statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
frobenius_eq_map_frobenius :
@frobenius p R _ _ = map (_root_.frobenius R p) | begin
ext x n,
simp only [coeff_frobenius_char_p, map_coeff, frobenius_def],
end | lemma | witt_vector.frobenius_eq_map_frobenius | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius.lean | [
"data.nat.multiplicity",
"data.zmod.algebra",
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly",
"field_theory.perfect_closure"
] | [
"frobenius",
"frobenius_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_zmodp (x : 𝕎 (zmod p)) :
(frobenius x) = x | by simp only [ext_iff, coeff_frobenius_char_p, zmod.pow_card, eq_self_iff_true, forall_const] | lemma | witt_vector.frobenius_zmodp | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius.lean | [
"data.nat.multiplicity",
"data.zmod.algebra",
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly",
"field_theory.perfect_closure"
] | [
"forall_const",
"frobenius",
"zmod",
"zmod.pow_card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_equiv [perfect_ring R p] : witt_vector p R ≃+* witt_vector p R | { to_fun := witt_vector.frobenius,
inv_fun := map (pth_root R p),
left_inv := λ f, ext $ λ n, by { rw frobenius_eq_map_frobenius, exact pth_root_frobenius _ },
right_inv := λ f, ext $ λ n, by { rw frobenius_eq_map_frobenius, exact frobenius_pth_root _ },
..(witt_vector.frobenius : witt_vector p R →+* witt_vect... | def | witt_vector.frobenius_equiv | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius.lean | [
"data.nat.multiplicity",
"data.zmod.algebra",
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly",
"field_theory.perfect_closure"
] | [
"frobenius_equiv",
"frobenius_pth_root",
"inv_fun",
"perfect_ring",
"pth_root",
"pth_root_frobenius",
"witt_vector",
"witt_vector.frobenius"
] | `witt_vector.frobenius` as an equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_bijective [perfect_ring R p] :
function.bijective (@witt_vector.frobenius p R _ _) | (frobenius_equiv p R).bijective | lemma | witt_vector.frobenius_bijective | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius.lean | [
"data.nat.multiplicity",
"data.zmod.algebra",
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly",
"field_theory.perfect_closure"
] | [
"frobenius_equiv",
"perfect_ring",
"witt_vector.frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_nth_defining_poly (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : fin (n+1) → k) : polynomial k | X^p * C (a₁.coeff 0 ^ (p^(n+1))) - X * C (a₂.coeff 0 ^ (p^(n+1)))
+ C (a₁.coeff (n+1) * ((bs 0)^p)^(p^(n+1)) +
nth_remainder p n (λ v, (bs v)^p) (truncate_fun (n+1) a₁) -
a₂.coeff (n+1) * (bs 0)^p^(n+1) - nth_remainder p n bs (truncate_fun (n+1) a₂)) | def | witt_vector.recursion_main.succ_nth_defining_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"polynomial"
] | The root of this polynomial determines the `n+1`st coefficient of our solution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_nth_defining_poly_degree [is_domain k] (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : fin (n+1) → k)
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
(succ_nth_defining_poly p n a₁ a₂ bs).degree = p | begin
have : (X ^ p * C (a₁.coeff 0 ^ p ^ (n+1))).degree = p,
{ rw [degree_mul, degree_C],
{ simp only [nat.cast_with_bot, add_zero, degree_X, degree_pow, nat.smul_one_eq_coe] },
{ exact pow_ne_zero _ ha₁ } },
have : (X ^ p * C (a₁.coeff 0 ^ p ^ (n+1)) - X * C (a₂.coeff 0 ^ p ^ (n+1))).degree = p,
{ rw ... | lemma | witt_vector.recursion_main.succ_nth_defining_poly_degree | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"is_domain",
"nat.cast_with_bot",
"nat.smul_one_eq_coe",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
root_exists (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : fin (n+1) → k)
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
∃ b : k, (succ_nth_defining_poly p n a₁ a₂ bs).is_root b | is_alg_closed.exists_root _ $
by simp only [(succ_nth_defining_poly_degree p n a₁ a₂ bs ha₁ ha₂), hp.out.ne_zero,
with_top.coe_eq_zero, ne.def, not_false_iff] | lemma | witt_vector.recursion_main.root_exists | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"is_alg_closed.exists_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_nth_val (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : fin (n+1) → k)
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) : k | classical.some (root_exists p n a₁ a₂ bs ha₁ ha₂) | def | witt_vector.recursion_main.succ_nth_val | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [] | This is the `n+1`st coefficient of our solution, projected from `root_exists`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_nth_val_spec (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : fin (n+1) → k)
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
(succ_nth_defining_poly p n a₁ a₂ bs).is_root (succ_nth_val p n a₁ a₂ bs ha₁ ha₂) | classical.some_spec (root_exists p n a₁ a₂ bs ha₁ ha₂) | lemma | witt_vector.recursion_main.succ_nth_val_spec | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_nth_val_spec' (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : fin (n+1) → k)
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
(succ_nth_val p n a₁ a₂ bs ha₁ ha₂)^p * a₁.coeff 0 ^ (p^(n+1)) +
a₁.coeff (n+1) * ((bs 0)^p)^(p^(n+1)) +
nth_remainder p n (λ v, (bs v)^p) (truncate_fun (n+1) a₁)
= (succ_nth_val p n a₁ a₂ bs ha... | begin
rw ← sub_eq_zero,
have := succ_nth_val_spec p n a₁ a₂ bs ha₁ ha₂,
simp only [polynomial.map_add, polynomial.eval_X, polynomial.map_pow, polynomial.eval_C,
polynomial.eval_pow, succ_nth_defining_poly, polynomial.eval_mul, polynomial.eval_add,
polynomial.eval_sub, polynomial.map_mul, polynomial.map_su... | lemma | witt_vector.recursion_main.succ_nth_val_spec' | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"polynomial.eval_C",
"polynomial.eval_X",
"polynomial.eval_add",
"polynomial.eval_mul",
"polynomial.eval_pow",
"polynomial.eval_sub",
"polynomial.is_root.def",
"polynomial.map_add",
"polynomial.map_mul",
"polynomial.map_pow",
"polynomial.map_sub",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
solution_pow (a₁ a₂ : 𝕎 k) :
∃ x : k, x^(p-1) = a₂.coeff 0 / a₁.coeff 0 | is_alg_closed.exists_pow_nat_eq _ $ by linarith [hp.out.one_lt, le_of_lt hp.out.one_lt] | lemma | witt_vector.recursion_base.solution_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"is_alg_closed.exists_pow_nat_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
solution (a₁ a₂ : 𝕎 k) : k | classical.some $ solution_pow p a₁ a₂ | def | witt_vector.recursion_base.solution | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [] | The base case (0th coefficient) of our solution vector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
solution_spec (a₁ a₂ : 𝕎 k) :
(solution p a₁ a₂)^(p-1) = a₂.coeff 0 / a₁.coeff 0 | classical.some_spec $ solution_pow p a₁ a₂ | lemma | witt_vector.recursion_base.solution_spec | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
solution_nonzero {a₁ a₂ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
solution p a₁ a₂ ≠ 0 | begin
intro h,
have := solution_spec p a₁ a₂,
rw [h, zero_pow] at this,
{ simpa [ha₁, ha₂] using _root_.div_eq_zero_iff.mp this.symm },
{ linarith [hp.out.one_lt, le_of_lt hp.out.one_lt] }
end | lemma | witt_vector.recursion_base.solution_nonzero | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
solution_spec' {a₁ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (a₂ : 𝕎 k) :
(solution p a₁ a₂)^p * a₁.coeff 0 = (solution p a₁ a₂) * a₂.coeff 0 | begin
have := solution_spec p a₁ a₂,
cases nat.exists_eq_succ_of_ne_zero hp.out.ne_zero with q hq,
have hq' : q = p - 1 := by simp only [hq, tsub_zero, nat.succ_sub_succ_eq_sub],
conv_lhs {congr, congr, skip, rw hq},
rw [pow_succ', hq', this],
field_simp [ha₁, mul_comm],
end | lemma | witt_vector.recursion_base.solution_spec' | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"mul_comm",
"pow_succ'",
"tsub_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_rotation_coeff {a₁ a₂ : 𝕎 k}
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) : ℕ → k | | 0 := solution p a₁ a₂
| (n + 1) := succ_nth_val p n a₁ a₂ (λ i, frobenius_rotation_coeff i.val) ha₁ ha₂
using_well_founded { dec_tac := `[apply fin.is_lt] } | def | witt_vector.frobenius_rotation_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"fin.is_lt"
] | Recursively defines the sequence of coefficients for `witt_vector.frobenius_rotation`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_rotation {a₁ a₂ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) : 𝕎 k | witt_vector.mk p (frobenius_rotation_coeff p ha₁ ha₂) | def | witt_vector.frobenius_rotation | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [] | For nonzero `a₁` and `a₂`, `frobenius_rotation a₁ a₂` is a Witt vector that satisfies the
equation `frobenius (frobenius_rotation a₁ a₂) * a₁ = (frobenius_rotation a₁ a₂) * a₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_rotation_nonzero {a₁ a₂ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
frobenius_rotation p ha₁ ha₂ ≠ 0 | begin
intro h,
apply solution_nonzero p ha₁ ha₂,
simpa [← h, frobenius_rotation, frobenius_rotation_coeff] using witt_vector.zero_coeff p k 0
end | lemma | witt_vector.frobenius_rotation_nonzero | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"witt_vector.zero_coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_frobenius_rotation {a₁ a₂ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
frobenius (frobenius_rotation p ha₁ ha₂) * a₁ = (frobenius_rotation p ha₁ ha₂) * a₂ | begin
ext n,
induction n with n ih,
{ simp only [witt_vector.mul_coeff_zero, witt_vector.coeff_frobenius_char_p,
frobenius_rotation, frobenius_rotation_coeff],
apply solution_spec' _ ha₁ },
{ simp only [nth_remainder_spec, witt_vector.coeff_frobenius_char_p, frobenius_rotation_coeff,
frobenius_r... | lemma | witt_vector.frobenius_frobenius_rotation | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"fin.val_eq_coe",
"frobenius",
"ih",
"truncated_witt_vector.ext",
"witt_vector.coeff_frobenius_char_p",
"witt_vector.coeff_truncate_fun",
"witt_vector.mul_coeff_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_frobenius_solution_fraction_ring_aux
(m n : ℕ) (r' q' : 𝕎 k) (hr' : r'.coeff 0 ≠ 0) (hq' : q'.coeff 0 ≠ 0)
(hq : ↑p ^ n * q' ∈ non_zero_divisors (𝕎 k)) :
let b : 𝕎 k | frobenius_rotation p hr' hq' in
is_fraction_ring.field_equiv_of_ring_equiv
(frobenius_equiv p k)
(algebra_map (𝕎 k) (fraction_ring (𝕎 k)) b) *
localization.mk (↑p ^ m * r') ⟨↑p ^ n * q', hq⟩ =
↑p ^ (m - n : ℤ) * algebra_map (𝕎 k) (fraction_ring (𝕎 k)) b :=
begin
intros b,
have key : witt_vec... | lemma | witt_vector.exists_frobenius_solution_fraction_ring_aux | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"algebra_map",
"fraction_ring",
"frobenius_equiv",
"is_fraction_ring.field_equiv_of_ring_equiv",
"is_fraction_ring.injective",
"is_localization.ring_equiv_of_ring_equiv_eq",
"localization.mk",
"map_nat_cast",
"non_zero_divisors",
"ring",
"ring_equiv.coe_of_bijective",
"ring_hom.map_mul",
"ri... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_frobenius_solution_fraction_ring {a : fraction_ring (𝕎 k)} (ha : a ≠ 0) :
∃ (b : fraction_ring (𝕎 k)) (hb : b ≠ 0) (m : ℤ), φ b * a = p ^ m * b | begin
revert ha,
refine localization.induction_on a _,
rintros ⟨r, q, hq⟩ hrq,
have hq0 : q ≠ 0 := mem_non_zero_divisors_iff_ne_zero.1 hq,
have hr0 : r ≠ 0 := λ h, hrq (by simp [h]),
obtain ⟨m, r', hr', rfl⟩ := exists_eq_pow_p_mul r hr0,
obtain ⟨n, q', hq', rfl⟩ := exists_eq_pow_p_mul q hq0,
let b := fr... | lemma | witt_vector.exists_frobenius_solution_fraction_ring | ring_theory.witt_vector | src/ring_theory/witt_vector/frobenius_fraction_field.lean | [
"data.nat.cast.with_top",
"field_theory.is_alg_closed.basic",
"ring_theory.witt_vector.discrete_valuation_ring"
] | [
"fraction_ring",
"is_fraction_ring.injective",
"localization.induction_on",
"witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_verschiebung (x : 𝕎 R) :
frobenius (verschiebung x) = x * p | by { ghost_calc x, ghost_simp [mul_comm] } | lemma | witt_vector.frobenius_verschiebung | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"frobenius",
"mul_comm"
] | The composition of Frobenius and Verschiebung is multiplication by `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
verschiebung_zmod (x : 𝕎 (zmod p)) :
verschiebung x = x * p | by rw [← frobenius_verschiebung, frobenius_zmodp] | lemma | witt_vector.verschiebung_zmod | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"zmod"
] | Verschiebung is the same as multiplication by `p` on the ring of Witt vectors of `zmod p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_p_pow [char_p R p] (i : ℕ) : (p ^ i : 𝕎 R).coeff i = 1 | begin
induction i with i h,
{ simp only [one_coeff_zero, ne.def, pow_zero] },
{ rw [pow_succ', ← frobenius_verschiebung, coeff_frobenius_char_p,
verschiebung_coeff_succ, h, one_pow], }
end | lemma | witt_vector.coeff_p_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"one_pow",
"pow_succ'",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_p_pow_eq_zero [char_p R p] {i j : ℕ} (hj : j ≠ i) : (p ^ i : 𝕎 R).coeff j = 0 | begin
induction i with i hi generalizing j,
{ rw [pow_zero, one_coeff_eq_of_pos],
exact nat.pos_of_ne_zero hj },
{ rw [pow_succ', ← frobenius_verschiebung, coeff_frobenius_char_p],
cases j,
{ rw [verschiebung_coeff_zero, zero_pow],
exact nat.prime.pos hp.out },
{ rw [verschiebung_coeff_succ,... | lemma | witt_vector.coeff_p_pow_eq_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"nat.prime.pos",
"ne_of_apply_ne",
"pow_succ'",
"pow_zero",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_p [char_p R p] (i : ℕ) : (p : 𝕎 R).coeff i = if i = 1 then 1 else 0 | begin
split_ifs with hi,
{ simpa only [hi, pow_one] using coeff_p_pow p R 1, },
{ simpa only [pow_one] using coeff_p_pow_eq_zero p R hi, }
end | lemma | witt_vector.coeff_p | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_p_zero [char_p R p] : (p : 𝕎 R).coeff 0 = 0 | by { rw [coeff_p, if_neg], exact zero_ne_one } | lemma | witt_vector.coeff_p_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_p_one [char_p R p] : (p : 𝕎 R).coeff 1 = 1 | by rw [coeff_p, if_pos rfl] | lemma | witt_vector.coeff_p_one | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
p_nonzero [nontrivial R] [char_p R p] : (p : 𝕎 R) ≠ 0 | by { intros h, simpa only [h, zero_coeff, zero_ne_one] using coeff_p_one p R } | lemma | witt_vector.p_nonzero | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"nontrivial",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fraction_ring.p_nonzero [nontrivial R] [char_p R p] :
(p : fraction_ring (𝕎 R)) ≠ 0 | by simpa using (is_fraction_ring.injective (𝕎 R) (fraction_ring (𝕎 R))).ne (p_nonzero _ _) | lemma | witt_vector.fraction_ring.p_nonzero | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"fraction_ring",
"is_fraction_ring.injective",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_mul_frobenius (x y : 𝕎 R) :
verschiebung (x * frobenius y) = verschiebung x * y | by { ghost_calc x y, rintro ⟨⟩; ghost_simp [mul_assoc] } | lemma | witt_vector.verschiebung_mul_frobenius | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"frobenius",
"mul_assoc"
] | The “projection formula” for Frobenius and Verschiebung. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_char_p_coeff_zero [char_p R p] (x : 𝕎 R) : (x * p).coeff 0 = 0 | begin
rw [← frobenius_verschiebung, coeff_frobenius_char_p, verschiebung_coeff_zero, zero_pow],
exact nat.prime.pos hp.out
end | lemma | witt_vector.mul_char_p_coeff_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"nat.prime.pos",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_char_p_coeff_succ [char_p R p] (x : 𝕎 R) (i : ℕ) :
(x * p).coeff (i + 1) = (x.coeff i)^p | by rw [← frobenius_verschiebung, coeff_frobenius_char_p, verschiebung_coeff_succ] | lemma | witt_vector.mul_char_p_coeff_succ | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_frobenius [char_p R p] (x : 𝕎 R) :
verschiebung (frobenius x) = x * p | begin
ext ⟨i⟩,
{ rw [mul_char_p_coeff_zero, verschiebung_coeff_zero], },
{ rw [mul_char_p_coeff_succ, verschiebung_coeff_succ, coeff_frobenius_char_p], }
end | lemma | witt_vector.verschiebung_frobenius | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_frobenius_comm [char_p R p] :
function.commute (verschiebung : 𝕎 R → 𝕎 R) frobenius | λ x, by rw [verschiebung_frobenius, frobenius_verschiebung] | lemma | witt_vector.verschiebung_frobenius_comm | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"char_p",
"frobenius",
"function.commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_verschiebung_coeff (x : 𝕎 R) (n k : ℕ) :
(verschiebung^[n] x).coeff (k + n) = x.coeff k | begin
induction n with k ih,
{ simp },
{ rw [iterate_succ_apply', nat.add_succ, verschiebung_coeff_succ],
exact ih }
end | lemma | witt_vector.iterate_verschiebung_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_verschiebung_mul_left (x y : 𝕎 R) (i : ℕ) :
(verschiebung^[i] x) * y = (verschiebung^[i] (x * (frobenius^[i] y))) | begin
induction i with i ih generalizing y,
{ simp },
{ rw [iterate_succ_apply', ← verschiebung_mul_frobenius, ih, iterate_succ_apply'], refl }
end | lemma | witt_vector.iterate_verschiebung_mul_left | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"frobenius",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_verschiebung_mul (x y : 𝕎 R) (i j : ℕ) :
(verschiebung^[i] x) * (verschiebung^[j] y) =
(verschiebung^[i + j] ((frobenius^[j] x) * (frobenius^[i] y))) | begin
calc
_ = (verschiebung^[i] (x * (frobenius^[i] ((verschiebung^[j] y))))) : _
... = (verschiebung^[i] (x * (verschiebung^[j] ((frobenius^[i] y))))) : _
... = (verschiebung^[i] ((verschiebung^[j] ((frobenius^[i] y)) * x))) : _
... = (verschiebung^[i] ((verschiebung^[j] ((frobenius^[i] y) * (frobenius^[j] x)))))... | lemma | witt_vector.iterate_verschiebung_mul | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"frobenius",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_frobenius_coeff (x : 𝕎 R) (i k : ℕ) :
((frobenius^[i] x)).coeff k = (x.coeff k)^(p^i) | begin
induction i with i ih,
{ simp },
{ rw [iterate_succ_apply', coeff_frobenius_char_p, ih],
ring_exp }
end | lemma | witt_vector.iterate_frobenius_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"frobenius",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_verschiebung_mul_coeff (x y : 𝕎 R) (i j : ℕ) :
((verschiebung^[i] x) * (verschiebung^[j] y)).coeff (i + j) =
(x.coeff 0)^(p ^ j) * (y.coeff 0)^(p ^ i) | begin
calc
_ = (verschiebung^[i + j] ((frobenius^[j] x) * (frobenius^[i] y))).coeff (i + j) : _
... = ((frobenius^[j] x) * (frobenius^[i] y)).coeff 0 : _
... = (frobenius^[j] x).coeff 0 * ((frobenius^[i] y)).coeff 0 : _
... = _ : _,
{ rw iterate_verschiebung_mul },
{ convert iterate_verschiebung_coeff _ _ _ usi... | lemma | witt_vector.iterate_verschiebung_mul_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/identities.lean | [
"ring_theory.witt_vector.frobenius",
"ring_theory.witt_vector.verschiebung",
"ring_theory.witt_vector.mul_p"
] | [
"frobenius"
] | This is a slightly specialized form of [Hazewinkel, *Witt Vectors*][Haze09] 6.2 equation 5. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
init_ring (assert : parse (tk "using" *> parser.pexpr)?) : tactic unit | do
`[rw ext_iff,
intros i,
simp only [init, select, coeff_mk],
split_ifs with hi; try {refl}],
match assert with
| none := skip
| some e := do
`[simp only [add_coeff, mul_coeff, neg_coeff, sub_coeff, nsmul_coeff, zsmul_coeff, pow_coeff],
apply eval₂_hom_congr' (ring_hom.ext_int _ _) _ rfl,
rintro ⟨b, k⟩... | def | tactic.interactive.init_ring | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"finset.mem_product",
"finset.mem_range",
"finset.mem_univ",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons",
"ring_hom.ext_int",
"tactic.replace"
] | `init_ring` is an auxiliary tactic that discharges goals factoring `init` over ring operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
select (P : ℕ → Prop) (x : 𝕎 R) : 𝕎 R | mk p (λ n, if P n then x.coeff n else 0) | def | witt_vector.select | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | `witt_vector.select P x`, for a predicate `P : ℕ → Prop` is the Witt vector
whose `n`-th coefficient is `x.coeff n` if `P n` is true, and `0` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
select_poly (n : ℕ) : mv_polynomial ℕ ℤ | if P n then X n else 0 | def | witt_vector.select_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"mv_polynomial"
] | The polynomial that witnesses that `witt_vector.select` is a polynomial function.
`select_poly n` is `X n` if `P n` holds, and `0` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_select (x : 𝕎 R) (n : ℕ) :
(select P x).coeff n = aeval x.coeff (select_poly P n) | begin
dsimp [select, select_poly],
split_ifs with hi,
{ rw aeval_X },
{ rw alg_hom.map_zero }
end | lemma | witt_vector.coeff_select | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"alg_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
select_is_poly (P : ℕ → Prop) :
is_poly p (λ R _Rcr x, by exactI select P x) | begin
use (select_poly P),
rintro R _Rcr x,
funext i,
apply coeff_select
end | lemma | witt_vector.select_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"is_poly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
select_add_select_not :
∀ (x : 𝕎 R), select P x + select (λ i, ¬ P i) x = x | begin
ghost_calc _,
intro n,
simp only [ring_hom.map_add],
suffices : (bind₁ (select_poly P)) (witt_polynomial p ℤ n) +
(bind₁ (select_poly (λ i, ¬P i))) (witt_polynomial p ℤ n) = witt_polynomial p ℤ n,
{ apply_fun (aeval x.coeff) at this,
simpa only [alg_hom.map_add, aeval_bind₁, ← coeff_sel... | lemma | witt_vector.select_add_select_not | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"alg_hom.map_pow",
"alg_hom.map_sum",
"ite_pow",
"pow_pos",
"ring_hom.map_add",
"witt_polynomial",
"witt_polynomial_eq_sum_C_mul_X_pow",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_add_of_disjoint (x y : 𝕎 R) (h : ∀ n, x.coeff n = 0 ∨ y.coeff n = 0) :
(x + y).coeff n = x.coeff n + y.coeff n | begin
let P : ℕ → Prop := λ n, y.coeff n = 0,
haveI : decidable_pred P := classical.dec_pred P,
set z := mk p (λ n, if P n then x.coeff n else y.coeff n) with hz,
have hx : select P z = x,
{ ext1 n, rw [select, coeff_mk, coeff_mk],
split_ifs with hn, { refl }, { rw (h n).resolve_right hn } },
have hy : ... | lemma | witt_vector.coeff_add_of_disjoint | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"classical.dec_pred"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init (n : ℕ) : 𝕎 R → 𝕎 R | select (λ i, i < n) | def | witt_vector.init | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | `witt_vector.init n x` is the Witt vector of which the first `n` coefficients are those from `x`
and all other coefficients are `0`.
See `witt_vector.tail` for the complementary part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tail (n : ℕ) : 𝕎 R → 𝕎 R | select (λ i, n ≤ i) | def | witt_vector.tail | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | `witt_vector.tail n x` is the Witt vector of which the first `n` coefficients are `0`
and all other coefficients are those from `x`.
See `witt_vector.init` for the complementary part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
init_add_tail (x : 𝕎 R) (n : ℕ) :
init n x + tail n x = x | by simp only [init, tail, ← not_lt, select_add_select_not] | lemma | witt_vector.init_add_tail | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_init (x : 𝕎 R) (n : ℕ) :
init n (init n x) = init n x | by init_ring | lemma | witt_vector.init_init | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_add (x y : 𝕎 R) (n : ℕ) :
init n (x + y) = init n (init n x + init n y) | by init_ring using witt_add_vars | lemma | witt_vector.init_add | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_mul (x y : 𝕎 R) (n : ℕ) :
init n (x * y) = init n (init n x * init n y) | by init_ring using witt_mul_vars | lemma | witt_vector.init_mul | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_neg (x : 𝕎 R) (n : ℕ) :
init n (-x) = init n (-init n x) | by init_ring using witt_neg_vars | lemma | witt_vector.init_neg | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_sub (x y : 𝕎 R) (n : ℕ) :
init n (x - y) = init n (init n x - init n y) | by init_ring using witt_sub_vars | lemma | witt_vector.init_sub | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_nsmul (m : ℕ) (x : 𝕎 R) (n : ℕ) :
init n (m • x) = init n (m • init n x) | by init_ring using (λ p [fact (nat.prime p)] n, by exactI witt_nsmul_vars p m n) | lemma | witt_vector.init_nsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"fact",
"nat.prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_zsmul (m : ℤ) (x : 𝕎 R) (n : ℕ) :
init n (m • x) = init n (m • init n x) | by init_ring using (λ p [fact (nat.prime p)] n, by exactI witt_zsmul_vars p m n) | lemma | witt_vector.init_zsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"fact",
"nat.prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_pow (m : ℕ) (x : 𝕎 R) (n : ℕ) :
init n (x ^ m) = init n (init n x ^ m) | by init_ring using (λ p [fact (nat.prime p)] n, by exactI witt_pow_vars p m n) | lemma | witt_vector.init_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"fact",
"nat.prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
init_is_poly (n : ℕ) : is_poly p (λ R _Rcr, by exactI init n) | select_is_poly (λ i, i < n) | lemma | witt_vector.init_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/init_tail.lean | [
"ring_theory.witt_vector.basic",
"ring_theory.witt_vector.is_poly"
] | [
"is_poly"
] | `witt_vector.init n x` is polynomial in the coefficients of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fraction_ring.frobenius : K(p, k) ≃+* K(p, k) | is_fraction_ring.field_equiv_of_ring_equiv (frobenius_equiv p k) | def | witt_vector.fraction_ring.frobenius | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [
"frobenius_equiv",
"is_fraction_ring.field_equiv_of_ring_equiv"
] | The Frobenius automorphism of `k` induces an automorphism of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fraction_ring.frobenius_ring_hom : K(p, k) →+* K(p, k) | fraction_ring.frobenius p k | def | witt_vector.fraction_ring.frobenius_ring_hom | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [] | The Frobenius automorphism of `k` induces an endomorphism of `K`. For notation purposes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_pair₁ : ring_hom_inv_pair (φ(p, k)) _ | ring_hom_inv_pair.of_ring_equiv (fraction_ring.frobenius p k) | instance | witt_vector.inv_pair₁ | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [
"ring_hom_inv_pair",
"ring_hom_inv_pair.of_ring_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pair₂ :
ring_hom_inv_pair ((fraction_ring.frobenius p k).symm : K(p, k) →+* K(p, k)) _ | ring_hom_inv_pair.of_ring_equiv (fraction_ring.frobenius p k).symm | instance | witt_vector.inv_pair₂ | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [
"ring_hom_inv_pair",
"ring_hom_inv_pair.of_ring_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isocrystal (V : Type*) [add_comm_group V] extends module K(p, k) V | ( frob : V ≃ᶠˡ[p, k] V ) | class | witt_vector.isocrystal | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [
"add_comm_group",
"module"
] | An isocrystal is a vector space over the field `K(p, k)` additionally equipped with a
Frobenius-linear automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isocrystal.frobenius : V ≃ᶠˡ[p, k] V | @isocrystal.frob p _ k _ _ _ _ _ _ _ | def | witt_vector.isocrystal.frobenius | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [] | Project the Frobenius automorphism from an isocrystal. Denoted by `Φ(p, k)` when V can be inferred. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isocrystal_hom extends V →ₗ[K(p, k)] V₂ | ( frob_equivariant : ∀ x : V, Φ(p, k) (to_linear_map x) = to_linear_map (Φ(p, k) x) ) | structure | witt_vector.isocrystal_hom | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [] | A homomorphism between isocrystals respects the Frobenius map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isocrystal_equiv extends V ≃ₗ[K(p, k)] V₂ | ( frob_equivariant : ∀ x : V, Φ(p, k) (to_linear_equiv x) = to_linear_equiv (Φ(p, k) x) ) | structure | witt_vector.isocrystal_equiv | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [] | An isomorphism between isocrystals respects the Frobenius map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fraction_ring.module : module K(p, k) K(p, k) | semiring.to_module | def | witt_vector.fraction_ring.module | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [
"module",
"semiring.to_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
standard_one_dim_isocrystal (m : ℤ) : Type* | K(p, k) | def | witt_vector.standard_one_dim_isocrystal | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [] | Type synonym for `K(p, k)` to carry the standard 1-dimensional isocrystal structure
of slope `m : ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
standard_one_dim_isocrystal.frobenius_apply (m : ℤ)
(x : standard_one_dim_isocrystal p k m) :
Φ(p, k) x = (p:K(p, k)) ^ m • φ(p, k) x | rfl | lemma | witt_vector.standard_one_dim_isocrystal.frobenius_apply | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isocrystal_classification
(k : Type*) [field k] [is_alg_closed k] [char_p k p]
(V : Type*) [add_comm_group V] [isocrystal p k V]
(h_dim : finrank K(p, k) V = 1) :
∃ (m : ℤ), nonempty (standard_one_dim_isocrystal p k m ≃ᶠⁱ[p, k] V) | begin
haveI : nontrivial V := finite_dimensional.nontrivial_of_finrank_eq_succ h_dim,
obtain ⟨x, hx⟩ : ∃ x : V, x ≠ 0 := exists_ne 0,
have : Φ(p, k) x ≠ 0 := by simpa only [map_zero] using Φ(p,k).injective.ne hx,
obtain ⟨a, ha, hax⟩ : ∃ a : K(p, k), a ≠ 0 ∧ Φ(p, k) x = a • x,
{ rw finrank_eq_one_iff_of_nonzer... | theorem | witt_vector.isocrystal_classification | ring_theory.witt_vector | src/ring_theory/witt_vector/isocrystal.lean | [
"ring_theory.witt_vector.frobenius_fraction_field"
] | [
"add_comm_group",
"algebra.id.smul_eq_mul",
"char_p",
"exists_ne",
"field",
"finite_dimensional.nontrivial_of_finrank_eq_succ",
"finrank_eq_one_iff_of_nonzero",
"finrank_eq_one_iff_of_nonzero'",
"is_alg_closed",
"linear_equiv.map_smul",
"linear_equiv.map_smulₛₗ",
"linear_equiv.of_bijective",
... | A one-dimensional isocrystal over an algebraically closed field
admits an isomorphism to one of the standard (indexed by `m : ℤ`) one-dimensional isocrystals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_simp (lems : parse simp_arg_list) : tactic unit | do tactic.try tactic.intro1,
simp none none tt
(lems ++ [simp_arg_type.symm_expr ``(sub_eq_add_neg)])
[`ghost_simps] (loc.ns [none]) | def | tactic.interactive.ghost_simp | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [] | A macro for a common simplification when rewriting with ghost component equations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_calc (ids' : parse ident_*) : tactic unit | do ids ← ids'.mmap $ λ n, get_local n <|> tactic.intro n,
`(@eq (witt_vector _ %%R) _ _) ← target,
match ids with
| [x] := refine ```(is_poly.ext _ _ _ _ %%x)
| [x, y] := refine ```(is_poly₂.ext _ _ _ _ %%x %%y)
| _ := fail "ghost_calc takes one or two arguments"
end,
nm ← match R with
| expr.lo... | def | tactic.interactive.ghost_calc | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"tactic.clear'"
] | `ghost_calc` is a tactic for proving identities between polynomial functions.
Typically, when faced with a goal like
```lean
∀ (x y : 𝕎 R), verschiebung (x * frobenius y) = verschiebung x * y
```
you can
1. call `ghost_calc`
2. do a small amount of manual work -- maybe nothing, maybe `rintro`, etc
3. call `ghost_simp`... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
poly_eq_of_witt_polynomial_bind_eq' (f g : ℕ → mv_polynomial (idx × ℕ) ℤ)
(h : ∀ n, bind₁ f (witt_polynomial p _ n) = bind₁ g (witt_polynomial p _ n)) :
f = g | begin
ext1 n,
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
rw ← function.funext_iff at h,
replace h := congr_arg
(λ fam, bind₁ (mv_polynomial.map (int.cast_ring_hom ℚ) ∘ fam)
(X_in_terms_of_W p ℚ n)) h,
simpa only [function.comp, map_bind₁, map_witt_polynomial,
← bin... | lemma | witt_vector.poly_eq_of_witt_polynomial_bind_eq' | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"X_in_terms_of_W",
"bind₁_witt_polynomial_X_in_terms_of_W",
"function.funext_iff",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_polynomial",
"mv_polynomial",
"mv_polynomial.map",
"mv_polynomial.map_injective",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
poly_eq_of_witt_polynomial_bind_eq (f g : ℕ → mv_polynomial ℕ ℤ)
(h : ∀ n, bind₁ f (witt_polynomial p _ n) = bind₁ g (witt_polynomial p _ n)) :
f = g | begin
ext1 n,
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
rw ← function.funext_iff at h,
replace h := congr_arg
(λ fam, bind₁ (mv_polynomial.map (int.cast_ring_hom ℚ) ∘ fam)
(X_in_terms_of_W p ℚ n)) h,
simpa only [function.comp, map_bind₁, map_witt_polynomial,
← bin... | lemma | witt_vector.poly_eq_of_witt_polynomial_bind_eq | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"X_in_terms_of_W",
"bind₁_witt_polynomial_X_in_terms_of_W",
"function.funext_iff",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_polynomial",
"mv_polynomial",
"mv_polynomial.map",
"mv_polynomial.map_injective",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_poly (f : Π ⦃R⦄ [comm_ring R], witt_vector p R → 𝕎 R) : Prop | mk' :: (poly : ∃ φ : ℕ → mv_polynomial ℕ ℤ, ∀ ⦃R⦄ [comm_ring R] (x : 𝕎 R),
by exactI (f x).coeff = λ n, aeval x.coeff (φ n)) | class | witt_vector.is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"comm_ring",
"is_poly",
"mk'",
"mv_polynomial",
"poly",
"witt_vector"
] | A function `f : Π R, 𝕎 R → 𝕎 R` that maps Witt vectors to Witt vectors over arbitrary base rings
is said to be *polynomial* if there is a family of polynomials `φₙ` over `ℤ` such that the `n`th
coefficient of `f x` is given by evaluating `φₙ` at the coefficients of `x`.
See also `witt_vector.is_poly₂` for the binary... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_is_poly : is_poly p (λ _ _, id) | ⟨⟨X, by { introsI, simp only [aeval_X, id] }⟩⟩ | instance | witt_vector.id_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly"
] | The identity function on Witt vectors is a polynomial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_is_poly_i' : is_poly p (λ _ _ a, a) | witt_vector.id_is_poly _ | instance | witt_vector.id_is_poly_i' | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly",
"witt_vector.id_is_poly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g} (hf : is_poly p f) (hg : is_poly p g)
(h : ∀ (R : Type u) [_Rcr : comm_ring R] (x : 𝕎 R) (n : ℕ),
by exactI ghost_component n (f x) = ghost_component n (g x)) :
∀ (R : Type u) [_Rcr : comm_ring R] (x : 𝕎 R), by exactI f x = g x | begin
unfreezingI
{ obtain ⟨φ, hf⟩ := hf,
obtain ⟨ψ, hg⟩ := hg },
intros,
ext n,
rw [hf, hg, poly_eq_of_witt_polynomial_bind_eq p φ ψ],
intro k,
apply mv_polynomial.funext,
intro x,
simp only [hom_bind₁],
specialize h (ulift ℤ) (mk p $ λ i, ⟨x i⟩) k,
simp only [ghost_component_apply, aeval_eq_... | lemma | witt_vector.is_poly.ext | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"comm_ring",
"is_poly",
"mv_polynomial.eval",
"mv_polynomial.funext",
"ring_hom.ext_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g f} (hg : is_poly p g) (hf : is_poly p f) :
is_poly p (λ R _Rcr, @g R _Rcr ∘ @f R _Rcr) | begin
unfreezingI
{ obtain ⟨φ, hf⟩ := hf,
obtain ⟨ψ, hg⟩ := hg },
use (λ n, bind₁ φ (ψ n)),
intros,
simp only [aeval_bind₁, function.comp, hg, hf]
end | lemma | witt_vector.is_poly.comp | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly"
] | The composition of polynomial functions is polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_poly₂ (f : Π ⦃R⦄ [comm_ring R], witt_vector p R → 𝕎 R → 𝕎 R) : Prop | mk' :: (poly : ∃ φ : ℕ → mv_polynomial (fin 2 × ℕ) ℤ, ∀ ⦃R⦄ [comm_ring R] (x y : 𝕎 R),
by exactI (f x y).coeff = λ n, peval (φ n) ![x.coeff, y.coeff]) | class | witt_vector.is_poly₂ | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"comm_ring",
"mk'",
"mv_polynomial",
"poly",
"witt_vector"
] | A binary function `f : Π R, 𝕎 R → 𝕎 R → 𝕎 R` on Witt vectors
is said to be *polynomial* if there is a family of polynomials `φₙ` over `ℤ` such that the `n`th
coefficient of `f x y` is given by evaluating `φₙ` at the coefficients of `x` and `y`.
See also `witt_vector.is_poly` for the unary variant.
The `ghost_calc`... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_poly₂.comp {h f g} (hh : is_poly₂ p h) (hf : is_poly p f) (hg : is_poly p g) :
is_poly₂ p (λ R _Rcr x y, by exactI h (f x) (g y)) | begin
unfreezingI
{ obtain ⟨φ, hf⟩ := hf,
obtain ⟨ψ, hg⟩ := hg,
obtain ⟨χ, hh⟩ := hh },
refine ⟨⟨(λ n, bind₁ (uncurry $
![λ k, rename (prod.mk (0 : fin 2)) (φ k),
λ k, rename (prod.mk (1 : fin 2)) (ψ k)]) (χ n)), _⟩⟩,
intros,
funext n,
simp only [peval, aeval_bind₁, function.co... | lemma | witt_vector.is_poly₂.comp | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons"
] | The composition of polynomial functions is polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_poly.comp₂ {g f} (hg : is_poly p g) (hf : is_poly₂ p f) :
is_poly₂ p (λ R _Rcr x y, by exactI g (f x y)) | begin
unfreezingI
{ obtain ⟨φ, hf⟩ := hf,
obtain ⟨ψ, hg⟩ := hg },
use (λ n, bind₁ φ (ψ n)),
intros,
simp only [peval, aeval_bind₁, function.comp, hg, hf]
end | lemma | witt_vector.is_poly.comp₂ | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly"
] | The composition of a polynomial function with a binary polynomial function is polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_poly₂.diag {f} (hf : is_poly₂ p f) :
is_poly p (λ R _Rcr x, by exactI f x x) | begin
unfreezingI {obtain ⟨φ, hf⟩ := hf},
refine ⟨⟨λ n, bind₁ (uncurry ![X, X]) (φ n), _⟩⟩,
intros, funext n,
simp only [hf, peval, uncurry, aeval_bind₁],
apply eval₂_hom_congr rfl _ rfl,
ext ⟨i, k⟩, fin_cases i;
simp only [matrix.head_cons, aeval_X, matrix.cons_val_zero, matrix.cons_val_one],
end | lemma | witt_vector.is_poly₂.diag | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons"
] | The diagonal `λ x, f x x` of a polynomial function `f` is polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_poly_comp_lemmas (n : name) (vars : list expr) (p : expr) : tactic unit | do c ← mk_const n,
let appd := vars.foldl expr.app c,
tgt_bod ← to_expr ``(λ f [hf : is_poly %%p f], is_poly.comp %%appd hf) >>=
replace_univ_metas_with_univ_params,
tgt_bod ← lambdas vars tgt_bod,
tgt_tp ← infer_type tgt_bod,
let nm := n <.> "comp_i",
add_decl $ mk_definition nm tgt_tp.collect_... | def | witt_vector.tactic.mk_poly_comp_lemmas | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"set_attribute"
] | If `n` is the name of a lemma with opened type `∀ vars, is_poly p _`,
`mk_poly_comp_lemmas n vars p` adds composition instances to the environment
`n.comp_i` and `n.comp₂_i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_poly₂_comp_lemmas (n : name) (vars : list expr) (p : expr) : tactic unit | do c ← mk_const n,
let appd := vars.foldl expr.app c,
tgt_bod ← to_expr ``(λ {f g} [hf : is_poly %%p f] [hg : is_poly %%p g],
is_poly₂.comp %%appd hf hg) >>= replace_univ_metas_with_univ_params,
tgt_bod ← lambdas vars tgt_bod,
tgt_tp ← infer_type tgt_bod >>= simp_lemmas.mk.dsimplify,
let nm := n <.... | def | witt_vector.tactic.mk_poly₂_comp_lemmas | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"set_attribute"
] | If `n` is the name of a lemma with opened type `∀ vars, is_poly₂ p _`,
`mk_poly₂_comp_lemmas n vars p` adds composition instances to the environment
`n.comp₂_i` and `n.comp_diag`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_comp_lemmas (n : name) : tactic unit | do d ← get_decl n,
(vars, tp) ← open_pis d.type,
match tp with
| `(is_poly %%p _) := mk_poly_comp_lemmas n vars p
| `(is_poly₂ %%p _) := mk_poly₂_comp_lemmas n vars p
| _ := fail "@[is_poly] should only be applied to terms of type `is_poly _ _` or `is_poly₂ _ _`"
end | def | witt_vector.tactic.mk_comp_lemmas | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [] | The `after_set` function for `@[is_poly]`. Calls `mk_poly(₂)_comp_lemmas`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_poly_attr : user_attribute | { name := `is_poly,
descr := "Lemmas with this attribute describe the polynomial structure of functions",
after_set := some $ λ n _ _, mk_comp_lemmas n } | def | witt_vector.tactic.is_poly_attr | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly"
] | `@[is_poly]` is applied to lemmas of the form `is_poly f φ` or `is_poly₂ f φ`.
These lemmas should *not* be tagged as instances, and only atomic `is_poly` defs should be tagged:
composition lemmas should not. Roughly speaking, lemmas that take `is_poly` proofs as arguments
should not be tagged.
Type class inference st... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_is_poly : is_poly p (λ R _, by exactI @has_neg.neg (𝕎 R) _) | ⟨⟨λ n, rename prod.snd (witt_neg p n),
begin
introsI, funext n,
rw [neg_coeff, aeval_eq_eval₂_hom, eval₂_hom_rename],
apply eval₂_hom_congr rfl _ rfl,
ext ⟨i, k⟩, fin_cases i, refl,
end⟩⟩ | lemma | witt_vector.neg_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly"
] | The additive negation is a polynomial function on Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_is_poly : is_poly p (λ _ _ _, by exactI 0) | ⟨⟨0, by { introsI, funext n, simp only [pi.zero_apply, alg_hom.map_zero, zero_coeff] }⟩⟩ | instance | witt_vector.zero_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"alg_hom.map_zero",
"is_poly"
] | The function that is constantly zero on Witt vectors is a polynomial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind₁_zero_witt_polynomial (n : ℕ) :
bind₁ (0 : ℕ → mv_polynomial ℕ R) (witt_polynomial p R n) = 0 | by rw [← aeval_eq_bind₁, aeval_zero, constant_coeff_witt_polynomial, ring_hom.map_zero] | lemma | witt_vector.bind₁_zero_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"constant_coeff_witt_polynomial",
"mv_polynomial",
"ring_hom.map_zero",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_poly (n : ℕ) : mv_polynomial ℕ ℤ | if n = 0 then 1 else 0 | def | witt_vector.one_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"mv_polynomial"
] | The coefficients of `1 : 𝕎 R` as polynomials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind₁_one_poly_witt_polynomial (n : ℕ) :
bind₁ one_poly (witt_polynomial p ℤ n) = 1 | begin
rw [witt_polynomial_eq_sum_C_mul_X_pow, alg_hom.map_sum, finset.sum_eq_single 0],
{ simp only [one_poly, one_pow, one_mul, alg_hom.map_pow, C_1, pow_zero, bind₁_X_right,
if_true, eq_self_iff_true], },
{ intros i hi hi0,
simp only [one_poly, if_neg hi0, zero_pow (pow_pos hp.1.pos _), mul_zero,
... | lemma | witt_vector.bind₁_one_poly_witt_polynomial | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"alg_hom.map_mul",
"alg_hom.map_pow",
"alg_hom.map_sum",
"finset.mem_range",
"mul_zero",
"one_mul",
"one_pow",
"pow_pos",
"pow_zero",
"witt_polynomial",
"witt_polynomial_eq_sum_C_mul_X_pow",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_is_poly : is_poly p (λ _ _ _, by exactI 1) | ⟨⟨one_poly,
begin
introsI, funext n, cases n,
{ simp only [one_poly, if_true, eq_self_iff_true, one_coeff_zero, alg_hom.map_one], },
{ simp only [one_poly, nat.succ_pos', one_coeff_eq_of_pos,
if_neg n.succ_ne_zero, alg_hom.map_zero] }
end⟩⟩ | instance | witt_vector.one_is_poly | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"alg_hom.map_one",
"alg_hom.map_zero",
"is_poly",
"nat.succ_pos'"
] | The function that is constantly one on Witt vectors is a polynomial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_is_poly₂ [fact p.prime] : is_poly₂ p (λ _ _, by exactI (+)) | ⟨⟨witt_add p, by { introsI, dunfold witt_vector.has_add, simp [eval] }⟩⟩ | lemma | witt_vector.add_is_poly₂ | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"fact"
] | Addition of Witt vectors is a polynomial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_is_poly₂ [fact p.prime] : is_poly₂ p (λ _ _, by exactI (*)) | ⟨⟨witt_mul p, by { introsI, dunfold witt_vector.has_mul, simp [eval] }⟩⟩ | lemma | witt_vector.mul_is_poly₂ | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"fact"
] | Multiplication of Witt vectors is a polynomial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_poly.map {f} (hf : is_poly p f) (g : R →+* S) (x : 𝕎 R) :
map g (f x) = f (map g x) | begin
-- this could be turned into a tactic “macro” (taking `hf` as parameter)
-- so that applications do not have to worry about the universe issue
-- see `is_poly₂.map` for a slightly more general proof strategy
unfreezingI {obtain ⟨φ, hf⟩ := hf},
ext n,
simp only [map_coeff, hf, map_aeval],
apply eval₂... | lemma | witt_vector.is_poly.map | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly",
"ring_hom.ext_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_left {g f} (hg : is_poly₂ p g) (hf : is_poly p f) :
is_poly₂ p (λ R _Rcr x y, by exactI g (f x) y) | hg.comp hf (witt_vector.id_is_poly _) | lemma | witt_vector.is_poly₂.comp_left | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly",
"witt_vector.id_is_poly"
] | The composition of a binary polynomial function
with a unary polynomial function in the first argument is polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_right {g f} (hg : is_poly₂ p g) (hf : is_poly p f) :
is_poly₂ p (λ R _Rcr x y, by exactI g x (f y)) | hg.comp (witt_vector.id_is_poly p) hf | lemma | witt_vector.is_poly₂.comp_right | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"is_poly",
"witt_vector.id_is_poly"
] | The composition of a binary polynomial function
with a unary polynomial function in the second argument is polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {f g} (hf : is_poly₂ p f) (hg : is_poly₂ p g)
(h : ∀ (R : Type u) [_Rcr : comm_ring R] (x y : 𝕎 R) (n : ℕ),
by exactI ghost_component n (f x y) = ghost_component n (g x y)) :
∀ (R) [_Rcr : comm_ring R] (x y : 𝕎 R), by exactI f x y = g x y | begin
unfreezingI
{ obtain ⟨φ, hf⟩ := hf,
obtain ⟨ψ, hg⟩ := hg },
intros,
ext n,
rw [hf, hg, poly_eq_of_witt_polynomial_bind_eq' p φ ψ],
clear x y,
intro k,
apply mv_polynomial.funext,
intro x,
simp only [hom_bind₁],
specialize h (ulift ℤ) (mk p $ λ i, ⟨x (0, i)⟩) (mk p $ λ i, ⟨x (1, i)⟩) k,
... | lemma | witt_vector.is_poly₂.ext | ring_theory.witt_vector | src/ring_theory/witt_vector/is_poly.lean | [
"algebra.ring.ulift",
"ring_theory.witt_vector.basic",
"data.mv_polynomial.funext"
] | [
"comm_ring",
"mv_polynomial.eval",
"mv_polynomial.funext",
"ring_hom.ext_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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