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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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infi_ker_truncate : (⨅ i : ℕ, (@witt_vector.truncate p _ i R _).ker) = ⊥ | begin
rw [submodule.eq_bot_iff],
intros x hx,
ext,
simp only [witt_vector.mem_ker_truncate, ideal.mem_infi, witt_vector.zero_coeff] at hx ⊢,
exact hx _ _ (nat.lt_succ_self _)
end | lemma | truncated_witt_vector.infi_ker_truncate | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"ideal.mem_infi",
"submodule.eq_bot_iff",
"witt_vector.mem_ker_truncate",
"witt_vector.zero_coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_fun (s : S) : 𝕎 R | witt_vector.mk p $ λ k, truncated_witt_vector.coeff (fin.last k) (f (k+1) s) | def | witt_vector.lift_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"fin.last",
"truncated_witt_vector.coeff"
] | Given a family `fₖ : S → truncated_witt_vector p k R` and `s : S`, we produce a Witt vector by
defining the `k`th entry to be the final entry of `fₖ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_lift_fun (s : S) :
witt_vector.truncate n (lift_fun f s) = f n s | begin
ext i,
simp only [lift_fun, truncated_witt_vector.coeff_mk, witt_vector.truncate_mk],
rw [← f_compat (i+1) n i.is_lt, ring_hom.comp_apply, truncated_witt_vector.coeff_truncate],
-- this is a bit unfortunate
congr' with _,
simp only [fin.coe_last, fin.coe_cast_le],
end | lemma | witt_vector.truncate_lift_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"fin.coe_cast_le",
"fin.coe_last",
"ring_hom.comp_apply",
"truncated_witt_vector.coeff_mk",
"truncated_witt_vector.coeff_truncate",
"witt_vector.truncate_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : S →+* 𝕎 R | by refine_struct { to_fun := lift_fun f };
{ intros,
rw [← sub_eq_zero, ← ideal.mem_bot, ← infi_ker_truncate, ideal.mem_infi],
simp [ring_hom.mem_ker, f_compat] } | def | witt_vector.lift | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"ideal.mem_bot",
"ideal.mem_infi",
"lift",
"ring_hom.mem_ker"
] | Given compatible ring homs from `S` into `truncated_witt_vector n` for each `n`, we can lift these
to a ring hom `S → 𝕎 R`.
`lift` defines the universal property of `𝕎 R` as the inverse limit of `truncated_witt_vector n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_lift (s : S) :
witt_vector.truncate n (lift _ f_compat s) = f n s | truncate_lift_fun _ f_compat s | lemma | witt_vector.truncate_lift | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_comp_lift :
(witt_vector.truncate n).comp (lift _ f_compat) = f n | by { ext1, rw [ring_hom.comp_apply, truncate_lift] } | lemma | witt_vector.truncate_comp_lift | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"lift",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (g : S →+* 𝕎 R) (g_compat : ∀ k, (witt_vector.truncate k).comp g = f k) :
lift _ f_compat = g | begin
ext1 x,
rw [← sub_eq_zero, ← ideal.mem_bot, ← infi_ker_truncate, ideal.mem_infi],
intro i,
simp only [ring_hom.mem_ker, g_compat, ←ring_hom.comp_apply,
truncate_comp_lift, ring_hom.map_sub, sub_self],
end | lemma | witt_vector.lift_unique | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"ideal.mem_bot",
"ideal.mem_infi",
"lift",
"lift_unique",
"ring_hom.map_sub",
"ring_hom.mem_ker"
] | The uniqueness part of the universal property of `𝕎 R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_equiv : {f : Π k, S →+* truncated_witt_vector p k R // ∀ k₁ k₂ (hk : k₁ ≤ k₂),
(truncated_witt_vector.truncate hk).comp (f k₂) = f k₁} ≃ (S →+* 𝕎 R) | { to_fun := λ f, lift f.1 f.2,
inv_fun := λ g, ⟨λ k, (truncate k).comp g,
by { intros _ _ h, simp only [←ring_hom.comp_assoc, truncate_comp_witt_vector_truncate] }⟩,
left_inv := by { rintro ⟨f, hf⟩, simp only [truncate_comp_lift] },
right_inv := λ g, lift_unique _ _ $ λ _, rfl } | def | witt_vector.lift_equiv | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"inv_fun",
"lift",
"lift_unique",
"truncated_witt_vector",
"truncated_witt_vector.truncate"
] | The universal property of `𝕎 R` as projective limit of truncated Witt vector rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_ext (g₁ g₂ : S →+* 𝕎 R) (h : ∀ k, (truncate k).comp g₁ = (truncate k).comp g₂) :
g₁ = g₂ | lift_equiv.symm.injective $ subtype.ext $ funext h | lemma | witt_vector.hom_ext | ring_theory.witt_vector | src/ring_theory/witt_vector/truncated.lean | [
"ring_theory.witt_vector.init_tail"
] | [
"hom_ext",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_fun (x : 𝕎 R) : 𝕎 R | mk p $ λ n, if n = 0 then 0 else x.coeff (n - 1) | def | witt_vector.verschiebung_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | `verschiebung_fun x` shifts the coefficients of `x` up by one,
by inserting 0 as the 0th coefficient.
`x.coeff i` then becomes `(verchiebung_fun x).coeff (i + 1)`.
`verschiebung_fun` is the underlying function of the additive monoid hom `witt_vector.verschiebung`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
verschiebung_fun_coeff (x : 𝕎 R) (n : ℕ) :
(verschiebung_fun x).coeff n = if n = 0 then 0 else x.coeff (n - 1) | by rw [verschiebung_fun, coeff_mk] | lemma | witt_vector.verschiebung_fun_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_fun_coeff_zero (x : 𝕎 R) :
(verschiebung_fun x).coeff 0 = 0 | by rw [verschiebung_fun_coeff, if_pos rfl] | lemma | witt_vector.verschiebung_fun_coeff_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_fun_coeff_succ (x : 𝕎 R) (n : ℕ) :
(verschiebung_fun x).coeff n.succ = x.coeff n | rfl | lemma | witt_vector.verschiebung_fun_coeff_succ | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_component_zero_verschiebung_fun (x : 𝕎 R) :
ghost_component 0 (verschiebung_fun x) = 0 | by rw [ghost_component_apply, aeval_witt_polynomial, finset.range_one, finset.sum_singleton,
verschiebung_fun_coeff_zero, pow_zero, pow_zero, pow_one, one_mul] | lemma | witt_vector.ghost_component_zero_verschiebung_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"aeval_witt_polynomial",
"finset.range_one",
"one_mul",
"pow_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_component_verschiebung_fun (x : 𝕎 R) (n : ℕ) :
ghost_component (n + 1) (verschiebung_fun x) = p * ghost_component n x | begin
simp only [ghost_component_apply, aeval_witt_polynomial],
rw [finset.sum_range_succ', verschiebung_fun_coeff, if_pos rfl, zero_pow (pow_pos hp.1.pos _),
mul_zero, add_zero, finset.mul_sum, finset.sum_congr rfl],
rintro i -,
simp only [pow_succ, mul_assoc, verschiebung_fun_coeff, if_neg (nat.succ_ne_... | lemma | witt_vector.ghost_component_verschiebung_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"aeval_witt_polynomial",
"finset.mul_sum",
"mul_assoc",
"mul_zero",
"pow_pos",
"pow_succ",
"tsub_zero",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_poly (n : ℕ) : mv_polynomial ℕ ℤ | if n = 0 then 0 else X (n-1) | def | witt_vector.verschiebung_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"mv_polynomial"
] | The 0th Verschiebung polynomial is 0. For `n > 0`, the `n`th Verschiebung polynomial is the
variable `X (n-1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
verschiebung_poly_zero :
verschiebung_poly 0 = 0 | rfl | lemma | witt_vector.verschiebung_poly_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_verschiebung_poly' (x : 𝕎 R) (n : ℕ) :
aeval x.coeff (verschiebung_poly n) = (verschiebung_fun x).coeff n | begin
cases n,
{ simp only [verschiebung_poly, verschiebung_fun_coeff_zero, if_pos rfl, alg_hom.map_zero] },
{ rw [verschiebung_poly, verschiebung_fun_coeff_succ, if_neg (n.succ_ne_zero),
aeval_X, nat.succ_eq_add_one, add_tsub_cancel_right] }
end | lemma | witt_vector.aeval_verschiebung_poly' | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"add_tsub_cancel_right",
"alg_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_fun_is_poly : is_poly p (λ R _Rcr, @verschiebung_fun p R _Rcr) | begin
use verschiebung_poly,
simp only [aeval_verschiebung_poly', eq_self_iff_true, forall_3_true_iff]
end | lemma | witt_vector.verschiebung_fun_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"forall_3_true_iff",
"is_poly"
] | `witt_vector.verschiebung` has polynomial structure given by `witt_vector.verschiebung_poly`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
verschiebung : 𝕎 R →+ 𝕎 R | { to_fun := verschiebung_fun,
map_zero' :=
by ext ⟨⟩; rw [verschiebung_fun_coeff]; simp only [if_true, eq_self_iff_true, zero_coeff, if_t_t],
map_add' := by { ghost_calc _ _, rintro ⟨⟩; ghost_simp } } | def | witt_vector.verschiebung | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | `verschiebung x` shifts the coefficients of `x` up by one, by inserting 0 as the 0th coefficient.
`x.coeff i` then becomes `(verchiebung x).coeff (i + 1)`.
This is a additive monoid hom with underlying function `verschiebung_fun`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
verschiebung_is_poly : is_poly p (λ R _Rcr, @verschiebung p R hp _Rcr) | verschiebung_fun_is_poly p | lemma | witt_vector.verschiebung_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"is_poly"
] | `witt_vector.verschiebung` is a polynomial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_verschiebung (f : R →+* S) (x : 𝕎 R) :
map f (verschiebung x) = verschiebung (map f x) | by { ext ⟨-, -⟩, exact f.map_zero, refl } | lemma | witt_vector.map_verschiebung | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | verschiebung is a natural transformation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_component_zero_verschiebung (x : 𝕎 R) :
ghost_component 0 (verschiebung x) = 0 | ghost_component_zero_verschiebung_fun _ | lemma | witt_vector.ghost_component_zero_verschiebung | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_component_verschiebung (x : 𝕎 R) (n : ℕ) :
ghost_component (n + 1) (verschiebung x) = p * ghost_component n x | ghost_component_verschiebung_fun _ _ | lemma | witt_vector.ghost_component_verschiebung | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_coeff_zero (x : 𝕎 R) :
(verschiebung x).coeff 0 = 0 | rfl | lemma | witt_vector.verschiebung_coeff_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_coeff_add_one (x : 𝕎 R) (n : ℕ) :
(verschiebung x).coeff (n + 1) = x.coeff n | rfl | lemma | witt_vector.verschiebung_coeff_add_one | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_coeff_succ (x : 𝕎 R) (n : ℕ) :
(verschiebung x).coeff n.succ = x.coeff n | rfl | lemma | witt_vector.verschiebung_coeff_succ | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_verschiebung_poly (x : 𝕎 R) (n : ℕ) :
aeval x.coeff (verschiebung_poly n) = (verschiebung x).coeff n | aeval_verschiebung_poly' x n | lemma | witt_vector.aeval_verschiebung_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind₁_verschiebung_poly_witt_polynomial (n : ℕ) :
bind₁ verschiebung_poly (witt_polynomial p ℤ n) =
if n = 0 then 0 else p * witt_polynomial p ℤ (n-1) | begin
apply mv_polynomial.funext,
intro x,
split_ifs with hn,
{ simp only [hn, verschiebung_poly_zero, witt_polynomial_zero, bind₁_X_right] },
{ obtain ⟨n, rfl⟩ := nat.exists_eq_succ_of_ne_zero hn,
rw [nat.succ_eq_add_one, add_tsub_cancel_right, ring_hom.map_mul,
map_nat_cast, hom_bind₁],
calc... | lemma | witt_vector.bind₁_verschiebung_poly_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/verschiebung.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"add_tsub_cancel_right",
"map_nat_cast",
"mv_polynomial.funext",
"ring_hom.ext_int",
"ring_hom.map_mul",
"witt_polynomial",
"witt_polynomial_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_polynomial (n : ℕ) : mv_polynomial ℕ R | ∑ i in range (n+1), monomial (single i (p ^ (n - i))) (p ^ i : R) | def | witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"mv_polynomial"
] | `witt_polynomial p R n` is the `n`-th Witt polynomial
with respect to a prime `p` with coefficients in a commutative ring `R`.
It is defined as:
`∑_{i ≤ n} p^i X_i^{p^{n-i}} ∈ R[X_0, X_1, X_2, …]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_polynomial_eq_sum_C_mul_X_pow (n : ℕ) :
witt_polynomial p R n = ∑ i in range (n+1), C (p ^ i : R) * X i ^ (p ^ (n - i)) | begin
apply sum_congr rfl,
rintro i -,
rw [monomial_eq, finsupp.prod_single_index],
rw pow_zero,
end | lemma | witt_polynomial_eq_sum_C_mul_X_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"finsupp.prod_single_index",
"pow_zero",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_witt_polynomial (f : R →+* S) (n : ℕ) :
map f (W n) = W n | begin
rw [witt_polynomial, ring_hom.map_sum, witt_polynomial, sum_congr rfl],
intros i hi,
rw [map_monomial, ring_hom.map_pow, map_nat_cast],
end | lemma | map_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"map_nat_cast",
"ring_hom.map_pow",
"ring_hom.map_sum",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_polynomial [hp : fact p.prime] (n : ℕ) :
constant_coeff (witt_polynomial p R n) = 0 | begin
simp only [witt_polynomial, ring_hom.map_sum, constant_coeff_monomial],
rw [sum_eq_zero],
rintro i hi,
rw [if_neg],
rw [finsupp.single_eq_zero],
exact ne_of_gt (pow_pos hp.1.pos _)
end | lemma | constant_coeff_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"fact",
"finsupp.single_eq_zero",
"pow_pos",
"ring_hom.map_sum",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_polynomial_zero : witt_polynomial p R 0 = X 0 | by simp only [witt_polynomial, X, sum_singleton, range_one, pow_zero] | lemma | witt_polynomial_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"pow_zero",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_polynomial_one : witt_polynomial p R 1 = C ↑p * X 1 + (X 0) ^ p | by simp only [witt_polynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, range_one,
sum_singleton, one_mul, pow_one, C_1, pow_zero] | lemma | witt_polynomial_one | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"one_mul",
"pow_one",
"pow_zero",
"witt_polynomial",
"witt_polynomial_eq_sum_C_mul_X_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_witt_polynomial {A : Type*} [comm_ring A] [algebra R A] (f : ℕ → A) (n : ℕ) :
aeval f (W_ R n) = ∑ i in range (n+1), p^i * (f i) ^ (p ^ (n-i)) | by simp [witt_polynomial, alg_hom.map_sum, aeval_monomial, finsupp.prod_single_index] | lemma | aeval_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"alg_hom.map_sum",
"algebra",
"comm_ring",
"finsupp.prod_single_index",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_polynomial_zmod_self (n : ℕ) :
W_ (zmod (p ^ (n + 1))) (n + 1) = expand p (W_ (zmod (p^(n + 1))) n) | begin
simp only [witt_polynomial_eq_sum_C_mul_X_pow],
rw [sum_range_succ, ← nat.cast_pow, char_p.cast_eq_zero (zmod (p^(n+1))) (p^(n+1)), C_0, zero_mul,
add_zero, alg_hom.map_sum, sum_congr rfl],
intros k hk,
rw [alg_hom.map_mul, alg_hom.map_pow, expand_X, alg_hom_C, ← pow_mul, ← pow_succ],
congr,
rw ... | lemma | witt_polynomial_zmod_self | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"add_tsub_assoc_of_le",
"alg_hom.map_mul",
"alg_hom.map_pow",
"alg_hom.map_sum",
"char_p.cast_eq_zero",
"nat.cast_pow",
"pow_mul",
"pow_succ",
"witt_polynomial_eq_sum_C_mul_X_pow",
"zero_mul",
"zmod"
] | Over the ring `zmod (p^(n+1))`, we produce the `n+1`st Witt polynomial
by expanding the `n`th Witt polynomial by `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_polynomial_vars [char_zero R] (n : ℕ) :
(witt_polynomial p R n).vars = range (n + 1) | begin
have : ∀ i, (monomial (finsupp.single i (p ^ (n - i))) (p ^ i : R)).vars = {i},
{ intro i,
refine vars_monomial_single i (pow_ne_zero _ hp.1) _,
rw [← nat.cast_pow, nat.cast_ne_zero],
exact pow_ne_zero i hp.1 },
rw [witt_polynomial, vars_sum_of_disjoint],
{ simp only [this, bUnion_singleton_eq... | lemma | witt_polynomial_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"char_zero",
"finsupp.single",
"nat.cast_ne_zero",
"nat.cast_pow",
"pow_ne_zero",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_polynomial_vars_subset (n : ℕ) :
(witt_polynomial p R n).vars ⊆ range (n + 1) | begin
rw [← map_witt_polynomial p (int.cast_ring_hom R), ← witt_polynomial_vars p ℤ],
apply vars_map,
end | lemma | witt_polynomial_vars_subset | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"int.cast_ring_hom",
"map_witt_polynomial",
"witt_polynomial",
"witt_polynomial_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_in_terms_of_W [invertible (p : R)] :
ℕ → mv_polynomial ℕ R | | n := (X n - (∑ i : fin n,
have _ := i.2, (C (p^(i : ℕ) : R) * (X_in_terms_of_W i)^(p^(n-i))))) * C (⅟p ^ n : R) | def | X_in_terms_of_W | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"invertible",
"mv_polynomial"
] | The `X_in_terms_of_W p R n` is the polynomial on the basis of Witt polynomials
that corresponds to the ordinary `X n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_in_terms_of_W_eq [invertible (p : R)] {n : ℕ} :
X_in_terms_of_W p R n =
(X n - (∑ i in range n, C (p^i : R) * X_in_terms_of_W p R i ^ p ^ (n - i))) * C (⅟p ^ n : R) | by { rw [X_in_terms_of_W, ← fin.sum_univ_eq_sum_range] } | lemma | X_in_terms_of_W_eq | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"X_in_terms_of_W",
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_X_in_terms_of_W [hp : fact p.prime] [invertible (p : R)] (n : ℕ) :
constant_coeff (X_in_terms_of_W p R n) = 0 | begin
apply nat.strong_induction_on n; clear n,
intros n IH,
rw [X_in_terms_of_W_eq, mul_comm, ring_hom.map_mul, ring_hom.map_sub, ring_hom.map_sum,
constant_coeff_C, sum_eq_zero],
{ simp only [constant_coeff_X, sub_zero, mul_zero] },
{ intros m H,
rw mem_range at H,
simp only [ring_hom.map_mul, r... | lemma | constant_coeff_X_in_terms_of_W | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"X_in_terms_of_W",
"X_in_terms_of_W_eq",
"fact",
"invertible",
"mul_comm",
"mul_zero",
"pow_pos",
"ring_hom.map_mul",
"ring_hom.map_pow",
"ring_hom.map_sub",
"ring_hom.map_sum",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_in_terms_of_W_zero [invertible (p : R)] :
X_in_terms_of_W p R 0 = X 0 | by rw [X_in_terms_of_W_eq, range_zero, sum_empty, pow_zero, C_1, mul_one, sub_zero] | lemma | X_in_terms_of_W_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"X_in_terms_of_W",
"X_in_terms_of_W_eq",
"invertible",
"mul_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_in_terms_of_W_vars_aux (n : ℕ) :
n ∈ (X_in_terms_of_W p ℚ n).vars ∧
(X_in_terms_of_W p ℚ n).vars ⊆ range (n + 1) | begin
apply nat.strong_induction_on n, clear n,
intros n ih,
rw [X_in_terms_of_W_eq, mul_comm, vars_C_mul, vars_sub_of_disjoint, vars_X, range_succ,
insert_eq],
swap 3, { apply nonzero_of_invertible },
work_on_goal 1
{ simp only [true_and, true_or, eq_self_iff_true,
mem_union, mem_singleton],
... | lemma | X_in_terms_of_W_vars_aux | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"X_in_terms_of_W",
"X_in_terms_of_W_eq",
"ih",
"mul_comm",
"nonzero_of_invertible",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_in_terms_of_W_vars_subset (n : ℕ) :
(X_in_terms_of_W p ℚ n).vars ⊆ range (n + 1) | (X_in_terms_of_W_vars_aux p n).2 | lemma | X_in_terms_of_W_vars_subset | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"X_in_terms_of_W",
"X_in_terms_of_W_vars_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_in_terms_of_W_aux [invertible (p : R)] (n : ℕ) :
X_in_terms_of_W p R n * C (p^n : R) =
X n - ∑ i in range n, C (p^i : R) * (X_in_terms_of_W p R i)^p^(n-i) | by rw [X_in_terms_of_W_eq, mul_assoc, ← C_mul, ← mul_pow, inv_of_mul_self, one_pow, C_1, mul_one] | lemma | X_in_terms_of_W_aux | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"X_in_terms_of_W",
"X_in_terms_of_W_eq",
"inv_of_mul_self",
"invertible",
"mul_assoc",
"mul_one",
"mul_pow",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind₁_X_in_terms_of_W_witt_polynomial [invertible (p : R)] (k : ℕ) :
bind₁ (X_in_terms_of_W p R) (W_ R k) = X k | begin
rw [witt_polynomial_eq_sum_C_mul_X_pow, alg_hom.map_sum],
simp only [alg_hom.map_pow, C_pow, alg_hom.map_mul, alg_hom_C],
rw [sum_range_succ_comm, tsub_self, pow_zero, pow_one, bind₁_X_right,
mul_comm, ← C_pow, X_in_terms_of_W_aux],
simp only [C_pow, bind₁_X_right, sub_add_cancel],
end | lemma | bind₁_X_in_terms_of_W_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"X_in_terms_of_W",
"X_in_terms_of_W_aux",
"alg_hom.map_mul",
"alg_hom.map_pow",
"alg_hom.map_sum",
"invertible",
"mul_comm",
"pow_one",
"pow_zero",
"tsub_self",
"witt_polynomial_eq_sum_C_mul_X_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind₁_witt_polynomial_X_in_terms_of_W [invertible (p : R)] (n : ℕ) :
bind₁ (W_ R) (X_in_terms_of_W p R n) = X n | begin
apply nat.strong_induction_on n,
clear n, intros n H,
rw [X_in_terms_of_W_eq, alg_hom.map_mul, alg_hom.map_sub, bind₁_X_right, alg_hom_C,
alg_hom.map_sum],
have : W_ R n - ∑ i in range n, C (p ^ i : R) * (X i) ^ p ^ (n - i) = C (p ^ n : R) * X n,
by simp only [witt_polynomial_eq_sum_C_mul_X_pow, t... | lemma | bind₁_witt_polynomial_X_in_terms_of_W | ring_theory.witt_vector | src/ring_theory/witt_vector/witt_polynomial.lean | [
"algebra.char_p.invertible",
"data.fintype.big_operators",
"data.mv_polynomial.variables",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.expand",
"data.zmod.basic"
] | [
"X_in_terms_of_W",
"X_in_terms_of_W_eq",
"alg_hom.map_mul",
"alg_hom.map_pow",
"alg_hom.map_sub",
"alg_hom.map_sum",
"invertible",
"mul_inv_of_self",
"mul_pow",
"mul_right_comm",
"one_mul",
"one_pow",
"pow_one",
"pow_zero",
"tsub_self",
"witt_polynomial_eq_sum_C_mul_X_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{u} lists' (α : Type u) : bool → Type u
| atom : α → lists' ff
| nil : lists' tt
| cons' {b} : lists' b → lists' tt → lists' tt | inductive | lists' | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [] | Prelists, helper type to define `lists`. `lists' α ff` are the "atoms", a copy of `α`.
`lists' α tt` are the "proper" ZFA prelists, inductively defined from the empty ZFA prelist and from
appending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything to an
atom while having only one appen... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lists (α : Type*) | Σ b, lists' α b | def | lists | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | Hereditarily finite list, aka ZFA list. A ZFA list is either an "atom" (`b = ff`), corresponding
to an element of `α`, or a "proper" ZFA list, inductively defined from the empty ZFA list and from
appending a ZFA list to a proper ZFA list. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cons : lists α → lists' α tt → lists' α tt | | ⟨b, a⟩ l := cons' a l | def | lists'.cons | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'"
] | Appending a ZFA list to a proper ZFA prelist. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_list : ∀ {b}, lists' α b → list (lists α) | | _ (atom a) := []
| _ nil := []
| _ (cons' a l) := ⟨_, a⟩ :: l.to_list | def | lists'.to_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'"
] | Converts a ZFA prelist to a `list` of ZFA lists. Atoms are sent to `[]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_list_cons (a : lists α) (l) :
to_list (cons a l) = a :: l.to_list | by cases a; simp [cons] | theorem | lists'.to_list_cons | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"to_list_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list : list (lists α) → lists' α tt | | [] := nil
| (a :: l) := cons a (of_list l) | def | lists'.of_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'"
] | Converts a `list` of ZFA lists to a proper ZFA prelist. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_of_list (l : list (lists α)) : to_list (of_list l) = l | by induction l; simp * | theorem | lists'.to_of_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_to_list : ∀ (l : lists' α tt), of_list (to_list l) = l | suffices ∀ b (h : tt = b) (l : lists' α b),
let l' : lists' α tt := by rw h; exact l in
of_list (to_list l') = l', from this _ rfl,
λ b h l, begin
induction l, {cases h}, {exact rfl},
case lists'.cons' : b a l IH₁ IH₂
{ intro, change l' with cons' a l,
simpa [cons] using IH₂ rfl }
end | theorem | lists'.of_to_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_def {b a} {l : lists' α b} :
a ∈ l ↔ ∃ a' ∈ l.to_list, a ~ a' | iff.rfl | theorem | lists'.mem_def | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l | by simp [mem_def, or_and_distrib_right, exists_or_distrib] | theorem | lists'.mem_cons | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"exists_or_distrib",
"mem_cons",
"or_and_distrib_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cons_subset {a} {l₁ l₂ : lists' α tt} :
lists'.cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂ | begin
refine ⟨λ h, _, λ ⟨⟨a', m, e⟩, s⟩, subset.cons e m s⟩,
generalize_hyp h' : lists'.cons a l₁ = l₁' at h,
cases h with l a' a'' l l' e m s, {cases a, cases h'},
cases a, cases a', cases h', exact ⟨⟨_, m, e⟩, s⟩
end | theorem | lists'.cons_subset | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'",
"lists'.cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_subset {l₁ l₂ : list (lists α)} (h : l₁ ⊆ l₂) :
lists'.of_list l₁ ⊆ lists'.of_list l₂ | begin
induction l₁, {exact subset.nil},
refine subset.cons (lists.equiv.refl _) _ (l₁_ih (list.subset_of_cons_subset h)),
simp at h, simp [h]
end | theorem | lists'.of_list_subset | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'.of_list"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset.refl {l : lists' α tt} : l ⊆ l | by rw ← lists'.of_to_list l; exact
of_list_subset (list.subset.refl _) | theorem | lists'.subset.refl | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'",
"lists'.of_to_list"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_nil {l : lists' α tt} :
l ⊆ lists'.nil → l = lists'.nil | begin
rw ← of_to_list l,
induction to_list l; intro h, {refl},
rcases cons_subset.1 h with ⟨⟨_, ⟨⟩, _⟩, _⟩
end | theorem | lists'.subset_nil | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_subset' {a} {l₁ l₂ : lists' α tt}
(s : l₁ ⊆ l₂) (h : a ∈ l₁.to_list) : a ∈ l₂ | begin
induction s with _ a a' l l' e m s IH, {cases h},
simp at h, rcases h with rfl|h,
exacts [⟨_, m, e⟩, IH h]
end | theorem | lists'.mem_of_subset' | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_def {l₁ l₂ : lists' α tt} :
l₁ ⊆ l₂ ↔ ∀ a ∈ l₁.to_list, a ∈ l₂ | ⟨λ H a, mem_of_subset' H, λ H, begin
rw ← of_to_list l₁,
revert H, induction to_list l₁; intro,
{ exact subset.nil },
{ simp at H, exact cons_subset.2 ⟨H.1, ih H.2⟩ }
end⟩ | theorem | lists'.subset_def | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"ih",
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
atom (a : α) : lists α | ⟨_, lists'.atom a⟩ | def | lists.atom | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | Sends `a : α` to the corresponding atom in `lists α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of' (l : lists' α tt) : lists α | ⟨_, l⟩ | def | lists.of' | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'"
] | Converts a proper ZFA prelist to a ZFA list. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_list : lists α → list (lists α) | | ⟨b, l⟩ := l.to_list | def | lists.to_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | Converts a ZFA list to a `list` of ZFA lists. Atoms are sent to `[]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_list (l : lists α) : Prop | l.1 | def | lists.is_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | Predicate stating that a ZFA list is proper. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_list (l : list (lists α)) : lists α | of' (lists'.of_list l) | def | lists.of_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'.of_list"
] | Converts a `list` of ZFA lists to a ZFA list. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_list_to_list (l : list (lists α)) : is_list (of_list l) | eq.refl _ | theorem | lists.is_list_to_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_of_list (l : list (lists α)) : to_list (of_list l) = l | by simp [of_list, of'] | theorem | lists.to_of_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_to_list : ∀ {l : lists α}, is_list l → of_list (to_list l) = l | | ⟨tt, l⟩ _ := by simp [of_list, of'] | theorem | lists.of_to_list | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_mut (C : lists α → Sort*) (D : lists' α tt → Sort*)
(C0 : ∀ a, C (atom a)) (C1 : ∀ l, D l → C (of' l))
(D0 : D lists'.nil) (D1 : ∀ a l, C a → D l → D (lists'.cons a l)) :
pprod (∀ l, C l) (∀ l, D l) | begin
suffices : ∀ {b} (l : lists' α b),
pprod (C ⟨_, l⟩) (match b, l with
| tt, l := D l
| ff, l := punit
end),
{ exact ⟨λ ⟨b, l⟩, (this _).1, λ l, (this l).2⟩ },
intros, induction l with a b a l IH₁ IH₂,
{ exact ⟨C0 _, ⟨⟩⟩ },
{ exact ⟨C1 _ D0, D0⟩ },
{ suffices, {exact ⟨C1 _ this, this⟩},
... | def | lists.induction_mut | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'",
"lists'.cons"
] | A recursion principle for pairs of ZFA lists and proper ZFA prelists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem (a : lists α) : lists α → Prop | | ⟨ff, l⟩ := false
| ⟨tt, l⟩ := a ∈ l | def | lists.mem | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | Membership of ZFA list. A ZFA list belongs to a proper ZFA list if it belongs to the latter as a
proper ZFA prelist. An atom has no members. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_list_of_mem {a : lists α} : ∀ {l : lists α}, a ∈ l → is_list l | | ⟨_, lists'.nil⟩ _ := rfl
| ⟨_, lists'.cons' _ _⟩ _ := rfl | theorem | lists.is_list_of_mem | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.antisymm_iff {l₁ l₂ : lists' α tt} :
of' l₁ ~ of' l₂ ↔ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁ | begin
refine ⟨λ h, _, λ ⟨h₁, h₂⟩, equiv.antisymm h₁ h₂⟩,
cases h with _ _ _ h₁ h₂,
{ simp [lists'.subset.refl] }, { exact ⟨h₁, h₂⟩ }
end | theorem | lists.equiv.antisymm_iff | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'",
"lists'.subset.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_atom {a} {l : lists α} : atom a ~ l ↔ atom a = l | ⟨λ h, by cases h; refl, λ h, h ▸ equiv.refl _⟩ | theorem | lists.equiv_atom | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"equiv.refl",
"lists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.symm {l₁ l₂ : lists α} (h : l₁ ~ l₂) : l₂ ~ l₁ | by cases h with _ _ _ h₁ h₂; [refl, exact equiv.antisymm h₂ h₁] | theorem | lists.equiv.symm | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"equiv.symm",
"lists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.trans : ∀ {l₁ l₂ l₃ : lists α}, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ | begin
let trans := λ (l₁ : lists α), ∀ ⦃l₂ l₃⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃,
suffices : pprod (∀ l₁, trans l₁)
(∀ (l : lists' α tt) (l' ∈ l.to_list), trans l'), {exact this.1},
apply induction_mut,
{ intros a l₂ l₃ h₁ h₂,
rwa ← equiv_atom.1 h₁ at h₂ },
{ intros l₁ IH l₂ l₃ h₁ h₂,
cases h₁ with _ _... | theorem | lists.equiv.trans | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"equiv.trans",
"lists",
"lists'",
"lists'.mem_of_subset'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.decidable_meas :
(psum (Σ' (l₁ : lists α), lists α) $
psum (Σ' (l₁ : lists' α tt), lists' α tt)
Σ' (a : lists α), lists' α tt) → ℕ | | (psum.inl ⟨l₁, l₂⟩) := sizeof l₁ + sizeof l₂
| (psum.inr $ psum.inl ⟨l₁, l₂⟩) := sizeof l₁ + sizeof l₂
| (psum.inr $ psum.inr ⟨l₁, l₂⟩) := sizeof l₁ + sizeof l₂ | def | lists.equiv.decidable_meas | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sizeof_pos {b} (l : lists' α b) : 0 < sizeof l | by cases l; unfold_sizeof; trivial_nat_lt | theorem | lists.sizeof_pos | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_sizeof_cons' {b} (a : lists' α b) (l) :
sizeof (⟨b, a⟩ : lists α) < sizeof (lists'.cons' a l) | by {unfold_sizeof, apply sizeof_pos} | theorem | lists.lt_sizeof_cons' | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists",
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_equiv_left {l : lists' α tt} :
∀ {a a'}, a ~ a' → (a ∈ l ↔ a' ∈ l) | suffices ∀ {a a'}, a ~ a' → a ∈ l → a' ∈ l,
from λ a a' e, ⟨this e, this e.symm⟩,
λ a₁ a₂ e₁ ⟨a₃, m₃, e₂⟩, ⟨_, m₃, e₁.symm.trans e₂⟩ | theorem | lists.lists'.mem_equiv_left | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_subset {a} {l₁ l₂ : lists' α tt}
(s : l₁ ⊆ l₂) : a ∈ l₁ → a ∈ l₂ | ⟨a', m, e⟩ | (mem_equiv_left e).2 (mem_of_subset' s m) | theorem | lists.lists'.mem_of_subset | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset.trans {l₁ l₂ l₃ : lists' α tt}
(h₁ : l₁ ⊆ l₂) (h₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ | subset_def.2 $ λ a₁ m₁, mem_of_subset h₂ $ mem_of_subset' h₁ m₁ | theorem | lists.lists'.subset.trans | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [
"lists'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finsets (α : Type*) | quotient (@lists.setoid α) | def | lists.finsets | set_theory | src/set_theory/lists.lean | [
"data.list.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cardinal.is_equivalent : setoid (Type u) | { r := λ α β, nonempty (α ≃ β),
iseqv := ⟨λ α,
⟨equiv.refl α⟩,
λ α β ⟨e⟩, ⟨e.symm⟩,
λ α β γ ⟨e₁⟩ ⟨e₂⟩, ⟨e₁.trans e₂⟩⟩ } | instance | cardinal.is_equivalent | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | The equivalence relation on types given by equivalence (bijective correspondence) of types.
Quotienting by this equivalence relation gives the cardinal numbers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cardinal : Type (u + 1) | quotient cardinal.is_equivalent | def | cardinal | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.is_equivalent"
] | `cardinal.{u}` is the type of cardinal numbers in `Type u`,
defined as the quotient of `Type u` by existence of an equivalence
(a bijection with explicit inverse). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk : Type u → cardinal | quotient.mk | def | cardinal.mk | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | The cardinal number of a type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
can_lift_cardinal_Type : can_lift cardinal.{u} (Type u) mk (λ _, true) | ⟨λ c _, quot.induction_on c $ λ α, ⟨α, rfl⟩⟩ | instance | cardinal.can_lift_cardinal_Type | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"can_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on {p : cardinal → Prop} (c : cardinal) (h : ∀ α, p (#α)) : p c | quotient.induction_on c h | lemma | cardinal.induction_on | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on₂ {p : cardinal → cardinal → Prop} (c₁ : cardinal) (c₂ : cardinal)
(h : ∀ α β, p (#α) (#β)) : p c₁ c₂ | quotient.induction_on₂ c₁ c₂ h | lemma | cardinal.induction_on₂ | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on₃ {p : cardinal → cardinal → cardinal → Prop} (c₁ : cardinal) (c₂ : cardinal)
(c₃ : cardinal) (h : ∀ α β γ, p (#α) (#β) (#γ)) : p c₁ c₂ c₃ | quotient.induction_on₃ c₁ c₂ c₃ h | lemma | cardinal.induction_on₃ | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq : #α = #β ↔ nonempty (α ≃ β) | quotient.eq | lemma | cardinal.eq | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"quotient.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_def (α : Type u) : @eq cardinal ⟦α⟧ (#α) | rfl | theorem | cardinal.mk_def | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_out (c : cardinal) : #(c.out) = c | quotient.out_eq _ | theorem | cardinal.mk_out | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"quotient.out_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_mk_equiv {α : Type v} : (#α).out ≃ α | nonempty.some $ cardinal.eq.mp (by simp) | def | cardinal.out_mk_equiv | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"nonempty.some"
] | The representative of the cardinal of a type is equivalent ot the original type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_congr (e : α ≃ β) : # α = # β | quot.sound ⟨e⟩ | lemma | cardinal.mk_congr | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : Type u → Type v) (hf : ∀ α β, α ≃ β → f α ≃ f β) :
cardinal.{u} → cardinal.{v} | quotient.map f (λ α β ⟨e⟩, ⟨hf α β e⟩) | def | cardinal.map | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"quotient.map"
] | Lift a function between `Type*`s to a function between `cardinal`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mk (f : Type u → Type v) (hf : ∀ α β, α ≃ β → f α ≃ f β) (α : Type u) :
map f hf (#α) = #(f α) | rfl | lemma | cardinal.map_mk | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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