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
value |
|---|---|---|---|---|---|---|---|---|---|---|
coeff_pow_p' (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) f ^ p = coeff R p n f | f.2 n | lemma | perfection.coeff_pow_p' | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_frobenius (f : ring.perfection R p) (n : ℕ) :
coeff R p (n + 1) (frobenius _ p f) = coeff R p n f | by apply coeff_pow_p f n | lemma | perfection.coeff_frobenius | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"frobenius",
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_iterate_frobenius (f : ring.perfection R p) (n m : ℕ) :
coeff R p (n + m) (frobenius _ p ^[m] f) = coeff R p n f | nat.rec_on m rfl $ λ m ih, by erw [function.iterate_succ_apply', coeff_frobenius, ih] | lemma | perfection.coeff_iterate_frobenius | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"frobenius",
"function.iterate_succ_apply'",
"ih",
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_iterate_frobenius' (f : ring.perfection R p) (n m : ℕ) (hmn : m ≤ n) :
coeff R p n (frobenius _ p ^[m] f) = coeff R p (n - m) f | eq.symm $ (coeff_iterate_frobenius _ _ m).symm.trans $ (tsub_add_cancel_of_le hmn).symm ▸ rfl | lemma | perfection.coeff_iterate_frobenius' | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"frobenius",
"ring.perfection",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pth_root_frobenius : (pth_root R p).comp (frobenius _ p) = ring_hom.id _ | ring_hom.ext $ λ x, ext $ λ n,
by rw [ring_hom.comp_apply, ring_hom.id_apply, coeff_pth_root, coeff_frobenius] | lemma | perfection.pth_root_frobenius | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"frobenius",
"pth_root",
"pth_root_frobenius",
"ring_hom.comp_apply",
"ring_hom.ext",
"ring_hom.id",
"ring_hom.id_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_pth_root : (frobenius _ p).comp (pth_root R p) = ring_hom.id _ | ring_hom.ext $ λ x, ext $ λ n,
by rw [ring_hom.comp_apply, ring_hom.id_apply, ring_hom.map_frobenius, coeff_pth_root,
← ring_hom.map_frobenius, coeff_frobenius] | lemma | perfection.frobenius_pth_root | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"frobenius",
"frobenius_pth_root",
"pth_root",
"ring_hom.comp_apply",
"ring_hom.ext",
"ring_hom.id",
"ring_hom.id_apply",
"ring_hom.map_frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_add_ne_zero {f : ring.perfection R p} {n : ℕ} (hfn : coeff R p n f ≠ 0) (k : ℕ) :
coeff R p (n + k) f ≠ 0 | nat.rec_on k hfn $ λ k ih h, ih $ by erw [← coeff_pow_p, ring_hom.map_pow, h, zero_pow hp.1.pos] | lemma | perfection.coeff_add_ne_zero | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ih",
"ring.perfection",
"ring_hom.map_pow",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_ne_zero_of_le {f : ring.perfection R p} {m n : ℕ} (hfm : coeff R p m f ≠ 0)
(hmn : m ≤ n) : coeff R p n f ≠ 0 | let ⟨k, hk⟩ := nat.exists_eq_add_of_le hmn in hk.symm ▸ coeff_add_ne_zero hfm k | lemma | perfection.coeff_ne_zero_of_le | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"nat.exists_eq_add_of_le",
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
perfect_ring : perfect_ring (ring.perfection R p) p | { pth_root' := pth_root R p,
frobenius_pth_root' := congr_fun $ congr_arg ring_hom.to_fun $ @frobenius_pth_root R _ p _ _,
pth_root_frobenius' := congr_fun $ congr_arg ring_hom.to_fun $ @pth_root_frobenius R _ p _ _ } | instance | perfection.perfect_ring | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"frobenius_pth_root",
"perfect_ring",
"pth_root",
"pth_root_frobenius",
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (R : Type u₁) [comm_semiring R] [char_p R p] [perfect_ring R p]
(S : Type u₂) [comm_semiring S] [char_p S p] :
(R →+* S) ≃ (R →+* ring.perfection S p) | { to_fun := λ f,
{ to_fun := λ r, ⟨λ n, f $ _root_.pth_root R p ^[n] r,
λ n, by rw [← f.map_pow, function.iterate_succ_apply', pth_root_pow_p]⟩,
map_one' := ext $ λ n, (congr_arg f $ ring_hom.iterate_map_one _ _).trans f.map_one,
map_mul' := λ x y, ext $ λ n, (congr_arg f $ ring_hom.iterate_map_mul _ _ ... | def | perfection.lift | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_semiring",
"function.iterate_succ_apply'",
"inv_fun",
"lift",
"perfect_ring",
"pth_root_pow_p",
"ring.perfection",
"ring_hom.comp",
"ring_hom.ext",
"ring_hom.iterate_map_add",
"ring_hom.iterate_map_mul",
"ring_hom.iterate_map_one",
"ring_hom.iterate_map_zero",
"ring_hom.m... | Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* perfection S p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_ext {R : Type u₁} [comm_semiring R] [char_p R p] [perfect_ring R p]
{S : Type u₂} [comm_semiring S] [char_p S p] {f g : R →+* ring.perfection S p}
(hfg : ∀ x, coeff S p 0 (f x) = coeff S p 0 (g x)) : f = g | (lift p R S).symm.injective $ ring_hom.ext hfg | lemma | perfection.hom_ext | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_semiring",
"hom_ext",
"lift",
"perfect_ring",
"ring.perfection",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (φ : R →+* S) : ring.perfection R p →+* ring.perfection S p | { to_fun := λ f, ⟨λ n, φ (coeff R p n f), λ n, by rw [← φ.map_pow, coeff_pow_p']⟩,
map_one' := subtype.eq $ funext $ λ n, φ.map_one,
map_mul' := λ f g, subtype.eq $ funext $ λ n, φ.map_mul _ _,
map_zero' := subtype.eq $ funext $ λ n, φ.map_zero,
map_add' := λ f g, subtype.eq $ funext $ λ n, φ.map_add _ _ } | def | perfection.map | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ring.perfection"
] | A ring homomorphism `R →+* S` induces `perfection R p →+* perfection S p` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_map (φ : R →+* S) (f : ring.perfection R p) (n : ℕ) :
coeff S p n (map p φ f) = φ (coeff R p n f) | rfl | lemma | perfection.coeff_map | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
perfection_map (p : ℕ) [fact p.prime]
{R : Type u₁} [comm_semiring R] [char_p R p]
{P : Type u₂} [comm_semiring P] [char_p P p] [perfect_ring P p] (π : P →+* R) : Prop | (injective : ∀ ⦃x y : P⦄, (∀ n, π (pth_root P p ^[n] x) = π (pth_root P p ^[n] y)) → x = y)
(surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) →
∃ x : P, ∀ n, π (pth_root P p ^[n] x) = f n) | structure | perfection_map | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_semiring",
"fact",
"perfect_ring",
"pth_root"
] | A perfection map to a ring of characteristic `p` is a map that is isomorphic
to its perfection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk' {f : P →+* R} (g : P ≃+* ring.perfection R p)
(hfg : perfection.lift p P R f = g) :
perfection_map p f | { injective := λ x y hxy, g.injective $ (ring_hom.ext_iff.1 hfg x).symm.trans $
eq.symm $ (ring_hom.ext_iff.1 hfg y).symm.trans $ perfection.ext $ λ n, (hxy n).symm,
surjective := λ y hy, let ⟨x, hx⟩ := g.surjective ⟨y, hy⟩ in
⟨x, λ n, show perfection.coeff R p n (perfection.lift p P R f x) =
perfecti... | lemma | perfection_map.mk' | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"coe_fn_coe_base",
"mk'",
"perfection.coeff",
"perfection.ext",
"perfection.lift",
"perfection_map",
"ring.perfection"
] | Create a `perfection_map` from an isomorphism to the perfection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of : perfection_map p (perfection.coeff R p 0) | mk' (ring_equiv.refl _) $ (equiv.apply_eq_iff_eq_symm_apply _).2 rfl | lemma | perfection_map.of | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"equiv.apply_eq_iff_eq_symm_apply",
"mk'",
"perfection.coeff",
"perfection_map",
"ring_equiv.refl"
] | The canonical perfection map from the perfection of a ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id [perfect_ring R p] : perfection_map p (ring_hom.id R) | { injective := λ x y hxy, hxy 0,
surjective := λ f hf, ⟨f 0, λ n, show pth_root R p ^[n] (f 0) = f n,
from nat.rec_on n rfl $ λ n ih, injective_pow_p p $
by rw [function.iterate_succ_apply', pth_root_pow_p _, ih, hf]⟩ } | lemma | perfection_map.id | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"function.iterate_succ_apply'",
"ih",
"injective_pow_p",
"perfect_ring",
"perfection_map",
"pth_root",
"pth_root_pow_p",
"ring_hom.id"
] | For a perfect ring, it itself is the perfection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv {π : P →+* R} (m : perfection_map p π) : P ≃+* ring.perfection R p | ring_equiv.of_bijective (perfection.lift p P R π)
⟨λ x y hxy, m.injective $ λ n, (congr_arg (perfection.coeff R p n) hxy : _),
λ f, let ⟨x, hx⟩ := m.surjective f.1 f.2 in ⟨x, perfection.ext $ hx⟩⟩ | def | perfection_map.equiv | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"equiv",
"perfection.coeff",
"perfection.ext",
"perfection.lift",
"perfection_map",
"ring.perfection",
"ring_equiv.of_bijective"
] | A perfection map induces an isomorphism to the prefection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_apply {π : P →+* R} (m : perfection_map p π) (x : P) :
m.equiv x = perfection.lift p P R π x | rfl | lemma | perfection_map.equiv_apply | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"perfection.lift",
"perfection_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_equiv {π : P →+* R} (m : perfection_map p π) (x : P) :
perfection.coeff R p 0 (m.equiv x) = π x | rfl | lemma | perfection_map.comp_equiv | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"perfection.coeff",
"perfection_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_equiv' {π : P →+* R} (m : perfection_map p π) :
(perfection.coeff R p 0).comp ↑m.equiv = π | ring_hom.ext $ λ x, rfl | lemma | perfection_map.comp_equiv' | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"perfection.coeff",
"perfection_map",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_symm_equiv {π : P →+* R} (m : perfection_map p π) (f : ring.perfection R p) :
π (m.equiv.symm f) = perfection.coeff R p 0 f | (m.comp_equiv _).symm.trans $ congr_arg _ $ m.equiv.apply_symm_apply f | lemma | perfection_map.comp_symm_equiv | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"perfection.coeff",
"perfection_map",
"ring.perfection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_symm_equiv' {π : P →+* R} (m : perfection_map p π) :
π.comp ↑m.equiv.symm = perfection.coeff R p 0 | ring_hom.ext m.comp_symm_equiv | lemma | perfection_map.comp_symm_equiv' | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"perfection.coeff",
"perfection_map",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift [perfect_ring R p] (S : Type u₂) [comm_semiring S] [char_p S p]
(P : Type u₃) [comm_semiring P] [char_p P p] [perfect_ring P p]
(π : P →+* S) (m : perfection_map p π) :
(R →+* S) ≃ (R →+* P) | { to_fun := λ f, ring_hom.comp ↑m.equiv.symm $ perfection.lift p R S f,
inv_fun := λ f, π.comp f,
left_inv := λ f, by { simp_rw [← ring_hom.comp_assoc, comp_symm_equiv'],
exact (perfection.lift p R S).symm_apply_apply f },
right_inv := λ f, ring_hom.ext $ λ x, m.equiv.injective $ (m.equiv.apply_symm_apply _).... | def | perfection_map.lift | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_semiring",
"inv_fun",
"lift",
"perfect_ring",
"perfection.lift",
"perfection_map",
"ring_hom.comp",
"ring_hom.comp_assoc",
"ring_hom.ext"
] | Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* P`,
where `P` is any perfection of `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_ext [perfect_ring R p] {S : Type u₂} [comm_semiring S] [char_p S p]
{P : Type u₃} [comm_semiring P] [char_p P p] [perfect_ring P p]
(π : P →+* S) (m : perfection_map p π) {f g : R →+* P}
(hfg : ∀ x, π (f x) = π (g x)) : f = g | (lift p R S P π m).symm.injective $ ring_hom.ext hfg | lemma | perfection_map.hom_ext | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"char_p",
"comm_semiring",
"hom_ext",
"lift",
"perfect_ring",
"perfection_map",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) : P →+* Q | lift p P S Q σ n $ φ.comp π | def | perfection_map.map | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"lift",
"perfection_map"
] | A ring homomorphism `R →+* S` induces `P →+* Q`, a map of the respective perfections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) : σ.comp (map p m n φ) = φ.comp π | (lift p P S Q σ n).symm_apply_apply _ | lemma | perfection_map.comp_map | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"lift",
"perfection_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map {π : P →+* R} (m : perfection_map p π) {σ : Q →+* S} (n : perfection_map p σ)
(φ : R →+* S) (x : P) : σ (map p m n φ x) = φ (π x) | ring_hom.ext_iff.1 (comp_map p m n φ) x | lemma | perfection_map.map_map | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"perfection_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_map (φ : R →+* S) :
@map p _ R _ _ _ _ _ _ S _ _ _ _ _ _ _ (of p R) _ (of p S) φ = perfection.map p φ | hom_ext _ (of p S) $ λ f, by rw [map_map, perfection.coeff_map] | lemma | perfection_map.map_eq_map | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"hom_ext",
"perfection.coeff_map",
"perfection.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_p | O ⧸ (ideal.span {p} : ideal O) | def | mod_p | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ideal",
"ideal.span"
] | `O/(p)` for `O`, ring of integers of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre_val (x : mod_p K v O hv p) : ℝ≥0 | if x = 0 then 0 else v (algebra_map O K x.out') | def | mod_p.pre_val | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"algebra_map",
"mod_p"
] | For a field `K` with valuation `v : K → ℝ≥0` and ring of integers `O`,
a function `O/(p) → ℝ≥0` that sends `0` to `0` and `x + (p)` to `v(x)` as long as `x ∉ (p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre_val_mk {x : O} (hx : (ideal.quotient.mk _ x : mod_p K v O hv p) ≠ 0) :
pre_val K v O hv p (ideal.quotient.mk _ x) = v (algebra_map O K x) | begin
obtain ⟨r, hr⟩ := ideal.mem_span_singleton'.1 (ideal.quotient.eq.1 $ quotient.sound' $
@quotient.mk_out' O (ideal.span {p} : ideal O).quotient_rel x),
refine (if_neg hx).trans (v.map_eq_of_sub_lt $ lt_of_not_le _),
erw [← ring_hom.map_sub, ← hr, hv.le_iff_dvd],
exact λ hprx, hx (ideal.quotient.eq_zero... | lemma | mod_p.pre_val_mk | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"algebra_map",
"dvd_of_mul_left_dvd",
"ideal",
"ideal.quotient.mk",
"ideal.span",
"lt_of_not_le",
"mod_p",
"quotient.mk_out'",
"quotient.sound'",
"ring_hom.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_val_zero : pre_val K v O hv p 0 = 0 | if_pos rfl | lemma | mod_p.pre_val_zero | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_val_mul {x y : mod_p K v O hv p} (hxy0 : x * y ≠ 0) :
pre_val K v O hv p (x * y) = pre_val K v O hv p x * pre_val K v O hv p y | begin
have hx0 : x ≠ 0 := mt (by { rintro rfl, rw zero_mul }) hxy0,
have hy0 : y ≠ 0 := mt (by { rintro rfl, rw mul_zero }) hxy0,
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective y,
rw ← ring_hom.map_mul at hxy0 ⊢,
rw [pre_val_mk hx0, pre_val_mk hy0, pre_val... | lemma | mod_p.pre_val_mul | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ideal.quotient.mk_surjective",
"mod_p",
"mul_zero",
"ring_hom.map_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_val_add (x y : mod_p K v O hv p) :
pre_val K v O hv p (x + y) ≤ max (pre_val K v O hv p x) (pre_val K v O hv p y) | begin
by_cases hx0 : x = 0, { rw [hx0, zero_add], exact le_max_right _ _ },
by_cases hy0 : y = 0, { rw [hy0, add_zero], exact le_max_left _ _ },
by_cases hxy0 : x + y = 0, { rw [hxy0, pre_val_zero], exact zero_le _ },
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surj... | lemma | mod_p.pre_val_add | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ideal.quotient.mk_surjective",
"mod_p",
"ring_hom.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
v_p_lt_pre_val {x : mod_p K v O hv p} : v p < pre_val K v O hv p x ↔ x ≠ 0 | begin
refine ⟨λ h hx, by { rw [hx, pre_val_zero] at h, exact not_lt_zero' h },
λ h, lt_of_not_le $ λ hp, h _⟩,
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
rw [pre_val_mk h, ← map_nat_cast (algebra_map O K) p, hv.le_iff_dvd] at hp,
rw [ideal.quotient.eq_zero_iff_mem, ideal.mem_span_singleton], exact h... | lemma | mod_p.v_p_lt_pre_val | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"algebra_map",
"ideal.mem_span_singleton",
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.mk_surjective",
"lt_of_not_le",
"map_nat_cast",
"mod_p",
"not_lt_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_val_eq_zero {x : mod_p K v O hv p} : pre_val K v O hv p x = 0 ↔ x = 0 | ⟨λ hvx, classical.by_contradiction $ λ hx0 : x ≠ 0,
by { rw [← v_p_lt_pre_val, hvx] at hx0, exact not_lt_zero' hx0 },
λ hx, hx.symm ▸ pre_val_zero⟩ | lemma | mod_p.pre_val_eq_zero | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"mod_p",
"not_lt_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
v_p_lt_val {x : O} :
v p < v (algebra_map O K x) ↔ (ideal.quotient.mk _ x : mod_p K v O hv p) ≠ 0 | by rw [lt_iff_not_le, not_iff_not, ← map_nat_cast (algebra_map O K) p, hv.le_iff_dvd,
ideal.quotient.eq_zero_iff_mem, ideal.mem_span_singleton] | lemma | mod_p.v_p_lt_val | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"algebra_map",
"ideal.mem_span_singleton",
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.mk",
"lt_iff_not_le",
"map_nat_cast",
"mod_p",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ne_zero_of_pow_p_ne_zero {x y : mod_p K v O hv p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0) :
x * y ≠ 0 | begin
obtain ⟨r, rfl⟩ := ideal.quotient.mk_surjective x,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective y,
have h1p : (0 : ℝ) < 1 / p := one_div_pos.2 (nat.cast_pos.2 hp.1.pos),
rw ← ring_hom.map_mul, rw ← ring_hom.map_pow at hx hy,
rw ← v_p_lt_val hv at hx hy ⊢,
rw [ring_hom.map_pow, v.map_pow, ← rpow_lt_... | lemma | mod_p.mul_ne_zero_of_pow_p_ne_zero | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"add_div",
"algebra_map",
"div_le_one",
"ideal.quotient.mk_surjective",
"map_nat_cast",
"mod_p",
"mul_lt_mul₀",
"mul_one_div_cancel",
"ring_hom.map_mul",
"ring_hom.map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_tilt | ring.perfection (mod_p K v O hv p) p | def | pre_tilt | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"mod_p",
"ring.perfection"
] | Perfection of `O/(p)` where `O` is the ring of integers of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
val_aux (f : pre_tilt K v O hv p) : ℝ≥0 | if h : ∃ n, coeff _ _ n f ≠ 0
then mod_p.pre_val K v O hv p (coeff _ _ (nat.find h) f) ^ (p ^ nat.find h)
else 0 | def | pre_tilt.val_aux | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"mod_p.pre_val",
"pre_tilt"
] | The valuation `Perfection(O/(p)) → ℝ≥0` as a function.
Given `f ∈ Perfection(O/(p))`, if `f = 0` then output `0`;
otherwise output `pre_val(f(n))^(p^n)` for any `n` such that `f(n) ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_nat_find_add_ne_zero {f : pre_tilt K v O hv p} {h : ∃ n, coeff _ _ n f ≠ 0} (k : ℕ) :
coeff _ _ (nat.find h + k) f ≠ 0 | coeff_add_ne_zero (nat.find_spec h) k | lemma | pre_tilt.coeff_nat_find_add_ne_zero | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"pre_tilt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val_aux_eq {f : pre_tilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0) :
val_aux K v O hv p f = mod_p.pre_val K v O hv p (coeff _ _ n f) ^ (p ^ n) | begin
have h : ∃ n, coeff _ _ n f ≠ 0 := ⟨n, hfn⟩,
rw [val_aux, dif_pos h],
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le (nat.find_min' h hfn),
induction k with k ih, { refl },
obtain ⟨x, hx⟩ := ideal.quotient.mk_surjective (coeff _ _ (nat.find h + k + 1) f),
have h1 : (ideal.quotient.mk _ x : mod_p K v O hv ... | lemma | pre_tilt.val_aux_eq | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ideal.quotient.mk",
"ideal.quotient.mk_surjective",
"ih",
"mod_p",
"mod_p.pre_val",
"mod_p.pre_val_mk",
"nat.exists_eq_add_of_le",
"pow_mul",
"pow_succ",
"pre_tilt",
"ring_hom.map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val_aux_zero : val_aux K v O hv p 0 = 0 | dif_neg $ λ ⟨n, hn⟩, hn rfl | lemma | pre_tilt.val_aux_zero | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val_aux_one : val_aux K v O hv p 1 = 1 | (val_aux_eq $ show coeff (mod_p K v O hv p) p 0 1 ≠ 0, from one_ne_zero).trans $
by { rw [pow_zero, pow_one, ring_hom.map_one, ← (ideal.quotient.mk _).map_one, mod_p.pre_val_mk,
ring_hom.map_one, v.map_one],
change (1 : mod_p K v O hv p) ≠ 0,
exact one_ne_zero } | lemma | pre_tilt.val_aux_one | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"ideal.quotient.mk",
"map_one",
"mod_p",
"mod_p.pre_val_mk",
"one_ne_zero",
"pow_one",
"pow_zero",
"ring_hom.map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val_aux_mul (f g : pre_tilt K v O hv p) :
val_aux K v O hv p (f * g) = val_aux K v O hv p f * val_aux K v O hv p g | begin
by_cases hf : f = 0, { rw [hf, zero_mul, val_aux_zero, zero_mul] },
by_cases hg : g = 0, { rw [hg, mul_zero, val_aux_zero, mul_zero] },
obtain ⟨m, hm⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, hf $ perfection.ext h),
obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 (λ h, hg $ perfection.ext h)... | lemma | pre_tilt.val_aux_mul | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"mod_p.mul_ne_zero_of_pow_p_ne_zero",
"mod_p.pre_val_mul",
"mul_pow",
"mul_zero",
"perfection.ext",
"pre_tilt",
"ring_hom.map_mul",
"ring_hom.map_pow",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val_aux_add (f g : pre_tilt K v O hv p) :
val_aux K v O hv p (f + g) ≤ max (val_aux K v O hv p f) (val_aux K v O hv p g) | begin
by_cases hf : f = 0, { rw [hf, zero_add, val_aux_zero, max_eq_right], exact zero_le _ },
by_cases hg : g = 0, { rw [hg, add_zero, val_aux_zero, max_eq_left], exact zero_le _ },
by_cases hfg : f + g = 0, { rw [hfg, val_aux_zero], exact zero_le _ },
replace hf : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, ... | lemma | pre_tilt.val_aux_add | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"le_max_of_le_left",
"le_max_of_le_right",
"mod_p.pre_val_add",
"perfection.ext",
"pow_le_pow_of_le_left'",
"pre_tilt",
"ring_hom.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val : valuation (pre_tilt K v O hv p) ℝ≥0 | { to_fun := val_aux K v O hv p,
map_one' := val_aux_one,
map_mul' := val_aux_mul,
map_zero' := val_aux_zero,
map_add_le_max' := val_aux_add } | def | pre_tilt.val | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"pre_tilt",
"valuation"
] | The valuation `Perfection(O/(p)) → ℝ≥0`.
Given `f ∈ Perfection(O/(p))`, if `f = 0` then output `0`;
otherwise output `pre_val(f(n))^(p^n)` for any `n` such that `f(n) ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_eq_zero {f : pre_tilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 | begin
by_cases hf0 : f = 0, { rw hf0, exact iff_of_true (valuation.map_zero _) rfl },
obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 (λ h, hf0 $ perfection.ext h),
show val_aux K v O hv p f = 0 ↔ f = 0, refine iff_of_false (λ hvf, hn _) hf0,
rw val_aux_eq hn at hvf, replace hvf := pow_eq_zero hvf, rwa ... | lemma | pre_tilt.map_eq_zero | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"iff_of_false",
"iff_of_true",
"map_eq_zero",
"mod_p.pre_val_eq_zero",
"perfection.ext",
"pow_eq_zero",
"pre_tilt",
"valuation.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tilt | fraction_ring (pre_tilt K v O hv p) | def | tilt | ring_theory | src/ring_theory/perfection.lean | [
"algebra.char_p.pi",
"algebra.char_p.quotient",
"algebra.char_p.subring",
"algebra.ring.pi",
"analysis.special_functions.pow.nnreal",
"field_theory.perfect_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.subring.basic",
"ring_theory.valuation.integers"
] | [
"fraction_ring",
"pre_tilt"
] | The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in
[scholze2011perfectoid]. Given a field `K` with valuation `K → ℝ≥0` and ring of integers `O`,
this is implemented as the fraction field of the perfection of `O/(p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_bilinear : A →ₗ[A] R[X] →ₗ[R] A[X] | linear_map.to_span_singleton A _ (aeval (polynomial.X : A[X])).to_linear_map | def | poly_equiv_tensor.to_fun_bilinear | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"linear_map.to_span_singleton",
"polynomial.X"
] | (Implementation detail).
The function underlying `A ⊗[R] R[X] →ₐ[R] A[X]`,
as a bilinear function of two arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_bilinear_apply_eq_sum (a : A) (p : R[X]) :
to_fun_bilinear R A a p = p.sum (λ n r, monomial n (a * algebra_map R A r)) | begin
simp only [to_fun_bilinear_apply_apply, aeval_def, eval₂_eq_sum, polynomial.sum, finset.smul_sum],
congr' with i : 1,
rw [← algebra.smul_def, ←C_mul', mul_smul_comm, C_mul_X_pow_eq_monomial, ←algebra.commutes,
← algebra.smul_def, smul_monomial],
end | lemma | poly_equiv_tensor.to_fun_bilinear_apply_eq_sum | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra.smul_def",
"algebra_map",
"finset.smul_sum",
"mul_smul_comm",
"polynomial.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_linear : A ⊗[R] R[X] →ₗ[R] A[X] | tensor_product.lift (to_fun_bilinear R A) | def | poly_equiv_tensor.to_fun_linear | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"tensor_product.lift"
] | (Implementation detail).
The function underlying `A ⊗[R] R[X] →ₐ[R] A[X]`,
as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_linear_tmul_apply (a : A) (p : R[X]) :
to_fun_linear R A (a ⊗ₜ[R] p) = to_fun_bilinear R A a p | rfl | lemma | poly_equiv_tensor.to_fun_linear_tmul_apply | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_linear_mul_tmul_mul_aux_1
(p : R[X]) (k : ℕ) (h : decidable (¬p.coeff k = 0)) (a : A) :
ite (¬coeff p k = 0) (a * (algebra_map R A) (coeff p k)) 0 = a * (algebra_map R A) (coeff p k) | by { classical, split_ifs; simp *, } | lemma | poly_equiv_tensor.to_fun_linear_mul_tmul_mul_aux_1 | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_linear_mul_tmul_mul_aux_2 (k : ℕ) (a₁ a₂ : A) (p₁ p₂ : R[X]) :
a₁ * a₂ * (algebra_map R A) ((p₁ * p₂).coeff k) =
(finset.nat.antidiagonal k).sum
(λ x, a₁ * (algebra_map R A) (coeff p₁ x.1) * (a₂ * (algebra_map R A) (coeff p₂ x.2))) | begin
simp_rw [mul_assoc, algebra.commutes, ←finset.mul_sum, mul_assoc, ←finset.mul_sum],
congr,
simp_rw [algebra.commutes (coeff p₂ _), coeff_mul, ring_hom.map_sum, ring_hom.map_mul],
end | lemma | poly_equiv_tensor.to_fun_linear_mul_tmul_mul_aux_2 | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra.commutes",
"algebra_map",
"finset.nat.antidiagonal",
"mul_assoc",
"ring_hom.map_mul",
"ring_hom.map_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_linear_mul_tmul_mul (a₁ a₂ : A) (p₁ p₂ : R[X]) :
(to_fun_linear R A) ((a₁ * a₂) ⊗ₜ[R] (p₁ * p₂)) =
(to_fun_linear R A) (a₁ ⊗ₜ[R] p₁) * (to_fun_linear R A) (a₂ ⊗ₜ[R] p₂) | begin
classical,
simp only [to_fun_linear_tmul_apply, to_fun_bilinear_apply_eq_sum],
ext k,
simp_rw [coeff_sum, coeff_monomial, sum_def, finset.sum_ite_eq', mem_support_iff, ne.def],
conv_rhs { rw [coeff_mul] },
simp_rw [finset_sum_coeff, coeff_monomial,
finset.sum_ite_eq', mem_support_iff, ne.def,
... | lemma | poly_equiv_tensor.to_fun_linear_mul_tmul_mul | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra_map",
"ite_mul",
"ite_mul_zero_left",
"ite_mul_zero_right",
"mul_ite",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_linear_algebra_map_tmul_one (r : R) :
(to_fun_linear R A) ((algebra_map R A) r ⊗ₜ[R] 1) = (algebra_map R A[X]) r | by rw [to_fun_linear_tmul_apply, to_fun_bilinear_apply_apply, polynomial.aeval_one,
algebra_map_smul, algebra.algebra_map_eq_smul_one] | lemma | poly_equiv_tensor.to_fun_linear_algebra_map_tmul_one | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"algebra_map_smul",
"polynomial.aeval_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_alg_hom : A ⊗[R] R[X] →ₐ[R] A[X] | alg_hom_of_linear_map_tensor_product
(to_fun_linear R A)
(to_fun_linear_mul_tmul_mul R A)
(to_fun_linear_algebra_map_tmul_one R A) | def | poly_equiv_tensor.to_fun_alg_hom | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [] | (Implementation detail).
The algebra homomorphism `A ⊗[R] R[X] →ₐ[R] A[X]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_alg_hom_apply_tmul (a : A) (p : R[X]) :
to_fun_alg_hom R A (a ⊗ₜ[R] p) = p.sum (λ n r, monomial n (a * (algebra_map R A) r)) | to_fun_bilinear_apply_eq_sum R A _ _ | lemma | poly_equiv_tensor.to_fun_alg_hom_apply_tmul | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun (p : A[X]) : A ⊗[R] R[X] | p.eval₂
(include_left : A →ₐ[R] A ⊗[R] R[X])
((1 : A) ⊗ₜ[R] (X : R[X])) | def | poly_equiv_tensor.inv_fun | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"inv_fun"
] | (Implementation detail.)
The bare function `A[X] → A ⊗[R] R[X]`.
(We don't need to show that it's an algebra map, thankfully --- just that it's an inverse.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_fun_add {p q} : inv_fun R A (p + q) = inv_fun R A p + inv_fun R A q | by simp only [inv_fun, eval₂_add] | lemma | poly_equiv_tensor.inv_fun_add | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_monomial (n : ℕ) (a : A) :
inv_fun R A (monomial n a) = include_left a * ((1 : A) ⊗ₜ[R] (X : R[X])) ^ n | eval₂_monomial _ _ | lemma | poly_equiv_tensor.inv_fun_monomial | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv (x : A ⊗ R[X]) :
inv_fun R A ((to_fun_alg_hom R A) x) = x | begin
apply tensor_product.induction_on x,
{ simp [inv_fun], },
{ intros a p, dsimp only [inv_fun],
rw [to_fun_alg_hom_apply_tmul, eval₂_sum],
simp_rw [eval₂_monomial, alg_hom.coe_to_ring_hom, algebra.tensor_product.tmul_pow, one_pow,
algebra.tensor_product.include_left_apply, algebra.tensor_product... | lemma | poly_equiv_tensor.left_inv | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"alg_hom.coe_to_ring_hom",
"alg_hom.map_add",
"algebra.smul_def",
"algebra.tensor_product.include_left_apply",
"algebra.tensor_product.tmul_mul_tmul",
"algebra.tensor_product.tmul_pow",
"inv_fun",
"mul_one",
"one_mul",
"one_pow",
"tensor_product.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inv (x : A[X]) :
(to_fun_alg_hom R A) (inv_fun R A x) = x | begin
apply polynomial.induction_on' x,
{ intros p q hp hq, simp only [inv_fun_add, alg_hom.map_add, hp, hq], },
{ intros n a,
rw [inv_fun_monomial, algebra.tensor_product.include_left_apply,
algebra.tensor_product.tmul_pow, one_pow, algebra.tensor_product.tmul_mul_tmul,
mul_one, one_mul, to_fun_a... | lemma | poly_equiv_tensor.right_inv | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"alg_hom.map_add",
"algebra.tensor_product.include_left_apply",
"algebra.tensor_product.tmul_mul_tmul",
"algebra.tensor_product.tmul_pow",
"inv_fun",
"mul_one",
"one_mul",
"one_pow",
"polynomial.induction_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv : (A ⊗[R] R[X]) ≃ A[X] | { to_fun := to_fun_alg_hom R A,
inv_fun := inv_fun R A,
left_inv := left_inv R A,
right_inv := right_inv R A, } | def | poly_equiv_tensor.equiv | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"equiv",
"inv_fun"
] | (Implementation detail)
The equivalence, ignoring the algebra structure, `(A ⊗[R] R[X]) ≃ A[X]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
poly_equiv_tensor : A[X] ≃ₐ[R] (A ⊗[R] R[X]) | alg_equiv.symm
{ ..(poly_equiv_tensor.to_fun_alg_hom R A), ..(poly_equiv_tensor.equiv R A) } | def | poly_equiv_tensor | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"alg_equiv.symm",
"poly_equiv_tensor.equiv",
"poly_equiv_tensor.to_fun_alg_hom"
] | The `R`-algebra isomorphism `A[X] ≃ₐ[R] (A ⊗[R] R[X])`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
poly_equiv_tensor_apply (p : A[X]) :
poly_equiv_tensor R A p =
p.eval₂ (include_left : A →ₐ[R] A ⊗[R] R[X]) ((1 : A) ⊗ₜ[R] (X : R[X])) | rfl | lemma | poly_equiv_tensor_apply | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"poly_equiv_tensor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
poly_equiv_tensor_symm_apply_tmul (a : A) (p : R[X]) :
(poly_equiv_tensor R A).symm (a ⊗ₜ p) = p.sum (λ n r, monomial n (a * algebra_map R A r)) | to_fun_alg_hom_apply_tmul _ _ _ _ | lemma | poly_equiv_tensor_symm_apply_tmul | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra_map",
"poly_equiv_tensor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mat_poly_equiv :
matrix n n R[X] ≃ₐ[R] (matrix n n R)[X] | (((matrix_equiv_tensor R R[X] n)).trans
(algebra.tensor_product.comm R _ _)).trans
(poly_equiv_tensor R (matrix n n R)).symm | def | mat_poly_equiv | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra.tensor_product.comm",
"matrix",
"matrix_equiv_tensor",
"poly_equiv_tensor"
] | The algebra isomorphism stating "matrices of polynomials are the same as polynomials of matrices".
(You probably shouldn't attempt to use this underlying definition ---
it's an algebra equivalence, and characterised extensionally by the lemma
`mat_poly_equiv_coeff_apply` below.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mat_poly_equiv_coeff_apply_aux_1 (i j : n) (k : ℕ) (x : R) :
mat_poly_equiv (std_basis_matrix i j $ monomial k x) =
monomial k (std_basis_matrix i j x) | begin
simp only [mat_poly_equiv, alg_equiv.trans_apply,
matrix_equiv_tensor_apply_std_basis],
apply (poly_equiv_tensor R (matrix n n R)).injective,
simp only [alg_equiv.apply_symm_apply],
convert algebra.tensor_product.comm_tmul _ _ _ _ _,
simp only [poly_equiv_tensor_apply],
convert eval₂_monomial _ _,... | lemma | mat_poly_equiv_coeff_apply_aux_1 | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"alg_equiv.apply_symm_apply",
"alg_equiv.trans_apply",
"algebra.tensor_product.comm_tmul",
"algebra.tensor_product.include_left_apply",
"algebra.tensor_product.tmul_mul_tmul",
"algebra.tensor_product.tmul_pow",
"mat_poly_equiv",
"matrix",
"matrix.mul_one",
"matrix_equiv_tensor_apply_std_basis",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mat_poly_equiv_coeff_apply_aux_2
(i j : n) (p : R[X]) (k : ℕ) :
coeff (mat_poly_equiv (std_basis_matrix i j p)) k =
std_basis_matrix i j (coeff p k) | begin
apply polynomial.induction_on' p,
{ intros p q hp hq, ext,
simp [hp, hq, coeff_add, add_apply, std_basis_matrix_add], },
{ intros k x,
simp only [mat_poly_equiv_coeff_apply_aux_1, coeff_monomial],
split_ifs; { funext, simp, }, }
end | lemma | mat_poly_equiv_coeff_apply_aux_2 | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"mat_poly_equiv",
"mat_poly_equiv_coeff_apply_aux_1",
"polynomial.induction_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mat_poly_equiv_coeff_apply
(m : matrix n n R[X]) (k : ℕ) (i j : n) :
coeff (mat_poly_equiv m) k i j = coeff (m i j) k | begin
apply matrix.induction_on' m,
{ simp, },
{ intros p q hp hq, simp [hp, hq], },
{ intros i' j' x,
erw mat_poly_equiv_coeff_apply_aux_2,
dsimp [std_basis_matrix],
split_ifs,
{ rcases h with ⟨rfl, rfl⟩, simp [std_basis_matrix], },
{ simp [std_basis_matrix, h], }, },
end | lemma | mat_poly_equiv_coeff_apply | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"mat_poly_equiv",
"mat_poly_equiv_coeff_apply_aux_2",
"matrix",
"matrix.induction_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mat_poly_equiv_symm_apply_coeff
(p : (matrix n n R)[X]) (i j : n) (k : ℕ) :
coeff (mat_poly_equiv.symm p i j) k = coeff p k i j | begin
have t : p = mat_poly_equiv
(mat_poly_equiv.symm p) := by simp,
conv_rhs { rw t, },
simp only [mat_poly_equiv_coeff_apply],
end | lemma | mat_poly_equiv_symm_apply_coeff | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"mat_poly_equiv",
"mat_poly_equiv_coeff_apply",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mat_poly_equiv_smul_one (p : R[X]) :
mat_poly_equiv (p • 1) = p.map (algebra_map R (matrix n n R)) | begin
ext m i j,
simp only [coeff_map, one_apply, algebra_map_matrix_apply, mul_boole,
pi.smul_apply, mat_poly_equiv_coeff_apply],
split_ifs; simp,
end | lemma | mat_poly_equiv_smul_one | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"algebra_map",
"mat_poly_equiv",
"mat_poly_equiv_coeff_apply",
"matrix",
"mul_boole",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_subset_support_mat_poly_equiv
(m : matrix n n R[X]) (i j : n) :
support (m i j) ⊆ support (mat_poly_equiv m) | begin
assume k,
contrapose,
simp only [not_mem_support_iff],
assume hk,
rw [← mat_poly_equiv_coeff_apply, hk],
refl
end | lemma | support_subset_support_mat_poly_equiv | ring_theory | src/ring_theory/polynomial_algebra.lean | [
"ring_theory.matrix_algebra",
"data.polynomial.algebra_map",
"data.matrix.basis",
"data.matrix.dmatrix"
] | [
"mat_poly_equiv",
"mat_poly_equiv_coeff_apply",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_basis (R S : Type*) [comm_ring R] [ring S] [algebra R S] | (gen : S)
(dim : ℕ)
(basis : basis (fin dim) R S)
(basis_eq_pow : ∀ i, basis i = gen ^ (i : ℕ))
-- this is usually not needed because of `basis_eq_pow` but can be needed in some cases;
-- in such circumstances, add it manually using `@[simps dim gen basis]`.
initialize_simps_projections power_basis (-basis) | structure | power_basis | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"algebra",
"basis",
"comm_ring",
"ring"
] | `pb : power_basis R S` states that `1, pb.gen, ..., pb.gen ^ (pb.dim - 1)`
is a basis for the `R`-algebra `S` (viewed as `R`-module).
This is a structure, not a class, since the same algebra can have many power bases.
For the common case where `S` is defined by adjoining an integral element to `R`,
the canonical power... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_basis (pb : power_basis R S) :
⇑pb.basis = λ (i : fin pb.dim), pb.gen ^ (i : ℕ) | funext pb.basis_eq_pow | lemma | power_basis.coe_basis | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_dimensional [algebra K S] (pb : power_basis K S) : finite_dimensional K S | finite_dimensional.of_fintype_basis pb.basis | lemma | power_basis.finite_dimensional | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"algebra",
"finite_dimensional",
"finite_dimensional.of_fintype_basis",
"power_basis"
] | Cannot be an instance because `power_basis` cannot be a class. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finrank [algebra K S] (pb : power_basis K S) : finite_dimensional.finrank K S = pb.dim | by rw [finite_dimensional.finrank_eq_card_basis pb.basis, fintype.card_fin] | lemma | power_basis.finrank | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"algebra",
"finite_dimensional.finrank",
"finite_dimensional.finrank_eq_card_basis",
"fintype.card_fin",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_pow' {x y : S} {d : ℕ} :
y ∈ submodule.span R (set.range (λ (i : fin d), x ^ (i : ℕ))) ↔
∃ f : R[X], f.degree < d ∧ y = aeval x f | begin
have : set.range (λ (i : fin d), x ^ (i : ℕ)) = (λ (i : ℕ), x ^ i) '' ↑(finset.range d),
{ ext n,
simp_rw [set.mem_range, set.mem_image, finset.mem_coe, finset.mem_range],
exact ⟨λ ⟨⟨i, hi⟩, hy⟩, ⟨i, hi, hy⟩, λ ⟨i, hi, hy⟩, ⟨⟨i, hi⟩, hy⟩⟩ },
simp only [this, finsupp.mem_span_image_iff_total, degree_... | lemma | power_basis.mem_span_pow' | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"alg_hom.coe_mk",
"algebra.smul_def",
"exists_prop",
"finset.mem_coe",
"finset.mem_range",
"finset.range",
"finsupp.coe_lsum",
"finsupp.mem_span_image_iff_total",
"finsupp.mem_supported'",
"finsupp.total",
"linear_map.coe_smul_right",
"linear_map.id_coe",
"polynomial.sum",
"set.mem_image",... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_pow {x y : S} {d : ℕ} (hd : d ≠ 0) :
y ∈ submodule.span R (set.range (λ (i : fin d), x ^ (i : ℕ))) ↔
∃ f : R[X], f.nat_degree < d ∧ y = aeval x f | begin
rw mem_span_pow',
split;
{ rintros ⟨f, h, hy⟩,
refine ⟨f, _, hy⟩,
by_cases hf : f = 0,
{ simp only [hf, nat_degree_zero, degree_zero] at h ⊢,
exact lt_of_le_of_ne (nat.zero_le d) hd.symm <|> exact with_bot.bot_lt_coe d },
simpa only [degree_eq_nat_degree hf, with_bot.coe_lt_coe] using ... | lemma | power_basis.mem_span_pow | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"set.range",
"submodule.span",
"with_bot.bot_lt_coe",
"with_bot.coe_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dim_ne_zero [h : nontrivial S] (pb : power_basis R S) : pb.dim ≠ 0 | λ h, not_nonempty_iff.mpr (h.symm ▸ fin.is_empty : is_empty (fin pb.dim)) pb.basis.index_nonempty | lemma | power_basis.dim_ne_zero | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"is_empty",
"nontrivial",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dim_pos [nontrivial S] (pb : power_basis R S) : 0 < pb.dim | nat.pos_of_ne_zero pb.dim_ne_zero | lemma | power_basis.dim_pos | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"nontrivial",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_aeval [nontrivial S] (pb : power_basis R S) (y : S) :
∃ f : R[X], f.nat_degree < pb.dim ∧ y = aeval pb.gen f | (mem_span_pow pb.dim_ne_zero).mp (by simpa using pb.basis.mem_span y) | lemma | power_basis.exists_eq_aeval | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"nontrivial",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_aeval' (pb : power_basis R S) (y : S) :
∃ f : R[X], y = aeval pb.gen f | begin
nontriviality S,
obtain ⟨f, _, hf⟩ := exists_eq_aeval pb y,
exact ⟨f, hf⟩
end | lemma | power_basis.exists_eq_aeval' | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom_ext {S' : Type*} [semiring S'] [algebra R S']
(pb : power_basis R S) ⦃f g : S →ₐ[R] S'⦄ (h : f pb.gen = g pb.gen) :
f = g | begin
ext x,
obtain ⟨f, rfl⟩ := pb.exists_eq_aeval' x,
rw [← polynomial.aeval_alg_hom_apply, ← polynomial.aeval_alg_hom_apply, h]
end | lemma | power_basis.alg_hom_ext | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"algebra",
"polynomial.aeval_alg_hom_apply",
"power_basis",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_gen (pb : power_basis A S) : A[X] | X ^ pb.dim -
∑ (i : fin pb.dim), C (pb.basis.repr (pb.gen ^ pb.dim) i) * X ^ (i : ℕ) | def | power_basis.minpoly_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"power_basis"
] | `pb.minpoly_gen` is the minimal polynomial for `pb.gen`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aeval_minpoly_gen (pb : power_basis A S) : aeval pb.gen (minpoly_gen pb) = 0 | begin
simp_rw [minpoly_gen, alg_hom.map_sub, alg_hom.map_sum, alg_hom.map_mul, alg_hom.map_pow,
aeval_C, ← algebra.smul_def, aeval_X],
refine sub_eq_zero.mpr ((pb.basis.total_repr (pb.gen ^ pb.dim)).symm.trans _),
rw [finsupp.total_apply, finsupp.sum_fintype];
simp only [pb.coe_basis, zero_smul, eq... | lemma | power_basis.aeval_minpoly_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"alg_hom.map_mul",
"alg_hom.map_pow",
"alg_hom.map_sub",
"alg_hom.map_sum",
"algebra.smul_def",
"finsupp.total_apply",
"power_basis",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_gen_monic (pb : power_basis A S) : monic (minpoly_gen pb) | begin
nontriviality A,
apply (monic_X_pow _).sub_of_left _,
rw degree_X_pow,
exact degree_sum_fin_lt _
end | lemma | power_basis.minpoly_gen_monic | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dim_le_nat_degree_of_root (pb : power_basis A S) {p : A[X]}
(ne_zero : p ≠ 0) (root : aeval pb.gen p = 0) :
pb.dim ≤ p.nat_degree | begin
refine le_of_not_lt (λ hlt, ne_zero _),
rw [p.as_sum_range' _ hlt, finset.sum_range],
refine fintype.sum_eq_zero _ (λ i, _),
simp_rw [aeval_eq_sum_range' hlt, finset.sum_range, ← pb.basis_eq_pow] at root,
have := fintype.linear_independent_iff.1 pb.basis.linear_independent _ root,
dsimp only at this,
... | lemma | power_basis.dim_le_nat_degree_of_root | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"ne_zero",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dim_le_degree_of_root (h : power_basis A S) {p : A[X]}
(ne_zero : p ≠ 0) (root : aeval h.gen p = 0) :
↑h.dim ≤ p.degree | by { rw [degree_eq_nat_degree ne_zero, with_bot.coe_le_coe],
exact h.dim_le_nat_degree_of_root ne_zero root } | lemma | power_basis.dim_le_degree_of_root | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"ne_zero",
"power_basis",
"with_bot.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_minpoly_gen [nontrivial A] (pb : power_basis A S) :
degree (minpoly_gen pb) = pb.dim | begin
unfold minpoly_gen,
rw degree_sub_eq_left_of_degree_lt; rw degree_X_pow,
apply degree_sum_fin_lt
end | lemma | power_basis.degree_minpoly_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"nontrivial",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_minpoly_gen [nontrivial A] (pb : power_basis A S) :
nat_degree (minpoly_gen pb) = pb.dim | nat_degree_eq_of_degree_eq_some pb.degree_minpoly_gen | lemma | power_basis.nat_degree_minpoly_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"nontrivial",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_gen_eq (pb : power_basis A S) : pb.minpoly_gen = minpoly A pb.gen | begin
nontriviality A,
refine minpoly.unique' A _ pb.minpoly_gen_monic
pb.aeval_minpoly_gen (λ q hq, or_iff_not_imp_left.2 $ λ hn0 h0, _),
exact (pb.dim_le_degree_of_root hn0 h0).not_lt (pb.degree_minpoly_gen ▸ hq),
end | lemma | power_basis.minpoly_gen_eq | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"minpoly.unique'",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_gen (pb : power_basis A S) : is_integral A pb.gen | ⟨minpoly_gen pb, minpoly_gen_monic pb, aeval_minpoly_gen pb⟩ | lemma | power_basis.is_integral_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"is_integral",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_minpoly [nontrivial A] (pb : power_basis A S) : degree (minpoly A pb.gen) = pb.dim | by rw [← minpoly_gen_eq, degree_minpoly_gen] | lemma | power_basis.degree_minpoly | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"nontrivial",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_minpoly [nontrivial A] (pb : power_basis A S) :
(minpoly A pb.gen).nat_degree = pb.dim | by rw [← minpoly_gen_eq, nat_degree_minpoly_gen] | lemma | power_basis.nat_degree_minpoly | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"nontrivial",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_mul_matrix (pb : power_basis A S) :
algebra.left_mul_matrix pb.basis pb.gen = matrix.of
(λ i j, if ↑j + 1 = pb.dim then -pb.minpoly_gen.coeff ↑i else if ↑i = ↑j + 1 then 1 else 0) | begin
casesI subsingleton_or_nontrivial A, { apply subsingleton.elim },
rw [algebra.left_mul_matrix_apply, ← linear_equiv.eq_symm_apply, linear_map.to_matrix_symm],
refine pb.basis.ext (λ k, _),
simp_rw [matrix.to_lin_self, matrix.of_apply, pb.basis_eq_pow],
apply (pow_succ _ _).symm.trans,
split_ifs with h... | lemma | power_basis.left_mul_matrix | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"algebra.left_mul_matrix",
"algebra.left_mul_matrix_apply",
"fin.ext",
"linear_equiv.eq_symm_apply",
"linear_map.to_matrix_symm",
"matrix.of",
"matrix.of_apply",
"matrix.to_lin_self",
"neg_smul",
"one_smul",
"pow_succ",
"power_basis",
"subsingleton_or_nontrivial",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constr_pow_aeval (pb : power_basis A S) {y : S'}
(hy : aeval y (minpoly A pb.gen) = 0) (f : A[X]) :
pb.basis.constr A (λ i, y ^ (i : ℕ)) (aeval pb.gen f) = aeval y f | begin
casesI subsingleton_or_nontrivial A,
{ rw [(subsingleton.elim _ _ : f = 0), aeval_zero, map_zero, aeval_zero] },
rw [← aeval_mod_by_monic_eq_self_of_root (minpoly.monic pb.is_integral_gen) (minpoly.aeval _ _),
← @aeval_mod_by_monic_eq_self_of_root _ _ _ _ _ f _ (minpoly.monic pb.is_integral_gen) y hy]... | lemma | power_basis.constr_pow_aeval | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"alg_hom.map_zero",
"basis.constr_basis",
"fin.coe_mk",
"finset.mem_range",
"finset.range",
"linear_map.map_smul",
"linear_map.map_sum",
"linear_map.map_zero",
"minpoly",
"minpoly.aeval",
"minpoly.monic",
"power_basis",
"subsingleton_or_nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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