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
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le_span_singleton_mul_iff {x : P} {I J : fractional_ideal S P} :
I ≤ span_singleton S x * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI | show (∀ {zI} (hzI : zI ∈ I), zI ∈ span_singleton _ x * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI,
by simp only [mem_singleton_mul, eq_comm] | lemma | fractional_ideal.le_span_singleton_mul_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_le_iff {x : P} {I J : fractional_ideal S P} :
span_singleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | begin
simp only [mul_le, mem_singleton_mul, mem_span_singleton],
split,
{ intros h zI hzI,
exact h x ⟨1, one_smul _ _⟩ zI hzI },
{ rintros h _ ⟨z, rfl⟩ zI hzI,
rw [algebra.smul_mul_assoc],
exact submodule.smul_mem J.1 _ (h zI hzI) },
end | lemma | fractional_ideal.span_singleton_mul_le_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_mul_assoc",
"fractional_ideal",
"one_smul",
"submodule.smul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_span_singleton_mul {x : P} {I J : fractional_ideal S P} :
I = span_singleton _ x * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I | by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] | lemma | fractional_ideal.eq_span_singleton_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_zero : is_noetherian R₁ (0 : fractional_ideal R₁⁰ K) | is_noetherian_submodule.mpr (λ I (hI : I ≤ (0 : fractional_ideal R₁⁰ K)),
by { rw coe_zero at hI, rw le_bot_iff.mp hI, exact fg_bot }) | lemma | fractional_ideal.is_noetherian_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"is_noetherian"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_iff {I : fractional_ideal R₁⁰ K} :
is_noetherian R₁ I ↔ ∀ J ≤ I, (J : submodule R₁ K).fg | is_noetherian_submodule.trans ⟨λ h J hJ, h _ hJ, λ h J hJ, h ⟨J, is_fractional_of_le hJ⟩ hJ⟩ | lemma | fractional_ideal.is_noetherian_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"is_noetherian",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_coe_ideal [_root_.is_noetherian_ring R₁] (I : ideal R₁) :
is_noetherian R₁ (I : fractional_ideal R₁⁰ K) | begin
rw is_noetherian_iff,
intros J hJ,
obtain ⟨J, rfl⟩ := le_one_iff_exists_coe_ideal.mp (le_trans hJ coe_ideal_le_one),
exact (is_noetherian.noetherian J).map _,
end | lemma | fractional_ideal.is_noetherian_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"is_noetherian"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_span_singleton_inv_to_map_mul (x : R₁) {I : fractional_ideal R₁⁰ K}
(hI : is_noetherian R₁ I) :
is_noetherian R₁ (span_singleton R₁⁰ (algebra_map R₁ K x)⁻¹ * I : fractional_ideal R₁⁰ K) | begin
by_cases hx : x = 0,
{ rw [hx, ring_hom.map_zero, _root_.inv_zero, span_singleton_zero, zero_mul],
exact is_noetherian_zero },
have h_gx : algebra_map R₁ K x ≠ 0,
from mt ((injective_iff_map_eq_zero (algebra_map R₁ K)).mp
(is_fraction_ring.injective _ _) x) hx,
have h_spanx : span_singleton R₁⁰ ... | lemma | fractional_ideal.is_noetherian_span_singleton_inv_to_map_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"finset.coe_mul",
"finset.coe_singleton",
"fractional_ideal",
"is_fraction_ring.injective",
"is_noetherian",
"le_div_iff_mul_le",
"mul_assoc",
"mul_inv_cancel",
"mul_one",
"ring_hom.map_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian [_root_.is_noetherian_ring R₁] (I : fractional_ideal R₁⁰ K) :
is_noetherian R₁ I | begin
obtain ⟨d, J, h_nzd, rfl⟩ := exists_eq_span_singleton_mul I,
apply is_noetherian_span_singleton_inv_to_map_mul,
apply is_noetherian_coe_ideal
end | theorem | fractional_ideal.is_noetherian | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"is_noetherian"
] | Every fractional ideal of a noetherian integral domain is noetherian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fractional_adjoin_integral :
is_fractional S (algebra.adjoin R ({x} : set P)).to_submodule | is_fractional_of_fg (fg_adjoin_singleton_of_integral x hx) | lemma | fractional_ideal.is_fractional_adjoin_integral | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.adjoin",
"fg_adjoin_singleton_of_integral",
"is_fractional"
] | `A[x]` is a fractional ideal for every integral `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_integral : fractional_ideal S P | ⟨_, is_fractional_adjoin_integral S x hx⟩ | def | fractional_ideal.adjoin_integral | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | `fractional_ideal.adjoin_integral (S : submonoid R) x hx` is `R[x]` as a fractional ideal,
where `hx` is a proof that `x : P` is integral over `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_adjoin_integral_self :
x ∈ adjoin_integral S x hx | algebra.subset_adjoin (set.mem_singleton x) | lemma | fractional_ideal.mem_adjoin_integral_self | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.subset_adjoin",
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_comm_ring (α : Type u) : Type u | free_abelian_group $ multiplicative $ multiset α | def | free_comm_ring | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_abelian_group",
"multiplicative",
"multiset"
] | `free_comm_ring α` is the free commutative ring on the type `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (x : α) : free_comm_ring α | free_abelian_group.of $ multiplicative.of_add ({x} : multiset α) | def | free_comm_ring.of | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_abelian_group.of",
"free_comm_ring",
"multiplicative.of_add",
"multiset"
] | The canonical map from `α` to the free commutative ring on `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_injective : function.injective (of : α → free_comm_ring α) | free_abelian_group.of_injective.comp (λ x y,
(multiset.coe_eq_coe.trans list.singleton_perm_singleton).mp) | lemma | free_comm_ring.of_injective | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"list.singleton_perm_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on
{C : free_comm_ring α → Prop} (z : free_comm_ring α)
(hn1 : C (-1)) (hb : ∀ b, C (of b))
(ha : ∀ x y, C x → C y → C (x + y))
(hm : ∀ x y, C x → C y → C (x * y)) : C z | have hn : ∀ x, C x → C (-x), from λ x ih, neg_one_mul x ▸ hm _ _ hn1 ih,
have h1 : C 1, from neg_neg (1 : free_comm_ring α) ▸ hn _ hn1,
free_abelian_group.induction_on z
(add_left_neg (1 : free_comm_ring α) ▸ ha _ _ hn1 h1)
(λ m, multiset.induction_on m h1 $ λ a m ih, hm _ _ (hb a) ih)
(λ m ih, hn _ ih)
ha | lemma | free_comm_ring.induction_on | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_abelian_group.induction_on",
"free_comm_ring",
"ih",
"multiset.induction_on",
"neg_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_to_multiset : (α → R) ≃ (multiplicative (multiset α) →* R) | { to_fun := λ f,
{ to_fun := λ s, (s.to_add.map f).prod,
map_mul' := λ x y, calc _ = multiset.prod ((multiset.map f x) + (multiset.map f y)) :
by {congr' 1, exact multiset.map_add _ _ _}
... = _ : multiset.prod_add _ _,
map_one' := rfl},
inv_fun ... | def | free_comm_ring.lift_to_multiset | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"inv_fun",
"monoid_hom.ext",
"multiplicative",
"multiplicative.of_add",
"multiset",
"multiset.map",
"multiset.map_add",
"multiset.prod",
"multiset.prod_add",
"multiset.sum_map_singleton"
] | A helper to implement `lift`. This is essentially `free_comm_monoid.lift`, but this does not
currently exist. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : (α → R) ≃ (free_comm_ring α →+* R) | equiv.trans lift_to_multiset free_abelian_group.lift_monoid | def | free_comm_ring.lift | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"equiv.trans",
"free_abelian_group.lift_monoid",
"free_comm_ring",
"lift"
] | Lift a map `α → R` to a additive group homomorphism `free_comm_ring α → R`.
For a version producing a bundled homomorphism, see `lift_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of (x : α) : lift f (of x) = f x | (free_abelian_group.lift.of _ _).trans $ mul_one _ | lemma | free_comm_ring.lift_of | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_abelian_group.lift.of",
"lift",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of (f : free_comm_ring α →+* R) : lift (f ∘ of) = f | ring_hom.ext $ λ x, free_comm_ring.induction_on x
(by rw [ring_hom.map_neg, ring_hom.map_one, f.map_neg, f.map_one])
(lift_of _)
(λ x y ihx ihy, by rw [ring_hom.map_add, f.map_add, ihx, ihy])
(λ x y ihx ihy, by rw [ring_hom.map_mul, f.map_mul, ihx, ihy]) | lemma | free_comm_ring.lift_comp_of | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_comm_ring.induction_on",
"lift",
"ring_hom.ext",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_neg",
"ring_hom.map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext ⦃f g : free_comm_ring α →+* R⦄ (h : ∀ x, f (of x) = g (of x)) :
f = g | lift.symm.injective (funext h) | lemma | free_comm_ring.hom_ext | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map : free_comm_ring α →+* free_comm_ring β | lift $ of ∘ f | def | free_comm_ring.map | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"lift"
] | A map `f : α → β` produces a ring homomorphism `free_comm_ring α →+* free_comm_ring β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_of (x : α) : map f (of x) = of (f x) | lift_of _ _ | lemma | free_comm_ring.map_of | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported (x : free_comm_ring α) (s : set α) : Prop | x ∈ subring.closure (of '' s) | def | free_comm_ring.is_supported | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"subring.closure"
] | `is_supported x s` means that all monomials showing up in `x` have variables in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_supported_upwards (hs : is_supported x s) (hst : s ⊆ t) :
is_supported x t | subring.closure_mono (set.monotone_image hst) hs | theorem | free_comm_ring.is_supported_upwards | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"set.monotone_image",
"subring.closure_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported_add (hxs : is_supported x s) (hys : is_supported y s) :
is_supported (x + y) s | subring.add_mem _ hxs hys | theorem | free_comm_ring.is_supported_add | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"subring.add_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported_neg (hxs : is_supported x s) :
is_supported (-x) s | subring.neg_mem _ hxs | theorem | free_comm_ring.is_supported_neg | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"subring.neg_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported_sub (hxs : is_supported x s) (hys : is_supported y s) :
is_supported (x - y) s | subring.sub_mem _ hxs hys | theorem | free_comm_ring.is_supported_sub | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"subring.sub_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported_mul (hxs : is_supported x s) (hys : is_supported y s) :
is_supported (x * y) s | subring.mul_mem _ hxs hys | theorem | free_comm_ring.is_supported_mul | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"subring.mul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported_zero : is_supported 0 s | subring.zero_mem _ | theorem | free_comm_ring.is_supported_zero | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"subring.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported_one : is_supported 1 s | subring.one_mem _ | theorem | free_comm_ring.is_supported_one | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"subring.one_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported_int {i : ℤ} {s : set α} : is_supported ↑i s | int.induction_on i is_supported_zero
(λ i hi, by rw [int.cast_add, int.cast_one]; exact is_supported_add hi is_supported_one)
(λ i hi, by rw [int.cast_sub, int.cast_one]; exact is_supported_sub hi is_supported_one) | theorem | free_comm_ring.is_supported_int | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"int.cast_add",
"int.cast_one",
"int.cast_sub",
"int.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restriction (s : set α) [decidable_pred (∈ s)] : free_comm_ring α →+* free_comm_ring s | lift (λ p, if H : p ∈ s then of (⟨p, H⟩ : s) else 0) | def | free_comm_ring.restriction | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"lift"
] | The restriction map from `free_comm_ring α` to `free_comm_ring s` where `s : set α`, defined
by sending all variables not in `s` to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restriction_of (p) :
restriction s (of p) = if H : p ∈ s then of ⟨p, H⟩ else 0 | lift_of _ _ | lemma | free_comm_ring.restriction_of | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_supported_of {p} {s : set α} : is_supported (of p) s ↔ p ∈ s | suffices is_supported (of p) s → p ∈ s, from ⟨this, λ hps, subring.subset_closure ⟨p, hps, rfl⟩⟩,
assume hps : is_supported (of p) s, begin
haveI := classical.dec_pred s,
have : ∀ x, is_supported x s →
∃ (n : ℤ), lift (λ a, if a ∈ s then (0 : ℤ[X]) else polynomial.X) x = n,
{ intros x hx, refine subring.in_cl... | theorem | free_comm_ring.is_supported_of | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"classical.dec_pred",
"int.cast_neg",
"int.cast_one",
"lift",
"one_ne_zero",
"polynomial.C",
"polynomial.X",
"polynomial.coeff_C",
"polynomial.coeff_X",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_neg",
"ring_hom.map_one",
"subring.in_closure.rec_on",
"subring.subset_closure",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_subtype_val_restriction {x} (s : set α) [decidable_pred (∈ s)]
(hxs : is_supported x s) :
map (subtype.val : s → α) (restriction s x) = x | begin
refine subring.in_closure.rec_on hxs _ _ _ _,
{ rw ring_hom.map_one, refl },
{ rw [ring_hom.map_neg, ring_hom.map_neg, ring_hom.map_one], refl },
{ rintros _ ⟨p, hps, rfl⟩ n ih,
rw [ring_hom.map_mul, restriction_of, dif_pos hps, ring_hom.map_mul, map_of, ih] },
{ intros x y ihx ihy, rw [ring_hom.map... | theorem | free_comm_ring.map_subtype_val_restriction | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"ih",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_neg",
"ring_hom.map_one",
"subring.in_closure.rec_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_finite_support (x : free_comm_ring α) :
∃ s : set α, set.finite s ∧ is_supported x s | free_comm_ring.induction_on x
⟨∅, set.finite_empty, is_supported_neg is_supported_one⟩
(λ p, ⟨{p}, set.finite_singleton p, is_supported_of.2 $ set.mem_singleton _⟩)
(λ x y ⟨s, hfs, hxs⟩ ⟨t, hft, hxt⟩, ⟨s ∪ t, hfs.union hft, is_supported_add
(is_supported_upwards hxs $ set.subset_union_left s t)
(is_suppor... | theorem | free_comm_ring.exists_finite_support | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_comm_ring.induction_on",
"set.finite",
"set.finite_empty",
"set.finite_singleton",
"set.mem_singleton",
"set.subset_union_left",
"set.subset_union_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_finset_support (x : free_comm_ring α) : ∃ s : finset α, is_supported x ↑s | let ⟨s, hfs, hxs⟩ := exists_finite_support x in ⟨hfs.to_finset, by rwa set.finite.coe_to_finset⟩ | theorem | free_comm_ring.exists_finset_support | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"finset",
"free_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_free_comm_ring {α} : free_ring α →+* free_comm_ring α | free_ring.lift free_comm_ring.of | def | free_ring.to_free_comm_ring | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_comm_ring.of",
"free_ring",
"free_ring.lift"
] | The canonical ring homomorphism from the free ring generated by `α` to the free commutative ring
generated by `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_ring_hom : free_ring α →+* free_comm_ring α | to_free_comm_ring | def | free_ring.coe_ring_hom | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_ring"
] | The natural map `free_ring α → free_comm_ring α`, as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_zero : ↑(0 : free_ring α) = (0 : free_comm_ring α) | rfl | lemma | free_ring.coe_zero | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ↑(1 : free_ring α) = (1 : free_comm_ring α) | rfl | lemma | free_ring.coe_one | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (a : α) : ↑(free_ring.of a) = free_comm_ring.of a | free_ring.lift_of _ _ | lemma | free_ring.coe_of | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring.of",
"free_ring.lift_of",
"free_ring.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (x : free_ring α) : ↑(-x) = -(x : free_comm_ring α) | (free_ring.lift _).map_neg _ | lemma | free_ring.coe_neg | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_ring",
"free_ring.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (x y : free_ring α) :
↑(x + y) = (x : free_comm_ring α) + y | (free_ring.lift _).map_add _ _ | lemma | free_ring.coe_add | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_ring",
"free_ring.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (x y : free_ring α) :
↑(x - y) = (x : free_comm_ring α) - y | (free_ring.lift _).map_sub _ _ | lemma | free_ring.coe_sub | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_ring",
"free_ring.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (x y : free_ring α) :
↑(x * y) = (x : free_comm_ring α) * y | (free_ring.lift _).map_mul _ _ | lemma | free_ring.coe_mul | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_ring",
"free_ring.lift",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_surjective : surjective (coe : free_ring α → free_comm_ring α) | λ x,
begin
apply free_comm_ring.induction_on x,
{ use -1, refl },
{ intro x, use free_ring.of x, refl },
{ rintros _ _ ⟨x, rfl⟩ ⟨y, rfl⟩, use x + y, exact (free_ring.lift _).map_add _ _ },
{ rintros _ _ ⟨x, rfl⟩ ⟨y, rfl⟩, use x * y, exact (free_ring.lift _).map_mul _ _ }
end | lemma | free_ring.coe_surjective | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_comm_ring.induction_on",
"free_ring",
"free_ring.lift",
"free_ring.of",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq :
(coe : free_ring α → free_comm_ring α) =
@functor.map free_abelian_group _ _ _ (λ (l : list α), (l : multiset α)) | funext $ λ x, free_abelian_group.lift.unique _ _ $ λ L,
by { simp_rw [free_abelian_group.lift.of, (∘)], exact free_monoid.rec_on L rfl
(λ hd tl ih, by { rw [(free_monoid.lift _).map_mul, free_monoid.lift_eval_of, ih], refl }) } | lemma | free_ring.coe_eq | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_abelian_group",
"free_abelian_group.lift.of",
"free_abelian_group.lift.unique",
"free_comm_ring",
"free_monoid.lift",
"free_monoid.lift_eval_of",
"free_monoid.rec_on",
"free_ring",
"ih",
"map_mul",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_equiv_free_comm_ring [subsingleton α] :
free_ring α ≃+* free_comm_ring α | ring_equiv.of_bijective (coe_ring_hom _)
begin
have : (coe_ring_hom _ : free_ring α → free_comm_ring α) =
(functor.map_equiv free_abelian_group (multiset.subsingleton_equiv α)) := coe_eq α,
rw this,
apply equiv.bijective,
end | def | free_ring.subsingleton_equiv_free_comm_ring | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"equiv.bijective",
"free_abelian_group",
"free_comm_ring",
"free_ring",
"functor.map_equiv",
"multiset.subsingleton_equiv",
"ring_equiv.of_bijective"
] | If α has size at most 1 then the natural map from the free ring on `α` to the
free commutative ring on `α` is an isomorphism of rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_comm_ring_equiv_mv_polynomial_int :
free_comm_ring α ≃+* mv_polynomial α ℤ | ring_equiv.of_hom_inv
(free_comm_ring.lift $ (λ a, mv_polynomial.X a : α → mv_polynomial α ℤ))
(mv_polynomial.eval₂_hom (int.cast_ring_hom (free_comm_ring α)) free_comm_ring.of)
(by { ext, simp })
(by ext; simp) | def | free_comm_ring_equiv_mv_polynomial_int | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_comm_ring.lift",
"free_comm_ring.of",
"int.cast_ring_hom",
"mv_polynomial",
"mv_polynomial.X",
"mv_polynomial.eval₂_hom",
"ring_equiv.of_hom_inv"
] | The free commutative ring on `α` is isomorphic to the polynomial ring over ℤ with
variables in `α` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_comm_ring_pempty_equiv_int : free_comm_ring pempty.{u+1} ≃+* ℤ | ring_equiv.trans (free_comm_ring_equiv_mv_polynomial_int _)
(mv_polynomial.is_empty_ring_equiv _ pempty) | def | free_comm_ring_pempty_equiv_int | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_comm_ring_equiv_mv_polynomial_int",
"mv_polynomial.is_empty_ring_equiv",
"pempty",
"ring_equiv.trans"
] | The free commutative ring on the empty type is isomorphic to `ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_comm_ring_punit_equiv_polynomial_int : free_comm_ring punit.{u+1} ≃+* ℤ[X] | (free_comm_ring_equiv_mv_polynomial_int _).trans (mv_polynomial.punit_alg_equiv ℤ).to_ring_equiv | def | free_comm_ring_punit_equiv_polynomial_int | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring",
"free_comm_ring_equiv_mv_polynomial_int",
"mv_polynomial.punit_alg_equiv"
] | The free commutative ring on a type with one term is isomorphic to `ℤ[X]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_ring_pempty_equiv_int : free_ring pempty.{u+1} ≃+* ℤ | ring_equiv.trans (subsingleton_equiv_free_comm_ring _) free_comm_ring_pempty_equiv_int | def | free_ring_pempty_equiv_int | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring_pempty_equiv_int",
"free_ring",
"ring_equiv.trans"
] | The free ring on the empty type is isomorphic to `ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_ring_punit_equiv_polynomial_int : free_ring punit.{u+1} ≃+* ℤ[X] | ring_equiv.trans (subsingleton_equiv_free_comm_ring _) free_comm_ring_punit_equiv_polynomial_int | def | free_ring_punit_equiv_polynomial_int | ring_theory | src/ring_theory/free_comm_ring.lean | [
"data.mv_polynomial.equiv",
"data.mv_polynomial.comm_ring",
"logic.equiv.functor",
"ring_theory.free_ring"
] | [
"free_comm_ring_punit_equiv_polynomial_int",
"free_ring",
"ring_equiv.trans"
] | The free ring on a type with one term is isomorphic to `ℤ[X]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_ring (α : Type u) : Type u | free_abelian_group $ free_monoid α | def | free_ring | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"free_abelian_group",
"free_monoid"
] | The free ring over a type `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (x : α) : free_ring α | free_abelian_group.of (free_monoid.of x) | def | free_ring.of | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"free_abelian_group.of",
"free_monoid.of",
"free_ring"
] | The canonical map from α to `free_ring α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_injective : function.injective (of : α → free_ring α) | free_abelian_group.of_injective.comp free_monoid.of_injective | lemma | free_ring.of_injective | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"free_monoid.of_injective",
"free_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on
{C : free_ring α → Prop} (z : free_ring α)
(hn1 : C (-1)) (hb : ∀ b, C (of b))
(ha : ∀ x y, C x → C y → C (x + y))
(hm : ∀ x y, C x → C y → C (x * y)) : C z | have hn : ∀ x, C x → C (-x), from λ x ih, neg_one_mul x ▸ hm _ _ hn1 ih,
have h1 : C 1, from neg_neg (1 : free_ring α) ▸ hn _ hn1,
free_abelian_group.induction_on z
(add_left_neg (1 : free_ring α) ▸ ha _ _ hn1 h1)
(λ m, list.rec_on m h1 $ λ a m ih, hm _ _ (hb a) ih)
(λ m ih, hn _ ih)
ha | lemma | free_ring.induction_on | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"free_abelian_group.induction_on",
"free_ring",
"ih",
"neg_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : (α → R) ≃ (free_ring α →+* R) | free_monoid.lift.trans free_abelian_group.lift_monoid | def | free_ring.lift | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"free_abelian_group.lift_monoid",
"free_ring",
"lift"
] | The ring homomorphism `free_ring α →+* R` induced from a map `α → R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of (x : α) : lift f (of x) = f x | congr_fun (lift.left_inv f) x | lemma | free_ring.lift_of | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of (f : free_ring α →+* R) : lift (f ∘ of) = f | lift.right_inv f | lemma | free_ring.lift_comp_of | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"free_ring",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext ⦃f g : free_ring α →+* R⦄ (h : ∀ x, f (of x) = g (of x)) :
f = g | lift.symm.injective (funext h) | lemma | free_ring.hom_ext | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"free_ring",
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map : free_ring α →+* free_ring β | lift $ of ∘ f | def | free_ring.map | ring_theory | src/ring_theory/free_ring.lean | [
"group_theory.free_abelian_group"
] | [
"free_ring",
"lift"
] | The canonical ring homomorphism `free_ring α →+* free_ring β` generated by a map `α → β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hahn_series (Γ : Type*) (R : Type*) [partial_order Γ] [has_zero R] | (coeff : Γ → R)
(is_pwo_support' : (support coeff).is_pwo) | structure | hahn_series | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | If `Γ` is linearly ordered and `R` has zero, then `hahn_series Γ R` consists of
formal series over `Γ` with coefficients in `R`, whose supports are well-founded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_injective : injective (coeff : hahn_series Γ R → (Γ → R)) | ext | lemma | hahn_series.coeff_injective | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_inj {x y : hahn_series Γ R} : x.coeff = y.coeff ↔ x = y | coeff_injective.eq_iff | lemma | hahn_series.coeff_inj | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support (x : hahn_series Γ R) : set Γ | support x.coeff | def | hahn_series.support | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | The support of a Hahn series is just the set of indices whose coefficients are nonzero.
Notably, it is well-founded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pwo_support (x : hahn_series Γ R) : x.support.is_pwo | x.is_pwo_support' | lemma | hahn_series.is_pwo_support | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_wf_support (x : hahn_series Γ R) : x.support.is_wf | x.is_pwo_support.is_wf | lemma | hahn_series.is_wf_support | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support (x : hahn_series Γ R) (a : Γ) : a ∈ x.support ↔ x.coeff a ≠ 0 | iff.refl _ | lemma | hahn_series.mem_support | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_coeff {a : Γ} : (0 : hahn_series Γ R).coeff a = 0 | rfl | lemma | hahn_series.zero_coeff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_fun_eq_zero_iff {x : hahn_series Γ R} : x.coeff = 0 ↔ x = 0 | coeff_injective.eq_iff' rfl | lemma | hahn_series.coeff_fun_eq_zero_iff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_coeff_ne_zero {x : hahn_series Γ R} {g : Γ} (h : x.coeff g ≠ 0) :
x ≠ 0 | mt (λ x0, (x0.symm ▸ zero_coeff : x.coeff g = 0)) h | lemma | hahn_series.ne_zero_of_coeff_ne_zero | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_zero : support (0 : hahn_series Γ R) = ∅ | function.support_zero | lemma | hahn_series.support_zero | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_nonempty_iff {x : hahn_series Γ R} : x.support.nonempty ↔ x ≠ 0 | by rw [support, support_nonempty_iff, ne.def, coeff_fun_eq_zero_iff] | lemma | hahn_series.support_nonempty_iff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_eq_empty_iff {x : hahn_series Γ R} : x.support = ∅ ↔ x = 0 | support_eq_empty_iff.trans coeff_fun_eq_zero_iff | lemma | hahn_series.support_eq_empty_iff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single (a : Γ) : zero_hom R (hahn_series Γ R) | { to_fun := λ r, { coeff := pi.single a r,
is_pwo_support' := (set.is_pwo_singleton a).mono pi.support_single_subset },
map_zero' := ext _ _ (pi.single_zero _) } | def | hahn_series.single | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series",
"set.is_pwo_singleton",
"zero_hom"
] | `single a r` is the Hahn series which has coefficient `r` at `a` and zero otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single_coeff_same (a : Γ) (r : R) : (single a r).coeff a = r | pi.single_eq_same a r | theorem | hahn_series.single_coeff_same | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_coeff_of_ne (h : b ≠ a) : (single a r).coeff b = 0 | pi.single_eq_of_ne h r | theorem | hahn_series.single_coeff_of_ne | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_coeff : (single a r).coeff b = if (b = a) then r else 0 | by { split_ifs with h; simp [h] } | theorem | hahn_series.single_coeff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_single_of_ne (h : r ≠ 0) : support (single a r) = {a} | pi.support_single_of_ne h | lemma | hahn_series.support_single_of_ne | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_single_subset : support (single a r) ⊆ {a} | pi.support_single_subset | lemma | hahn_series.support_single_subset | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_mem_support_single {b : Γ} (h : b ∈ support (single a r)) : b = a | support_single_subset h | lemma | hahn_series.eq_of_mem_support_single | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_eq_zero : (single a (0 : R)) = 0 | (single a).map_zero | lemma | hahn_series.single_eq_zero | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_injective (a : Γ) : function.injective (single a : R → hahn_series Γ R) | λ r s rs, by rw [← single_coeff_same a r, ← single_coeff_same a s, rs] | lemma | hahn_series.single_injective | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_ne_zero (h : r ≠ 0) : single a r ≠ 0 | λ con, h (single_injective a (con.trans single_eq_zero.symm)) | lemma | hahn_series.single_ne_zero | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"con",
"con.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_eq_zero_iff {a : Γ} {r : R} :
single a r = 0 ↔ r = 0 | begin
split,
{ contrapose!,
exact single_ne_zero },
{ simp {contextual := tt} }
end | lemma | hahn_series.single_eq_zero_iff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order (x : hahn_series Γ R) : Γ | if h : x = 0 then 0 else x.is_wf_support.min (support_nonempty_iff.2 h) | def | hahn_series.order | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | The order of a nonzero Hahn series `x` is a minimal element of `Γ` where `x` has a
nonzero coefficient, the order of 0 is 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_zero : order (0 : hahn_series Γ R) = 0 | dif_pos rfl | lemma | hahn_series.order_zero | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_of_ne {x : hahn_series Γ R} (hx : x ≠ 0) :
order x = x.is_wf_support.min (support_nonempty_iff.2 hx) | dif_neg hx | lemma | hahn_series.order_of_ne | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_order_ne_zero {x : hahn_series Γ R} (hx : x ≠ 0) :
x.coeff x.order ≠ 0 | begin
rw order_of_ne hx,
exact x.is_wf_support.min_mem (support_nonempty_iff.2 hx)
end | lemma | hahn_series.coeff_order_ne_zero | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_le_of_coeff_ne_zero {Γ} [linear_ordered_cancel_add_comm_monoid Γ]
{x : hahn_series Γ R} {g : Γ} (h : x.coeff g ≠ 0) :
x.order ≤ g | le_trans (le_of_eq (order_of_ne (ne_zero_of_coeff_ne_zero h)))
(set.is_wf.min_le _ _ ((mem_support _ _).2 h)) | lemma | hahn_series.order_le_of_coeff_ne_zero | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series",
"linear_ordered_cancel_add_comm_monoid",
"set.is_wf.min_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_single (h : r ≠ 0) : (single a r).order = a | (order_of_ne (single_ne_zero h)).trans (support_single_subset ((single a r).is_wf_support.min_mem
(support_nonempty_iff.2 (single_ne_zero h)))) | lemma | hahn_series.order_single | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_eq_zero_of_lt_order {x : hahn_series Γ R} {i : Γ} (hi : i < x.order) : x.coeff i = 0 | begin
rcases eq_or_ne x 0 with rfl|hx,
{ simp },
contrapose! hi,
rw [←ne.def, ←mem_support] at hi,
rw [order_of_ne hx],
exact set.is_wf.not_lt_min _ _ hi
end | lemma | hahn_series.coeff_eq_zero_of_lt_order | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"eq_or_ne",
"hahn_series",
"set.is_wf.not_lt_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain (f : Γ ↪o Γ') : hahn_series Γ R → hahn_series Γ' R | λ x, { coeff := λ (b : Γ'),
if h : b ∈ f '' x.support then x.coeff (classical.some h) else 0,
is_pwo_support' := (x.is_pwo_support.image_of_monotone f.monotone).mono (λ b hb, begin
contrapose! hb,
rw [function.mem_support, dif_neg hb, not_not],
end) } | def | hahn_series.emb_domain | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series",
"not_not"
] | Extends the domain of a `hahn_series` by an `order_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emb_domain_coeff {f : Γ ↪o Γ'} {x : hahn_series Γ R} {a : Γ} :
(emb_domain f x).coeff (f a) = x.coeff a | begin
rw emb_domain,
dsimp only,
by_cases ha : a ∈ x.support,
{ rw dif_pos (set.mem_image_of_mem f ha),
exact congr rfl (f.injective (classical.some_spec (set.mem_image_of_mem f ha)).2) },
{ rw [dif_neg, not_not.1 (λ c, ha ((mem_support _ _).2 c))],
contrapose! ha,
obtain ⟨b, hb1, hb2⟩ := (set.mem... | lemma | hahn_series.emb_domain_coeff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series",
"set.mem_image",
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_mk_coeff {f : Γ → Γ'}
(hfi : function.injective f) (hf : ∀ g g' : Γ, f g ≤ f g' ↔ g ≤ g')
{x : hahn_series Γ R} {a : Γ} :
(emb_domain ⟨⟨f, hfi⟩, hf⟩ x).coeff (f a) = x.coeff a | emb_domain_coeff | lemma | hahn_series.emb_domain_mk_coeff | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_notin_image_support {f : Γ ↪o Γ'} {x : hahn_series Γ R} {b : Γ'}
(hb : b ∉ f '' x.support) : (emb_domain f x).coeff b = 0 | dif_neg hb | lemma | hahn_series.emb_domain_notin_image_support | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_emb_domain_subset {f : Γ ↪o Γ'} {x : hahn_series Γ R} :
support (emb_domain f x) ⊆ f '' x.support | begin
intros g hg,
contrapose! hg,
rw [mem_support, emb_domain_notin_image_support hg, not_not],
end | lemma | hahn_series.support_emb_domain_subset | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"hahn_series",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_notin_range {f : Γ ↪o Γ'} {x : hahn_series Γ R} {b : Γ'}
(hb : b ∉ set.range f) : (emb_domain f x).coeff b = 0 | emb_domain_notin_image_support (λ con, hb (set.image_subset_range _ _ con)) | lemma | hahn_series.emb_domain_notin_range | ring_theory | src/ring_theory/hahn_series.lean | [
"order.well_founded_set",
"algebra.big_operators.finprod",
"ring_theory.valuation.basic",
"ring_theory.power_series.basic",
"data.finsupp.pwo",
"data.finset.mul_antidiagonal",
"algebra.order.group.with_top"
] | [
"con",
"hahn_series",
"set.image_subset_range",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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