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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|---|---|---|---|---|---|---|---|---|---|---|
bit0_ne_zero (a : cardinal) : ¬bit0 a = 0 ↔ ¬a = 0 | by simp [bit0] | lemma | cardinal.bit0_ne_zero | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"bit0_ne_zero",
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_ne_zero (a : cardinal) : ¬bit1 a = 0 | by simp [bit1] | lemma | cardinal.bit1_ne_zero | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_bit0 (a : cardinal) : 0 < bit0 a ↔ 0 < a | by { rw ←not_iff_not, simp [bit0], } | lemma | cardinal.zero_lt_bit0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"zero_lt_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_bit1 (a : cardinal) : 0 < bit1 a | zero_lt_one.trans_le (self_le_add_left _ _) | lemma | cardinal.zero_lt_bit1 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_bit0 (a : cardinal) : 1 ≤ bit0 a ↔ 0 < a | ⟨λ h, (zero_lt_bit0 a).mp (zero_lt_one.trans_le h),
λ h, (one_le_iff_pos.mpr h).trans (self_le_add_left a a)⟩ | lemma | cardinal.one_le_bit0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"zero_lt_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_bit1 (a : cardinal) : 1 ≤ bit1 a | self_le_add_left _ _ | lemma | cardinal.one_le_bit1 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"one_le_bit1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_eq_self {c : cardinal} (h : ℵ₀ ≤ c) : bit0 c = c | add_eq_self h | theorem | cardinal.bit0_eq_self | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_lt_aleph_0 {c : cardinal} : bit0 c < ℵ₀ ↔ c < ℵ₀ | by simp [bit0, add_lt_aleph_0_iff] | theorem | cardinal.bit0_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_bit0 {c : cardinal} : ℵ₀ ≤ bit0 c ↔ ℵ₀ ≤ c | by { rw ←not_iff_not, simp } | theorem | cardinal.aleph_0_le_bit0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_eq_self_iff {c : cardinal} : bit1 c = c ↔ ℵ₀ ≤ c | begin
by_cases h : ℵ₀ ≤ c,
{ simp only [bit1, bit0_eq_self h, h, eq_self_iff_true, add_one_of_aleph_0_le] },
{ refine iff_of_false (ne_of_gt _) h,
rcases lt_aleph_0.1 (not_le.1 h) with ⟨n, rfl⟩,
norm_cast,
dsimp [bit1, bit0],
linarith }
end | theorem | cardinal.bit1_eq_self_iff | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"iff_of_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_lt_aleph_0 {c : cardinal} : bit1 c < ℵ₀ ↔ c < ℵ₀ | by simp [bit1, bit0, add_lt_aleph_0_iff, one_lt_aleph_0] | theorem | cardinal.bit1_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_bit1 {c : cardinal} : ℵ₀ ≤ bit1 c ↔ ℵ₀ ≤ c | by { rw ←not_iff_not, simp } | theorem | cardinal.aleph_0_le_bit1 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_le_bit0 {a b : cardinal} : bit0 a ≤ bit0 b ↔ a ≤ b | begin
cases le_or_lt ℵ₀ a with ha ha; cases le_or_lt ℵ₀ b with hb hb,
{ rw [bit0_eq_self ha, bit0_eq_self hb] },
{ rw bit0_eq_self ha,
refine iff_of_false (λ h, _) (hb.trans_le ha).not_le,
have A : bit0 b < ℵ₀, by simpa using hb,
exact lt_irrefl _ ((A.trans_le ha).trans_le h) },
{ rw bit0_eq_self hb... | lemma | cardinal.bit0_le_bit0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"bit0_le_bit0",
"cardinal",
"iff_of_false",
"iff_of_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_le_bit1 {a b : cardinal} : bit0 a ≤ bit1 b ↔ a ≤ b | begin
cases le_or_lt ℵ₀ a with ha ha; cases le_or_lt ℵ₀ b with hb hb,
{ rw [bit0_eq_self ha, bit1_eq_self_iff.2 hb] },
{ rw bit0_eq_self ha,
refine iff_of_false (λ h, _) (hb.trans_le ha).not_le,
have A : bit1 b < ℵ₀, by simpa using hb,
exact lt_irrefl _ ((A.trans_le ha).trans_le h) },
{ rw bit1_eq_s... | lemma | cardinal.bit0_le_bit1 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"iff_of_false",
"iff_of_true",
"nat.bit0_le_bit1_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_le_bit1 {a b : cardinal} : bit1 a ≤ bit1 b ↔ a ≤ b | ⟨λ h, bit0_le_bit1.1 ((self_le_add_right (bit0 a) 1).trans h), λ h,
(add_le_add_right (add_le_add_left h a) 1).trans (add_le_add_right (add_le_add_right h b) 1)⟩ | lemma | cardinal.bit1_le_bit1 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"bit1_le_bit1",
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_le_bit0 {a b : cardinal} : bit1 a ≤ bit0 b ↔ (a < b ∨ (a ≤ b ∧ ℵ₀ ≤ a)) | begin
cases le_or_lt ℵ₀ a with ha ha; cases le_or_lt ℵ₀ b with hb hb,
{ simp only [bit1_eq_self_iff.mpr ha, bit0_eq_self hb, ha, and_true],
refine ⟨λ h, or.inr h, λ h, _⟩,
cases h,
{ exact le_of_lt h },
{ exact h } },
{ rw bit1_eq_self_iff.2 ha,
refine iff_of_false (λ h, _) (λ h, _),
{ hav... | lemma | cardinal.bit1_le_bit0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"iff_of_false",
"iff_of_true",
"not_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_lt_bit0 {a b : cardinal} : bit0 a < bit0 b ↔ a < b | begin
cases le_or_lt ℵ₀ a with ha ha; cases le_or_lt ℵ₀ b with hb hb,
{ rw [bit0_eq_self ha, bit0_eq_self hb] },
{ rw bit0_eq_self ha,
refine iff_of_false (λ h, _) (hb.le.trans ha).not_lt,
have A : bit0 b < ℵ₀, by simpa using hb,
exact lt_irrefl _ ((A.trans_le ha).trans h) },
{ rw bit0_eq_self hb,
... | lemma | cardinal.bit0_lt_bit0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"bit0_lt_bit0",
"cardinal",
"iff_of_false",
"iff_of_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_lt_bit0 {a b : cardinal} : bit1 a < bit0 b ↔ a < b | begin
cases le_or_lt ℵ₀ a with ha ha; cases le_or_lt ℵ₀ b with hb hb,
{ rw [bit1_eq_self_iff.2 ha, bit0_eq_self hb] },
{ rw bit1_eq_self_iff.2 ha,
refine iff_of_false (λ h, _) (hb.le.trans ha).not_lt,
have A : bit0 b < ℵ₀, by simpa using hb,
exact lt_irrefl _ ((A.trans_le ha).trans h) },
{ rw bit0_e... | lemma | cardinal.bit1_lt_bit0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"iff_of_false",
"iff_of_true",
"nat.bit1_lt_bit0_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_lt_bit1 {a b : cardinal} : bit1 a < bit1 b ↔ a < b | begin
cases le_or_lt ℵ₀ a with ha ha; cases le_or_lt ℵ₀ b with hb hb,
{ rw [bit1_eq_self_iff.2 ha, bit1_eq_self_iff.2 hb] },
{ rw bit1_eq_self_iff.2 ha,
refine iff_of_false (λ h, _) (hb.le.trans ha).not_lt,
have A : bit1 b < ℵ₀, by simpa using hb,
exact lt_irrefl _ ((A.trans_le ha).trans h) },
{ rw ... | lemma | cardinal.bit1_lt_bit1 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"bit1_lt_bit1",
"cardinal",
"iff_of_false",
"iff_of_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_lt_bit1 {a b : cardinal} : bit0 a < bit1 b ↔ (a < b ∨ (a ≤ b ∧ a < ℵ₀)) | begin
cases le_or_lt ℵ₀ a with ha ha; cases le_or_lt ℵ₀ b with hb hb,
{ simp [bit0_eq_self ha, bit1_eq_self_iff.2 hb, not_lt.mpr ha] },
{ rw bit0_eq_self ha,
refine iff_of_false (λ h, _) (λ h, _),
{ have A : bit1 b < ℵ₀, by simpa using hb,
exact lt_irrefl _ ((A.trans_le ha).trans h) },
{ exact (... | lemma | cardinal.bit0_lt_bit1 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"iff_of_false",
"iff_of_true",
"nat.bit0_lt_bit1_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_two : (1 : cardinal) < 2 | -- This strategy works generally to prove inequalities between numerals in `cardinality`.
by { norm_cast, norm_num } | lemma | cardinal.one_lt_two | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"one_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_bit0 {a : cardinal} : 1 < bit0 a ↔ 0 < a | by simp [←bit1_zero] | lemma | cardinal.one_lt_bit0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_bit1 (a : cardinal) : 1 < bit1 a ↔ 0 < a | by simp [←bit1_zero] | lemma | cardinal.one_lt_bit1 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"one_lt_bit1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
schroeder_bernstein {f : α → β} {g : β → α}
(hf : function.injective f) (hg : function.injective g) : ∃ h : α → β, bijective h | begin
casesI is_empty_or_nonempty β with hβ hβ,
{ haveI : is_empty α, from function.is_empty f,
exact ⟨_, ((equiv.equiv_empty α).trans (equiv.equiv_empty β).symm).bijective⟩ },
set F : set α →o set α :=
{ to_fun := λ s, (g '' (f '' s)ᶜ)ᶜ,
monotone' := λ s t hst, compl_subset_compl.mpr $ image_subset _ $... | theorem | function.embedding.schroeder_bernstein | set_theory.cardinal | src/set_theory/cardinal/schroeder_bernstein.lean | [
"order.fixed_points",
"order.zorn"
] | [
"compl_injective",
"equiv.equiv_empty",
"function.is_empty",
"inv_fun",
"is_empty",
"is_empty_or_nonempty",
"left_inverse_inv_fun"
] | **The Schröder-Bernstein Theorem**:
Given injections `α → β` and `β → α`, we can get a bijection `α → β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antisymm : (α ↪ β) → (β ↪ α) → nonempty (α ≃ β) | | ⟨e₁, h₁⟩ ⟨e₂, h₂⟩ :=
let ⟨f, hf⟩ := schroeder_bernstein h₁ h₂ in
⟨equiv.of_bijective f hf⟩ | theorem | function.embedding.antisymm | set_theory.cardinal | src/set_theory/cardinal/schroeder_bernstein.lean | [
"order.fixed_points",
"order.zorn"
] | [] | **The Schröder-Bernstein Theorem**: Given embeddings `α ↪ β` and `β ↪ α`, there exists an
equivalence `α ≃ β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sets | {s : set (∀ i, β i) |
∀ (x ∈ s) (y ∈ s) i, (x : ∀ i, β i) i = y i → x = y} | def | function.embedding.sets | set_theory.cardinal | src/set_theory/cardinal/schroeder_bernstein.lean | [
"order.fixed_points",
"order.zorn"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_injective [I : nonempty ι] : ∃ i, nonempty (∀ j, β i ↪ β j) | let ⟨s, hs, ms⟩ := show ∃ s ∈ sets, ∀ a ∈ sets, s ⊆ a → a = s, from
zorn_subset sets (λ c hc hcc, ⟨⋃₀ c,
λ x ⟨p, hpc, hxp⟩ y ⟨q, hqc, hyq⟩ i hi, (hcc.total hpc hqc).elim
(λ h, hc hqc x (h hxp) y hyq i hi) (λ h, hc hpc x hxp y (h hyq) i hi),
λ _, subset_sUnion_of_mem⟩) in
let ⟨i, e⟩ := show ∃ i, ∀ y, ∃ x ∈... | theorem | function.embedding.min_injective | set_theory.cardinal | src/set_theory/cardinal/schroeder_bernstein.lean | [
"order.fixed_points",
"order.zorn"
] | [
"not_exists",
"not_forall",
"zorn_subset"
] | The cardinals are well-ordered. We express it here by the fact that in any set of cardinals
there is an element that injects into the others. See `cardinal.linear_order` for (one of) the
lattice instances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
total (α : Type u) (β : Type v) : nonempty (α ↪ β) ∨ nonempty (β ↪ α) | match @min_injective bool (λ b, cond b (ulift α) (ulift.{(max u v) v} β)) ⟨tt⟩ with
| ⟨tt, ⟨h⟩⟩ := let ⟨f, hf⟩ := h ff in or.inl ⟨embedding.congr equiv.ulift equiv.ulift ⟨f, hf⟩⟩
| ⟨ff, ⟨h⟩⟩ := let ⟨f, hf⟩ := h tt in or.inr ⟨embedding.congr equiv.ulift equiv.ulift ⟨f, hf⟩⟩
end | theorem | function.embedding.total | set_theory.cardinal | src/set_theory/cardinal/schroeder_bernstein.lean | [
"order.fixed_points",
"order.zorn"
] | [
"equiv.ulift"
] | The cardinals are totally ordered. See `cardinal.linear_order` for (one of) the lattice
instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pgame.setoid : setoid pgame | ⟨(≈), equiv_refl, @pgame.equiv.symm, @pgame.equiv.trans⟩ | instance | pgame.setoid | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame",
"pgame.equiv.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
game | quotient pgame.setoid | abbreviation | game | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame.setoid"
] | The type of combinatorial games. In ZFC, a combinatorial game is constructed from
two sets of combinatorial games that have been constructed at an earlier
stage. To do this in type theory, we say that a combinatorial pre-game is built
inductively from two families of combinatorial games indexed over any type
in... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lf : game → game → Prop | quotient.lift₂ lf (λ x₁ y₁ x₂ y₂ hx hy, propext (lf_congr hx hy)) | def | game.lf | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"game"
] | The less or fuzzy relation on games.
If `0 ⧏ x` (less or fuzzy with), then Left can win `x` as the first player. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_le : ∀ {x y : game}, ¬ x ≤ y ↔ y ⧏ x | by { rintro ⟨x⟩ ⟨y⟩, exact pgame.not_le } | theorem | game.not_le | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"game",
"pgame.not_le"
] | On `game`, simp-normal inequalities should use as few negations as possible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_lf : ∀ {x y : game}, ¬ x ⧏ y ↔ y ≤ x | by { rintro ⟨x⟩ ⟨y⟩, exact not_lf } | theorem | game.not_lf | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"game"
] | On `game`, simp-normal inequalities should use as few negations as possible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.pgame.le_iff_game_le {x y : pgame} : x ≤ y ↔ ⟦x⟧ ≤ ⟦y⟧ | iff.rfl | theorem | pgame.le_iff_game_le | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pgame.lf_iff_game_lf {x y : pgame} : pgame.lf x y ↔ ⟦x⟧ ⧏ ⟦y⟧ | iff.rfl | theorem | pgame.lf_iff_game_lf | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame",
"pgame.lf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pgame.lt_iff_game_lt {x y : pgame} : x < y ↔ ⟦x⟧ < ⟦y⟧ | iff.rfl | theorem | pgame.lt_iff_game_lt | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pgame.equiv_iff_game_eq {x y : pgame} : x ≈ y ↔ ⟦x⟧ = ⟦y⟧ | (@quotient.eq _ _ x y).symm | theorem | pgame.equiv_iff_game_eq | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame",
"quotient.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fuzzy : game → game → Prop | quotient.lift₂ fuzzy (λ x₁ y₁ x₂ y₂ hx hy, propext (fuzzy_congr hx hy)) | def | game.fuzzy | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"game"
] | The fuzzy, confused, or incomparable relation on games.
If `x ‖ 0`, then the first player can always win `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.pgame.fuzzy_iff_game_fuzzy {x y : pgame} : pgame.fuzzy x y ↔ ⟦x⟧ ‖ ⟦y⟧ | iff.rfl | theorem | pgame.fuzzy_iff_game_fuzzy | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame",
"pgame.fuzzy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_add_le : covariant_class game game (+) (≤) | ⟨by { rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ h, exact @add_le_add_left _ _ _ _ b c h a }⟩ | instance | game.covariant_class_add_le | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"covariant_class",
"game"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_add_le : covariant_class game game (swap (+)) (≤) | ⟨by { rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ h, exact @add_le_add_right _ _ _ _ b c h a }⟩ | instance | game.covariant_class_swap_add_le | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"covariant_class",
"game"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_add_lt : covariant_class game game (+) (<) | ⟨by { rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ h, exact @add_lt_add_left _ _ _ _ b c h a }⟩ | instance | game.covariant_class_add_lt | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"covariant_class",
"game"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_add_lt : covariant_class game game (swap (+)) (<) | ⟨by { rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ h, exact @add_lt_add_right _ _ _ _ b c h a }⟩ | instance | game.covariant_class_swap_add_lt | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"covariant_class",
"game"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lf_add_right : ∀ {b c : game} (h : b ⧏ c) (a), b + a ⧏ c + a | by { rintro ⟨b⟩ ⟨c⟩ h ⟨a⟩, apply add_lf_add_right h } | theorem | game.add_lf_add_right | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"game"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lf_add_left : ∀ {b c : game} (h : b ⧏ c) (a), a + b ⧏ a + c | by { rintro ⟨b⟩ ⟨c⟩ h ⟨a⟩, apply add_lf_add_left h } | theorem | game.add_lf_add_left | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"game"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_add_comm_group : ordered_add_comm_group game | { add_le_add_left := @add_le_add_left _ _ _ game.covariant_class_add_le,
..game.add_comm_group_with_one,
..game.partial_order } | instance | game.ordered_add_comm_group | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"game",
"game.covariant_class_add_le",
"ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_of_small (s : set game.{u}) [small.{u} s] : bdd_above s | ⟨_, λ i hi, by simpa using pgame.le_iff_game_le.1
(upper_bound_mem_upper_bounds _ (set.mem_image_of_mem quotient.out hi))⟩ | lemma | game.bdd_above_of_small | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"bdd_above",
"quotient.out",
"set.mem_image_of_mem"
] | A small set `s` of games is bounded above. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_below_of_small (s : set game.{u}) [small.{u} s] : bdd_below s | ⟨_, λ i hi, by simpa using pgame.le_iff_game_le.1
(lower_bound_mem_lower_bounds _ (set.mem_image_of_mem quotient.out hi))⟩ | lemma | game.bdd_below_of_small | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"bdd_below",
"quotient.out",
"set.mem_image_of_mem"
] | A small set `s` of games is bounded below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_neg (a : pgame) : ⟦-a⟧ = -⟦a⟧ | rfl | lemma | pgame.quot_neg | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_add (a b : pgame) : ⟦a + b⟧ = ⟦a⟧ + ⟦b⟧ | rfl | lemma | pgame.quot_add | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_sub (a b : pgame) : ⟦a - b⟧ = ⟦a⟧ - ⟦b⟧ | rfl | lemma | pgame.quot_sub | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_eq_of_mk_quot_eq {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⟧ | by { simp_rw [quotient.eq] at hl hr, exact quot.sound (equiv_of_mk_equiv L R hl hr) } | theorem | pgame.quot_eq_of_mk_quot_eq | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame",
"quotient.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_moves_mul : ∀ (x y : pgame.{u}), (x * y).left_moves
= (x.left_moves × y.left_moves ⊕ x.right_moves × y.right_moves) | | ⟨_, _, _, _⟩ ⟨_, _, _, _⟩ := rfl | theorem | pgame.left_moves_mul | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_moves_mul : ∀ (x y : pgame.{u}), (x * y).right_moves
= (x.left_moves × y.right_moves ⊕ x.right_moves × y.left_moves) | | ⟨_, _, _, _⟩ ⟨_, _, _, _⟩ := rfl | theorem | pgame.right_moves_mul | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_left_moves_mul {x y : pgame} :
x.left_moves × y.left_moves ⊕ x.right_moves × y.right_moves ≃ (x * y).left_moves | equiv.cast (left_moves_mul x y).symm | def | pgame.to_left_moves_mul | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv.cast",
"pgame"
] | Turns two left or right moves for `x` and `y` into a left move for `x * y` and vice versa.
Even though these types are the same (not definitionally so), this is the preferred way to convert
between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_right_moves_mul {x y : pgame} :
x.left_moves × y.right_moves ⊕ x.right_moves × y.left_moves ≃ (x * y).right_moves | equiv.cast (right_moves_mul x y).symm | def | pgame.to_right_moves_mul | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv.cast",
"pgame"
] | Turns a left and a right move for `x` and `y` into a right move for `x * y` and vice versa.
Even though these types are the same (not definitionally so), this is the preferred way to convert
between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_mul_move_left_inl {xl xr yl yr} {xL xR yL yR} {i j} :
(mk xl xr xL xR * mk yl yr yL yR).move_left (sum.inl (i, j)) =
xL i * mk yl yr yL yR + mk xl xr xL xR * yL j - xL i * yL j | rfl | lemma | pgame.mk_mul_move_left_inl | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_move_left_inl {x y : pgame} {i j} :
(x * y).move_left (to_left_moves_mul (sum.inl (i, j))) =
x.move_left i * y + x * y.move_left j - x.move_left i * y.move_left j | by { cases x, cases y, refl } | lemma | pgame.mul_move_left_inl | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_move_left_inr {xl xr yl yr} {xL xR yL yR} {i j} :
(mk xl xr xL xR * mk yl yr yL yR).move_left (sum.inr (i, j)) =
xR i * mk yl yr yL yR + mk xl xr xL xR * yR j - xR i * yR j | rfl | lemma | pgame.mk_mul_move_left_inr | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_move_left_inr {x y : pgame} {i j} :
(x * y).move_left (to_left_moves_mul (sum.inr (i, j))) =
x.move_right i * y + x * y.move_right j - x.move_right i * y.move_right j | by { cases x, cases y, refl } | lemma | pgame.mul_move_left_inr | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_move_right_inl {xl xr yl yr} {xL xR yL yR} {i j} :
(mk xl xr xL xR * mk yl yr yL yR).move_right (sum.inl (i, j)) =
xL i * mk yl yr yL yR + mk xl xr xL xR * yR j - xL i * yR j | rfl | lemma | pgame.mk_mul_move_right_inl | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_move_right_inl {x y : pgame} {i j} :
(x * y).move_right (to_right_moves_mul (sum.inl (i, j))) =
x.move_left i * y + x * y.move_right j - x.move_left i * y.move_right j | by { cases x, cases y, refl } | lemma | pgame.mul_move_right_inl | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_move_right_inr {xl xr yl yr} {xL xR yL yR} {i j} :
(mk xl xr xL xR * mk yl yr yL yR).move_right (sum.inr (i, j)) =
xR i * mk yl yr yL yR + mk xl xr xL xR * yL j - xR i * yL j | rfl | lemma | pgame.mk_mul_move_right_inr | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_move_right_inr {x y : pgame} {i j} :
(x * y).move_right (to_right_moves_mul (sum.inr (i, j))) =
x.move_right i * y + x * y.move_left j - x.move_right i * y.move_left j | by { cases x, cases y, refl } | lemma | pgame.mul_move_right_inr | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mk_mul_move_left_inl {xl xr yl yr} {xL xR yL yR} {i j} :
(-(mk xl xr xL xR * mk yl yr yL yR)).move_left (sum.inl (i, j)) =
-(xL i * mk yl yr yL yR + mk xl xr xL xR * yR j - xL i * yR j) | rfl | lemma | pgame.neg_mk_mul_move_left_inl | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mk_mul_move_left_inr {xl xr yl yr} {xL xR yL yR} {i j} :
(-(mk xl xr xL xR * mk yl yr yL yR)).move_left (sum.inr (i, j)) =
-(xR i * mk yl yr yL yR + mk xl xr xL xR * yL j - xR i * yL j) | rfl | lemma | pgame.neg_mk_mul_move_left_inr | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mk_mul_move_right_inl {xl xr yl yr} {xL xR yL yR} {i j} :
(-(mk xl xr xL xR * mk yl yr yL yR)).move_right (sum.inl (i, j)) =
-(xL i * mk yl yr yL yR + mk xl xr xL xR * yL j - xL i * yL j) | rfl | lemma | pgame.neg_mk_mul_move_right_inl | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mk_mul_move_right_inr {xl xr yl yr} {xL xR yL yR} {i j} :
(-(mk xl xr xL xR * mk yl yr yL yR)).move_right (sum.inr (i, j)) =
-(xR i * mk yl yr yL yR + mk xl xr xL xR * yR j - xR i * yR j) | rfl | lemma | pgame.neg_mk_mul_move_right_inr | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_moves_mul_cases {x y : pgame} (k) {P : (x * y).left_moves → Prop}
(hl : ∀ ix iy, P $ to_left_moves_mul (sum.inl ⟨ix, iy⟩))
(hr : ∀ jx jy, P $ to_left_moves_mul (sum.inr ⟨jx, jy⟩)) : P k | begin
rw ←to_left_moves_mul.apply_symm_apply k,
rcases to_left_moves_mul.symm k with ⟨ix, iy⟩ | ⟨jx, jy⟩,
{ apply hl },
{ apply hr }
end | lemma | pgame.left_moves_mul_cases | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_moves_mul_cases {x y : pgame} (k) {P : (x * y).right_moves → Prop}
(hl : ∀ ix jy, P $ to_right_moves_mul (sum.inl ⟨ix, jy⟩))
(hr : ∀ jx iy, P $ to_right_moves_mul (sum.inr ⟨jx, iy⟩)) : P k | begin
rw ←to_right_moves_mul.apply_symm_apply k,
rcases to_right_moves_mul.symm k with ⟨ix, iy⟩ | ⟨jx, jy⟩,
{ apply hl },
{ apply hr }
end | lemma | pgame.right_moves_mul_cases | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comm_relabelling : Π (x y : pgame.{u}), x * y ≡r y * x | | ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ := begin
refine ⟨equiv.sum_congr (equiv.prod_comm _ _) (equiv.prod_comm _ _),
(equiv.sum_comm _ _).trans (equiv.sum_congr (equiv.prod_comm _ _) (equiv.prod_comm _ _)), _, _⟩;
rintro (⟨i, j⟩ | ⟨i, j⟩);
dsimp;
exact ((add_comm_relabelling _ _).trans $ (mul_comm_relabelling ... | def | pgame.mul_comm_relabelling | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv.prod_comm",
"equiv.sum_comm",
"equiv.sum_congr"
] | `x * y` and `y * x` have the same moves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mul_comm (x y : pgame.{u}) : ⟦x * y⟧ = ⟦y * x⟧ | quot.sound (mul_comm_relabelling x y).equiv | theorem | pgame.quot_mul_comm | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comm_equiv (x y : pgame) : x * y ≈ y * x | quotient.exact $ quot_mul_comm _ _ | theorem | pgame.mul_comm_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `x * y` is equivalent to `y * x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_empty_mul_zero_left_moves (x : pgame.{u}) : is_empty (x * 0).left_moves | by { cases x, apply sum.is_empty } | instance | pgame.is_empty_mul_zero_left_moves | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_empty_mul_zero_right_moves (x : pgame.{u}) : is_empty (x * 0).right_moves | by { cases x, apply sum.is_empty } | instance | pgame.is_empty_mul_zero_right_moves | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_empty_zero_mul_left_moves (x : pgame.{u}) : is_empty (0 * x).left_moves | by { cases x, apply sum.is_empty } | instance | pgame.is_empty_zero_mul_left_moves | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_empty_zero_mul_right_moves (x : pgame.{u}) : is_empty (0 * x).right_moves | by { cases x, apply sum.is_empty } | instance | pgame.is_empty_zero_mul_right_moves | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero_relabelling (x : pgame) : x * 0 ≡r 0 | relabelling.is_empty _ | def | pgame.mul_zero_relabelling | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `x * 0` has exactly the same moves as `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_zero_equiv (x : pgame) : x * 0 ≈ 0 | (mul_zero_relabelling x).equiv | theorem | pgame.mul_zero_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv",
"pgame"
] | `x * 0` is equivalent to `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mul_zero (x : pgame) : ⟦x * 0⟧ = ⟦0⟧ | @quotient.sound _ _ (x * 0) _ x.mul_zero_equiv | theorem | pgame.quot_mul_zero | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mul_relabelling (x : pgame) : 0 * x ≡r 0 | relabelling.is_empty _ | def | pgame.zero_mul_relabelling | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `0 * x` has exactly the same moves as `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mul_equiv (x : pgame) : 0 * x ≈ 0 | (zero_mul_relabelling x).equiv | theorem | pgame.zero_mul_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv",
"pgame"
] | `0 * x` is equivalent to `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_zero_mul (x : pgame) : ⟦0 * x⟧ = ⟦0⟧ | @quotient.sound _ _ (0 * x) _ x.zero_mul_equiv | theorem | pgame.quot_zero_mul | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mul_relabelling : Π (x y : pgame.{u}), -x * y ≡r -(x * y) | | ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ := begin
refine ⟨equiv.sum_comm _ _, equiv.sum_comm _ _, _, _⟩;
rintro (⟨i, j⟩ | ⟨i, j⟩);
dsimp;
apply ((neg_add_relabelling _ _).trans _).symm;
apply ((neg_add_relabelling _ _).trans (relabelling.add_congr _ _)).sub_congr;
exact (neg_mul_relabelling _ _).symm
end
using_w... | def | pgame.neg_mul_relabelling | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv.sum_comm"
] | `-x * y` and `-(x * y)` have the same moves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_neg_mul (x y : pgame) : ⟦-x * y⟧ = -⟦x * y⟧ | quot.sound (neg_mul_relabelling x y).equiv | theorem | pgame.quot_neg_mul | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_neg_relabelling (x y : pgame) : x * -y ≡r -(x * y) | (mul_comm_relabelling x _).trans $
(neg_mul_relabelling _ x).trans (mul_comm_relabelling y x).neg_congr | def | pgame.mul_neg_relabelling | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `x * -y` and `-(x * y)` have the same moves. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mul_neg (x y : pgame) : ⟦x * -y⟧ = -⟦x * y⟧ | quot.sound (mul_neg_relabelling x y).equiv | theorem | pgame.quot_mul_neg | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_left_distrib : Π (x y z : pgame), ⟦x * (y + z)⟧ = ⟦x * y⟧ + ⟦x * z⟧ | | (mk xl xr xL xR) (mk yl yr yL yR) (mk zl zr zL zR) :=
begin
let x := mk xl xr xL xR,
let y := mk yl yr yL yR,
let z := mk zl zr zL zR,
refine quot_eq_of_mk_quot_eq _ _ _ _,
{ fsplit,
{ rintro (⟨_, _ | _⟩ | ⟨_, _ | _⟩);
solve_by_elim [sum.inl, sum.inr, prod.mk] { max_depth := 5 } },
{ rintro (⟨... | theorem | pgame.quot_left_distrib | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_distrib_equiv (x y z : pgame) : x * (y + z) ≈ x * y + x * z | quotient.exact $ quot_left_distrib _ _ _ | theorem | pgame.left_distrib_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 + x * z.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_left_distrib_sub (x y z : pgame) : ⟦x * (y - z)⟧ = ⟦x * y⟧ - ⟦x * z⟧ | by { change ⟦x * (y + -z)⟧ = ⟦x * y⟧ + -⟦x * z⟧, rw [quot_left_distrib, quot_mul_neg] } | theorem | pgame.quot_left_distrib_sub | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_right_distrib (x y z : pgame) : ⟦(x + y) * z⟧ = ⟦x * z⟧ + ⟦y * z⟧ | by simp only [quot_mul_comm, quot_left_distrib] | theorem | pgame.quot_right_distrib | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_distrib_equiv (x y z : pgame) : (x + y) * z ≈ x * z + y * z | quotient.exact $ quot_right_distrib _ _ _ | theorem | pgame.right_distrib_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `(x + y) * z` is equivalent to `x * z + y * z.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_right_distrib_sub (x y z : pgame) : ⟦(y - z) * x⟧ = ⟦y * x⟧ - ⟦z * x⟧ | by { change ⟦(y + -z) * x⟧ = ⟦y * x⟧ + -⟦z * x⟧, rw [quot_right_distrib, quot_neg_mul] } | theorem | pgame.quot_right_distrib_sub | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_relabelling : Π (x : pgame.{u}), x * 1 ≡r x | | ⟨xl, xr, xL, xR⟩ := begin
unfold has_one.one,
refine ⟨(equiv.sum_empty _ _).trans (equiv.prod_punit _),
(equiv.empty_sum _ _).trans (equiv.prod_punit _), _, _⟩;
try { rintro (⟨i, ⟨ ⟩⟩ | ⟨i, ⟨ ⟩⟩) }; try { intro i };
dsimp;
apply (relabelling.sub_congr (relabelling.refl _) (mul_zero_relabelling _)).trans... | def | pgame.mul_one_relabelling | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"equiv.empty_sum",
"equiv.prod_punit",
"equiv.sum_empty"
] | `x * 1` has the same moves as `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mul_one (x : pgame) : ⟦x * 1⟧ = ⟦x⟧ | quot.sound $ mul_one_relabelling x | theorem | pgame.quot_mul_one | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_equiv (x : pgame) : x * 1 ≈ x | quotient.exact $ quot_mul_one x | theorem | pgame.mul_one_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `x * 1` is equivalent to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_mul_relabelling (x : pgame) : 1 * x ≡r x | (mul_comm_relabelling 1 x).trans $ mul_one_relabelling x | def | pgame.one_mul_relabelling | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `1 * x` has the same moves as `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_one_mul (x : pgame) : ⟦1 * x⟧ = ⟦x⟧ | quot.sound $ one_mul_relabelling x | theorem | pgame.quot_one_mul | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mul_equiv (x : pgame) : 1 * x ≈ x | quotient.exact $ quot_one_mul x | theorem | pgame.one_mul_equiv | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | `1 * x` is equivalent to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mul_assoc : Π (x y z : pgame), ⟦x * y * z⟧ = ⟦x * (y * z)⟧ | | (mk xl xr xL xR) (mk yl yr yL yR) (mk zl zr zL zR) :=
begin
let x := mk xl xr xL xR,
let y := mk yl yr yL yR,
let z := mk zl zr zL zR,
refine quot_eq_of_mk_quot_eq _ _ _ _,
{ fsplit,
{ rintro (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩);
solve_by_elim [sum.inl, sum.inr, prod.mk] { max_depth := 7 ... | theorem | pgame.quot_mul_assoc | set_theory.game | src/set_theory/game/basic.lean | [
"set_theory.game.pgame",
"tactic.abel"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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