statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
emb_domain_zero {f : Γ ↪o Γ'} : emb_domain f (0 : hahn_series Γ R) = 0 | by { ext, simp [emb_domain_notin_image_support] } | lemma | hahn_series.emb_domain_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 | |
emb_domain_single {f : Γ ↪o Γ'} {g : Γ} {r : R} :
emb_domain f (single g r) = single (f g) r | begin
ext g',
by_cases h : g' = f g,
{ simp [h] },
rw [emb_domain_notin_image_support, single_coeff_of_ne h],
by_cases hr : r = 0,
{ simp [hr] },
rwa [support_single_of_ne hr, set.image_singleton, set.mem_singleton_iff],
end | lemma | hahn_series.emb_domain_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"
] | [
"set.image_singleton",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_injective {f : Γ ↪o Γ'} :
function.injective (emb_domain f : hahn_series Γ R → hahn_series Γ' R) | λ x y xy, begin
ext g,
rw [ext_iff, function.funext_iff] at xy,
have xyg := xy (f g),
rwa [emb_domain_coeff, emb_domain_coeff] at xyg,
end | lemma | hahn_series.emb_domain_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"
] | [
"function.funext_iff",
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_coeff' {x y : hahn_series Γ R} :
(x + y).coeff = x.coeff + y.coeff | rfl | lemma | hahn_series.add_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 | |
add_coeff {x y : hahn_series Γ R} {a : Γ} :
(x + y).coeff a = x.coeff a + y.coeff a | rfl | lemma | hahn_series.add_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 | |
support_add_subset {x y : hahn_series Γ R} :
support (x + y) ⊆ support x ∪ support y | λ a ha, begin
rw [mem_support, add_coeff] at ha,
rw [set.mem_union, mem_support, mem_support],
contrapose! ha,
rw [ha.1, ha.2, add_zero],
end | lemma | hahn_series.support_add_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",
"set.mem_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_order_le_order_add {Γ} [linear_ordered_cancel_add_comm_monoid Γ] {x y : hahn_series Γ R}
(hxy : x + y ≠ 0) :
min x.order y.order ≤ (x + y).order | begin
by_cases hx : x = 0, { simp [hx], },
by_cases hy : y = 0, { simp [hy], },
rw [order_of_ne hx, order_of_ne hy, order_of_ne hxy],
refine le_trans _ (set.is_wf.min_le_min_of_subset support_add_subset),
{ exact x.is_wf_support.union y.is_wf_support },
{ exact set.nonempty.mono (set.subset_union_left _ _) ... | lemma | hahn_series.min_order_le_order_add | 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_min_of_subset",
"set.is_wf.min_union",
"set.nonempty.mono",
"set.subset_union_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single.add_monoid_hom (a : Γ) : R →+ (hahn_series Γ R) | { map_add' := λ x y, by { ext b, by_cases h : b = a; simp [h] },
..single a } | def | hahn_series.single.add_monoid_hom | 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"
] | `single` as an additive monoid/group homomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff.add_monoid_hom (g : Γ) : (hahn_series Γ R) →+ R | { to_fun := λ f, f.coeff g,
map_zero' := zero_coeff,
map_add' := λ x y, add_coeff } | def | hahn_series.coeff.add_monoid_hom | 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"
] | `coeff g` as an additive monoid/group homomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emb_domain_add (f : Γ ↪o Γ') (x y : hahn_series Γ R) :
emb_domain f (x + y) = emb_domain f x + emb_domain f y | begin
ext g,
by_cases hg : g ∈ set.range f,
{ obtain ⟨a, rfl⟩ := hg,
simp },
{ simp [emb_domain_notin_range, hg] }
end | lemma | hahn_series.emb_domain_add | 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.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_coeff' {x : hahn_series Γ R} : (- x).coeff = - x.coeff | rfl | lemma | hahn_series.neg_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 | |
neg_coeff {x : hahn_series Γ R} {a : Γ} : (- x).coeff a = - x.coeff a | rfl | lemma | hahn_series.neg_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 | |
support_neg {x : hahn_series Γ R} : (- x).support = x.support | by { ext, simp } | lemma | hahn_series.support_neg | 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 | |
sub_coeff' {x y : hahn_series Γ R} :
(x - y).coeff = x.coeff - y.coeff | by { ext, simp [sub_eq_add_neg] } | lemma | hahn_series.sub_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 | |
sub_coeff {x y : hahn_series Γ R} {a : Γ} :
(x - y).coeff a = x.coeff a - y.coeff a | by simp | lemma | hahn_series.sub_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 | |
order_neg [has_zero Γ] {f : hahn_series Γ R} : (- f).order = f.order | by { by_cases hf : f = 0, { simp only [hf, neg_zero] },
simp only [order, support_neg, neg_eq_zero] } | lemma | hahn_series.order_neg | 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 | |
smul_coeff {r : R} {x : hahn_series Γ V} {a : Γ} : (r • x).coeff a = r • (x.coeff a) | rfl | lemma | hahn_series.smul_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 | |
single.linear_map (a : Γ) : R →ₗ[R] (hahn_series Γ R) | { map_smul' := λ r s, by { ext b, by_cases h : b = a; simp [h] },
..single.add_monoid_hom a } | def | hahn_series.single.linear_map | 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"
] | `single` as a linear map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff.linear_map (g : Γ) : (hahn_series Γ R) →ₗ[R] R | { map_smul' := λ r s, rfl,
..coeff.add_monoid_hom g } | def | hahn_series.coeff.linear_map | 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"
] | `coeff g` as a linear map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emb_domain_smul (f : Γ ↪o Γ') (r : R) (x : hahn_series Γ R) :
emb_domain f (r • x) = r • emb_domain f x | begin
ext g,
by_cases hg : g ∈ set.range f,
{ obtain ⟨a, rfl⟩ := hg,
simp },
{ simp [emb_domain_notin_range, hg] }
end | lemma | hahn_series.emb_domain_smul | 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.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_linear_map (f : Γ ↪o Γ') : hahn_series Γ R →ₗ[R] hahn_series Γ' R | { to_fun := emb_domain f, map_add' := emb_domain_add f, map_smul' := emb_domain_smul f } | def | hahn_series.emb_domain_linear_map | 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"
] | Extending the domain of Hahn series is a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_coeff [has_zero R] [has_one R] {a : Γ} :
(1 : hahn_series Γ R).coeff a = if a = 0 then 1 else 0 | single_coeff | lemma | hahn_series.one_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 | |
single_zero_one [has_zero R] [has_one R] : (single 0 (1 : R)) = 1 | rfl | lemma | hahn_series.single_zero_one | 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_one [mul_zero_one_class R] [nontrivial R] :
support (1 : hahn_series Γ R) = {0} | support_single_of_ne one_ne_zero | lemma | hahn_series.support_one | 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",
"mul_zero_one_class",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_one [mul_zero_one_class R] :
order (1 : hahn_series Γ R) = 0 | begin
cases subsingleton_or_nontrivial R with h h; haveI := h,
{ rw [subsingleton.elim (1 : hahn_series Γ R) 0, order_zero] },
{ exact order_single one_ne_zero }
end | lemma | hahn_series.order_one | 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",
"mul_zero_one_class",
"one_ne_zero",
"subsingleton_or_nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coeff [non_unital_non_assoc_semiring R] {x y : hahn_series Γ R} {a : Γ} :
(x * y).coeff a = ∑ ij in (add_antidiagonal x.is_pwo_support y.is_pwo_support a),
x.coeff ij.fst * y.coeff ij.snd | rfl | lemma | hahn_series.mul_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",
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coeff_right' [non_unital_non_assoc_semiring R] {x y : hahn_series Γ R} {a : Γ} {s : set Γ}
(hs : s.is_pwo) (hys : y.support ⊆ s) :
(x * y).coeff a = ∑ ij in (add_antidiagonal x.is_pwo_support hs a),
x.coeff ij.fst * y.coeff ij.snd | begin
rw mul_coeff,
apply sum_subset_zero_on_sdiff (add_antidiagonal_mono_right hys) _ (λ _ _, rfl),
intros b hb,
simp only [not_and, mem_sdiff, mem_add_antidiagonal, mem_support, not_imp_not] at hb,
rw [hb.2 hb.1.1 hb.1.2.2, mul_zero],
end | lemma | hahn_series.mul_coeff_right' | 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",
"mul_zero",
"non_unital_non_assoc_semiring",
"not_and",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coeff_left' [non_unital_non_assoc_semiring R] {x y : hahn_series Γ R} {a : Γ} {s : set Γ}
(hs : s.is_pwo) (hxs : x.support ⊆ s) :
(x * y).coeff a = ∑ ij in (add_antidiagonal hs y.is_pwo_support a),
x.coeff ij.fst * y.coeff ij.snd | begin
rw mul_coeff,
apply sum_subset_zero_on_sdiff (add_antidiagonal_mono_left hxs) _ (λ _ _, rfl),
intros b hb,
simp only [not_and', mem_sdiff, mem_add_antidiagonal, mem_support, not_ne_iff] at hb,
rw [hb.2 ⟨hb.1.2.1, hb.1.2.2⟩, zero_mul],
end | lemma | hahn_series.mul_coeff_left' | 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",
"non_unital_non_assoc_semiring",
"not_and'",
"not_ne_iff",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_mul_coeff_add [non_unital_non_assoc_semiring R] {r : R} {x : hahn_series Γ R} {a : Γ}
{b : Γ} :
((single b r) * x).coeff (a + b) = r * x.coeff a | begin
by_cases hr : r = 0,
{ simp [hr] },
simp only [hr, smul_coeff, mul_coeff, support_single_of_ne, ne.def, not_false_iff, smul_eq_mul],
by_cases hx : x.coeff a = 0,
{ simp only [hx, mul_zero],
rw [sum_congr _ (λ _ _, rfl), sum_empty],
ext ⟨a1, a2⟩,
simp only [not_mem_empty, not_and, set.mem_sin... | lemma | hahn_series.single_mul_coeff_add | 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",
"mul_zero",
"non_unital_non_assoc_semiring",
"not_and",
"not_not",
"prod.mk.inj_iff",
"set.mem_singleton_iff",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_single_coeff_add [non_unital_non_assoc_semiring R] {r : R} {x : hahn_series Γ R} {a : Γ}
{b : Γ} :
(x * (single b r)).coeff (a + b) = x.coeff a * r | begin
by_cases hr : r = 0,
{ simp [hr] },
simp only [hr, smul_coeff, mul_coeff, support_single_of_ne, ne.def, not_false_iff, smul_eq_mul],
by_cases hx : x.coeff a = 0,
{ simp only [hx, zero_mul],
rw [sum_congr _ (λ _ _, rfl), sum_empty],
ext ⟨a1, a2⟩,
simp only [not_mem_empty, not_and, set.mem_sin... | lemma | hahn_series.mul_single_coeff_add | 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",
"non_unital_non_assoc_semiring",
"not_and",
"not_not",
"prod.mk.inj_iff",
"set.mem_singleton_iff",
"smul_eq_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_single_zero_coeff [non_unital_non_assoc_semiring R] {r : R} {x : hahn_series Γ R}
{a : Γ} :
(x * (single 0 r)).coeff a = x.coeff a * r | by rw [← add_zero a, mul_single_coeff_add, add_zero] | lemma | hahn_series.mul_single_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",
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_zero_mul_coeff [non_unital_non_assoc_semiring R] {r : R} {x : hahn_series Γ R}
{a : Γ} :
((single 0 r) * x).coeff a = r * x.coeff a | by rw [← add_zero a, single_mul_coeff_add, add_zero] | lemma | hahn_series.single_zero_mul_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",
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_zero_mul_eq_smul [semiring R] {r : R} {x : hahn_series Γ R} :
(single 0 r) * x = r • x | by { ext, exact single_zero_mul_coeff } | lemma | hahn_series.single_zero_mul_eq_smul | 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",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_mul_subset_add_support [non_unital_non_assoc_semiring R] {x y : hahn_series Γ R} :
support (x * y) ⊆ support x + support y | begin
apply set.subset.trans (λ x hx, _) support_add_antidiagonal_subset_add,
{ exact x.is_pwo_support },
{ exact y.is_pwo_support },
contrapose! hx,
simp only [not_nonempty_iff_eq_empty, ne.def, set.mem_set_of_eq] at hx,
simp [hx],
end | theorem | hahn_series.support_mul_subset_add_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",
"non_unital_non_assoc_semiring",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coeff_order_add_order {Γ} [linear_ordered_cancel_add_comm_monoid Γ]
[non_unital_non_assoc_semiring R]
(x y : hahn_series Γ R) :
(x * y).coeff (x.order + y.order) = x.coeff x.order * y.coeff y.order | begin
by_cases hx : x = 0, { simp [hx], },
by_cases hy : y = 0, { simp [hy], },
rw [order_of_ne hx, order_of_ne hy, mul_coeff, finset.add_antidiagonal_min_add_min,
finset.sum_singleton],
end | lemma | hahn_series.mul_coeff_order_add_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",
"linear_ordered_cancel_add_comm_monoid",
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_assoc' [non_unital_semiring R] (x y z : hahn_series Γ R) :
x * y * z = x * (y * z) | begin
ext b,
rw [mul_coeff_left' (x.is_pwo_support.add y.is_pwo_support) support_mul_subset_add_support,
mul_coeff_right' (y.is_pwo_support.add z.is_pwo_support) support_mul_subset_add_support],
simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma'],
refine sum_bij_ne_zero (λ a has ha0, ⟨⟨a.2.1, a... | lemma | hahn_series.mul_assoc' | 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",
"exists_prop",
"hahn_series",
"heq_iff_eq",
"mul_assoc",
"non_unital_semiring",
"prod.mk.inj_iff",
"set.image_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_mul {Γ} [linear_ordered_cancel_add_comm_monoid Γ] [non_unital_non_assoc_semiring R]
[no_zero_divisors R] {x y : hahn_series Γ R} (hx : x ≠ 0) (hy : y ≠ 0) :
(x * y).order = x.order + y.order | begin
apply le_antisymm,
{ apply order_le_of_coeff_ne_zero,
rw [mul_coeff_order_add_order x y],
exact mul_ne_zero (coeff_order_ne_zero hx) (coeff_order_ne_zero hy) },
{ rw [order_of_ne hx, order_of_ne hy, order_of_ne (mul_ne_zero hx hy), ← set.is_wf.min_add],
exact set.is_wf.min_le_min_of_subset (supp... | lemma | hahn_series.order_mul | 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",
"mul_ne_zero",
"no_zero_divisors",
"non_unital_non_assoc_semiring",
"set.is_wf.min_le_min_of_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_pow {Γ} [linear_ordered_cancel_add_comm_monoid Γ] [semiring R] [no_zero_divisors R]
(x : hahn_series Γ R) (n : ℕ) : (x ^ n).order = n • x.order | begin
induction n with h IH,
{ simp },
rcases eq_or_ne x 0 with rfl|hx,
{ simp },
rw [pow_succ', order_mul (pow_ne_zero _ hx) hx, succ_nsmul', IH]
end | lemma | hahn_series.order_pow | 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",
"linear_ordered_cancel_add_comm_monoid",
"no_zero_divisors",
"pow_ne_zero",
"pow_succ'",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_mul_single {a b : Γ} {r s : R} :
single a r * single b s = single (a + b) (r * s) | begin
ext x,
by_cases h : x = a + b,
{ rw [h, mul_single_coeff_add],
simp },
{ rw [single_coeff_of_ne h, mul_coeff, sum_eq_zero],
simp_rw mem_add_antidiagonal,
rintro ⟨y, z⟩ ⟨hy, hz, rfl⟩,
rw [eq_of_mem_support_single hy, eq_of_mem_support_single hz] at h,
exact (h rfl).elim }
end | lemma | hahn_series.single_mul_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 | |
C : R →+* (hahn_series Γ R) | { to_fun := single 0,
map_zero' := single_eq_zero,
map_one' := rfl,
map_add' := λ x y, by { ext a, by_cases h : a = 0; simp [h] },
map_mul' := λ x y, by rw [single_mul_single, zero_add] } | def | hahn_series.C | 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"
] | `C a` is the constant Hahn Series `a`. `C` is provided as a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
C_zero : C (0 : R) = (0 : hahn_series Γ R) | C.map_zero | lemma | hahn_series.C_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 | |
C_one : C (1 : R) = (1 : hahn_series Γ R) | C.map_one | lemma | hahn_series.C_one | 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 | |
C_injective : function.injective (C : R → hahn_series Γ R) | begin
intros r s rs,
rw [ext_iff, function.funext_iff] at rs,
have h := rs 0,
rwa [C_apply, single_coeff_same, C_apply, single_coeff_same] at h,
end | lemma | hahn_series.C_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"
] | [
"function.funext_iff",
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_ne_zero {r : R} (h : r ≠ 0) : (C r : hahn_series Γ R) ≠ 0 | begin
contrapose! h,
rw ← C_zero at h,
exact C_injective h,
end | lemma | hahn_series.C_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_C {r : R} : order (C r : hahn_series Γ R) = 0 | begin
by_cases h : r = 0,
{ rw [h, C_zero, order_zero] },
{ exact order_single h }
end | lemma | hahn_series.order_C | 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 | |
C_mul_eq_smul {r : R} {x : hahn_series Γ R} : C r * x = r • x | single_zero_mul_eq_smul | lemma | hahn_series.C_mul_eq_smul | 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_mul [non_unital_non_assoc_semiring R]
(f : Γ ↪o Γ') (hf : ∀ x y, f (x + y) = f x + f y) (x y : hahn_series Γ R) :
emb_domain f (x * y) = emb_domain f x * emb_domain f y | begin
ext g,
by_cases hg : g ∈ set.range f,
{ obtain ⟨g, rfl⟩ := hg,
simp only [mul_coeff, emb_domain_coeff],
transitivity ∑ ij in (add_antidiagonal x.is_pwo_support y.is_pwo_support g).map
(function.embedding.prod_map f.to_embedding f.to_embedding),
(emb_domain f x).coeff (ij.1) *
(emb_... | lemma | hahn_series.emb_domain_mul | 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"
] | [
"exists_prop",
"function.embedding.coe_prod_map",
"function.embedding.prod_map",
"hahn_series",
"mem_map",
"ne_zero_and_ne_zero_of_mul",
"non_unital_non_assoc_semiring",
"order_embedding.eq_iff_eq",
"prod.mk.inj_iff",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_one [non_assoc_semiring R] (f : Γ ↪o Γ') (hf : f 0 = 0):
emb_domain f (1 : hahn_series Γ R) = (1 : hahn_series Γ' R) | emb_domain_single.trans $ hf.symm ▸ rfl | lemma | hahn_series.emb_domain_one | 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",
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_ring_hom [non_assoc_semiring R] (f : Γ →+ Γ') (hfi : function.injective f)
(hf : ∀ g g' : Γ, f g ≤ f g' ↔ g ≤ g') :
hahn_series Γ R →+* hahn_series Γ' R | { to_fun := emb_domain ⟨⟨f, hfi⟩, hf⟩,
map_one' := emb_domain_one _ f.map_zero,
map_mul' := emb_domain_mul _ f.map_add,
map_zero' := emb_domain_zero,
map_add' := emb_domain_add _} | def | hahn_series.emb_domain_ring_hom | 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",
"non_assoc_semiring"
] | Extending the domain of Hahn series is a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
emb_domain_ring_hom_C [non_assoc_semiring R] {f : Γ →+ Γ'} {hfi : function.injective f}
{hf : ∀ g g' : Γ, f g ≤ f g' ↔ g ≤ g'} {r : R} :
emb_domain_ring_hom f hfi hf (C r) = C r | emb_domain_single.trans (by simp) | lemma | hahn_series.emb_domain_ring_hom_C | 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"
] | [
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_eq_algebra_map : C = (algebra_map R (hahn_series Γ R)) | rfl | theorem | hahn_series.C_eq_algebra_map | 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"
] | [
"algebra_map",
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply {r : R} :
algebra_map R (hahn_series Γ A) r = C (algebra_map R A r) | rfl | theorem | hahn_series.algebra_map_apply | 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"
] | [
"algebra_map",
"algebra_map_apply",
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
emb_domain_alg_hom (f : Γ →+ Γ') (hfi : function.injective f)
(hf : ∀ g g' : Γ, f g ≤ f g' ↔ g ≤ g') :
hahn_series Γ A →ₐ[R] hahn_series Γ' A | { commutes' := λ r, emb_domain_ring_hom_C,
.. emb_domain_ring_hom f hfi hf } | def | hahn_series.emb_domain_alg_hom | 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"
] | Extending the domain of Hahn series is an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_power_series : hahn_series ℕ R ≃+* power_series R | { to_fun := λ f, power_series.mk f.coeff,
inv_fun := λ f, ⟨λ n, power_series.coeff R n f, (nat.lt_wf.is_wf _).is_pwo⟩,
left_inv := λ f, by { ext, simp },
right_inv := λ f, by { ext, simp },
map_add' := λ f g, by { ext, simp },
map_mul' := λ f g, begin
ext n,
simp only [power_series.coeff_mul, power_se... | def | hahn_series.to_power_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"
] | [
"and.congr_left_iff",
"hahn_series",
"inv_fun",
"left_ne_zero_of_mul",
"power_series",
"power_series.coeff",
"power_series.coeff_mk",
"power_series.coeff_mul",
"power_series.mk",
"right_ne_zero_of_mul"
] | The ring `hahn_series ℕ R` is isomorphic to `power_series R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_to_power_series {f : hahn_series ℕ R} {n : ℕ} :
power_series.coeff R n f.to_power_series = f.coeff n | power_series.coeff_mk _ _ | lemma | hahn_series.coeff_to_power_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"
] | [
"hahn_series",
"power_series.coeff",
"power_series.coeff_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_to_power_series_symm {f : power_series R} {n : ℕ} :
(hahn_series.to_power_series.symm f).coeff n = power_series.coeff R n f | rfl | lemma | hahn_series.coeff_to_power_series_symm | 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"
] | [
"power_series",
"power_series.coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_power_series : (power_series R) →+* hahn_series Γ R | (hahn_series.emb_domain_ring_hom (nat.cast_add_monoid_hom Γ) nat.strict_mono_cast.injective
(λ _ _, nat.cast_le)).comp
(ring_equiv.to_ring_hom to_power_series.symm) | def | hahn_series.of_power_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"
] | [
"hahn_series",
"hahn_series.emb_domain_ring_hom",
"nat.cast_add_monoid_hom",
"nat.cast_le",
"power_series",
"ring_equiv.to_ring_hom"
] | Casts a power series as a Hahn series with coefficients from an `strict_ordered_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_power_series_injective : function.injective (of_power_series Γ R) | emb_domain_injective.comp to_power_series.symm.injective | lemma | hahn_series.of_power_series_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_power_series_apply (x : power_series R) :
of_power_series Γ R x = hahn_series.emb_domain
⟨⟨(coe : ℕ → Γ), nat.strict_mono_cast.injective⟩, λ a b, begin
simp only [function.embedding.coe_fn_mk],
exact nat.cast_le,
end⟩ (to_power_series.symm x) | rfl | lemma | hahn_series.of_power_series_apply | 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"
] | [
"function.embedding.coe_fn_mk",
"hahn_series.emb_domain",
"nat.cast_le",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_power_series_apply_coeff (x : power_series R) (n : ℕ) :
(of_power_series Γ R x).coeff n = power_series.coeff R n x | by simp | lemma | hahn_series.of_power_series_apply_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"
] | [
"power_series",
"power_series.coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_power_series_C (r : R) :
of_power_series Γ R (power_series.C R r) = hahn_series.C r | begin
ext n,
simp only [C, single_coeff, of_power_series_apply, ring_hom.coe_mk],
split_ifs with hn hn,
{ subst hn,
convert @emb_domain_coeff _ _ _ _ _ _ _ _ 0; simp },
{ rw emb_domain_notin_image_support,
simp only [not_exists, set.mem_image, to_power_series_symm_apply_coeff, mem_support,
... | lemma | hahn_series.of_power_series_C | 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.C",
"not_exists",
"power_series.C",
"power_series.coeff_C",
"ring_hom.coe_mk",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_power_series_X :
of_power_series Γ R power_series.X = single 1 1 | begin
ext n,
simp only [single_coeff, of_power_series_apply, ring_hom.coe_mk],
split_ifs with hn hn,
{ rw hn,
convert @emb_domain_coeff _ _ _ _ _ _ _ _ 1;
simp },
{ rw emb_domain_notin_image_support,
simp only [not_exists, set.mem_image, to_power_series_symm_apply_coeff, mem_support,
... | lemma | hahn_series.of_power_series_X | 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"
] | [
"not_exists",
"power_series.X",
"power_series.coeff_X",
"ring_hom.coe_mk",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_power_series_X_pow {R} [comm_semiring R] (n : ℕ) :
of_power_series Γ R (power_series.X ^ n) = single (n : Γ) 1 | begin
rw ring_hom.map_pow,
induction n with n ih,
{ simp, refl },
rw [pow_succ, ih, of_power_series_X, mul_comm, single_mul_single, one_mul, nat.cast_succ]
end | lemma | hahn_series.of_power_series_X_pow | 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"
] | [
"comm_semiring",
"ih",
"mul_comm",
"nat.cast_succ",
"one_mul",
"pow_succ",
"power_series.X",
"ring_hom.map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mv_power_series {σ : Type*} [fintype σ] :
hahn_series (σ →₀ ℕ) R ≃+* mv_power_series σ R | { to_fun := λ f, f.coeff,
inv_fun := λ f, ⟨(f : (σ →₀ ℕ) → R), finsupp.is_pwo _⟩,
left_inv := λ f, by { ext, simp },
right_inv := λ f, by { ext, simp },
map_add' := λ f g, by { ext, simp },
map_mul' := λ f g, begin
ext n,
simp only [mv_power_series.coeff_mul],
classical,
change (f * g).coeff n... | def | hahn_series.to_mv_power_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"
] | [
"and.congr_left_iff",
"finsupp.is_pwo",
"finsupp.mem_antidiagonal",
"fintype",
"hahn_series",
"inv_fun",
"left_ne_zero_of_mul",
"mv_power_series",
"mv_power_series.coeff_mul",
"right_ne_zero_of_mul"
] | The ring `hahn_series (σ →₀ ℕ) R` is isomorphic to `mv_power_series σ R` for a `fintype` `σ`.
We take the index set of the hahn series to be `finsupp` rather than `pi`,
even though we assume `fintype σ` as this is more natural for alignment with `mv_power_series`.
After importing `algebra.order.pi` the ring `hahn_serie... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_to_mv_power_series {f : hahn_series (σ →₀ ℕ) R} {n : σ →₀ ℕ} :
mv_power_series.coeff R n f.to_mv_power_series = f.coeff n | rfl | lemma | hahn_series.coeff_to_mv_power_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"
] | [
"hahn_series",
"mv_power_series.coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_to_mv_power_series_symm {f : mv_power_series σ R} {n : σ →₀ ℕ} :
(hahn_series.to_mv_power_series.symm f).coeff n = mv_power_series.coeff R n f | rfl | lemma | hahn_series.coeff_to_mv_power_series_symm | 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"
] | [
"mv_power_series",
"mv_power_series.coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_power_series_alg : (hahn_series ℕ A) ≃ₐ[R] power_series A | { commutes' := λ r, begin
ext n,
simp only [algebra_map_apply, power_series.algebra_map_apply, ring_equiv.to_fun_eq_coe, C_apply,
coeff_to_power_series],
cases n,
{ simp only [power_series.coeff_zero_eq_constant_coeff, single_coeff_same],
refl },
{ simp only [n.succ_ne_zero, ne.def, not_... | def | hahn_series.to_power_series_alg | 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"
] | [
"algebra_map_apply",
"hahn_series",
"power_series",
"power_series.algebra_map_apply",
"power_series.coeff_C",
"power_series.coeff_zero_eq_constant_coeff",
"ring_equiv.to_fun_eq_coe"
] | The `R`-algebra `hahn_series ℕ A` is isomorphic to `power_series A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_power_series_alg : (power_series A) →ₐ[R] hahn_series Γ A | (hahn_series.emb_domain_alg_hom (nat.cast_add_monoid_hom Γ) nat.strict_mono_cast.injective
(λ _ _, nat.cast_le)).comp
(alg_equiv.to_alg_hom (to_power_series_alg R).symm) | def | hahn_series.of_power_series_alg | 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"
] | [
"alg_equiv.to_alg_hom",
"hahn_series",
"hahn_series.emb_domain_alg_hom",
"nat.cast_add_monoid_hom",
"nat.cast_le",
"power_series"
] | Casting a power series as a Hahn series with coefficients from an `strict_ordered_semiring`
is an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_series_algebra {S : Type*} [comm_semiring S] [algebra S (power_series R)] :
algebra S (hahn_series Γ R) | ring_hom.to_algebra $ (of_power_series Γ R).comp (algebra_map S (power_series R)) | instance | hahn_series.power_series_algebra | 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"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"hahn_series",
"power_series",
"ring_hom.to_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply' (x : S) :
algebra_map S (hahn_series Γ R) x = of_power_series Γ R (algebra_map S (power_series R) x) | rfl | lemma | hahn_series.algebra_map_apply' | 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"
] | [
"algebra_map",
"hahn_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.polynomial.algebra_map_hahn_series_apply (f : R[X]) :
algebra_map R[X] (hahn_series Γ R) f = of_power_series Γ R f | rfl | lemma | polynomial.algebra_map_hahn_series_apply | 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"
] | [
"algebra_map",
"hahn_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.polynomial.algebra_map_hahn_series_injective :
function.injective (algebra_map R[X] (hahn_series Γ R)) | of_power_series_injective.comp (polynomial.coe_injective R) | lemma | polynomial.algebra_map_hahn_series_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"
] | [
"algebra_map",
"hahn_series",
"polynomial.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val : add_valuation (hahn_series Γ R) (with_top Γ) | add_valuation.of (λ x, if x = (0 : hahn_series Γ R) then (⊤ : with_top Γ) else x.order)
(if_pos rfl)
((if_neg one_ne_zero).trans (by simp [order_of_ne]))
(λ x y, begin
by_cases hx : x = 0,
{ by_cases hy : y = 0; { simp [hx, hy] } },
{ by_cases hy : y = 0,
{ simp [hx, hy] },
{ simp only [hx... | def | hahn_series.add_val | 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"
] | [
"add_valuation",
"add_valuation.of",
"hahn_series",
"mul_ne_zero",
"one_ne_zero",
"with_top",
"with_top.coe_add",
"with_top.coe_eq_coe",
"with_top.coe_le_coe",
"with_top.coe_min"
] | The additive valuation on `hahn_series Γ R`, returning the smallest index at which
a Hahn Series has a nonzero coefficient, or `⊤` for the 0 series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_val_apply {x : hahn_series Γ R} :
add_val Γ R x = if x = (0 : hahn_series Γ R) then (⊤ : with_top Γ) else x.order | add_valuation.of_apply _ | lemma | hahn_series.add_val_apply | 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"
] | [
"add_valuation.of_apply",
"hahn_series",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_apply_of_ne {x : hahn_series Γ R} (hx : x ≠ 0) :
add_val Γ R x = x.order | if_neg hx | lemma | hahn_series.add_val_apply_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 | |
add_val_le_of_coeff_ne_zero {x : hahn_series Γ R} {g : Γ} (h : x.coeff g ≠ 0) :
add_val Γ R x ≤ g | begin
rw [add_val_apply_of_ne (ne_zero_of_coeff_ne_zero h), with_top.coe_le_coe],
exact order_le_of_coeff_ne_zero h
end | lemma | hahn_series.add_val_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",
"with_top.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pwo_Union_support_powers
[linear_ordered_cancel_add_comm_monoid Γ] [ring R] [is_domain R]
{x : hahn_series Γ R} (hx : 0 < add_val Γ R x) :
(⋃ n : ℕ, (x ^ n).support).is_pwo | begin
apply (x.is_wf_support.is_pwo.add_submonoid_closure (λ g hg, _)).mono _,
{ exact with_top.coe_le_coe.1 (le_trans (le_of_lt hx) (add_val_le_of_coeff_ne_zero hg)) },
refine set.Union_subset (λ n, _),
induction n with n ih;
intros g hn,
{ simp only [exists_prop, and_true, set.mem_singleton_iff, set.set_o... | lemma | hahn_series.is_pwo_Union_support_powers | 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"
] | [
"exists_prop",
"hahn_series",
"ih",
"is_domain",
"ite_eq_right_iff",
"linear_ordered_cancel_add_comm_monoid",
"not_forall",
"one_ne_zero",
"pow_zero",
"ring",
"set.Union_subset",
"set.mem_singleton_iff",
"set.set_of_eq_eq_singleton",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_family (α : Type*) | (to_fun : α → hahn_series Γ R)
(is_pwo_Union_support' : set.is_pwo (⋃ (a : α), (to_fun a).support))
(finite_co_support' : ∀ (g : Γ), ({a | (to_fun a).coeff g ≠ 0}).finite) | structure | hahn_series.summable_family | 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"
] | [
"finite",
"hahn_series",
"set.is_pwo"
] | An infinite family of Hahn series which has a formal coefficient-wise sum.
The requirements for this are that the union of the supports of the series is well-founded,
and that only finitely many series are nonzero at any given coefficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pwo_Union_support (s : summable_family Γ R α) : set.is_pwo (⋃ (a : α), (s a).support) | s.is_pwo_Union_support' | lemma | hahn_series.summable_family.is_pwo_Union_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"
] | [
"set.is_pwo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_co_support (s : summable_family Γ R α) (g : Γ) :
(function.support (λ a, (s a).coeff g)).finite | s.finite_co_support' g | lemma | hahn_series.summable_family.finite_co_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"
] | [
"finite",
"function.support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @function.injective (summable_family Γ R α) (α → hahn_series Γ R) coe_fn | | ⟨f1, hU1, hf1⟩ ⟨f2, hU2, hf2⟩ h :=
begin
change f1 = f2 at h,
subst h,
end | lemma | hahn_series.summable_family.coe_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 | |
ext {s t : summable_family Γ R α} (h : ∀ (a : α), s a = t a) : s = t | coe_injective $ funext h | lemma | hahn_series.summable_family.ext | 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 | |
coe_add {s t : summable_family Γ R α} : ⇑(s + t) = s + t | rfl | lemma | hahn_series.summable_family.coe_add | 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 | |
add_apply {s t : summable_family Γ R α} {a : α} : (s + t) a = s a + t a | rfl | lemma | hahn_series.summable_family.add_apply | 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 | |
coe_zero : ((0 : summable_family Γ R α) : α → hahn_series Γ R) = 0 | rfl | lemma | hahn_series.summable_family.coe_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 | |
zero_apply {a : α} : (0 : summable_family Γ R α) a = 0 | rfl | lemma | hahn_series.summable_family.zero_apply | 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 | |
hsum (s : summable_family Γ R α) :
hahn_series Γ R | { coeff := λ g, ∑ᶠ i, (s i).coeff g,
is_pwo_support' := s.is_pwo_Union_support.mono (λ g, begin
contrapose,
rw [set.mem_Union, not_exists, function.mem_support, not_not],
simp_rw [mem_support, not_not],
intro h,
rw [finsum_congr h, finsum_zero],
end) } | def | hahn_series.summable_family.hsum | 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_exists",
"not_not",
"set.mem_Union"
] | The infinite sum of a `summable_family` of Hahn series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hsum_coeff {s : summable_family Γ R α} {g : Γ} :
s.hsum.coeff g = ∑ᶠ i, (s i).coeff g | rfl | lemma | hahn_series.summable_family.hsum_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_hsum_subset {s : summable_family Γ R α} :
s.hsum.support ⊆ ⋃ (a : α), (s a).support | λ g hg, begin
rw [mem_support, hsum_coeff, finsum_eq_sum _ (s.finite_co_support _)] at hg,
obtain ⟨a, h1, h2⟩ := exists_ne_zero_of_sum_ne_zero hg,
rw [set.mem_Union],
exact ⟨a, h2⟩,
end | lemma | hahn_series.summable_family.support_hsum_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"
] | [
"set.mem_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hsum_add {s t : summable_family Γ R α} : (s + t).hsum = s.hsum + t.hsum | begin
ext g,
simp only [hsum_coeff, add_coeff, add_apply],
exact finsum_add_distrib (s.finite_co_support _) (t.finite_co_support _)
end | lemma | hahn_series.summable_family.hsum_add | 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 | |
coe_neg : ⇑(-s) = - s | rfl | lemma | hahn_series.summable_family.coe_neg | 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 | |
neg_apply : (-s) a = - (s a) | rfl | lemma | hahn_series.summable_family.neg_apply | 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 | |
coe_sub : ⇑(s - t) = s - t | rfl | lemma | hahn_series.summable_family.coe_sub | 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 | |
sub_apply : (s - t) a = s a - t a | rfl | lemma | hahn_series.summable_family.sub_apply | 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 | |
smul_apply {x : hahn_series Γ R} {s : summable_family Γ R α} {a : α} :
(x • s) a = x * (s a) | rfl | lemma | hahn_series.summable_family.smul_apply | 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 | |
hsum_smul {x : hahn_series Γ R} {s : summable_family Γ R α} :
(x • s).hsum = x * s.hsum | begin
ext g,
simp only [mul_coeff, hsum_coeff, smul_apply],
have h : ∀ i, (s i).support ⊆ ⋃ j, (s j).support := set.subset_Union _,
refine (eq.trans (finsum_congr (λ a, _))
(finsum_sum_comm (add_antidiagonal x.is_pwo_support s.is_pwo_Union_support g)
(λ i ij, x.coeff (prod.fst ij) * (s i).coeff ij.snd) ... | lemma | hahn_series.summable_family.hsum_smul | 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",
"mul_finsum",
"mul_zero",
"not_not",
"set.mem_Union",
"set.subset_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsum : (summable_family Γ R α) →ₗ[hahn_series Γ R] (hahn_series Γ R) | { to_fun := hsum, map_add' := λ _ _, hsum_add, map_smul' := λ _ _, hsum_smul } | def | hahn_series.summable_family.lsum | 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 summation of a `summable_family` as a `linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hsum_sub {R : Type*} [ring R] {s t : summable_family Γ R α} :
(s - t).hsum = s.hsum - t.hsum | by rw [← lsum_apply, linear_map.map_sub, lsum_apply, lsum_apply] | lemma | hahn_series.summable_family.hsum_sub | 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"
] | [
"linear_map.map_sub",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_finsupp (f : α →₀ (hahn_series Γ R)) :
summable_family Γ R α | { to_fun := f,
is_pwo_Union_support' := begin
apply (f.support.is_pwo_bUnion.2 $ λ a ha, (f a).is_pwo_support).mono,
refine set.Union_subset_iff.2 (λ a g hg, _),
have haf : a ∈ f.support,
{ rw [finsupp.mem_support_iff, ← support_nonempty_iff],
exact ⟨g, hg⟩ },
exact set.mem_bUnio... | def | hahn_series.summable_family.of_finsupp | 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"
] | [
"finsupp.mem_support_iff",
"hahn_series",
"set.mem_bUnion"
] | A family with only finitely many nonzero elements is summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_finsupp {f : α →₀ (hahn_series Γ R)} : ⇑(summable_family.of_finsupp f) = f | rfl | lemma | hahn_series.summable_family.coe_of_finsupp | 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 |
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