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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|---|---|---|---|---|---|---|---|---|---|---|
wf_subsequent : well_founded subsequent | wf_is_option.trans_gen | theorem | pgame.wf_subsequent | 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 | |
subsequent.move_left {x : pgame} (i : x.left_moves) : subsequent (x.move_left i) x | trans_gen.single (is_option.move_left i) | lemma | pgame.subsequent.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 | |
subsequent.move_right {x : pgame} (j : x.right_moves) : subsequent (x.move_right j) x | trans_gen.single (is_option.move_right j) | lemma | pgame.subsequent.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 | |
subsequent.mk_left {xl xr} (xL : xl → pgame) (xR : xr → pgame) (i : xl) :
subsequent (xL i) (mk xl xr xL xR) | @subsequent.move_left (mk _ _ _ _) i | lemma | pgame.subsequent.mk_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 | |
subsequent.mk_right {xl xr} (xL : xl → pgame) (xR : xr → pgame) (j : xr) :
subsequent (xR j) (mk xl xr xL xR) | @subsequent.move_right (mk _ _ _ _) j | lemma | pgame.subsequent.mk_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 | |
pgame_wf_tac | `[solve_by_elim
[psigma.lex.left, psigma.lex.right, subsequent.move_left, subsequent.move_right,
subsequent.mk_left, subsequent.mk_right, subsequent.trans]
{ max_depth := 6 }] | def | pgame.pgame_wf_tac | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | A local tactic for proving well-foundedness of recursive definitions involving pregames. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_left_moves : left_moves 0 = pempty | rfl | lemma | pgame.zero_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"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_right_moves : right_moves 0 = pempty | rfl | lemma | pgame.zero_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"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_empty_zero_left_moves : is_empty (left_moves 0) | pempty.is_empty | instance | pgame.is_empty_zero_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"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_empty_zero_right_moves : is_empty (right_moves 0) | pempty.is_empty | instance | pgame.is_empty_zero_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_left_moves : left_moves 1 = punit | rfl | lemma | pgame.one_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 | |
one_move_left (x) : move_left 1 x = 0 | rfl | lemma | pgame.one_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 | |
one_right_moves : right_moves 1 = pempty | rfl | lemma | pgame.one_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"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_one_left_moves : unique (left_moves 1) | punit.unique | instance | pgame.unique_one_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 | |
is_empty_one_right_moves : is_empty (right_moves 1) | pempty.is_empty | instance | pgame.is_empty_one_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf (x y : pgame) : Prop | ¬ y ≤ x | def | pgame.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"
] | The less or fuzzy relation on pre-games.
If `0 ⧏ x`, then Left can win `x` as the first player. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_le {x y : pgame} : ¬ x ≤ y ↔ y ⧏ x | iff.rfl | theorem | pgame.not_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_lf {x y : pgame} : ¬ x ⧏ y ↔ y ≤ x | not_not | theorem | pgame.not_lf | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"not_not",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.has_le.le.not_gf {x y : pgame} : x ≤ y → ¬ y ⧏ x | not_lf.2 | theorem | has_le.le.not_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 | |
lf.not_ge {x y : pgame} : x ⧏ y → ¬ y ≤ x | id | theorem | pgame.lf.not_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 | |
le_iff_forall_lf {x y : pgame} :
x ≤ y ↔ (∀ i, x.move_left i ⧏ y) ∧ ∀ j, x ⧏ y.move_right j | by { unfold has_le.le, rw sym2.game_add.fix_eq, refl } | theorem | pgame.le_iff_forall_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",
"sym2.game_add.fix_eq"
] | Definition of `x ≤ y` on pre-games, in terms of `⧏`.
The ordering here is chosen so that `and.left` refer to moves by Left, and `and.right` refer to
moves by Right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_le_mk {xl xr xL xR yl yr yL yR} :
mk xl xr xL xR ≤ mk yl yr yL yR ↔
(∀ i, xL i ⧏ mk yl yr yL yR) ∧ ∀ j, mk xl xr xL xR ⧏ yR j | le_iff_forall_lf | theorem | pgame.mk_le_mk | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | Definition of `x ≤ y` on pre-games built using the constructor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_forall_lf {x y : pgame} (h₁ : ∀ i, x.move_left i ⧏ y) (h₂ : ∀ j, x ⧏ y.move_right j) :
x ≤ y | le_iff_forall_lf.2 ⟨h₁, h₂⟩ | theorem | pgame.le_of_forall_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 | |
lf_iff_exists_le {x y : pgame} :
x ⧏ y ↔ (∃ i, x ≤ y.move_left i) ∨ ∃ j, x.move_right j ≤ y | by { rw [lf, le_iff_forall_lf, not_and_distrib], simp } | theorem | pgame.lf_iff_exists_le | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"not_and_distrib",
"pgame"
] | Definition of `x ⧏ y` on pre-games, in terms of `≤`.
The ordering here is chosen so that `or.inl` refer to moves by Left, and `or.inr` refer to
moves by Right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_lf_mk {xl xr xL xR yl yr yL yR} :
mk xl xr xL xR ⧏ mk yl yr yL yR ↔
(∃ i, mk xl xr xL xR ≤ yL i) ∨ ∃ j, xR j ≤ mk yl yr yL yR | lf_iff_exists_le | theorem | pgame.mk_lf_mk | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [] | Definition of `x ⧏ y` on pre-games built using the constructor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_or_gf (x y : pgame) : x ≤ y ∨ y ⧏ x | by { rw ←pgame.not_le, apply em } | theorem | pgame.le_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"
] | [
"em",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_lf_of_le {x y : pgame} (h : x ≤ y) (i) : x.move_left i ⧏ y | (le_iff_forall_lf.1 h).1 i | theorem | pgame.move_left_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 | |
lf_move_right_of_le {x y : pgame} (h : x ≤ y) (j) : x ⧏ y.move_right j | (le_iff_forall_lf.1 h).2 j | theorem | pgame.lf_move_right_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 | |
lf_of_move_right_le {x y : pgame} {j} (h : x.move_right j ≤ y) : x ⧏ y | lf_iff_exists_le.2 $ or.inr ⟨j, h⟩ | theorem | pgame.lf_of_move_right_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 | |
lf_of_le_move_left {x y : pgame} {i} (h : x ≤ y.move_left i) : x ⧏ y | lf_iff_exists_le.2 $ or.inl ⟨i, h⟩ | theorem | pgame.lf_of_le_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 | |
lf_of_le_mk {xl xr xL xR y} : mk xl xr xL xR ≤ y → ∀ i, xL i ⧏ y | move_left_lf_of_le | theorem | pgame.lf_of_le_mk | 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 | |
lf_of_mk_le {x yl yr yL yR} : x ≤ mk yl yr yL yR → ∀ j, x ⧏ yR j | lf_move_right_of_le | theorem | pgame.lf_of_mk_le | 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_lf_of_le {xl xr y j} (xL) {xR : xr → pgame} : xR j ≤ y → mk xl xr xL xR ⧏ y | @lf_of_move_right_le (mk _ _ _ _) y j | theorem | pgame.mk_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 | |
lf_mk_of_le {x yl yr} {yL : yl → pgame} (yR) {i} : x ≤ yL i → x ⧏ mk yl yr yL yR | @lf_of_le_move_left x (mk _ _ _ _) i | theorem | pgame.lf_mk_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 | |
le_trans_aux {x y z : pgame}
(h₁ : ∀ {i}, y ≤ z → z ≤ x.move_left i → y ≤ x.move_left i)
(h₂ : ∀ {j}, z.move_right j ≤ x → x ≤ y → z.move_right j ≤ y)
(hxy : x ≤ y) (hyz : y ≤ z) : x ≤ z | le_of_forall_lf
(λ i, pgame.not_le.1 $ λ h, (h₁ hyz h).not_gf $ hxy.move_left_lf i)
(λ j, pgame.not_le.1 $ λ h, (h₂ h hxy).not_gf $ hyz.lf_move_right j) | theorem | pgame.le_trans_aux | 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_le_and_lf {x y : pgame} : x < y ↔ x ≤ y ∧ x ⧏ y | iff.rfl | theorem | pgame.lt_iff_le_and_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_of_le_of_lf {x y : pgame} (h₁ : x ≤ y) (h₂ : x ⧏ y) : x < y | ⟨h₁, h₂⟩ | theorem | pgame.lt_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 | |
lf_of_lt {x y : pgame} (h : x < y) : x ⧏ y | h.2 | theorem | pgame.lf_of_lt | 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_irrefl (x : pgame) : ¬ x ⧏ x | le_rfl.not_gf | theorem | pgame.lf_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_of_le_of_lf {x y z : pgame} (h₁ : x ≤ y) (h₂ : y ⧏ z) : x ⧏ z | by { rw ←pgame.not_le at h₂ ⊢, exact λ h₃, h₂ (h₃.trans h₁) } | theorem | pgame.lf_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 | |
lf_of_lf_of_le {x y z : pgame} (h₁ : x ⧏ y) (h₂ : y ≤ z) : x ⧏ z | by { rw ←pgame.not_le at h₁ ⊢, exact λ h₃, h₁ (h₂.trans h₃) } | theorem | pgame.lf_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 | |
lf_of_lt_of_lf {x y z : pgame} (h₁ : x < y) (h₂ : y ⧏ z) : x ⧏ z | h₁.le.trans_lf h₂ | theorem | pgame.lf_of_lt_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 | |
lf_of_lf_of_lt {x y z : pgame} (h₁ : x ⧏ y) (h₂ : y < z) : x ⧏ z | h₁.trans_le h₂.le | theorem | pgame.lf_of_lf_of_lt | 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_lf {x : pgame} : ∀ i, x.move_left i ⧏ x | le_rfl.move_left_lf | theorem | pgame.move_left_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 | |
lf_move_right {x : pgame} : ∀ j, x ⧏ x.move_right j | le_rfl.lf_move_right | theorem | pgame.lf_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 | |
lf_mk {xl xr} (xL : xl → pgame) (xR : xr → pgame) (i) : xL i ⧏ mk xl xr xL xR | @move_left_lf (mk _ _ _ _) i | theorem | pgame.lf_mk | 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_lf {xl xr} (xL : xl → pgame) (xR : xr → pgame) (j) : mk xl xr xL xR ⧏ xR j | @lf_move_right (mk _ _ _ _) j | theorem | pgame.mk_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 | |
le_of_forall_lt {x y : pgame} (h₁ : ∀ i, x.move_left i < y) (h₂ : ∀ j, x < y.move_right j) :
x ≤ y | le_of_forall_lf (λ i, (h₁ i).lf) (λ i, (h₂ i).lf) | theorem | pgame.le_of_forall_lt | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"le_of_forall_lt",
"pgame"
] | This special case of `pgame.le_of_forall_lf` is useful when dealing with surreals, where `<` is
preferred over `⧏`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_def {x y : pgame} : x ≤ y ↔
(∀ i, (∃ i', x.move_left i ≤ y.move_left i') ∨ ∃ j, (x.move_left i).move_right j ≤ y) ∧
∀ j, (∃ i, x ≤ (y.move_right j).move_left i) ∨ ∃ j', x.move_right j' ≤ y.move_right j | by { rw le_iff_forall_lf, conv { to_lhs, simp only [lf_iff_exists_le] } } | theorem | pgame.le_def | 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 definition of `x ≤ y` on pre-games, in terms of `≤` two moves later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lf_def {x y : pgame} : x ⧏ y ↔
(∃ i, (∀ i', x.move_left i' ⧏ y.move_left i) ∧ ∀ j, x ⧏ (y.move_left i).move_right j) ∨
∃ j, (∀ i, (x.move_right j).move_left i ⧏ y) ∧ ∀ j', x.move_right j ⧏ y.move_right j' | by { rw lf_iff_exists_le, conv { to_lhs, simp only [le_iff_forall_lf] } } | theorem | pgame.lf_def | 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 definition of `x ⧏ y` on pre-games, in terms of `⧏` two moves later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_le_lf {x : pgame} : 0 ≤ x ↔ ∀ j, 0 ⧏ x.move_right j | by { rw le_iff_forall_lf, simp } | theorem | pgame.zero_le_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"
] | The definition of `0 ≤ x` on pre-games, in terms of `0 ⧏`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_zero_lf {x : pgame} : x ≤ 0 ↔ ∀ i, x.move_left i ⧏ 0 | by { rw le_iff_forall_lf, simp } | theorem | pgame.le_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"
] | The definition of `x ≤ 0` on pre-games, in terms of `⧏ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lf_le {x : pgame} : 0 ⧏ x ↔ ∃ i, 0 ≤ x.move_left i | by { rw lf_iff_exists_le, simp } | theorem | pgame.zero_lf_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"
] | The definition of `0 ⧏ x` on pre-games, in terms of `0 ≤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lf_zero_le {x : pgame} : x ⧏ 0 ↔ ∃ j, x.move_right j ≤ 0 | by { rw lf_iff_exists_le, simp } | theorem | pgame.lf_zero_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"
] | The definition of `x ⧏ 0` on pre-games, in terms of `≤ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_le {x : pgame} : 0 ≤ x ↔ ∀ j, ∃ i, 0 ≤ (x.move_right j).move_left i | by { rw le_def, simp } | theorem | pgame.zero_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"
] | The definition of `0 ≤ x` on pre-games, in terms of `0 ≤` two moves later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_zero {x : pgame} : x ≤ 0 ↔ ∀ i, ∃ j, (x.move_left i).move_right j ≤ 0 | by { rw le_def, simp } | theorem | pgame.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"
] | The definition of `x ≤ 0` on pre-games, in terms of `≤ 0` two moves later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_lf {x : pgame} : 0 ⧏ x ↔ ∃ i, ∀ j, 0 ⧏ (x.move_left i).move_right j | by { rw lf_def, simp } | theorem | pgame.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"
] | The definition of `0 ⧏ x` on pre-games, in terms of `0 ⧏` two moves later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lf_zero {x : pgame} : x ⧏ 0 ↔ ∃ j, ∀ i, (x.move_right j).move_left i ⧏ 0 | by { rw lf_def, simp } | theorem | pgame.lf_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"
] | The definition of `x ⧏ 0` on pre-games, in terms of `⧏ 0` two moves later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_le_of_is_empty_right_moves (x : pgame) [is_empty x.right_moves] : 0 ≤ x | zero_le.2 is_empty_elim | theorem | pgame.zero_le_of_is_empty_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",
"is_empty_elim",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_zero_of_is_empty_left_moves (x : pgame) [is_empty x.left_moves] : x ≤ 0 | le_zero.2 is_empty_elim | theorem | pgame.le_zero_of_is_empty_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"
] | [
"is_empty",
"is_empty_elim",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_response {x : pgame} (h : x ≤ 0) (i : x.left_moves) :
(x.move_left i).right_moves | classical.some $ (le_zero.1 h) i | def | pgame.right_response | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | Given a game won by the right player when they play second, provide a response to any move by
left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_response_spec {x : pgame} (h : x ≤ 0) (i : x.left_moves) :
(x.move_left i).move_right (right_response h i) ≤ 0 | classical.some_spec $ (le_zero.1 h) i | lemma | pgame.right_response_spec | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | Show that the response for right provided by `right_response` preserves the right-player-wins
condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_response {x : pgame} (h : 0 ≤ x) (j : x.right_moves) :
(x.move_right j).left_moves | classical.some $ (zero_le.1 h) j | def | pgame.left_response | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | Given a game won by the left player when they play second, provide a response to any move by
right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_response_spec {x : pgame} (h : 0 ≤ x) (j : x.right_moves) :
0 ≤ (x.move_right j).move_left (left_response h j) | classical.some_spec $ (zero_le.1 h) j | lemma | pgame.left_response_spec | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pgame"
] | Show that the response for left provided by `left_response` preserves the left-player-wins
condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_bound {ι : Type u} (f : ι → pgame.{u}) : pgame | ⟨Σ i, (f i).left_moves, pempty, λ x, move_left _ x.2, pempty.elim⟩ | def | pgame.upper_bound | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pempty",
"pgame"
] | An explicit upper bound for a family of pre-games, whose left moves are the union of the left
moves of all the pre-games in the family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_bound_right_moves_empty {ι : Type u} (f : ι → pgame.{u}) :
is_empty (upper_bound f).right_moves | pempty.is_empty | instance | pgame.upper_bound_right_moves_empty | 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 | |
le_upper_bound {ι : Type u} (f : ι → pgame.{u}) (i : ι) : f i ≤ upper_bound f | begin
rw [upper_bound, le_iff_forall_lf],
dsimp,
simp only [and_true, is_empty.forall_iff],
exact λ j, @move_left_lf (upper_bound f) ⟨i, j⟩
end | theorem | pgame.le_upper_bound | 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.forall_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_bound_mem_upper_bounds (s : set pgame.{u}) [small.{u} s] :
upper_bound (subtype.val ∘ (equiv_shrink s).symm) ∈ upper_bounds s | λ i hi, by simpa using
le_upper_bound (subtype.val ∘ (equiv_shrink s).symm) (equiv_shrink s ⟨i, hi⟩) | lemma | pgame.upper_bound_mem_upper_bounds | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv_shrink",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_of_small (s : set pgame.{u}) [small.{u} s] : bdd_above s | ⟨_, upper_bound_mem_upper_bounds s⟩ | lemma | pgame.bdd_above_of_small | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"bdd_above"
] | A small set `s` of pre-games is bounded above. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_bound {ι : Type u} (f : ι → pgame.{u}) : pgame | ⟨pempty, Σ i, (f i).right_moves, pempty.elim, λ x, move_right _ x.2⟩ | def | pgame.lower_bound | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"pempty.elim",
"pgame"
] | An explicit lower bound for a family of pre-games, whose right moves are the union of the right
moves of all the pre-games in the family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_bound_left_moves_empty {ι : Type u} (f : ι → pgame.{u}) :
is_empty (lower_bound f).left_moves | pempty.is_empty | instance | pgame.lower_bound_left_moves_empty | 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 | |
lower_bound_le {ι : Type u} (f : ι → pgame.{u}) (i : ι) : lower_bound f ≤ f i | begin
rw [lower_bound, le_iff_forall_lf],
dsimp,
simp only [is_empty.forall_iff, true_and],
exact λ j, @lf_move_right (lower_bound f) ⟨i, j⟩
end | theorem | pgame.lower_bound_le | 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.forall_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_bound_mem_lower_bounds (s : set pgame.{u}) [small.{u} s] :
lower_bound (subtype.val ∘ (equiv_shrink s).symm) ∈ lower_bounds s | λ i hi, by simpa using
lower_bound_le (subtype.val ∘ (equiv_shrink s).symm) (equiv_shrink s ⟨i, hi⟩) | lemma | pgame.lower_bound_mem_lower_bounds | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv_shrink",
"lower_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_of_small (s : set pgame.{u}) [small.{u} s] : bdd_below s | ⟨_, lower_bound_mem_lower_bounds s⟩ | lemma | pgame.bdd_below_of_small | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"bdd_below"
] | A small set `s` of pre-games is bounded below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv (x y : pgame) : Prop | x ≤ y ∧ y ≤ x | def | pgame.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"
] | The equivalence relation on pre-games. Two pre-games `x`, `y` are equivalent if `x ≤ y` and
`y ≤ x`.
If `x ≈ 0`, then the second player can always win `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv.le {x y : pgame} (h : x ≈ y) : x ≤ y | h.1 | theorem | pgame.equiv.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 | |
equiv.ge {x y : pgame} (h : x ≈ y) : y ≤ x | h.2 | theorem | pgame.equiv.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_rfl {x} : x ≈ x | refl x | theorem | pgame.equiv_rfl | 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_refl (x) : x ≈ x | refl x | theorem | pgame.equiv_refl | 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.symm {x y} : x ≈ y → y ≈ x | symm | theorem | pgame.equiv.symm | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.trans {x y z} : x ≈ y → y ≈ z → x ≈ z | trans | theorem | pgame.equiv.trans | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"equiv.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_comm {x y} : x ≈ y ↔ y ≈ x | comm | theorem | pgame.equiv_comm | set_theory.game | src/set_theory/game/pgame.lean | [
"data.fin.basic",
"data.list.basic",
"logic.relation",
"logic.small.basic",
"order.game_add"
] | [
"comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_eq {x y} (h : x = y) : x ≈ y | by subst h | theorem | pgame.equiv_of_eq | 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 | |
le_of_le_of_equiv {x y z} (h₁ : x ≤ y) (h₂ : y ≈ z) : x ≤ z | h₁.trans h₂.1 | theorem | pgame.le_of_le_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 | |
le_of_equiv_of_le {x y z} (h₁ : x ≈ y) : y ≤ z → x ≤ z | h₁.1.trans | theorem | pgame.le_of_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf.not_equiv {x y} (h : x ⧏ y) : ¬ x ≈ y | λ h', h.not_ge h'.2 | theorem | pgame.lf.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf.not_equiv' {x y} (h : x ⧏ y) : ¬ y ≈ x | λ h', h.not_ge h'.1 | theorem | pgame.lf.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf.not_gt {x y} (h : x ⧏ y) : ¬ y < x | λ h', h.not_ge h'.le | theorem | pgame.lf.not_gt | 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 | |
le_congr_imp {x₁ y₁ x₂ y₂} (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) (h : x₁ ≤ y₁) : x₂ ≤ y₂ | hx.2.trans (h.trans hy.1) | theorem | pgame.le_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 | |
le_congr {x₁ y₁ x₂ y₂} (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ ≤ y₁ ↔ x₂ ≤ y₂ | ⟨le_congr_imp hx hy, le_congr_imp hx.symm hy.symm⟩ | theorem | pgame.le_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 | |
le_congr_left {x₁ x₂ y} (hx : x₁ ≈ x₂) : x₁ ≤ y ↔ x₂ ≤ y | le_congr hx equiv_rfl | theorem | pgame.le_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 | |
le_congr_right {x y₁ y₂} (hy : y₁ ≈ y₂) : x ≤ y₁ ↔ x ≤ y₂ | le_congr equiv_rfl hy | theorem | pgame.le_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 | |
lf_congr {x₁ y₁ x₂ y₂} (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ ⧏ y₁ ↔ x₂ ⧏ y₂ | pgame.not_le.symm.trans $ (not_congr (le_congr hy hx)).trans pgame.not_le | theorem | pgame.lf_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.not_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf_congr_imp {x₁ y₁ x₂ y₂} (hx : x₁ ≈ x₂) (hy : y₁ ≈ y₂) : x₁ ⧏ y₁ → x₂ ⧏ y₂ | (lf_congr hx hy).1 | theorem | pgame.lf_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 | |
lf_congr_left {x₁ x₂ y} (hx : x₁ ≈ x₂) : x₁ ⧏ y ↔ x₂ ⧏ y | lf_congr hx equiv_rfl | theorem | pgame.lf_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 | |
lf_congr_right {x y₁ y₂} (hy : y₁ ≈ y₂) : x ⧏ y₁ ↔ x ⧏ y₂ | lf_congr equiv_rfl hy | theorem | pgame.lf_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 | |
lf_of_lf_of_equiv {x y z} (h₁ : x ⧏ y) (h₂ : y ≈ z) : x ⧏ z | lf_congr_imp equiv_rfl h₂ h₁ | theorem | pgame.lf_of_lf_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 | |
lf_of_equiv_of_lf {x y z} (h₁ : x ≈ y) : y ⧏ z → x ⧏ z | lf_congr_imp h₁.symm equiv_rfl | theorem | pgame.lf_of_equiv_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_lt_of_equiv {x y z} (h₁ : x < y) (h₂ : y ≈ z) : x < z | h₁.trans_le h₂.1 | theorem | pgame.lt_of_lt_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 | |
lt_of_equiv_of_lt {x y z} (h₁ : x ≈ y) : y < z → x < z | h₁.1.trans_lt | theorem | pgame.lt_of_equiv_of_lt | 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 |
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