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 |
|---|---|---|---|---|---|---|---|---|---|---|
coe_unit_group_mul_equiv_apply (a : A.unit_group) :
(A.unit_group_mul_equiv a : K) = a | rfl | lemma | valuation_subring.coe_unit_group_mul_equiv_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_unit_group_mul_equiv_symm_apply (a : Aˣ) :
(A.unit_group_mul_equiv.symm a : K) = a | rfl | lemma | valuation_subring.coe_unit_group_mul_equiv_symm_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_group_le_unit_group {A B : valuation_subring K} :
A.unit_group ≤ B.unit_group ↔ A ≤ B | begin
split,
{ intros h x hx,
rw [← A.valuation_le_one_iff x, le_iff_lt_or_eq] at hx,
by_cases h_1 : x = 0, { simp only [h_1, zero_mem] },
by_cases h_2 : 1 + x = 0,
{ simp only [← add_eq_zero_iff_neg_eq.1 h_2, neg_mem _ _ (one_mem _)] },
cases hx,
{ have := h (show (units.mk0 _ h_2) ∈ A.un... | lemma | valuation_subring.unit_group_le_unit_group | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set_like.coe_mem",
"units.mk0",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_group_injective : function.injective (unit_group : valuation_subring K → subgroup _) | λ A B h, by { simpa only [le_antisymm_iff, unit_group_le_unit_group] using h} | lemma | valuation_subring.unit_group_injective | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subgroup",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff_unit_group {A B : valuation_subring K} :
A = B ↔ A.unit_group = B.unit_group | unit_group_injective.eq_iff.symm | lemma | valuation_subring.eq_iff_unit_group | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_group_order_embedding : valuation_subring K ↪o subgroup Kˣ | { to_fun := λ A, A.unit_group,
inj' := unit_group_injective,
map_rel_iff' := λ A B, unit_group_le_unit_group } | def | valuation_subring.unit_group_order_embedding | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subgroup",
"valuation_subring"
] | The map on valuation subrings to their unit groups is an order embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_group_strict_mono : strict_mono (unit_group : valuation_subring K → subgroup _) | unit_group_order_embedding.strict_mono | lemma | valuation_subring.unit_group_strict_mono | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"strict_mono",
"subgroup",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonunits : subsemigroup K | { carrier := { x | A.valuation x < 1 },
mul_mem' := λ a b ha hb, (mul_lt_mul₀ ha hb).trans_eq $ mul_one _ } | def | valuation_subring.nonunits | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"mul_lt_mul₀",
"mul_one",
"nonunits",
"subsemigroup"
] | The nonunits of a valuation subring of `K`, as a subsemigroup of `K` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nonunits_iff {x : K} : x ∈ A.nonunits ↔ A.valuation x < 1 | iff.rfl | lemma | valuation_subring.mem_nonunits_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"mem_nonunits_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonunits_le_nonunits {A B : valuation_subring K} :
B.nonunits ≤ A.nonunits ↔ A ≤ B | begin
split,
{ intros h x hx,
by_cases h_1 : x = 0, { simp only [h_1, zero_mem] },
rw [← valuation_le_one_iff, ← not_lt, valuation.one_lt_val_iff _ h_1] at hx ⊢,
by_contra h_2, from hx (h h_2) },
{ intros h x hx,
by_contra h_1, from not_lt.2 (monotone_map_of_le _ _ h (not_lt.1 h_1)) hx }
end | lemma | valuation_subring.nonunits_le_nonunits | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"by_contra",
"valuation.one_lt_val_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonunits_injective :
function.injective (nonunits : valuation_subring K → subsemigroup _) | λ A B h, by { simpa only [le_antisymm_iff, nonunits_le_nonunits] using h.symm } | lemma | valuation_subring.nonunits_injective | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"nonunits",
"subsemigroup",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonunits_inj {A B : valuation_subring K} : A.nonunits = B.nonunits ↔ A = B | nonunits_injective.eq_iff | lemma | valuation_subring.nonunits_inj | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonunits_order_embedding :
valuation_subring K ↪o (subsemigroup K)ᵒᵈ | { to_fun := λ A, A.nonunits,
inj' := nonunits_injective,
map_rel_iff' := λ A B, nonunits_le_nonunits } | def | valuation_subring.nonunits_order_embedding | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subsemigroup",
"valuation_subring"
] | The map on valuation subrings to their nonunits is a dual order embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mem_nonunits_iff {a : A} : (a : K) ∈ A.nonunits ↔ a ∈ local_ring.maximal_ideal A | (valuation_lt_one_iff _ _).symm | theorem | valuation_subring.coe_mem_nonunits_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"local_ring.maximal_ideal"
] | The elements of `A.nonunits` are those of the maximal ideal of `A` after coercion to `K`.
See also `mem_nonunits_iff_exists_mem_maximal_ideal`, which gets rid of the coercion to `K`,
at the expense of a more complicated right hand side. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonunits_le : A.nonunits ≤ A.to_subring.to_submonoid.to_subsemigroup | λ a ha, (A.valuation_le_one_iff _).mp (A.mem_nonunits_iff.mp ha).le | lemma | valuation_subring.nonunits_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonunits_subset : (A.nonunits : set K) ⊆ A | nonunits_le | lemma | valuation_subring.nonunits_subset | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nonunits_iff_exists_mem_maximal_ideal {a : K} :
a ∈ A.nonunits ↔ ∃ ha, (⟨a, ha⟩ : A) ∈ local_ring.maximal_ideal A | ⟨λ h, ⟨nonunits_subset h, coe_mem_nonunits_iff.mp h⟩,
λ ⟨ha, h⟩, coe_mem_nonunits_iff.mpr h⟩ | theorem | valuation_subring.mem_nonunits_iff_exists_mem_maximal_ideal | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"local_ring.maximal_ideal"
] | The elements of `A.nonunits` are those of the maximal ideal of `A`.
See also `coe_mem_nonunits_iff`, which has a simpler right hand side but requires the element
to be in `A` already. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_maximal_ideal : (coe : A → K) '' local_ring.maximal_ideal A = A.nonunits | begin
ext a,
simp only [set.mem_image, set_like.mem_coe, mem_nonunits_iff_exists_mem_maximal_ideal],
erw subtype.exists,
simp_rw [subtype.coe_mk, exists_and_distrib_right, exists_eq_right],
end | theorem | valuation_subring.image_maximal_ideal | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"exists_and_distrib_right",
"exists_eq_right",
"local_ring.maximal_ideal",
"set.mem_image",
"set_like.mem_coe",
"subtype.coe_mk"
] | `A.nonunits` agrees with the maximal ideal of `A`, after taking its image in `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
principal_unit_group : subgroup Kˣ | { carrier := { x | A.valuation (x - 1) < 1 },
mul_mem' := begin
intros a b ha hb,
refine lt_of_le_of_lt _ (max_lt hb ha),
rw [← one_mul (A.valuation (b - 1)), ← A.valuation.map_one_add_of_lt ha, add_sub_cancel'_right,
← valuation.map_mul, mul_sub_one, ← sub_add_sub_cancel],
exact A.valuation.map... | def | valuation_subring.principal_unit_group | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"mul_one",
"mul_sub_one",
"one_mul",
"subgroup",
"units.inv_mul",
"valuation.map_mul",
"valuation.map_neg"
] | The principal unit group of a valuation subring, as a subgroup of `Kˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
principal_units_le_units : A.principal_unit_group ≤ A.unit_group | λ a h, by simpa only [add_sub_cancel'_right] using A.valuation.map_one_add_of_lt h | lemma | valuation_subring.principal_units_le_units | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_principal_unit_group_iff (x : Kˣ) :
x ∈ A.principal_unit_group ↔ A.valuation ((x : K) - 1) < 1 | iff.rfl | lemma | valuation_subring.mem_principal_unit_group_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_unit_group_le_principal_unit_group {A B : valuation_subring K} :
B.principal_unit_group ≤ A.principal_unit_group ↔ A ≤ B | begin
split,
{ intros h x hx,
by_cases h_1 : x = 0, { simp only [h_1, zero_mem] },
by_cases h_2 : x⁻¹ + 1 = 0,
{ rw [add_eq_zero_iff_eq_neg, inv_eq_iff_eq_inv, inv_neg, inv_one] at h_2,
simpa only [h_2] using B.neg_mem _ B.one_mem },
{ rw [← valuation_le_one_iff, ← not_lt, valuation.one_lt_val... | lemma | valuation_subring.principal_unit_group_le_principal_unit_group | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"by_contra",
"inv_eq_iff_eq_inv",
"inv_neg",
"inv_one",
"units.coe_mk0",
"units.mk0",
"valuation.one_lt_val_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_unit_group_injective :
function.injective (principal_unit_group : valuation_subring K → subgroup _) | λ A B h, by { simpa [le_antisymm_iff, principal_unit_group_le_principal_unit_group] using h.symm } | lemma | valuation_subring.principal_unit_group_injective | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subgroup",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff_principal_unit_group {A B : valuation_subring K} :
A = B ↔ A.principal_unit_group = B.principal_unit_group | principal_unit_group_injective.eq_iff.symm | lemma | valuation_subring.eq_iff_principal_unit_group | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_unit_group_order_embedding :
valuation_subring K ↪o (subgroup Kˣ)ᵒᵈ | { to_fun := λ A, A.principal_unit_group,
inj' := principal_unit_group_injective,
map_rel_iff' := λ A B, principal_unit_group_le_principal_unit_group } | def | valuation_subring.principal_unit_group_order_embedding | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subgroup",
"valuation_subring"
] | The map on valuation subrings to their principal unit groups is an order embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mem_principal_unit_group_iff {x : A.unit_group} :
(x : Kˣ) ∈ A.principal_unit_group ↔
A.unit_group_mul_equiv x ∈ (units.map (local_ring.residue A).to_monoid_hom).ker | begin
rw [monoid_hom.mem_ker, units.ext_iff],
let π := ideal.quotient.mk (local_ring.maximal_ideal A), convert_to _ ↔ π _ = 1,
rw [← π.map_one, ← sub_eq_zero, ← π.map_sub, ideal.quotient.eq_zero_iff_mem,
valuation_lt_one_iff],
simpa,
end | lemma | valuation_subring.coe_mem_principal_unit_group_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.mk",
"local_ring.maximal_ideal",
"local_ring.residue",
"monoid_hom.mem_ker",
"units.ext_iff",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_unit_group_equiv :
A.principal_unit_group ≃* (units.map (local_ring.residue A).to_monoid_hom).ker | { to_fun := λ x, ⟨A.unit_group_mul_equiv ⟨_, A.principal_units_le_units x.2⟩,
A.coe_mem_principal_unit_group_iff.1 x.2⟩,
inv_fun := λ x, ⟨A.unit_group_mul_equiv.symm x,
by { rw A.coe_mem_principal_unit_group_iff, simpa using set_like.coe_mem x }⟩,
left_inv := λ x, by simp,
right_inv := λ x, by simp,
map... | def | valuation_subring.principal_unit_group_equiv | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"inv_fun",
"local_ring.residue",
"set_like.coe_mem",
"units.map"
] | The principal unit group agrees with the kernel of the canonical map from
the units of `A` to the units of the residue field of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
principal_unit_group_equiv_apply (a : A.principal_unit_group) :
(principal_unit_group_equiv A a : K) = a | rfl | lemma | valuation_subring.principal_unit_group_equiv_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_unit_group_symm_apply
(a : (units.map (local_ring.residue A).to_monoid_hom).ker) :
(A.principal_unit_group_equiv.symm a : K) = a | rfl | lemma | valuation_subring.principal_unit_group_symm_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"local_ring.residue",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_group_to_residue_field_units :
A.unit_group →* (local_ring.residue_field A)ˣ | monoid_hom.comp (units.map $ (ideal.quotient.mk _).to_monoid_hom)
A.unit_group_mul_equiv.to_monoid_hom | def | valuation_subring.unit_group_to_residue_field_units | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal.quotient.mk",
"local_ring.residue_field",
"monoid_hom.comp",
"units.map"
] | The canonical map from the unit group of `A` to the units of the residue field of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_unit_group_to_residue_field_units_apply (x : A.unit_group) :
(A.unit_group_to_residue_field_units x : (local_ring.residue_field A) ) =
(ideal.quotient.mk _ (A.unit_group_mul_equiv x : A)) | rfl | lemma | valuation_subring.coe_unit_group_to_residue_field_units_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal.quotient.mk",
"local_ring.residue_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_unit_group_to_residue_field_units :
A.unit_group_to_residue_field_units.ker = A.principal_unit_group.comap A.unit_group.subtype | by { ext, simpa only [subgroup.mem_comap, subgroup.coe_subtype, coe_mem_principal_unit_group_iff] } | lemma | valuation_subring.ker_unit_group_to_residue_field_units | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subgroup.coe_subtype",
"subgroup.mem_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_unit_group_to_residue_field_units :
function.surjective A.unit_group_to_residue_field_units | (local_ring.surjective_units_map_of_local_ring_hom _
ideal.quotient.mk_surjective local_ring.is_local_ring_hom_residue).comp (mul_equiv.surjective _) | lemma | valuation_subring.surjective_unit_group_to_residue_field_units | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal.quotient.mk_surjective",
"local_ring.is_local_ring_hom_residue",
"local_ring.surjective_units_map_of_local_ring_hom",
"mul_equiv.surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_mod_principal_units_equiv_residue_field_units :
(A.unit_group ⧸ (A.principal_unit_group.comap A.unit_group.subtype)) ≃*
(local_ring.residue_field A)ˣ | (quotient_group.quotient_mul_equiv_of_eq A.ker_unit_group_to_residue_field_units.symm).trans
(quotient_group.quotient_ker_equiv_of_surjective _ A.surjective_unit_group_to_residue_field_units) | def | valuation_subring.units_mod_principal_units_equiv_residue_field_units | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"local_ring.residue_field",
"quotient_group.quotient_ker_equiv_of_surjective",
"quotient_group.quotient_mul_equiv_of_eq"
] | The quotient of the unit group of `A` by the principal unit group of `A` agrees with
the units of the residue field of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_mod_principal_units_equiv_residue_field_units_comp_quotient_group_mk :
A.units_mod_principal_units_equiv_residue_field_units.to_monoid_hom.comp
(quotient_group.mk' _) = A.unit_group_to_residue_field_units | rfl | lemma | valuation_subring.units_mod_principal_units_equiv_residue_field_units_comp_quotient_group_mk | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"quotient_group.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_mod_principal_units_equiv_residue_field_units_comp_quotient_group_mk_apply
(x : A.unit_group) :
A.units_mod_principal_units_equiv_residue_field_units.to_monoid_hom
(quotient_group.mk x) = A.unit_group_to_residue_field_units x | rfl | lemma | valuation_subring.units_mod_principal_units_equiv_residue_field_units_comp_quotient_group_mk_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"quotient_group.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_has_smul : has_smul G (valuation_subring K) | { smul := λ g S,
-- TODO: if we add `valuation_subring.map` at a later date, we should use it here
{ mem_or_inv_mem' := λ x, (mem_or_inv_mem S (g⁻¹ • x)).imp
(subring.mem_pointwise_smul_iff_inv_smul_mem.mpr)
(λ h, subring.mem_pointwise_smul_iff_inv_smul_mem.mpr $ by rwa smul_inv''),
.. g • S.to_subr... | def | valuation_subring.pointwise_has_smul | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"has_smul",
"smul_inv''",
"valuation_subring"
] | The action on a valuation subring corresponding to applying the action to every element.
This is available as an instance in the `pointwise` locale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pointwise_smul (g : G) (S : valuation_subring K) : ↑(g • S) = g • (S : set K) | rfl | lemma | valuation_subring.coe_pointwise_smul | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_to_subring (g : G) (S : valuation_subring K) :
(g • S).to_subring = g • S.to_subring | rfl | lemma | valuation_subring.pointwise_smul_to_subring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_mul_action : mul_action G (valuation_subring K) | to_subring_injective.mul_action to_subring pointwise_smul_to_subring | def | valuation_subring.pointwise_mul_action | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"mul_action",
"valuation_subring"
] | The action on a valuation subring corresponding to applying the action to every element.
This is available as an instance in the `pointwise` locale.
This is a stronger version of `valuation_subring.pointwise_has_smul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_mem_pointwise_smul (g : G) (x : K) (S : valuation_subring K) : x ∈ S → g • x ∈ g • S | (set.smul_mem_smul_set : _ → _ ∈ g • (S : set K)) | lemma | valuation_subring.smul_mem_pointwise_smul | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set.smul_mem_smul_set",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_smul_pointwise_iff_exists (g : G) (x : K) (S : valuation_subring K) :
x ∈ g • S ↔ ∃ (s : K), s ∈ S ∧ g • s = x | (set.mem_smul_set : x ∈ g • (S : set K) ↔ _) | lemma | valuation_subring.mem_smul_pointwise_iff_exists | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set.mem_smul_set",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_central_scalar [mul_semiring_action Gᵐᵒᵖ K] [is_central_scalar G K] :
is_central_scalar G (valuation_subring K) | ⟨λ g S, to_subring_injective $ by exact op_smul_eq_smul g S.to_subring⟩ | instance | valuation_subring.pointwise_central_scalar | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"is_central_scalar",
"mul_semiring_action",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_pointwise_smul_iff {g : G} {S : valuation_subring K} {x : K} :
g • x ∈ g • S ↔ x ∈ S | set.smul_mem_smul_set_iff | lemma | valuation_subring.smul_mem_pointwise_smul_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set.smul_mem_smul_set_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pointwise_smul_iff_inv_smul_mem {g : G} {S : valuation_subring K} {x : K} :
x ∈ g • S ↔ g⁻¹ • x ∈ S | set.mem_smul_set_iff_inv_smul_mem | lemma | valuation_subring.mem_pointwise_smul_iff_inv_smul_mem | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set.mem_smul_set_iff_inv_smul_mem",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inv_pointwise_smul_iff {g : G} {S : valuation_subring K} {x : K} :
x ∈ g⁻¹ • S ↔ g • x ∈ S | set.mem_inv_smul_set_iff | lemma | valuation_subring.mem_inv_pointwise_smul_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set.mem_inv_smul_set_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_le_pointwise_smul_iff {g : G} {S T : valuation_subring K} :
g • S ≤ g • T ↔ S ≤ T | set.set_smul_subset_set_smul_iff | lemma | valuation_subring.pointwise_smul_le_pointwise_smul_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set.set_smul_subset_set_smul_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pointwise_smul_subset_iff {g : G} {S T : valuation_subring K} : g • S ≤ T ↔ S ≤ g⁻¹ • T | set.set_smul_subset_iff | lemma | valuation_subring.pointwise_smul_subset_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set.set_smul_subset_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_pointwise_smul_iff {g : G} {S T : valuation_subring K} : S ≤ g • T ↔ g⁻¹ • S ≤ T | set.subset_set_smul_iff | lemma | valuation_subring.subset_pointwise_smul_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set.subset_set_smul_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (A : valuation_subring L) (f : K →+* L) :
valuation_subring K | { mem_or_inv_mem' := λ k, by simp [valuation_subring.mem_or_inv_mem],
..(A.to_subring.comap f) } | def | valuation_subring.comap | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring",
"valuation_subring.mem_or_inv_mem"
] | The pullback of a valuation subring `A` along a ring homomorphism `K →+* L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comap (A : valuation_subring L) (f : K →+* L) : (A.comap f : set K) = f ⁻¹' A | rfl | lemma | valuation_subring.coe_comap | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {A : valuation_subring L} {f : K →+* L} {x : K} : x ∈ A.comap f ↔ f x ∈ A | iff.rfl | lemma | valuation_subring.mem_comap | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap (A : valuation_subring J) (g : L →+* J) (f : K →+* L) :
(A.comap g).comap f = A.comap (g.comp f) | rfl | lemma | valuation_subring.comap_comap | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_unit_group_iff : x ∈ v.valuation_subring.unit_group ↔ v x = 1 | (valuation.is_equiv_iff_val_eq_one _ _).mp (valuation.is_equiv_valuation_valuation_subring _).symm | lemma | valuation.mem_unit_group_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation.is_equiv_iff_val_eq_one",
"valuation.is_equiv_valuation_valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_fun (f : α → β) : 𝕎 α → 𝕎 β | λ x, mk _ (f ∘ x.coeff) | def | witt_vector.map_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | `f : α → β` induces a map from `𝕎 α` to `𝕎 β` by applying `f` componentwise.
If `f` is a ring homomorphism, then so is `f`, see `witt_vector.map f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective (f : α → β) (hf : injective f) : injective (map_fun f : 𝕎 α → 𝕎 β) | λ x y h, ext $ λ n, hf (congr_arg (λ x, coeff x n) h : _) | lemma | witt_vector.map_fun.injective | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective (f : α → β) (hf : surjective f) : surjective (map_fun f : 𝕎 α → 𝕎 β) | λ x, ⟨mk _ (λ n, classical.some $ hf $ x.coeff n),
by { ext n, dsimp [map_fun], rw classical.some_spec (hf (x.coeff n)) }⟩ | lemma | witt_vector.map_fun.surjective | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_fun_tac : tactic unit | `[ext n,
show f (aeval _ _) = aeval _ _,
rw map_aeval,
apply eval₂_hom_congr (ring_hom.ext_int _ _) _ rfl,
ext ⟨i, k⟩,
fin_cases i; refl] | def | witt_vector.map_fun.map_fun_tac | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"ring_hom.ext_int"
] | Auxiliary tactic for showing that `map_fun` respects the ring operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero : map_fun f (0 : 𝕎 R) = 0 | by map_fun_tac | lemma | witt_vector.map_fun.zero | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one : map_fun f (1 : 𝕎 R) = 1 | by map_fun_tac | lemma | witt_vector.map_fun.one | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add : map_fun f (x + y) = map_fun f x + map_fun f y | by map_fun_tac | lemma | witt_vector.map_fun.add | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub : map_fun f (x - y) = map_fun f x - map_fun f y | by map_fun_tac | lemma | witt_vector.map_fun.sub | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul : map_fun f (x * y) = map_fun f x * map_fun f y | by map_fun_tac | lemma | witt_vector.map_fun.mul | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg : map_fun f (-x) = -map_fun f x | by map_fun_tac | lemma | witt_vector.map_fun.neg | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul (n : ℕ) : map_fun f (n • x) = n • map_fun f x | by map_fun_tac | lemma | witt_vector.map_fun.nsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul (z : ℤ) : map_fun f (z • x) = z • map_fun f x | by map_fun_tac | lemma | witt_vector.map_fun.zsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow (n : ℕ) : map_fun f (x^ n) = map_fun f x ^ n | by map_fun_tac | lemma | witt_vector.map_fun.pow | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast (n : ℕ) : map_fun f (n : 𝕎 R) = n | show map_fun f n.unary_cast = coe n,
by induction n; simp [*, nat.unary_cast, add, one, zero]; refl | lemma | witt_vector.map_fun.nat_cast | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"nat.unary_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast (n : ℤ) : map_fun f (n : 𝕎 R) = n | show map_fun f n.cast_def = coe n,
by cases n; simp [*, int.cast_def, add, one, neg, zero, nat_cast]; refl | lemma | witt_vector.map_fun.int_cast | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"int.cast_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tactic.interactive.ghost_fun_tac (φ fn : parse parser.pexpr) : tactic unit | do
fn ← to_expr ```(%%fn : fin _ → ℕ → R),
`(fin %%k → _ → _) ← infer_type fn,
`[ext n],
`[dunfold
witt_vector.has_zero witt_zero
witt_vector.has_one witt_one
witt_vector.has_neg witt_neg
witt_vector.has_mul witt_mul
witt_vector.has_sub witt_sub
witt_vector.has_add witt_add
witt_vector.has_nat_scalar witt... | def | tactic.interactive.ghost_fun_tac | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"witt_vector.has_int_scalar",
"witt_vector.has_nat_pow",
"witt_vector.has_nat_scalar"
] | An auxiliary tactic for proving that `ghost_fun` respects the ring operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_fun : 𝕎 R → (ℕ → R) | λ x n, aeval x.coeff (W_ ℤ n) | def | witt_vector.ghost_fun | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | Evaluates the `n`th Witt polynomial on the first `n` coefficients of `x`,
producing a value in `R`.
This function will be bundled as the ring homomorphism `witt_vector.ghost_map`
once the ring structure is available,
but we rely on it to set up the ring structure in the first place. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
matrix_vec_empty_coeff {R} (i j) :
@coeff p R (matrix.vec_empty i) j = (matrix.vec_empty i : ℕ → R) j | by rcases i with ⟨_ | _ | _ | _ | i_val, ⟨⟩⟩ | lemma | witt_vector.matrix_vec_empty_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"matrix.vec_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_zero : ghost_fun (0 : 𝕎 R) = 0 | by ghost_fun_tac 0 ![] | lemma | witt_vector.ghost_fun_zero | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_one : ghost_fun (1 : 𝕎 R) = 1 | by ghost_fun_tac 1 ![] | lemma | witt_vector.ghost_fun_one | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_add : ghost_fun (x + y) = ghost_fun x + ghost_fun y | by ghost_fun_tac (X 0 + X 1) ![x.coeff, y.coeff] | lemma | witt_vector.ghost_fun_add | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_nat_cast (i : ℕ) : ghost_fun (i : 𝕎 R) = i | show ghost_fun i.unary_cast = _,
by induction i; simp [*, nat.unary_cast, ghost_fun_zero, ghost_fun_one, ghost_fun_add, -pi.coe_nat] | lemma | witt_vector.ghost_fun_nat_cast | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"nat.unary_cast",
"pi.coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_sub : ghost_fun (x - y) = ghost_fun x - ghost_fun y | by ghost_fun_tac (X 0 - X 1) ![x.coeff, y.coeff] | lemma | witt_vector.ghost_fun_sub | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_mul : ghost_fun (x * y) = ghost_fun x * ghost_fun y | by ghost_fun_tac (X 0 * X 1) ![x.coeff, y.coeff] | lemma | witt_vector.ghost_fun_mul | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_neg : ghost_fun (-x) = - ghost_fun x | by ghost_fun_tac (-X 0) ![x.coeff] | lemma | witt_vector.ghost_fun_neg | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_int_cast (i : ℤ) : ghost_fun (i : 𝕎 R) = i | show ghost_fun i.cast_def = _,
by cases i; simp [*, int.cast_def, ghost_fun_nat_cast, ghost_fun_neg, -pi.coe_nat, -pi.coe_int] | lemma | witt_vector.ghost_fun_int_cast | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"int.cast_def",
"pi.coe_int",
"pi.coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_nsmul (m : ℕ) : ghost_fun (m • x) = m • ghost_fun x | by ghost_fun_tac (m • X 0) ![x.coeff] | lemma | witt_vector.ghost_fun_nsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_zsmul (m : ℤ) : ghost_fun (m • x) = m • ghost_fun x | by ghost_fun_tac (m • X 0) ![x.coeff] | lemma | witt_vector.ghost_fun_zsmul | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_fun_pow (m : ℕ) : ghost_fun (x ^ m) = ghost_fun x ^ m | by ghost_fun_tac (X 0 ^ m) ![x.coeff] | lemma | witt_vector.ghost_fun_pow | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_equiv' [invertible (p : R)] : 𝕎 R ≃ (ℕ → R) | { to_fun := ghost_fun,
inv_fun := λ x, mk p $ λ n, aeval x (X_in_terms_of_W p R n),
left_inv :=
begin
intro x,
ext n,
have := bind₁_witt_polynomial_X_in_terms_of_W p R n,
apply_fun (aeval x.coeff) at this,
simpa only [aeval_bind₁, aeval_X, ghost_fun, aeval_witt_polynomial]
end,
right_inv :... | def | witt_vector.ghost_equiv' | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"X_in_terms_of_W",
"aeval_witt_polynomial",
"bind₁_X_in_terms_of_W_witt_polynomial",
"bind₁_witt_polynomial_X_in_terms_of_W",
"inv_fun",
"invertible"
] | The bijection between `𝕎 R` and `ℕ → R`, under the assumption that `p` is invertible in `R`.
In `witt_vector.ghost_equiv` we upgrade this to an isomorphism of rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_ring_aux₁ : comm_ring (𝕎 (mv_polynomial R ℚ)) | by letI : comm_ring (mv_polynomial R ℚ) := mv_polynomial.comm_ring; exact
(ghost_equiv' p (mv_polynomial R ℚ)).injective.comm_ring (ghost_fun)
ghost_fun_zero ghost_fun_one ghost_fun_add ghost_fun_mul ghost_fun_neg ghost_fun_sub
ghost_fun_nsmul ghost_fun_zsmul ghost_fun_pow ghost_fun_nat_cast ghost_fun_int_cast | def | witt_vector.comm_ring_aux₁ | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"comm_ring",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring_aux₂ : comm_ring (𝕎 (mv_polynomial R ℤ)) | (map_fun.injective _ $ map_injective (int.cast_ring_hom ℚ) int.cast_injective).comm_ring _
(map_fun.zero _) (map_fun.one _) (map_fun.add _) (map_fun.mul _) (map_fun.neg _) (map_fun.sub _)
(map_fun.nsmul _) (map_fun.zsmul _) (map_fun.pow _) (map_fun.nat_cast _) (map_fun.int_cast _) | def | witt_vector.comm_ring_aux₂ | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"comm_ring",
"int.cast_injective",
"int.cast_ring_hom",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_injective (f : R →+* S) (hf : injective f) : injective (map f : 𝕎 R → 𝕎 S) | map_fun.injective f hf | lemma | witt_vector.map_injective | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_surjective (f : R →+* S) (hf : surjective f) : surjective (map f : 𝕎 R → 𝕎 S) | map_fun.surjective f hf | lemma | witt_vector.map_surjective | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coeff (f : R →+* S) (x : 𝕎 R) (n : ℕ) :
(map f x).coeff n = f (x.coeff n) | rfl | lemma | witt_vector.map_coeff | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_map : 𝕎 R →+* ℕ → R | { to_fun := ghost_fun,
map_zero' := ghost_fun_zero,
map_one' := ghost_fun_one,
map_add' := ghost_fun_add,
map_mul' := ghost_fun_mul } | def | witt_vector.ghost_map | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | `witt_vector.ghost_map` is a ring homomorphism that maps each Witt vector
to the sequence of its ghost components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_component (n : ℕ) : 𝕎 R →+* R | (pi.eval_ring_hom _ n).comp ghost_map | def | witt_vector.ghost_component | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [
"pi.eval_ring_hom"
] | Evaluates the `n`th Witt polynomial on the first `n` coefficients of `x`,
producing a value in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_component_apply (n : ℕ) (x : 𝕎 R) : ghost_component n x = aeval x.coeff (W_ ℤ n) | rfl | lemma | witt_vector.ghost_component_apply | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_map_apply (x : 𝕎 R) (n : ℕ) : ghost_map x n = ghost_component n x | rfl | lemma | witt_vector.ghost_map_apply | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_equiv : 𝕎 R ≃+* (ℕ → R) | { .. (ghost_map : 𝕎 R →+* (ℕ → R)), .. (ghost_equiv' p R) } | def | witt_vector.ghost_equiv | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | `witt_vector.ghost_map` is a ring isomorphism when `p` is invertible in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghost_equiv_coe : (ghost_equiv p R : 𝕎 R →+* (ℕ → R)) = ghost_map | rfl | lemma | witt_vector.ghost_equiv_coe | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ghost_map.bijective_of_invertible : function.bijective (ghost_map : 𝕎 R → ℕ → R) | (ghost_equiv p R).bijective | lemma | witt_vector.ghost_map.bijective_of_invertible | ring_theory.witt_vector | src/ring_theory/witt_vector/basic.lean | [
"data.mv_polynomial.counit",
"data.mv_polynomial.invertible",
"ring_theory.witt_vector.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_le_of_cast_pow_eq_zero [char_p R p] (i : ℕ) (hin : i ≤ n)
(hpi : (p ^ i : truncated_witt_vector p n R) = 0) :
i = n | begin
contrapose! hpi,
replace hin := lt_of_le_of_ne hin hpi, clear hpi,
have : (↑p ^ i : truncated_witt_vector p n R) = witt_vector.truncate n (↑p ^ i),
{ rw [ring_hom.map_pow, map_nat_cast] },
rw [this, ext_iff, not_forall], clear this,
use ⟨i, hin⟩,
rw [witt_vector.coeff_truncate, coeff_zero, fin.coe_m... | lemma | truncated_witt_vector.eq_of_le_of_cast_pow_eq_zero | 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"
] | [
"char_p",
"char_p.nontrivial_of_char_ne_one",
"fin.coe_mk",
"map_nat_cast",
"nontrivial",
"not_forall",
"one_ne_zero",
"ring_hom.map_pow",
"truncated_witt_vector",
"witt_vector.coeff_p_pow",
"witt_vector.coeff_truncate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_zmod : fintype.card (truncated_witt_vector p n (zmod p)) = p ^ n | by rw [card, zmod.card] | lemma | truncated_witt_vector.card_zmod | 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"
] | [
"fintype.card",
"truncated_witt_vector",
"zmod",
"zmod.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p_zmod : char_p (truncated_witt_vector p n (zmod p)) (p ^ n) | char_p_of_prime_pow_injective _ _ _ (card_zmod _ _)
(eq_of_le_of_cast_pow_eq_zero p n (zmod p)) | lemma | truncated_witt_vector.char_p_zmod | 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"
] | [
"char_p",
"char_p_of_prime_pow_injective",
"truncated_witt_vector",
"zmod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmod_equiv_trunc : zmod (p^n) ≃+* truncated_witt_vector p n (zmod p) | zmod.ring_equiv (truncated_witt_vector p n (zmod p)) (card_zmod _ _) | def | truncated_witt_vector.zmod_equiv_trunc | 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.ring_equiv"
] | The unique isomorphism between `zmod p^n` and `truncated_witt_vector p n (zmod p)`.
This isomorphism exists, because `truncated_witt_vector p n (zmod p)` is a finite ring
with characteristic and cardinality `p^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.