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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|---|---|---|---|---|---|---|---|---|---|---|
double_pow_half_succ_eq_pow_half (n : ℕ) : 2 • pow_half n.succ = pow_half n | by { rw two_nsmul, exact quotient.sound (pgame.add_pow_half_succ_self_eq_pow_half n) } | lemma | surreal.double_pow_half_succ_eq_pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"pgame.add_pow_half_succ_self_eq_pow_half"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_pow_two_pow_half (n : ℕ) : 2 ^ n • pow_half n = 1 | begin
induction n with n hn,
{ simp only [nsmul_one, pow_half_zero, nat.cast_one, pow_zero] },
{ rw [← hn, ← double_pow_half_succ_eq_pow_half n, smul_smul (2^n) 2 (pow_half n.succ),
mul_comm, pow_succ] }
end | lemma | surreal.nsmul_pow_two_pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"mul_comm",
"nat.cast_one",
"nsmul_one",
"pow_succ",
"pow_zero",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_pow_two_pow_half' (n k : ℕ) : 2 ^ n • pow_half (n + k) = pow_half k | begin
induction k with k hk,
{ simp only [add_zero, surreal.nsmul_pow_two_pow_half, nat.nat_zero_eq_zero, eq_self_iff_true,
surreal.pow_half_zero] },
{ rw [← double_pow_half_succ_eq_pow_half (n + k), ← double_pow_half_succ_eq_pow_half k,
smul_algebra_smul_comm] at hk,
rwa ← zsmul_eq_zsm... | lemma | surreal.nsmul_pow_two_pow_half' | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"smul_algebra_smul_comm",
"surreal.nsmul_pow_two_pow_half",
"surreal.pow_half_zero",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_pow_two_pow_half (m : ℤ) (n k : ℕ) :
(m * 2 ^ n) • pow_half (n + k) = m • pow_half k | begin
rw mul_zsmul,
congr,
norm_cast,
exact nsmul_pow_two_pow_half' n k
end | lemma | surreal.zsmul_pow_two_pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dyadic_aux {m₁ m₂ : ℤ} {y₁ y₂ : ℕ} (h₂ : m₁ * (2 ^ y₁) = m₂ * (2 ^ y₂)) :
m₁ • pow_half y₂ = m₂ • pow_half y₁ | begin
revert m₁ m₂,
wlog h : y₁ ≤ y₂,
{ intros m₁ m₂ aux, exact (this (le_of_not_le h) aux.symm).symm },
intros m₁ m₂ h₂,
obtain ⟨c, rfl⟩ := le_iff_exists_add.mp h,
rw [add_comm, pow_add, ← mul_assoc, mul_eq_mul_right_iff] at h₂,
cases h₂,
{ rw [h₂, add_comm, zsmul_pow_two_pow_half m₂ c y₁] },
{ have ... | lemma | surreal.dyadic_aux | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"aux",
"mul_assoc",
"mul_eq_mul_right_iff",
"nat.one_le_pow",
"nat.succ_pos'",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dyadic_map : localization.away (2 : ℤ) →+ surreal | { to_fun :=
λ x, localization.lift_on x (λ x y, x • pow_half (submonoid.log y)) $
begin
intros m₁ m₂ n₁ n₂ h₁,
obtain ⟨⟨n₃, y₃, hn₃⟩, h₂⟩ := localization.r_iff_exists.mp h₁,
simp only [subtype.coe_mk, mul_eq_mul_left_iff] at h₂,
cases h₂,
{ simp only,
obtain ⟨a₁, ha₁⟩ := n₁.prop,
obt... | def | surreal.dyadic_map | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"add_smul",
"int.pow_right_injective",
"localization.add_mk",
"localization.away",
"localization.induction_on₂",
"localization.lift_on",
"localization.lift_on_mk",
"localization.lift_on_zero",
"mul_comm",
"mul_eq_mul_left_iff",
"nat.one_le_pow",
"nat.succ_pos'",
"one_lt_two",
"submonoid.lo... | The additive monoid morphism `dyadic_map` sends ⟦⟨m, 2^n⟩⟧ to m • half ^ n. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dyadic_map_apply (m : ℤ) (p : submonoid.powers (2 : ℤ)) :
dyadic_map (is_localization.mk' (localization (submonoid.powers 2)) m p) =
m • pow_half (submonoid.log p) | by { rw ← localization.mk_eq_mk', refl } | lemma | surreal.dyadic_map_apply | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"is_localization.mk'",
"localization",
"localization.mk_eq_mk'",
"submonoid.log",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dyadic_map_apply_pow (m : ℤ) (n : ℕ) :
dyadic_map (is_localization.mk' (localization (submonoid.powers 2)) m (submonoid.pow 2 n)) =
m • pow_half n | by rw [dyadic_map_apply, @submonoid.log_pow_int_eq_self 2 one_lt_two] | lemma | surreal.dyadic_map_apply_pow | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"is_localization.mk'",
"localization",
"one_lt_two",
"submonoid.log_pow_int_eq_self",
"submonoid.pow",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dyadic : set surreal | set.range dyadic_map | def | surreal.dyadic | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"set.range",
"surreal"
] | We define dyadic surreals as the range of the map `dyadic_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arity (α : Type u) : ℕ → Type u | | 0 := α
| (n+1) := α → arity n | def | arity | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | The type of `n`-ary functions `α → α → ... → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arity_zero (α : Type u) : arity α 0 = α | rfl | theorem | arity_zero | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arity_succ (α : Type u) (n : ℕ) : arity α n.succ = (α → arity α n) | rfl | theorem | arity_succ | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const {α : Type u} (a : α) : ∀ n, arity α n | | 0 := a
| (n+1) := λ _, const n | def | arity.const | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | Constant `n`-ary function with value `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_zero {α : Type u} (a : α) : const a 0 = a | rfl | theorem | arity.const_zero | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_succ {α : Type u} (a : α) (n : ℕ) : const a n.succ = λ _, const a n | rfl | theorem | arity.const_succ | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_succ_apply {α : Type u} (a : α) (n : ℕ) (x : α) : const a n.succ x = const a n | rfl | theorem | arity.const_succ_apply | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arity.inhabited {α n} [inhabited α] : inhabited (arity α n) | ⟨const default _⟩ | instance | arity.arity.inhabited | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pSet : Type (u+1)
| mk (α : Type u) (A : α → pSet) : pSet | inductive | pSet | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | The type of pre-sets in universe `u`. A pre-set
is a family of pre-sets indexed by a type in `Type u`.
The ZFC universe is defined as a quotient of this
to ensure extensionality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type : pSet → Type u | | ⟨α, A⟩ := α | def | pSet.type | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The underlying type of a pre-set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
func : Π (x : pSet), x.type → pSet | | ⟨α, A⟩ := A | def | pSet.func | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The underlying pre-set family of a pre-set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_type (α A) : type ⟨α, A⟩ = α | rfl | theorem | pSet.mk_type | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_func (α A) : func ⟨α, A⟩ = A | rfl | theorem | pSet.mk_func | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eta : Π (x : pSet), mk x.type x.func = x | | ⟨α, A⟩ := rfl | theorem | pSet.eta | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv (x y : pSet) : Prop | pSet.rec (λ α z m ⟨β, B⟩, (∀ a, ∃ b, m a (B b)) ∧ (∀ b, ∃ a, m a (B b))) x y | def | pSet.equiv | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"pSet"
] | Two pre-sets are extensionally equivalent if every element of the first family is extensionally
equivalent to some element of the second family and vice-versa. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_iff : Π {x y : pSet}, equiv x y ↔
(∀ i, ∃ j, equiv (x.func i) (y.func j)) ∧ (∀ j, ∃ i, equiv (x.func i) (y.func j)) | | ⟨α, A⟩ ⟨β, B⟩ := iff.rfl | theorem | pSet.equiv_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.exists_left {x y : pSet} (h : equiv x y) : ∀ i, ∃ j, equiv (x.func i) (y.func j) | (equiv_iff.1 h).1 | theorem | pSet.equiv.exists_left | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.exists_right {x y : pSet} (h : equiv x y) : ∀ j, ∃ i, equiv (x.func i) (y.func j) | (equiv_iff.1 h).2 | theorem | pSet.equiv.exists_right | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.refl (x) : equiv x x | pSet.rec_on x $ λ α A IH, ⟨λ a, ⟨a, IH a⟩, λ a, ⟨a, IH a⟩⟩ | theorem | pSet.equiv.refl | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.rfl : ∀ {x}, equiv x x | equiv.refl | theorem | pSet.equiv.rfl | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.euc {x} : Π {y z}, equiv x y → equiv z y → equiv x z | pSet.rec_on x $ λ α A IH y, pSet.cases_on y $ λ β B ⟨γ, Γ⟩ ⟨αβ, βα⟩ ⟨γβ, βγ⟩,
⟨λ a, let ⟨b, ab⟩ := αβ a, ⟨c, bc⟩ := βγ b in ⟨c, IH a ab bc⟩,
λ c, let ⟨b, cb⟩ := γβ c, ⟨a, ba⟩ := βα b in ⟨a, IH a ba cb⟩⟩ | theorem | pSet.equiv.euc | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.symm {x y} : equiv x y → equiv y x | (equiv.refl y).euc | theorem | pSet.equiv.symm | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"equiv.refl",
"equiv.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.comm {x y} : equiv x y ↔ equiv y x | ⟨equiv.symm, equiv.symm⟩ | theorem | pSet.equiv.comm | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.trans {x y z} (h1 : equiv x y) (h2 : equiv y z) : equiv x z | h1.euc h2.symm | theorem | pSet.equiv.trans | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"equiv.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_is_empty (x y : pSet) [is_empty x.type] [is_empty y.type] : equiv x y | equiv_iff.2 $ by simp | theorem | pSet.equiv_of_is_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"is_empty",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
setoid : setoid pSet | ⟨pSet.equiv, equiv.refl, λ x y, equiv.symm, λ x y z, equiv.trans⟩ | instance | pSet.setoid | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv.refl",
"equiv.symm",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset (x y : pSet) : Prop | ∀ a, ∃ b, equiv (x.func a) (y.func b) | def | pSet.subset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"pSet"
] | A pre-set is a subset of another pre-set if every element of the first family is extensionally
equivalent to some element of the second family. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv.ext : Π (x y : pSet), equiv x y ↔ (x ⊆ y ∧ y ⊆ x) | | ⟨α, A⟩ ⟨β, B⟩ :=
⟨λ ⟨αβ, βα⟩, ⟨αβ, λ b, let ⟨a, h⟩ := βα b in ⟨a, equiv.symm h⟩⟩,
λ ⟨αβ, βα⟩, ⟨αβ, λ b, let ⟨a, h⟩ := βα b in ⟨a, equiv.symm h⟩⟩⟩ | theorem | pSet.equiv.ext | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"equiv.ext",
"equiv.symm",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset.congr_left : Π {x y z : pSet}, equiv x y → (x ⊆ z ↔ y ⊆ z) | | ⟨α, A⟩ ⟨β, B⟩ ⟨γ, Γ⟩ ⟨αβ, βα⟩ :=
⟨λ αγ b, let ⟨a, ba⟩ := βα b, ⟨c, ac⟩ := αγ a in ⟨c, (equiv.symm ba).trans ac⟩,
λ βγ a, let ⟨b, ab⟩ := αβ a, ⟨c, bc⟩ := βγ b in ⟨c, equiv.trans ab bc⟩⟩ | theorem | pSet.subset.congr_left | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"equiv.symm",
"equiv.trans",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset.congr_right : Π {x y z : pSet}, equiv x y → (z ⊆ x ↔ z ⊆ y) | | ⟨α, A⟩ ⟨β, B⟩ ⟨γ, Γ⟩ ⟨αβ, βα⟩ :=
⟨λ γα c, let ⟨a, ca⟩ := γα c, ⟨b, ab⟩ := αβ a in ⟨b, ca.trans ab⟩,
λ γβ c, let ⟨b, cb⟩ := γβ c, ⟨a, ab⟩ := βα b in ⟨a, cb.trans (equiv.symm ab)⟩⟩ | theorem | pSet.subset.congr_right | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"equiv.symm",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem (x y : pSet.{u}) : Prop | ∃ b, equiv x (y.func b) | def | pSet.mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | `x ∈ y` as pre-sets if `x` is extensionally equivalent to a member of the family `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem.mk {α : Type u} (A : α → pSet) (a : α) : A a ∈ mk α A | ⟨a, equiv.refl (A a)⟩ | theorem | pSet.mem.mk | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv.refl",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
func_mem (x : pSet) (i : x.type) : x.func i ∈ x | by { cases x, apply mem.mk } | theorem | pSet.func_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem.ext : Π {x y : pSet.{u}}, (∀ w : pSet.{u}, w ∈ x ↔ w ∈ y) → equiv x y | | ⟨α, A⟩ ⟨β, B⟩ h := ⟨λ a, (h (A a)).1 (mem.mk A a),
λ b, let ⟨a, ha⟩ := (h (B b)).2 (mem.mk B b) in ⟨a, ha.symm⟩⟩ | theorem | pSet.mem.ext | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem.congr_right : Π {x y : pSet.{u}}, equiv x y → (∀ {w : pSet.{u}}, w ∈ x ↔ w ∈ y) | | ⟨α, A⟩ ⟨β, B⟩ ⟨αβ, βα⟩ w :=
⟨λ ⟨a, ha⟩, let ⟨b, hb⟩ := αβ a in ⟨b, ha.trans hb⟩,
λ ⟨b, hb⟩, let ⟨a, ha⟩ := βα b in ⟨a, hb.euc ha⟩⟩ | theorem | pSet.mem.congr_right | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_iff_mem {x y : pSet.{u}} : equiv x y ↔ (∀ {w : pSet.{u}}, w ∈ x ↔ w ∈ y) | ⟨mem.congr_right, match x, y with
| ⟨α, A⟩, ⟨β, B⟩, h := ⟨λ a, h.1 (mem.mk A a), λ b,
let ⟨a, h⟩ := h.2 (mem.mk B b) in ⟨a, h.symm⟩⟩
end⟩ | theorem | pSet.equiv_iff_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem.congr_left : Π {x y : pSet.{u}}, equiv x y → (∀ {w : pSet.{u}}, x ∈ w ↔ y ∈ w) | | x y h ⟨α, A⟩ := ⟨λ ⟨a, ha⟩, ⟨a, h.symm.trans ha⟩, λ ⟨a, ha⟩, ⟨a, h.trans ha⟩⟩ | theorem | pSet.mem.congr_left | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_wf_aux : Π {x y : pSet.{u}}, equiv x y → acc (∈) y | | ⟨α, A⟩ ⟨β, B⟩ H := ⟨_, begin
rintros ⟨γ, C⟩ ⟨b, hc⟩,
cases H.exists_right b with a ha,
have H := ha.trans hc.symm,
rw mk_func at H,
exact mem_wf_aux H
end⟩ | theorem | pSet.mem_wf_aux | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_wf : @well_founded pSet (∈) | ⟨λ x, mem_wf_aux $ equiv.refl x⟩ | theorem | pSet.mem_wf | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv.refl",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_asymm {x y : pSet} : x ∈ y → y ∉ x | asymm | theorem | pSet.mem_asymm | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_irrefl (x : pSet) : x ∉ x | irrefl x | theorem | pSet.mem_irrefl | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set (u : pSet.{u}) : set pSet.{u} | {x | x ∈ u} | def | pSet.to_set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Convert a pre-set to a `set` of pre-sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_to_set (a u : pSet.{u}) : a ∈ u.to_set ↔ a ∈ u | iff.rfl | theorem | pSet.mem_to_set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty (u : pSet) : Prop | u.to_set.nonempty | def | pSet.nonempty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | A nonempty set is one that contains some element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_def (u : pSet) : u.nonempty ↔ ∃ x, x ∈ u | iff.rfl | theorem | pSet.nonempty_def | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_of_mem {x u : pSet} (h : x ∈ u) : u.nonempty | ⟨x, h⟩ | theorem | pSet.nonempty_of_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_to_set_iff {u : pSet} : u.to_set.nonempty ↔ u.nonempty | iff.rfl | theorem | pSet.nonempty_to_set_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_type_iff_nonempty {x : pSet} : nonempty x.type ↔ pSet.nonempty x | ⟨λ ⟨i⟩, ⟨_, func_mem _ i⟩, λ ⟨i, j, h⟩, ⟨j⟩⟩ | theorem | pSet.nonempty_type_iff_nonempty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet",
"pSet.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_of_nonempty_type (x : pSet) [h : nonempty x.type] : pSet.nonempty x | nonempty_type_iff_nonempty.1 h | theorem | pSet.nonempty_of_nonempty_type | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet",
"pSet.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.eq {x y : pSet} : equiv x y ↔ to_set x = to_set y | equiv_iff_mem.trans set.ext_iff.symm | theorem | pSet.equiv.eq | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"pSet"
] | Two pre-sets are equivalent iff they have the same members. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty : pSet | ⟨_, pempty.elim⟩ | def | pSet.empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The empty pre-set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_mem_empty (x : pSet.{u}) : x ∉ (∅ : pSet.{u}) | is_empty.exists_iff.1 | theorem | pSet.not_mem_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_empty : to_set ∅ = ∅ | by simp [to_set] | theorem | pSet.to_set_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_subset (x : pSet.{u}) : (∅ : pSet) ⊆ x | λ x, x.elim | theorem | pSet.empty_subset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_nonempty_empty : ¬ pSet.nonempty ∅ | by simp [pSet.nonempty] | theorem | pSet.not_nonempty_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_empty (x : pSet) [is_empty x.type] : equiv x ∅ | pSet.equiv_of_is_empty x _ | theorem | pSet.equiv_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"is_empty",
"pSet",
"pSet.equiv_of_is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
insert (x y : pSet) : pSet | ⟨option y.type, λ o, option.rec x y.func o⟩ | def | pSet.insert | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | Insert an element into a pre-set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_nat : ℕ → pSet | | 0 := ∅
| (n+1) := insert (of_nat n) (of_nat n) | def | pSet.of_nat | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The n-th von Neumann ordinal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
omega : pSet | ⟨ulift ℕ, λ n, of_nat n.down⟩ | def | pSet.omega | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The von Neumann ordinal ω | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sep (p : pSet → Prop) (x : pSet) : pSet | ⟨{a // p (x.func a)}, λ y, x.func y.1⟩ | def | pSet.sep | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The pre-set separation operation `{x ∈ a | p x}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
powerset (x : pSet) : pSet | ⟨set x.type, λ p, ⟨{a // p a}, λ y, x.func y.1⟩⟩ | def | pSet.powerset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The pre-set powerset operator | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_powerset : Π {x y : pSet}, y ∈ powerset x ↔ y ⊆ x | | ⟨α, A⟩ ⟨β, B⟩ := ⟨λ ⟨p, e⟩, (subset.congr_left e).2 $ λ ⟨a, pa⟩, ⟨a, equiv.refl (A a)⟩,
λ βα, ⟨{a | ∃ b, equiv (B b) (A a)}, λ b, let ⟨a, ba⟩ := βα b in ⟨⟨a, b, ba⟩, ba⟩,
λ ⟨a, b, ba⟩, ⟨b, ba⟩⟩⟩ | theorem | pSet.mem_powerset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv",
"equiv.refl",
"pSet"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion (a : pSet) : pSet | ⟨Σ x, (a.func x).type, λ ⟨x, y⟩, (a.func x).func y⟩ | def | pSet.sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The pre-set union operator | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_sUnion : Π {x y : pSet.{u}}, y ∈ ⋃₀ x ↔ ∃ z ∈ x, y ∈ z | | ⟨α, A⟩ y :=
⟨λ ⟨⟨a, c⟩, (e : equiv y ((A a).func c))⟩,
have func (A a) c ∈ mk (A a).type (A a).func, from mem.mk (A a).func c,
⟨_, mem.mk _ _, (mem.congr_left e).2 (by rwa eta at this)⟩,
λ ⟨⟨β, B⟩, ⟨a, (e : equiv (mk β B) (A a))⟩, ⟨b, yb⟩⟩,
by { rw ←(eta (A a)) at e, exact
let ⟨βt, tβ⟩ := e, ⟨c, b... | theorem | pSet.mem_sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set_sUnion (x : pSet.{u}) : (⋃₀ x).to_set = ⋃₀ (to_set '' x.to_set) | by { ext, simp } | theorem | pSet.to_set_sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image (f : pSet.{u} → pSet.{u}) (x : pSet.{u}) : pSet | ⟨x.type, f ∘ x.func⟩ | def | pSet.image | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | The image of a function from pre-sets to pre-sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_image {f : pSet.{u} → pSet.{u}} (H : ∀ {x y}, equiv x y → equiv (f x) (f y)) :
Π {x y : pSet.{u}}, y ∈ image f x ↔ ∃ z ∈ x, equiv y (f z) | | ⟨α, A⟩ y := ⟨λ ⟨a, ya⟩, ⟨A a, mem.mk A a, ya⟩, λ ⟨z, ⟨a, za⟩, yz⟩, ⟨a, yz.trans (H za)⟩⟩ | theorem | pSet.mem_image | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : pSet.{u} → pSet.{max u v} | | ⟨α, A⟩ := ⟨ulift α, λ ⟨x⟩, lift (A x)⟩ | def | pSet.lift | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"lift"
] | Universe lift operation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embed : pSet.{max (u+1) v} | ⟨ulift.{v u+1} pSet, λ ⟨x⟩, pSet.lift.{u (max (u+1) v)} x⟩ | def | pSet.embed | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"pSet"
] | Embedding of one universe in another | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mem_embed : Π (x : pSet.{u}), pSet.lift.{u (max (u+1) v)} x ∈ embed.{u v} | λ x, ⟨⟨x⟩, equiv.rfl⟩ | theorem | pSet.lift_mem_embed | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arity.equiv : Π {n}, arity pSet.{u} n → arity pSet.{u} n → Prop | | 0 a b := equiv a b
| (n+1) a b := ∀ x y, equiv x y → arity.equiv (a x) (b y) | def | pSet.arity.equiv | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity",
"equiv"
] | Function equivalence is defined so that `f ~ g` iff `∀ x y, x ~ y → f x ~ g y`. This extends to
equivalence of `n`-ary functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arity.equiv_const {a : pSet.{u}} : ∀ n, arity.equiv (arity.const a n) (arity.const a n) | | 0 := equiv.rfl
| (n+1) := λ x y h, arity.equiv_const _ | lemma | pSet.arity.equiv_const | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resp (n) | {x : arity pSet.{u} n // arity.equiv x x} | def | pSet.resp | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | `resp n` is the collection of n-ary functions on `pSet` that respect
equivalence, i.e. when the inputs are equivalent the output is as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
resp.inhabited {n} : inhabited (resp n) | ⟨⟨arity.const default _, arity.equiv_const _⟩⟩ | instance | pSet.resp.inhabited | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resp.f {n} (f : resp (n+1)) (x : pSet) : resp n | ⟨f.1 x, f.2 _ _ $ equiv.refl x⟩ | def | pSet.resp.f | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv.refl",
"pSet"
] | The `n`-ary image of a `(n + 1)`-ary function respecting equivalence as a function respecting
equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
resp.equiv {n} (a b : resp n) : Prop | arity.equiv a.1 b.1 | def | pSet.resp.equiv | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Function equivalence for functions respecting equivalence. See `pSet.arity.equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
resp.equiv.refl {n} (a : resp n) : resp.equiv a a | a.2 | theorem | pSet.resp.equiv.refl | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resp.equiv.euc : Π {n} {a b c : resp n},
resp.equiv a b → resp.equiv c b → resp.equiv a c | | 0 a b c hab hcb := equiv.euc hab hcb
| (n+1) a b c hab hcb := λ x y h,
@resp.equiv.euc n (a.f x) (b.f y) (c.f y) (hab _ _ h) (hcb _ _ $ equiv.refl y) | theorem | pSet.resp.equiv.euc | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resp.equiv.symm {n} {a b : resp n} : resp.equiv a b → resp.equiv b a | (resp.equiv.refl b).euc | theorem | pSet.resp.equiv.symm | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resp.equiv.trans {n} {x y z : resp n}
(h1 : resp.equiv x y) (h2 : resp.equiv y z) : resp.equiv x z | h1.euc h2.symm | theorem | pSet.resp.equiv.trans | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resp.setoid {n} : setoid (resp n) | ⟨resp.equiv, resp.equiv.refl, λ x y, resp.equiv.symm, λ x y z, resp.equiv.trans⟩ | instance | pSet.resp.setoid | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Set : Type (u+1) | quotient pSet.setoid.{u} | def | Set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | The ZFC universe of sets consists of the type of pre-sets,
quotiented by extensional equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_aux : Π {n}, {f : resp n → arity Set.{u} n // ∀ (a b : resp n), resp.equiv a b → f a = f b} | | 0 := ⟨λ a, ⟦a.1⟧, λ a b h, quotient.sound h⟩
| (n+1) := let F : resp (n + 1) → arity Set (n + 1) := λ a, @quotient.lift _ _ pSet.setoid
(λ x, eval_aux.1 (a.f x)) (λ b c h, eval_aux.2 _ _ (a.2 _ _ h)) in
⟨F, λ b c h, funext $ @quotient.ind _ _ (λ q, F b q = F c q) $ λ z,
eval_aux.2 (resp.f b z) (resp.f c z... | def | pSet.resp.eval_aux | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"arity",
"pSet.equiv.refl",
"pSet.setoid"
] | Helper function for `pSet.eval`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval (n) : resp n → arity Set.{u} n | eval_aux.1 | def | pSet.resp.eval | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | An equivalence-respecting function yields an n-ary ZFC set function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_val {n f x} : (@eval (n+1) f : Set → arity Set n) ⟦x⟧ = eval n (resp.f f x) | rfl | theorem | pSet.resp.eval_val | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"arity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
definable (n) : arity Set.{u} n → Type (u+1)
| mk (f) : definable (resp.eval n f) | class inductive | pSet.definable | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | A set function is "definable" if it is the image of some n-ary pre-set
function. This isn't exactly definability, but is useful as a sufficient
condition for functions that have a computable image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
definable.eq_mk {n} (f) : Π {s : arity Set.{u} n} (H : resp.eval _ f = s), definable n s | | ._ rfl := ⟨f⟩ | def | pSet.definable.eq_mk | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | The evaluation of a function respecting equivalence is definable, by that same function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
definable.resp {n} : Π (s : arity Set.{u} n) [definable n s], resp n | | ._ ⟨f⟩ := f | def | pSet.definable.resp | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | Turns a definable function into a function that respects equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
definable.eq {n} :
Π (s : arity Set.{u} n) [H : definable n s], (@definable.resp n s H).eval _ = s | | ._ ⟨f⟩ := rfl | theorem | pSet.definable.eq | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
all_definable : Π {n} (F : arity Set.{u} n), definable n F | | 0 F := let p := @quotient.exists_rep pSet _ F in
definable.eq_mk ⟨some p, equiv.rfl⟩ (some_spec p)
| (n+1) (F : arity Set.{u} (n + 1)) := begin
have I := λ x, (all_definable (F x)),
refine definable.eq_mk ⟨λ x : pSet, (@definable.resp _ _ (I ⟦x⟧)).1, _⟩ _,
{ dsimp [arity.equiv],
in... | def | classical.all_definable | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"arity",
"pSet",
"subtype.coe_eta",
"subtype.val_eq_coe"
] | All functions are classically definable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk : pSet → Set | quotient.mk | def | Set.mk | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"pSet"
] | Turns a pre-set into a ZFC set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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