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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|---|---|---|---|---|---|---|---|---|---|---|
zmod_equiv_trunc_apply {x : zmod (p^n)} :
zmod_equiv_trunc p n x = zmod.cast_hom (by refl) (truncated_witt_vector p n (zmod p)) x | rfl | lemma | truncated_witt_vector.zmod_equiv_trunc_apply | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"truncated_witt_vector",
"zmod",
"zmod.cast_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutes {m : ℕ} (hm : n ≤ m) :
(truncate hm).comp (zmod_equiv_trunc p m).to_ring_hom =
(zmod_equiv_trunc p n).to_ring_hom.comp (zmod.cast_hom (pow_dvd_pow p hm) _) | ring_hom.ext_zmod _ _ | lemma | truncated_witt_vector.commutes | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"pow_dvd_pow",
"ring_hom.ext_zmod",
"zmod.cast_hom"
] | The following diagram commutes:
```text
zmod (p^n) ----------------------------> zmod (p^m)
| |
| |
v v
truncated_witt_vector p n (zmod p) ----> truncated_wi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commutes' {m : ℕ} (hm : n ≤ m) (x : zmod (p^m)) :
truncate hm (zmod_equiv_trunc p m x) =
zmod_equiv_trunc p n (zmod.cast_hom (pow_dvd_pow p hm) _ x) | show (truncate hm).comp (zmod_equiv_trunc p m).to_ring_hom x = _,
by rw commutes _ _ hm; refl | lemma | truncated_witt_vector.commutes' | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"pow_dvd_pow",
"zmod",
"zmod.cast_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutes_symm' {m : ℕ} (hm : n ≤ m) (x : truncated_witt_vector p m (zmod p)) :
(zmod_equiv_trunc p n).symm (truncate hm x) =
zmod.cast_hom (pow_dvd_pow p hm) _ ((zmod_equiv_trunc p m).symm x) | begin
apply (zmod_equiv_trunc p n).injective,
rw ← commutes',
simp
end | lemma | truncated_witt_vector.commutes_symm' | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"pow_dvd_pow",
"truncated_witt_vector",
"zmod",
"zmod.cast_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutes_symm {m : ℕ} (hm : n ≤ m) :
(zmod_equiv_trunc p n).symm.to_ring_hom.comp (truncate hm) =
(zmod.cast_hom (pow_dvd_pow p hm) _).comp (zmod_equiv_trunc p m).symm.to_ring_hom | by ext; apply commutes_symm' | lemma | truncated_witt_vector.commutes_symm | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"pow_dvd_pow",
"zmod.cast_hom"
] | The following diagram commutes:
```text
truncated_witt_vector p n (zmod p) ----> truncated_witt_vector p m (zmod p)
| |
| |
v v
zmod (p^n) ------------------... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_zmod_pow (k : ℕ) : 𝕎 (zmod p) →+* zmod (p ^ k) | (zmod_equiv_trunc p k).symm.to_ring_hom.comp (truncate k) | def | witt_vector.to_zmod_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"zmod"
] | `to_zmod_pow` is a family of compatible ring homs. We get this family by composing
`truncated_witt_vector.zmod_equiv_trunc` (in right-to-left direction)
with `witt_vector.truncate`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_zmod_pow_compat (m n : ℕ) (h : m ≤ n) :
(zmod.cast_hom (pow_dvd_pow p h) (zmod (p ^ m))).comp (to_zmod_pow p n) = to_zmod_pow p m | calc (zmod.cast_hom _ (zmod (p ^ m))).comp
((zmod_equiv_trunc p n).symm.to_ring_hom.comp (truncate n)) =
((zmod_equiv_trunc p m).symm.to_ring_hom.comp
(truncated_witt_vector.truncate h)).comp (truncate n) :
by rw [commutes_symm, ring_hom.comp_assoc]
... = (zmod_equiv_trunc p m).symm.to_ring_hom.comp (trun... | lemma | witt_vector.to_zmod_pow_compat | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"pow_dvd_pow",
"ring_hom.comp_assoc",
"truncated_witt_vector.truncate",
"zmod",
"zmod.cast_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_padic_int : 𝕎 (zmod p) →+* ℤ_[p] | padic_int.lift $ to_zmod_pow_compat p | def | witt_vector.to_padic_int | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"padic_int.lift",
"zmod"
] | `to_padic_int` lifts `to_zmod_pow : 𝕎 (zmod p) →+* zmod (p ^ k)` to a ring hom to `ℤ_[p]`
using `padic_int.lift`, the universal property of `ℤ_[p]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zmod_equiv_trunc_compat (k₁ k₂ : ℕ) (hk : k₁ ≤ k₂) :
(truncated_witt_vector.truncate hk).comp
((zmod_equiv_trunc p k₂).to_ring_hom.comp
(padic_int.to_zmod_pow k₂)) =
(zmod_equiv_trunc p k₁).to_ring_hom.comp (padic_int.to_zmod_pow k₁) | by rw [← ring_hom.comp_assoc, commutes, ring_hom.comp_assoc, padic_int.zmod_cast_comp_to_zmod_pow] | lemma | witt_vector.zmod_equiv_trunc_compat | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"padic_int.to_zmod_pow",
"padic_int.zmod_cast_comp_to_zmod_pow",
"ring_hom.comp_assoc",
"truncated_witt_vector.truncate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_padic_int : ℤ_[p] →+* 𝕎 (zmod p) | witt_vector.lift (λ k, (zmod_equiv_trunc p k).to_ring_hom.comp (padic_int.to_zmod_pow k)) $
zmod_equiv_trunc_compat _ | def | witt_vector.from_padic_int | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"padic_int.to_zmod_pow",
"witt_vector.lift",
"zmod"
] | `from_padic_int` uses `witt_vector.lift` to lift `truncated_witt_vector.zmod_equiv_trunc`
composed with `padic_int.to_zmod_pow` to a ring hom `ℤ_[p] →+* 𝕎 (zmod p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_padic_int_comp_from_padic_int :
(to_padic_int p).comp (from_padic_int p) = ring_hom.id ℤ_[p] | begin
rw ← padic_int.to_zmod_pow_eq_iff_ext,
intro n,
rw [← ring_hom.comp_assoc, to_padic_int, padic_int.lift_spec],
simp only [from_padic_int, to_zmod_pow, ring_hom.comp_id],
rw [ring_hom.comp_assoc, truncate_comp_lift, ← ring_hom.comp_assoc],
simp only [ring_equiv.symm_to_ring_hom_comp_to_ring_hom, ring_h... | lemma | witt_vector.to_padic_int_comp_from_padic_int | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"padic_int.lift_spec",
"padic_int.to_zmod_pow_eq_iff_ext",
"ring_equiv.symm_to_ring_hom_comp_to_ring_hom",
"ring_hom.comp_assoc",
"ring_hom.comp_id",
"ring_hom.id",
"ring_hom.id_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_padic_int_comp_from_padic_int_ext (x) :
(to_padic_int p).comp (from_padic_int p) x = ring_hom.id ℤ_[p] x | by rw to_padic_int_comp_from_padic_int | lemma | witt_vector.to_padic_int_comp_from_padic_int_ext | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_padic_int_comp_to_padic_int :
(from_padic_int p).comp (to_padic_int p) = ring_hom.id (𝕎 (zmod p)) | begin
apply witt_vector.hom_ext,
intro n,
rw [from_padic_int, ← ring_hom.comp_assoc, truncate_comp_lift, ring_hom.comp_assoc],
simp only [to_padic_int, to_zmod_pow, ring_hom.comp_id, padic_int.lift_spec, ring_hom.id_comp,
← ring_hom.comp_assoc, ring_equiv.to_ring_hom_comp_symm_to_ring_hom]
end | lemma | witt_vector.from_padic_int_comp_to_padic_int | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"padic_int.lift_spec",
"ring_equiv.to_ring_hom_comp_symm_to_ring_hom",
"ring_hom.comp_assoc",
"ring_hom.comp_id",
"ring_hom.id",
"ring_hom.id_comp",
"witt_vector.hom_ext",
"zmod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_padic_int_comp_to_padic_int_ext (x) :
(from_padic_int p).comp (to_padic_int p) x = ring_hom.id (𝕎 (zmod p)) x | by rw from_padic_int_comp_to_padic_int | lemma | witt_vector.from_padic_int_comp_to_padic_int_ext | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"ring_hom.id",
"zmod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv : 𝕎 (zmod p) ≃+* ℤ_[p] | { to_fun := to_padic_int p,
inv_fun := from_padic_int p,
left_inv := from_padic_int_comp_to_padic_int_ext _,
right_inv := to_padic_int_comp_from_padic_int_ext _,
map_mul' := ring_hom.map_mul _,
map_add' := ring_hom.map_add _ } | def | witt_vector.equiv | ring_theory.witt_vector | src/ring_theory/witt_vector/compare.lean | [
"ring_theory.witt_vector.truncated",
"ring_theory.witt_vector.identities",
"number_theory.padics.ring_homs"
] | [
"equiv",
"inv_fun",
"ring_hom.map_add",
"ring_hom.map_mul",
"zmod"
] | The ring of Witt vectors over `zmod p` is isomorphic to the ring of `p`-adic integers. This
equivalence is witnessed by `witt_vector.to_padic_int` with inverse `witt_vector.from_padic_int`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_vector (p : ℕ) (R : Type*) | mk [] :: (coeff : ℕ → R) | structure | witt_vector | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | `witt_vector p R` is the ring of `p`-typical Witt vectors over the commutative ring `R`,
where `p` is a prime number.
If `p` is invertible in `R`, this ring is isomorphic to `ℕ → R` (the product of `ℕ` copies of `R`).
If `R` is a ring of characteristic `p`, then `witt_vector p R` is a ring of characteristic `0`.
The c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {x y : 𝕎 R} (h : ∀ n, x.coeff n = y.coeff n) : x = y | begin
cases x,
cases y,
simp only at h,
simp [function.funext_iff, h]
end | lemma | witt_vector.ext | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"function.funext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {x y : 𝕎 R} : x = y ↔ ∀ n, x.coeff n = y.coeff n | ⟨λ h n, by rw h, ext⟩ | lemma | witt_vector.ext_iff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mk (x : ℕ → R) :
(mk p x).coeff = x | rfl | lemma | witt_vector.coeff_mk | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_zero : ℕ → mv_polynomial (fin 0 × ℕ) ℤ | witt_structure_int p 0 | def | witt_vector.witt_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for defining the element `0` of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_one : ℕ → mv_polynomial (fin 0 × ℕ) ℤ | witt_structure_int p 1 | def | witt_vector.witt_one | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for defining the element `1` of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_add : ℕ → mv_polynomial (fin 2 × ℕ) ℤ | witt_structure_int p (X 0 + X 1) | def | witt_vector.witt_add | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for defining the addition of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_nsmul (n : ℕ) : ℕ → mv_polynomial (fin 1 × ℕ) ℤ | witt_structure_int p (n • X 0) | def | witt_vector.witt_nsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for defining repeated addition of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_zsmul (n : ℤ) : ℕ → mv_polynomial (fin 1 × ℕ) ℤ | witt_structure_int p (n • X 0) | def | witt_vector.witt_zsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for defining repeated addition of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_sub : ℕ → mv_polynomial (fin 2 × ℕ) ℤ | witt_structure_int p (X 0 - X 1) | def | witt_vector.witt_sub | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for describing the subtraction of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_mul : ℕ → mv_polynomial (fin 2 × ℕ) ℤ | witt_structure_int p (X 0 * X 1) | def | witt_vector.witt_mul | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for defining the multiplication of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_neg : ℕ → mv_polynomial (fin 1 × ℕ) ℤ | witt_structure_int p (-X 0) | def | witt_vector.witt_neg | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for defining the negation of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
witt_pow (n : ℕ) : ℕ → mv_polynomial (fin 1 × ℕ) ℤ | witt_structure_int p (X 0 ^ n) | def | witt_vector.witt_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial",
"witt_structure_int"
] | The polynomials used for defining repeated addition of the ring of Witt vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
peval {k : ℕ} (φ : mv_polynomial (fin k × ℕ) ℤ) (x : fin k → ℕ → R) : R | aeval (function.uncurry x) φ | def | witt_vector.peval | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial"
] | An auxiliary definition used in `witt_vector.eval`.
Evaluates a polynomial whose variables come from the disjoint union of `k` copies of `ℕ`,
with a curried evaluation `x`.
This can be defined more generally but we use only a specific instance here. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval {k : ℕ} (φ : ℕ → mv_polynomial (fin k × ℕ) ℤ) (x : fin k → 𝕎 R) : 𝕎 R | mk p $ λ n, peval (φ n) $ λ i, (x i).coeff | def | witt_vector.eval | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"mv_polynomial"
] | Let `φ` be a family of polynomials, indexed by natural numbers, whose variables come from the
disjoint union of `k` copies of `ℕ`, and let `xᵢ` be a Witt vector for `0 ≤ i < k`.
`eval φ x` evaluates `φ` mapping the variable `X_(i, n)` to the `n`th coefficient of `xᵢ`.
Instantiating `φ` with certain polynomials define... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_nat_scalar : has_smul ℕ (𝕎 R) | ⟨λ n x, eval (witt_nsmul p n) ![x]⟩ | instance | witt_vector.has_nat_scalar | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_int_scalar : has_smul ℤ (𝕎 R) | ⟨λ n x, eval (witt_zsmul p n) ![x]⟩ | instance | witt_vector.has_int_scalar | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_pow : has_pow (𝕎 R) ℕ | ⟨λ x n, eval (witt_pow p n) ![x]⟩ | instance | witt_vector.has_nat_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_zero_eq_zero (n : ℕ) : witt_zero p n = 0 | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [witt_zero, witt_structure_rat, bind₁, aeval_zero',
constant_coeff_X_in_terms_of_W, ring_hom.map_zero,
alg_hom.map_zero, map_witt_structure_int],
end | lemma | witt_vector.witt_zero_eq_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"alg_hom.map_zero",
"constant_coeff_X_in_terms_of_W",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial.map_injective",
"ring_hom.map_zero",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_one_zero_eq_one : witt_one p 0 = 1 | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [witt_one, witt_structure_rat, X_in_terms_of_W_zero, alg_hom.map_one,
ring_hom.map_one, bind₁_X_right, map_witt_structure_int]
end | lemma | witt_vector.witt_one_zero_eq_one | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"X_in_terms_of_W_zero",
"alg_hom.map_one",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial.map_injective",
"ring_hom.map_one",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_one_pos_eq_zero (n : ℕ) (hn : 0 < n) : witt_one p n = 0 | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [witt_one, witt_structure_rat, ring_hom.map_zero, alg_hom.map_one,
ring_hom.map_one, map_witt_structure_int],
revert hn, apply nat.strong_induction_on n, clear n,
intros n IH hn,
rw X_in_terms_of_W_eq,
simp only ... | lemma | witt_vector.witt_one_pos_eq_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"X_in_terms_of_W_eq",
"X_in_terms_of_W_zero",
"alg_hom.map_mul",
"alg_hom.map_one",
"alg_hom.map_pow",
"alg_hom.map_sub",
"alg_hom.map_sum",
"finset.mem_range",
"int.cast_injective",
"int.cast_ring_hom",
"inv_of_eq_inv",
"inv_pow",
"map_witt_structure_int",
"mul_zero",
"mv_polynomial.map... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_add_zero : witt_add p 0 = X (0,0) + X (1,0) | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [witt_add, witt_structure_rat, alg_hom.map_add, ring_hom.map_add,
rename_X, X_in_terms_of_W_zero, map_X,
witt_polynomial_zero, bind₁_X_right, map_witt_structure_int],
end | lemma | witt_vector.witt_add_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"X_in_terms_of_W_zero",
"alg_hom.map_add",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial.map_injective",
"ring_hom.map_add",
"witt_polynomial_zero",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_sub_zero : witt_sub p 0 = X (0,0) - X (1,0) | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [witt_sub, witt_structure_rat, alg_hom.map_sub, ring_hom.map_sub,
rename_X, X_in_terms_of_W_zero, map_X,
witt_polynomial_zero, bind₁_X_right, map_witt_structure_int],
end | lemma | witt_vector.witt_sub_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"X_in_terms_of_W_zero",
"alg_hom.map_sub",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial.map_injective",
"ring_hom.map_sub",
"witt_polynomial_zero",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_mul_zero : witt_mul p 0 = X (0,0) * X (1,0) | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [witt_mul, witt_structure_rat, rename_X, X_in_terms_of_W_zero, map_X,
witt_polynomial_zero, ring_hom.map_mul,
bind₁_X_right, alg_hom.map_mul, map_witt_structure_int]
end | lemma | witt_vector.witt_mul_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"X_in_terms_of_W_zero",
"alg_hom.map_mul",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial.map_injective",
"ring_hom.map_mul",
"witt_polynomial_zero",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_neg_zero : witt_neg p 0 = - X (0,0) | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [witt_neg, witt_structure_rat, rename_X, X_in_terms_of_W_zero, map_X,
witt_polynomial_zero, ring_hom.map_neg,
alg_hom.map_neg, bind₁_X_right, map_witt_structure_int]
end | lemma | witt_vector.witt_neg_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"X_in_terms_of_W_zero",
"alg_hom.map_neg",
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_structure_int",
"mv_polynomial.map_injective",
"ring_hom.map_neg",
"witt_polynomial_zero",
"witt_structure_rat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_add (n : ℕ) :
constant_coeff (witt_add p n) = 0 | begin
apply constant_coeff_witt_structure_int p _ _ n,
simp only [add_zero, ring_hom.map_add, constant_coeff_X],
end | lemma | witt_vector.constant_coeff_witt_add | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"constant_coeff_witt_structure_int",
"ring_hom.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_sub (n : ℕ) :
constant_coeff (witt_sub p n) = 0 | begin
apply constant_coeff_witt_structure_int p _ _ n,
simp only [sub_zero, ring_hom.map_sub, constant_coeff_X],
end | lemma | witt_vector.constant_coeff_witt_sub | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"constant_coeff_witt_structure_int",
"ring_hom.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_mul (n : ℕ) :
constant_coeff (witt_mul p n) = 0 | begin
apply constant_coeff_witt_structure_int p _ _ n,
simp only [mul_zero, ring_hom.map_mul, constant_coeff_X],
end | lemma | witt_vector.constant_coeff_witt_mul | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"constant_coeff_witt_structure_int",
"mul_zero",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_neg (n : ℕ) :
constant_coeff (witt_neg p n) = 0 | begin
apply constant_coeff_witt_structure_int p _ _ n,
simp only [neg_zero, ring_hom.map_neg, constant_coeff_X],
end | lemma | witt_vector.constant_coeff_witt_neg | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"constant_coeff_witt_structure_int",
"ring_hom.map_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_nsmul (m : ℕ) (n : ℕ):
constant_coeff (witt_nsmul p m n) = 0 | begin
apply constant_coeff_witt_structure_int p _ _ n,
simp only [smul_zero, map_nsmul, constant_coeff_X],
end | lemma | witt_vector.constant_coeff_witt_nsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"constant_coeff_witt_structure_int",
"map_nsmul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_witt_zsmul (z : ℤ) (n : ℕ):
constant_coeff (witt_zsmul p z n) = 0 | begin
apply constant_coeff_witt_structure_int p _ _ n,
simp only [smul_zero, map_zsmul, constant_coeff_X],
end | lemma | witt_vector.constant_coeff_witt_zsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"constant_coeff_witt_structure_int",
"map_zsmul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_coeff (n : ℕ) : (0 : 𝕎 R).coeff n = 0 | show (aeval _ (witt_zero p n) : R) = 0,
by simp only [witt_zero_eq_zero, alg_hom.map_zero] | lemma | witt_vector.zero_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"alg_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_coeff_zero : (1 : 𝕎 R).coeff 0 = 1 | show (aeval _ (witt_one p 0) : R) = 1,
by simp only [witt_one_zero_eq_one, alg_hom.map_one] | lemma | witt_vector.one_coeff_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"alg_hom.map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_coeff_eq_of_pos (n : ℕ) (hn : 0 < n) : coeff (1 : 𝕎 R) n = 0 | show (aeval _ (witt_one p n) : R) = 0,
by simp only [hn, witt_one_pos_eq_zero, alg_hom.map_zero] | lemma | witt_vector.one_coeff_eq_of_pos | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"alg_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
v2_coeff {p' R'} (x y : witt_vector p' R') (i : fin 2) :
(![x, y] i).coeff = ![x.coeff, y.coeff] i | by fin_cases i; simp | lemma | witt_vector.v2_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_coeff (x y : 𝕎 R) (n : ℕ) :
(x + y).coeff n = peval (witt_add p n) ![x.coeff, y.coeff] | by simp [(+), eval] | lemma | witt_vector.add_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_coeff (x y : 𝕎 R) (n : ℕ) :
(x - y).coeff n = peval (witt_sub p n) ![x.coeff, y.coeff] | by simp [has_sub.sub, eval] | lemma | witt_vector.sub_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coeff (x y : 𝕎 R) (n : ℕ) :
(x * y).coeff n = peval (witt_mul p n) ![x.coeff, y.coeff] | by simp [(*), eval] | lemma | witt_vector.mul_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_coeff (x : 𝕎 R) (n : ℕ) :
(-x).coeff n = peval (witt_neg p n) ![x.coeff] | by simp [has_neg.neg, eval, matrix.cons_fin_one] | lemma | witt_vector.neg_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"matrix.cons_fin_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_coeff (m : ℕ) (x : 𝕎 R) (n : ℕ) :
(m • x).coeff n = peval (witt_nsmul p m n) ![x.coeff] | by simp [has_smul.smul, eval, matrix.cons_fin_one] | lemma | witt_vector.nsmul_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"matrix.cons_fin_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_coeff (m : ℤ) (x : 𝕎 R) (n : ℕ) :
(m • x).coeff n = peval (witt_zsmul p m n) ![x.coeff] | by simp [has_smul.smul, eval, matrix.cons_fin_one] | lemma | witt_vector.zsmul_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"matrix.cons_fin_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_coeff (m : ℕ) (x : 𝕎 R) (n : ℕ) :
(x ^ m).coeff n = peval (witt_pow p m n) ![x.coeff] | by simp [has_pow.pow, eval, matrix.cons_fin_one] | lemma | witt_vector.pow_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"matrix.cons_fin_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_coeff_zero (x y : 𝕎 R) : (x + y).coeff 0 = x.coeff 0 + y.coeff 0 | by simp [add_coeff, peval] | lemma | witt_vector.add_coeff_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coeff_zero (x y : 𝕎 R) : (x * y).coeff 0 = x.coeff 0 * y.coeff 0 | by simp [mul_coeff, peval] | lemma | witt_vector.mul_coeff_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_add_vars (n : ℕ) : (witt_add p n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | witt_structure_int_vars _ _ _ | lemma | witt_vector.witt_add_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"finset.range",
"finset.univ",
"witt_structure_int_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_sub_vars (n : ℕ) : (witt_sub p n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | witt_structure_int_vars _ _ _ | lemma | witt_vector.witt_sub_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"finset.range",
"finset.univ",
"witt_structure_int_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_mul_vars (n : ℕ) : (witt_mul p n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | witt_structure_int_vars _ _ _ | lemma | witt_vector.witt_mul_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"finset.range",
"finset.univ",
"witt_structure_int_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_neg_vars (n : ℕ) : (witt_neg p n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | witt_structure_int_vars _ _ _ | lemma | witt_vector.witt_neg_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"finset.range",
"finset.univ",
"witt_structure_int_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_nsmul_vars (m : ℕ) (n : ℕ) :
(witt_nsmul p m n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | witt_structure_int_vars _ _ _ | lemma | witt_vector.witt_nsmul_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"finset.range",
"finset.univ",
"witt_structure_int_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_zsmul_vars (m : ℤ) (n : ℕ) :
(witt_zsmul p m n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | witt_structure_int_vars _ _ _ | lemma | witt_vector.witt_zsmul_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"finset.range",
"finset.univ",
"witt_structure_int_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
witt_pow_vars (m : ℕ) (n : ℕ) :
(witt_pow p m n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) | witt_structure_int_vars _ _ _ | lemma | witt_vector.witt_pow_vars | ring_theory.witt_vector | src/ring_theory/witt_vector/defs.lean | [
"ring_theory.witt_vector.structure_polynomial"
] | [
"finset.range",
"finset.univ",
"witt_structure_int_vars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_nth_val_units (n : ℕ) (a : units k) (A : 𝕎 k) (bs : fin (n+1) → k) : k | - ↑(a⁻¹ ^ (p^(n+1)))
* (A.coeff (n + 1) * ↑(a⁻¹ ^ (p^(n+1))) + nth_remainder p n (truncate_fun (n+1) A) bs) | def | witt_vector.succ_nth_val_units | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"units"
] | This is the `n+1`st coefficient of our inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_coeff (a : units k) (A : 𝕎 k) : ℕ → k | | 0 := ↑a⁻¹
| (n + 1) := succ_nth_val_units n a A (λ i, inverse_coeff i.val)
using_well_founded { dec_tac := `[apply fin.is_lt] } | def | witt_vector.inverse_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"fin.is_lt",
"units"
] | Recursively defines the sequence of coefficients for the inverse to a Witt vector whose first entry
is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_unit {a : units k} {A : 𝕎 k} (hA : A.coeff 0 = a) : units (𝕎 k) | units.mk_of_mul_eq_one A (witt_vector.mk p (inverse_coeff a A))
begin
ext n,
induction n with n ih,
{ simp [witt_vector.mul_coeff_zero, inverse_coeff, hA] },
let H_coeff := A.coeff (n + 1) * ↑(a⁻¹ ^ p ^ (n + 1))
+ nth_remainder p n (truncate_fun (n + 1) A) (λ (i : fin (n + 1)), inverse_coeff a A... | def | witt_vector.mk_unit | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"fin.val_eq_coe",
"ih",
"inv_pow",
"nat.succ_pos'",
"normalize",
"units",
"units.mk_of_mul_eq_one",
"units.mul_inv",
"witt_vector.mul_coeff_zero"
] | Upgrade a Witt vector `A` whose first entry `A.coeff 0` is a unit to be, itself, a unit in `𝕎 k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk_unit {a : units k} {A : 𝕎 k} (hA : A.coeff 0 = a) : (mk_unit hA : 𝕎 k) = A | rfl | lemma | witt_vector.coe_mk_unit | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_coeff_zero_ne_zero (x : 𝕎 k) (hx : x.coeff 0 ≠ 0) : is_unit x | begin
let y : kˣ := units.mk0 (x.coeff 0) hx,
have hy : x.coeff 0 = y := rfl,
exact (mk_unit hy).is_unit
end | lemma | witt_vector.is_unit_of_coeff_zero_ne_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"is_unit",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible : irreducible (p : 𝕎 k) | begin
have hp : ¬ is_unit (p : 𝕎 k),
{ intro hp,
simpa only [constant_coeff_apply, coeff_p_zero, not_is_unit_zero]
using (constant_coeff : witt_vector p k →+* _).is_unit_map hp, },
refine ⟨hp, λ a b hab, _⟩,
obtain ⟨ha0, hb0⟩ : a ≠ 0 ∧ b ≠ 0,
{ rw ← mul_ne_zero_iff, intro h, rw h at hab, exact p_no... | lemma | witt_vector.irreducible | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"function.iterate_succ'",
"irreducible",
"is_unit",
"mul_ne_zero_iff",
"not_is_unit_zero",
"one_ne_zero",
"witt_vector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_pow_p_mul (a : 𝕎 k) (ha : a ≠ 0) :
∃ (m : ℕ) (b : 𝕎 k), b.coeff 0 ≠ 0 ∧ a = p ^ m * b | begin
obtain ⟨m, c, hc, hcm⟩ := witt_vector.verschiebung_nonzero ha,
obtain ⟨b, rfl⟩ := (frobenius_bijective p k).surjective.iterate m c,
rw witt_vector.iterate_frobenius_coeff at hc,
have := congr_fun (witt_vector.verschiebung_frobenius_comm.comp_iterate m) b,
simp only [function.comp_app] at this,
rw ← th... | lemma | witt_vector.exists_eq_pow_p_mul | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"mul_comm",
"mul_left_iterate",
"nat.prime.pos",
"pow_pos",
"witt_vector.iterate_frobenius_coeff",
"witt_vector.verschiebung_frobenius",
"witt_vector.verschiebung_nonzero",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_pow_p_mul' (a : 𝕎 k) (ha : a ≠ 0) :
∃ (m : ℕ) (b : units (𝕎 k)), a = p ^ m * b | begin
obtain ⟨m, b, h₁, h₂⟩ := exists_eq_pow_p_mul a ha,
let b₀ := units.mk0 (b.coeff 0) h₁,
have hb₀ : b.coeff 0 = b₀ := rfl,
exact ⟨m, mk_unit hb₀, h₂⟩,
end | lemma | witt_vector.exists_eq_pow_p_mul' | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"units",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_valuation_ring : discrete_valuation_ring (𝕎 k) | discrete_valuation_ring.of_has_unit_mul_pow_irreducible_factorization
begin
refine ⟨p, irreducible p, λ x hx, _⟩,
obtain ⟨n, b, hb⟩ := exists_eq_pow_p_mul' x hx,
exact ⟨n, b, hb.symm⟩,
end | lemma | witt_vector.discrete_valuation_ring | ring_theory.witt_vector | src/ring_theory/witt_vector/discrete_valuation_ring.lean | [
"ring_theory.witt_vector.domain",
"ring_theory.witt_vector.mul_coeff",
"ring_theory.discrete_valuation_ring.basic",
"tactic.linear_combination"
] | [
"discrete_valuation_ring",
"discrete_valuation_ring.of_has_unit_mul_pow_irreducible_factorization",
"irreducible"
] | The ring of Witt Vectors of a perfect field of positive characteristic is a DVR. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
shift (x : 𝕎 R) (n : ℕ) : 𝕎 R | mk p (λ i, x.coeff (n + i)) | def | witt_vector.shift | ring_theory.witt_vector | src/ring_theory/witt_vector/domain.lean | [
"ring_theory.witt_vector.identities"
] | [] | `witt_vector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`witt_vector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `witt_vector.verschiebung_nonzero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) | rfl | lemma | witt_vector.shift_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/domain.lean | [
"ring_theory.witt_vector.identities"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k+1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k | begin
ext ⟨j⟩,
{ rw [verschiebung_coeff_zero, shift_coeff, h],
apply nat.lt_succ_self },
{ simp only [verschiebung_coeff_succ, shift],
congr' 1,
rw [nat.add_succ, add_comm, nat.add_succ, add_comm] }
end | lemma | witt_vector.verschiebung_shift | ring_theory.witt_vector | src/ring_theory/witt_vector/domain.lean | [
"ring_theory.witt_vector.identities"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = (verschiebung^[n] (x.shift n)) | begin
induction n with k ih,
{ cases x; simp [shift] },
{ dsimp, rw verschiebung_shift,
{ exact ih (λ i hi, h _ (hi.trans (nat.lt_succ_self _))), },
{ exact h } }
end | lemma | witt_vector.eq_iterate_verschiebung | ring_theory.witt_vector | src/ring_theory/witt_vector/domain.lean | [
"ring_theory.witt_vector.identities"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = (verschiebung^[n] x') | begin
have hex : ∃ k : ℕ, x.coeff k ≠ 0,
{ by_contra' hall,
apply hx,
ext i,
simp only [hall, zero_coeff] },
let n := nat.find hex,
use [n, x.shift n],
refine ⟨nat.find_spec hex, eq_iterate_verschiebung (λ i hi, not_not.mp _)⟩,
exact nat.find_min hex hi,
end | lemma | witt_vector.verschiebung_nonzero | ring_theory.witt_vector | src/ring_theory/witt_vector/domain.lean | [
"ring_theory.witt_vector.identities"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_poly_rat (n : ℕ) : mv_polynomial ℕ ℚ | bind₁ (witt_polynomial p ℚ ∘ λ n, n + 1) (X_in_terms_of_W p ℚ n) | def | witt_vector.frobenius_poly_rat | 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"
] | [
"X_in_terms_of_W",
"mv_polynomial",
"witt_polynomial"
] | The rational polynomials that give the coefficients of `frobenius x`,
in terms of the coefficients of `x`.
These polynomials actually have integral coefficients,
see `frobenius_poly` and `map_frobenius_poly`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bind₁_frobenius_poly_rat_witt_polynomial (n : ℕ) :
bind₁ (frobenius_poly_rat p) (witt_polynomial p ℚ n) = (witt_polynomial p ℚ (n+1)) | begin
delta frobenius_poly_rat,
rw [← bind₁_bind₁, bind₁_X_in_terms_of_W_witt_polynomial, bind₁_X_right],
end | lemma | witt_vector.bind₁_frobenius_poly_rat_witt_polynomial | 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"
] | [
"bind₁_X_in_terms_of_W_witt_polynomial",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pnat_multiplicity (n : ℕ+) : ℕ | (multiplicity p n).get $ multiplicity.finite_nat_iff.mpr $ ⟨ne_of_gt hp.1.one_lt, n.2⟩ | def | witt_vector.pnat_multiplicity | 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"
] | [
"multiplicity"
] | An auxiliary definition, to avoid an excessive amount of finiteness proofs
for `multiplicity p n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_poly_aux : ℕ → mv_polynomial ℕ ℤ | | n := X (n + 1) - ∑ i : fin n, have _ := i.is_lt,
∑ j in range (p ^ (n - i)),
(X i ^ p) ^ (p ^ (n - i) - (j + 1)) *
(frobenius_poly_aux i) ^ (j + 1) *
C ↑((p ^ (n - i)).choose (j + 1) / (p ^ (n - i - v p ⟨j + 1, nat.succ_pos j⟩)) *
↑p ^ (j - v p ⟨j + 1, nat.succ_pos j⟩) : ℕ) | def | witt_vector.frobenius_poly_aux | 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"
] | [
"mv_polynomial"
] | An auxiliary polynomial over the integers, that satisfies
`p * (frobenius_poly_aux p n) + X n ^ p = frobenius_poly p n`.
This makes it easy to show that `frobenius_poly p n` is congruent to `X n ^ p`
modulo `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_poly_aux_eq (n : ℕ) :
frobenius_poly_aux p n =
X (n + 1) - ∑ i in range n, ∑ j in range (p ^ (n - i)),
(X i ^ p) ^ (p ^ (n - i) - (j + 1)) *
(frobenius_poly_aux p i) ^ (j + 1) *
C ↑((p ^ (n - i)).choose (j + 1) / (p ^ (n - i - v p ⟨j + 1, nat.succ_pos j⟩)) *
↑p ^ (j - v p ⟨j + 1, nat.suc... | by { rw [frobenius_poly_aux, ← fin.sum_univ_eq_sum_range] } | lemma | witt_vector.frobenius_poly_aux_eq | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_poly (n : ℕ) : mv_polynomial ℕ ℤ | X n ^ p + C ↑p * (frobenius_poly_aux p n) | def | witt_vector.frobenius_poly | 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"
] | [
"mv_polynomial"
] | The polynomials that give the coefficients of `frobenius x`,
in terms of the coefficients of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_frobenius_poly.key₁ (n j : ℕ) (hj : j < p ^ (n)) :
p ^ (n - v p ⟨j + 1, j.succ_pos⟩) ∣ (p ^ n).choose (j + 1) | begin
apply multiplicity.pow_dvd_of_le_multiplicity,
rw [hp.out.multiplicity_choose_prime_pow hj j.succ_ne_zero],
refl,
end | lemma | witt_vector.map_frobenius_poly.key₁ | 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"
] | [
"multiplicity.pow_dvd_of_le_multiplicity"
] | A key divisibility fact for the proof of `witt_vector.map_frobenius_poly`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_frobenius_poly.key₂ {n i j : ℕ} (hi : i ≤ n) (hj : j < p ^ (n - i)) :
j - v p ⟨j + 1, j.succ_pos⟩ + n = i + j + (n - i - v p ⟨j + 1, j.succ_pos⟩) | begin
generalize h : (v p ⟨j + 1, j.succ_pos⟩) = m,
rsuffices ⟨h₁, h₂⟩ : m ≤ n - i ∧ m ≤ j,
{ rw [tsub_add_eq_add_tsub h₂, add_comm i j,
add_tsub_assoc_of_le (h₁.trans (nat.sub_le n i)), add_assoc, tsub_right_comm, add_comm i,
tsub_add_cancel_of_le (le_tsub_of_add_le_right ((le_tsub_iff_left hi).mp h₁... | lemma | witt_vector.map_frobenius_poly.key₂ | 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"
] | [
"add_tsub_assoc_of_le",
"le_tsub_iff_left",
"le_tsub_of_add_le_right",
"multiplicity.pow_multiplicity_dvd",
"nat.lt_pow_self",
"pow_le_pow_iff",
"tsub_add_cancel_of_le",
"tsub_add_eq_add_tsub",
"tsub_right_comm"
] | A key numerical identity needed for the proof of `witt_vector.map_frobenius_poly`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_frobenius_poly (n : ℕ) :
mv_polynomial.map (int.cast_ring_hom ℚ) (frobenius_poly p n) = frobenius_poly_rat p n | begin
rw [frobenius_poly, ring_hom.map_add, ring_hom.map_mul, ring_hom.map_pow, map_C, map_X,
eq_int_cast, int.cast_coe_nat, frobenius_poly_rat],
apply nat.strong_induction_on n, clear n,
intros n IH,
rw [X_in_terms_of_W_eq],
simp only [alg_hom.map_sum, alg_hom.map_sub, alg_hom.map_mul, alg_hom.map_pow,... | lemma | witt_vector.map_frobenius_poly | 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"
] | [
"X_in_terms_of_W_eq",
"add_pow",
"add_tsub_cancel_left",
"alg_hom.map_mul",
"alg_hom.map_pow",
"alg_hom.map_sub",
"alg_hom.map_sum",
"aux",
"eq_int_cast",
"int.cast_coe_nat",
"int.cast_mul",
"int.cast_ring_hom",
"inv_of_eq_inv",
"inv_pow",
"mul_assoc",
"mul_comm",
"mul_inv_of_self",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_poly_zmod (n : ℕ) :
mv_polynomial.map (int.cast_ring_hom (zmod p)) (frobenius_poly p n) = X n ^ p | begin
rw [frobenius_poly, ring_hom.map_add, ring_hom.map_pow, ring_hom.map_mul, map_X, map_C],
simp only [int.cast_coe_nat, add_zero, eq_int_cast, zmod.nat_cast_self, zero_mul, C_0],
end | lemma | witt_vector.frobenius_poly_zmod | 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"
] | [
"eq_int_cast",
"int.cast_coe_nat",
"int.cast_ring_hom",
"mv_polynomial.map",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_pow",
"zero_mul",
"zmod",
"zmod.nat_cast_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind₁_frobenius_poly_witt_polynomial (n : ℕ) :
bind₁ (frobenius_poly p) (witt_polynomial p ℤ n) = (witt_polynomial p ℤ (n+1)) | begin
apply mv_polynomial.map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [map_bind₁, map_frobenius_poly, bind₁_frobenius_poly_rat_witt_polynomial,
map_witt_polynomial],
end | lemma | witt_vector.bind₁_frobenius_poly_witt_polynomial | 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"
] | [
"int.cast_injective",
"int.cast_ring_hom",
"map_witt_polynomial",
"mv_polynomial.map_injective",
"witt_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_fun (x : 𝕎 R) : 𝕎 R | mk p $ λ n, mv_polynomial.aeval x.coeff (frobenius_poly p n) | def | witt_vector.frobenius_fun | 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"
] | [
"mv_polynomial.aeval"
] | `frobenius_fun` is the function underlying the ring endomorphism
`frobenius : 𝕎 R →+* frobenius 𝕎 R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_frobenius_fun (x : 𝕎 R) (n : ℕ) :
coeff (frobenius_fun x) n = mv_polynomial.aeval x.coeff (frobenius_poly p n) | by rw [frobenius_fun, coeff_mk] | lemma | witt_vector.coeff_frobenius_fun | 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"
] | [
"mv_polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_fun_is_poly : is_poly p (λ R _Rcr, @frobenius_fun p R _ _Rcr) | ⟨⟨frobenius_poly p, by { introsI, funext n, apply coeff_frobenius_fun }⟩⟩ | lemma | witt_vector.frobenius_fun_is_poly | 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"
] | [
"is_poly"
] | `frobenius_fun` is tautologically a polynomial function.
See also `frobenius_is_poly`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_component_frobenius_fun (n : ℕ) (x : 𝕎 R) :
ghost_component n (frobenius_fun x) = ghost_component (n + 1) x | by simp only [ghost_component_apply, frobenius_fun, coeff_mk,
← bind₁_frobenius_poly_witt_polynomial, aeval_bind₁] | lemma | witt_vector.ghost_component_frobenius_fun | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius : 𝕎 R →+* 𝕎 R | { to_fun := frobenius_fun,
map_zero' :=
begin
refine is_poly.ext
((frobenius_fun_is_poly p).comp (witt_vector.zero_is_poly))
((witt_vector.zero_is_poly).comp (frobenius_fun_is_poly p)) _ _ 0,
ghost_simp
end,
map_one' :=
begin
refine is_poly.ext
((frobenius_fun_is_poly p).comp (wi... | def | witt_vector.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",
"witt_vector.one_is_poly",
"witt_vector.zero_is_poly"
] | If `R` has characteristic `p`, then there is a ring endomorphism
that raises `r : R` to the power `p`.
By applying `witt_vector.map` to this endomorphism,
we obtain a ring endomorphism `frobenius R p : 𝕎 R →+* 𝕎 R`.
The underlying function of this morphism is `witt_vector.frobenius_fun`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_frobenius (x : 𝕎 R) (n : ℕ) :
coeff (frobenius x) n = mv_polynomial.aeval x.coeff (frobenius_poly p n) | coeff_frobenius_fun _ _ | lemma | witt_vector.coeff_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",
"mv_polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_component_frobenius (n : ℕ) (x : 𝕎 R) :
ghost_component n (frobenius x) = ghost_component (n + 1) x | ghost_component_frobenius_fun _ _ | lemma | witt_vector.ghost_component_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_is_poly : is_poly p (λ R _Rcr, @frobenius p R _ _Rcr) | frobenius_fun_is_poly _ | lemma | witt_vector.frobenius_is_poly | 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",
"is_poly"
] | `frobenius` is tautologically a polynomial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_frobenius_char_p (x : 𝕎 R) (n : ℕ) :
coeff (frobenius x) n = (x.coeff n) ^ p | begin
rw [coeff_frobenius],
letI : algebra (zmod p) R := zmod.algebra _ _,
-- outline of the calculation, proofs follow below
calc aeval (λ k, x.coeff k) (frobenius_poly p n)
= aeval (λ k, x.coeff k)
(mv_polynomial.map (int.cast_ring_hom (zmod p)) (frobenius_poly p n)) : _
... = aeval (λ k, x.... | lemma | witt_vector.coeff_frobenius_char_p | 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"
] | [
"alg_hom.map_pow",
"algebra",
"frobenius",
"int.cast_ring_hom",
"mv_polynomial",
"mv_polynomial.map",
"ring_hom.ext_int",
"zmod",
"zmod.algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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