI { cases x, cases y },
apply is_empty_sum.2 ⟨_, _⟩,
assumption'
end | instance | pgame.is_empty_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"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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