statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
_root_.finset.esymm_map_val {σ} (f : σ → R) (s : finset σ) (n : ℕ) :
(s.val.map f).esymm n = (s.powerset_len n).sum (λ t, t.prod f) | by simpa only [esymm, powerset_len_map, ← finset.map_val_val_powerset_len, map_map] | lemma | finset.esymm_map_val | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"finset",
"finset.map_val_val_powerset_len"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric [comm_semiring R] (φ : mv_polynomial σ R) : Prop | ∀ e : perm σ, rename e φ = φ | def | mv_polynomial.is_symmetric | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"comm_semiring",
"mv_polynomial"
] | A `mv_polynomial φ` is symmetric if it is invariant under
permutations of its variables by the `rename` operation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symmetric_subalgebra [comm_semiring R] : subalgebra R (mv_polynomial σ R) | { carrier := set_of is_symmetric,
algebra_map_mem' := λ r e, rename_C e r,
mul_mem' := λ a b ha hb e, by rw [alg_hom.map_mul, ha, hb],
add_mem' := λ a b ha hb e, by rw [alg_hom.map_add, ha, hb] } | def | mv_polynomial.symmetric_subalgebra | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"comm_semiring",
"mv_polynomial",
"subalgebra"
] | The subalgebra of symmetric `mv_polynomial`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_symmetric_subalgebra [comm_semiring R] (p : mv_polynomial σ R) :
p ∈ symmetric_subalgebra σ R ↔ p.is_symmetric | iff.rfl | lemma | mv_polynomial.mem_symmetric_subalgebra | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"comm_semiring",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C (r : R) : is_symmetric (C r : mv_polynomial σ R) | (symmetric_subalgebra σ R).algebra_map_mem r | lemma | mv_polynomial.is_symmetric.C | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero : is_symmetric (0 : mv_polynomial σ R) | (symmetric_subalgebra σ R).zero_mem | lemma | mv_polynomial.is_symmetric.zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one : is_symmetric (1 : mv_polynomial σ R) | (symmetric_subalgebra σ R).one_mem | lemma | mv_polynomial.is_symmetric.one | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add (hφ : is_symmetric φ) (hψ : is_symmetric ψ) : is_symmetric (φ + ψ) | (symmetric_subalgebra σ R).add_mem hφ hψ | lemma | mv_polynomial.is_symmetric.add | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul (hφ : is_symmetric φ) (hψ : is_symmetric ψ) : is_symmetric (φ * ψ) | (symmetric_subalgebra σ R).mul_mem hφ hψ | lemma | mv_polynomial.is_symmetric.mul | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul (r : R) (hφ : is_symmetric φ) : is_symmetric (r • φ) | (symmetric_subalgebra σ R).smul_mem hφ r | lemma | mv_polynomial.is_symmetric.smul | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (hφ : is_symmetric φ) (f : R →+* S) : is_symmetric (map f φ) | λ e, by rw [← map_rename, hφ] | lemma | mv_polynomial.is_symmetric.map | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg (hφ : is_symmetric φ) : is_symmetric (-φ) | (symmetric_subalgebra σ R).neg_mem hφ | lemma | mv_polynomial.is_symmetric.neg | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub (hφ : is_symmetric φ) (hψ : is_symmetric ψ) : is_symmetric (φ - ψ) | (symmetric_subalgebra σ R).sub_mem hφ hψ | lemma | mv_polynomial.is_symmetric.sub | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
esymm (n : ℕ) : mv_polynomial σ R | ∑ t in powerset_len n univ, ∏ i in t, X i | def | mv_polynomial.esymm | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"mv_polynomial"
] | The `n`th elementary symmetric `mv_polynomial σ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
esymm_eq_multiset_esymm : esymm σ R = (finset.univ.val.map X).esymm | funext $ λ n, (finset.univ.esymm_map_val X n).symm | lemma | mv_polynomial.esymm_eq_multiset_esymm | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | The `n`th elementary symmetric `mv_polynomial σ R` is obtained by evaluating the
`n`th elementary symmetric at the `multiset` of the monomials | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aeval_esymm_eq_multiset_esymm [algebra R S] (f : σ → S) (n : ℕ) :
aeval f (esymm σ R n) = (finset.univ.val.map f).esymm n | by simp_rw [esymm, aeval_sum, aeval_prod, aeval_X, esymm_map_val] | lemma | mv_polynomial.aeval_esymm_eq_multiset_esymm | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
esymm_eq_sum_subtype (n : ℕ) : esymm σ R n =
∑ t : {s : finset σ // s.card = n}, ∏ i in (t : finset σ), X i | sum_subtype _ (λ _, mem_powerset_len_univ_iff) _ | lemma | mv_polynomial.esymm_eq_sum_subtype | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"finset"
] | We can define `esymm σ R n` by summing over a subtype instead of over `powerset_len`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
esymm_eq_sum_monomial (n : ℕ) : esymm σ R n =
∑ t in powerset_len n univ, monomial (∑ i in t, finsupp.single i 1) 1 | begin
simp_rw monomial_sum_one,
refl,
end | lemma | mv_polynomial.esymm_eq_sum_monomial | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"finsupp.single"
] | We can define `esymm σ R n` as a sum over explicit monomials | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
esymm_zero : esymm σ R 0 = 1 | by simp only [esymm, powerset_len_zero, sum_singleton, prod_empty] | lemma | mv_polynomial.esymm_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_esymm (n : ℕ) (f : R →+* S) : map f (esymm σ R n) = esymm σ S n | by simp_rw [esymm, map_sum, map_prod, map_X] | lemma | mv_polynomial.map_esymm | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rename_esymm (n : ℕ) (e : σ ≃ τ) : rename e (esymm σ R n) = esymm τ R n | calc rename e (esymm σ R n)
= ∑ x in powerset_len n univ, ∏ i in x, X (e i)
: by simp_rw [esymm, map_sum, map_prod, rename_X]
... = ∑ t in powerset_len n (univ.map e.to_embedding), ∏ i in t, X i
: by simp [finset.powerset_len_map, -finset.map_univ_equiv]
... = ∑ t in powerset_len n univ, ∏ i in t, ... | lemma | mv_polynomial.rename_esymm | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"finset.map_univ_equiv",
"finset.powerset_len_map",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
esymm_is_symmetric (n : ℕ) : is_symmetric (esymm σ R n) | by { intro, rw rename_esymm } | lemma | mv_polynomial.esymm_is_symmetric | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_esymm'' (n : ℕ) [decidable_eq σ] [nontrivial R] :
(esymm σ R n).support = (powerset_len n (univ : finset σ)).bUnion
(λ t, (finsupp.single (∑ (i : σ) in t, finsupp.single i 1) (1:R)).support) | begin
rw esymm_eq_sum_monomial,
simp only [← single_eq_monomial],
convert finsupp.support_sum_eq_bUnion (powerset_len n (univ : finset σ)) _,
intros s t hst,
rw finset.disjoint_left,
simp only [finsupp.support_single_ne_zero _ one_ne_zero, mem_singleton],
rintro a h rfl,
have := congr_arg finsupp.suppor... | lemma | mv_polynomial.support_esymm'' | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"finset",
"finset.disjoint_left",
"finsupp.single",
"finsupp.support_single_disjoint",
"finsupp.support_single_ne_zero",
"finsupp.support_sum_eq_bUnion",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_esymm' (n : ℕ) [decidable_eq σ] [nontrivial R] :
(esymm σ R n).support =
(powerset_len n (univ : finset σ)).bUnion (λ t, {∑ (i : σ) in t, finsupp.single i 1}) | begin
rw support_esymm'',
congr,
funext,
exact finsupp.support_single_ne_zero _ one_ne_zero
end | lemma | mv_polynomial.support_esymm' | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"finset",
"finsupp.single",
"finsupp.support_single_ne_zero",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_esymm (n : ℕ) [decidable_eq σ] [nontrivial R] :
(esymm σ R n).support =
(powerset_len n (univ : finset σ)).image (λ t, ∑ (i : σ) in t, finsupp.single i 1) | by { rw support_esymm', exact bUnion_singleton } | lemma | mv_polynomial.support_esymm | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"finset",
"finsupp.single",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degrees_esymm [nontrivial R]
(n : ℕ) (hpos : 0 < n) (hn : n ≤ fintype.card σ) :
(esymm σ R n).degrees = (univ : finset σ).val | begin
classical,
have : (finsupp.to_multiset ∘ λ (t : finset σ), ∑ (i : σ) in t, finsupp.single i 1) = finset.val,
{ funext, simp [finsupp.to_multiset_sum_single] },
rw [degrees_def, support_esymm, sup_image, this, ←comp_sup_eq_sup_comp],
{ obtain ⟨k, rfl⟩ := nat.exists_eq_succ_of_ne_zero hpos.ne',
simpa ... | lemma | mv_polynomial.degrees_esymm | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/symmetric.lean | [
"data.mv_polynomial.rename",
"data.mv_polynomial.comm_ring",
"algebra.algebra.subalgebra.basic"
] | [
"finset",
"finsupp.single",
"finsupp.to_multiset",
"finsupp.to_multiset_sum_single",
"fintype.card",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_map_algebra_map (x : σ → B) (p : mv_polynomial σ R) :
aeval x (map (algebra_map R A) p) = aeval x p | by rw [aeval_def, aeval_def, eval₂_map, is_scalar_tower.algebra_map_eq R A B] | theorem | mv_polynomial.aeval_map_algebra_map | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/tower.lean | [
"algebra.algebra.tower",
"data.mv_polynomial.basic"
] | [
"algebra_map",
"is_scalar_tower.algebra_map_eq",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_algebra_map_apply (x : σ → A) (p : mv_polynomial σ R) :
aeval (algebra_map A B ∘ x) p = algebra_map A B (mv_polynomial.aeval x p) | by rw [aeval_def, aeval_def, ← coe_eval₂_hom, ← coe_eval₂_hom, map_eval₂_hom,
←is_scalar_tower.algebra_map_eq] | lemma | mv_polynomial.aeval_algebra_map_apply | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/tower.lean | [
"algebra.algebra.tower",
"data.mv_polynomial.basic"
] | [
"algebra_map",
"mv_polynomial",
"mv_polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_algebra_map_eq_zero_iff [no_zero_smul_divisors A B] [nontrivial B]
(x : σ → A) (p : mv_polynomial σ R) :
aeval (algebra_map A B ∘ x) p = 0 ↔ aeval x p = 0 | by rw [aeval_algebra_map_apply, algebra.algebra_map_eq_smul_one, smul_eq_zero,
iff_false_intro (one_ne_zero' B), or_false] | lemma | mv_polynomial.aeval_algebra_map_eq_zero_iff | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/tower.lean | [
"algebra.algebra.tower",
"data.mv_polynomial.basic"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"mv_polynomial",
"no_zero_smul_divisors",
"nontrivial",
"one_ne_zero'",
"smul_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_algebra_map_eq_zero_iff_of_injective
{x : σ → A} {p : mv_polynomial σ R}
(h : function.injective (algebra_map A B)) :
aeval (algebra_map A B ∘ x) p = 0 ↔ aeval x p = 0 | by rw [aeval_algebra_map_apply, ← (algebra_map A B).map_zero, h.eq_iff] | lemma | mv_polynomial.aeval_algebra_map_eq_zero_iff_of_injective | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/tower.lean | [
"algebra.algebra.tower",
"data.mv_polynomial.basic"
] | [
"algebra_map",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mv_polynomial_aeval_coe (S : subalgebra R A) (x : σ → S) (p : mv_polynomial σ R) :
aeval (λ i, (x i : A)) p = aeval x p | by convert aeval_algebra_map_apply A x p | lemma | subalgebra.mv_polynomial_aeval_coe | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/tower.lean | [
"algebra.algebra.tower",
"data.mv_polynomial.basic"
] | [
"mv_polynomial",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weighted_degree' (w : σ → M) : (σ →₀ ℕ) →+ M | (finsupp.total σ M ℕ w).to_add_monoid_hom | def | mv_polynomial.weighted_degree' | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.total"
] | The `weighted degree'` of the finitely supported function `s : σ →₀ ℕ` is the sum
`∑(s i)•(w i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_total_degree' (w : σ → M) (p : mv_polynomial σ R) : with_bot M | p.support.sup (λ s, weighted_degree' w s) | def | mv_polynomial.weighted_total_degree' | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial",
"with_bot"
] | The weighted total degree of a multivariate polynomial, taking values in `with_bot M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_total_degree'_eq_bot_iff (w : σ → M) (p : mv_polynomial σ R) :
weighted_total_degree' w p = ⊥ ↔ p = 0 | begin
simp only [weighted_total_degree',finset.sup_eq_bot_iff, mem_support_iff, with_bot.coe_ne_bot,
mv_polynomial.eq_zero_iff ],
exact forall_congr (λ _, not_not)
end | lemma | mv_polynomial.weighted_total_degree'_eq_bot_iff | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset.sup_eq_bot_iff",
"mv_polynomial",
"mv_polynomial.eq_zero_iff",
"not_not",
"with_bot.coe_ne_bot"
] | The `weighted_total_degree'` of a polynomial `p` is `⊥` if and only if `p = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_total_degree'_zero (w : σ → M) :
weighted_total_degree' w (0 : mv_polynomial σ R) = ⊥ | by simp only [weighted_total_degree', support_zero, finset.sup_empty] | lemma | mv_polynomial.weighted_total_degree'_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset.sup_empty",
"mv_polynomial"
] | The `weighted_total_degree'` of the zero polynomial is `⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_total_degree (w : σ → M) (p : mv_polynomial σ R) : M | p.support.sup (λ s, weighted_degree' w s) | def | mv_polynomial.weighted_total_degree | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | When `M` has a `⊥` element, we can define the weighted total degree of a multivariate
polynomial as a function taking values in `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_total_degree_coe (w : σ → M) (p : mv_polynomial σ R) (hp : p ≠ 0):
weighted_total_degree' w p = ↑(weighted_total_degree w p) | begin
rw [ne.def, ← weighted_total_degree'_eq_bot_iff w p, ← ne.def, with_bot.ne_bot_iff_exists] at hp,
obtain ⟨m, hm⟩ := hp,
apply le_antisymm,
{ simp only [weighted_total_degree, weighted_total_degree', finset.sup_le_iff,
with_bot.coe_le_coe],
intro b,
exact finset.le_sup },
{ simp only [weigh... | lemma | mv_polynomial.weighted_total_degree_coe | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset.le_sup",
"finset.sup_le_iff",
"mv_polynomial",
"with_bot.coe_le_coe",
"with_bot.ne_bot_iff_exists"
] | This lemma relates `weighted_total_degree` and `weighted_total_degree'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_total_degree_zero (w : σ → M) :
weighted_total_degree w (0 : mv_polynomial σ R) = ⊥ | by simp only [weighted_total_degree, support_zero, finset.sup_empty] | lemma | mv_polynomial.weighted_total_degree_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset.sup_empty",
"mv_polynomial"
] | The `weighted_total_degree` of the zero polynomial is `⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_weighted_total_degree (w : σ → M) {φ : mv_polynomial σ R} {d : σ →₀ ℕ}
(hd : d ∈ φ.support) : weighted_degree' w d ≤ φ.weighted_total_degree w | le_sup hd | lemma | mv_polynomial.le_weighted_total_degree | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_weighted_homogeneous (w : σ → M) (φ : mv_polynomial σ R) (m : M) : Prop | ∀ ⦃d⦄, coeff d φ ≠ 0 → weighted_degree' w d = m | def | mv_polynomial.is_weighted_homogeneous | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | A multivariate polynomial `φ` is weighted homogeneous of weighted degree `m` if all monomials
occuring in `φ` have weighted degree `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_homogeneous_submodule (w : σ → M) (m : M) :
submodule R (mv_polynomial σ R) | { carrier := { x | x.is_weighted_homogeneous w m },
smul_mem' := λ r a ha c hc, begin
rw coeff_smul at hc,
exact ha (right_ne_zero_of_mul hc),
end,
zero_mem' := λ d hd, false.elim (hd $ coeff_zero _),
add_mem' := λ a b ha hb c hc, begin
rw coeff_add at hc,
obtain h|h : coeff c a ≠ 0 ∨ coeff c b ... | def | mv_polynomial.weighted_homogeneous_submodule | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial",
"right_ne_zero_of_mul",
"submodule"
] | The submodule of homogeneous `mv_polynomial`s of degree `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_weighted_homogeneous_submodule (w : σ → M) (m : M) (p : mv_polynomial σ R) :
p ∈ weighted_homogeneous_submodule R w m ↔ p.is_weighted_homogeneous w m | iff.rfl | lemma | mv_polynomial.mem_weighted_homogeneous_submodule | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weighted_homogeneous_submodule_eq_finsupp_supported (w : σ → M) (m : M) :
weighted_homogeneous_submodule R w m =
finsupp.supported _ R {d | weighted_degree' w d = m} | begin
ext,
simp only [mem_supported, set.subset_def, finsupp.mem_support_iff, mem_coe],
refl,
end | lemma | mv_polynomial.weighted_homogeneous_submodule_eq_finsupp_supported | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.mem_support_iff",
"finsupp.supported",
"set.subset_def"
] | The submodule ` weighted_homogeneous_submodule R w m` of homogeneous `mv_polynomial`s of
degree `n` is equal to the `R`-submodule of all `p : (σ →₀ ℕ) →₀ R` such that
`p.support ⊆ {d | weighted_degree' w d = m}`. While equal, the former has a
convenient definitional reduction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_homogeneous_submodule_mul (w : σ → M) (m n : M) :
weighted_homogeneous_submodule R w m * weighted_homogeneous_submodule R w n ≤
weighted_homogeneous_submodule R w (m + n) | begin
rw submodule.mul_le,
intros φ hφ ψ hψ c hc,
rw [coeff_mul] at hc,
obtain ⟨⟨d, e⟩, hde, H⟩ := finset.exists_ne_zero_of_sum_ne_zero hc,
have aux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0,
{ contrapose! H,
by_cases h : coeff d φ = 0;
simp only [*, ne.def, not_false_iff, zero_mul, mul_zero] at * },
rw [← ... | lemma | mv_polynomial.weighted_homogeneous_submodule_mul | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"aux",
"mul_zero",
"submodule.mul_le",
"zero_mul"
] | The submodule generated by products `Pm *Pn` of weighted homogeneous polynomials of degrees `m`
and `n` is contained in the submodule of weighted homogeneous polynomials of degree `m + n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weighted_homogeneous_monomial (w : σ → M) (d : σ →₀ ℕ) (r : R) {m : M}
(hm : weighted_degree' w d = m) : is_weighted_homogeneous w (monomial d r) m | begin
classical,
intros c hc,
rw coeff_monomial at hc,
split_ifs at hc with h,
{ subst c, exact hm },
{ contradiction }
end | lemma | mv_polynomial.is_weighted_homogeneous_monomial | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | Monomials are weighted homogeneous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weighted_homogeneous_of_total_degree_zero [semilattice_sup M] [order_bot M]
(w : σ → M) {p : mv_polynomial σ R} (hp : weighted_total_degree w p = (⊥ : M)) :
is_weighted_homogeneous w p (⊥ : M) | begin
intros d hd,
have h := weighted_total_degree_coe w p (mv_polynomial.ne_zero_iff.mpr ⟨d, hd⟩),
simp only [weighted_total_degree', hp] at h,
rw [eq_bot_iff, ← with_bot.coe_le_coe, ← h],
exact finset.le_sup (mem_support_iff.mpr hd),
end | lemma | mv_polynomial.is_weighted_homogeneous_of_total_degree_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"eq_bot_iff",
"finset.le_sup",
"mv_polynomial",
"order_bot",
"semilattice_sup",
"with_bot.coe_le_coe"
] | A polynomial of weighted_total_degree `⊥` is weighted_homogeneous of degree `⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weighted_homogeneous_C (w : σ → M) (r : R) :
is_weighted_homogeneous w (C r : mv_polynomial σ R) 0 | is_weighted_homogeneous_monomial _ _ _ (map_zero _) | lemma | mv_polynomial.is_weighted_homogeneous_C | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | Constant polynomials are weighted homogeneous of degree 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weighted_homogeneous_zero (w : σ → M) (m : M) :
is_weighted_homogeneous w (0 : mv_polynomial σ R) m | (weighted_homogeneous_submodule R w m).zero_mem | lemma | mv_polynomial.is_weighted_homogeneous_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | 0 is weighted homogeneous of any degree. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weighted_homogeneous_one (w : σ → M) :
is_weighted_homogeneous w (1 : mv_polynomial σ R) 0 | is_weighted_homogeneous_C _ _ | lemma | mv_polynomial.is_weighted_homogeneous_one | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | 1 is weighted homogeneous of degree 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weighted_homogeneous_X (w : σ → M) (i : σ) :
is_weighted_homogeneous w (X i : mv_polynomial σ R) (w i) | begin
apply is_weighted_homogeneous_monomial,
simp only [weighted_degree', linear_map.to_add_monoid_hom_coe, total_single, one_nsmul],
end | lemma | mv_polynomial.is_weighted_homogeneous_X | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"linear_map.to_add_monoid_hom_coe",
"mv_polynomial"
] | An indeterminate `i : σ` is weighted homogeneous of degree `w i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_eq_zero {w : σ → M} (hφ : is_weighted_homogeneous w φ n) (d : σ →₀ ℕ)
(hd : weighted_degree' w d ≠ n) : coeff d φ = 0 | by { have aux := mt (@hφ d) hd, rwa not_not at aux } | lemma | mv_polynomial.is_weighted_homogeneous.coeff_eq_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"aux",
"not_not"
] | The weighted degree of a weighted homogeneous polynomial controls its support. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inj_right {w : σ → M} (hφ : φ ≠ 0) (hm : is_weighted_homogeneous w φ m)
(hn : is_weighted_homogeneous w φ n) : m = n | begin
obtain ⟨d, hd⟩ : ∃ d, coeff d φ ≠ 0 := exists_coeff_ne_zero hφ,
rw [← hm hd, ← hn hd]
end | lemma | mv_polynomial.is_weighted_homogeneous.inj_right | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | The weighted degree of a nonzero weighted homogeneous polynomial is well-defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add {w : σ → M} (hφ : is_weighted_homogeneous w φ n) (hψ : is_weighted_homogeneous w ψ n) :
is_weighted_homogeneous w (φ + ψ) n | (weighted_homogeneous_submodule R w n).add_mem hφ hψ | lemma | mv_polynomial.is_weighted_homogeneous.add | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | The sum of two weighted homogeneous polynomials of degree `n` is weighted homogeneous of
weighted degree `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum {ι : Type*} (s : finset ι) (φ : ι → mv_polynomial σ R) (n : M) {w : σ → M}
(h : ∀ i ∈ s, is_weighted_homogeneous w (φ i) n) :
is_weighted_homogeneous w (∑ i in s, φ i) n | (weighted_homogeneous_submodule R w n).sum_mem h | lemma | mv_polynomial.is_weighted_homogeneous.sum | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset",
"mv_polynomial"
] | The sum of weighted homogeneous polynomials of degree `n` is weighted homogeneous of
weighted degree `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul {w : σ → M} (hφ : is_weighted_homogeneous w φ m) (hψ : is_weighted_homogeneous w ψ n) :
is_weighted_homogeneous w (φ * ψ) (m + n) | weighted_homogeneous_submodule_mul w m n $ submodule.mul_mem_mul hφ hψ | lemma | mv_polynomial.is_weighted_homogeneous.mul | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"submodule.mul_mem_mul"
] | The product of weighted homogeneous polynomials of weighted degrees `m` and `n` is weighted
homogeneous of weighted degree `m + n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod {ι : Type*} (s : finset ι) (φ : ι → mv_polynomial σ R) (n : ι → M) {w : σ → M} :
(∀ i ∈ s, is_weighted_homogeneous w (φ i) (n i)) →
is_weighted_homogeneous w (∏ i in s, φ i) (∑ i in s, n i) | begin
classical,
apply finset.induction_on s,
{ intro, simp only [is_weighted_homogeneous_one, finset.sum_empty, finset.prod_empty] },
{ intros i s his IH h,
simp only [his, finset.prod_insert, finset.sum_insert, not_false_iff],
apply (h i (finset.mem_insert_self _ _)).mul (IH _),
intros j hjs,
... | lemma | mv_polynomial.is_weighted_homogeneous.prod | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.prod_empty",
"finset.prod_insert",
"mv_polynomial"
] | A product of weighted homogeneous polynomials is weighted homogeneous, with weighted degree
equal to the sum of the weighted degrees. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_total_degree [semilattice_sup M] {w : σ → M} (hφ : is_weighted_homogeneous w φ n)
(h : φ ≠ 0) : weighted_total_degree' w φ = n | begin
simp only [weighted_total_degree'],
apply le_antisymm,
{ simp only [finset.sup_le_iff, mem_support_iff, with_bot.coe_le_coe],
exact λ d hd, le_of_eq (hφ hd), },
{ obtain ⟨d, hd⟩ : ∃ d, coeff d φ ≠ 0 := exists_coeff_ne_zero h,
simp only [← hφ hd, finsupp.sum],
replace hd := finsupp.mem_support_... | lemma | mv_polynomial.is_weighted_homogeneous.weighted_total_degree | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finset.le_sup",
"finset.sup_le_iff",
"semilattice_sup",
"with_bot.coe_le_coe"
] | A non zero weighted homogeneous polynomial of weighted degree `n` has weighted total degree
`n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_homogeneous_submodule.gcomm_monoid {w : σ → M} :
set_like.graded_monoid (weighted_homogeneous_submodule R w) | { one_mem := is_weighted_homogeneous_one R w,
mul_mem := λ i j xi xj, is_weighted_homogeneous.mul } | instance | mv_polynomial.is_weighted_homogeneous.weighted_homogeneous_submodule.gcomm_monoid | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"set_like.graded_monoid"
] | The weighted homogeneous submodules form a graded monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_homogeneous_component (w : σ → M) (n : M) :
mv_polynomial σ R →ₗ[R] mv_polynomial σ R | (submodule.subtype _).comp $ finsupp.restrict_dom _ _ {d | weighted_degree' w d = n} | def | mv_polynomial.weighted_homogeneous_component | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.restrict_dom",
"mv_polynomial",
"submodule.subtype"
] | `weighted_homogeneous_component w n φ` is the part of `φ` that is weighted homogeneous of
weighted degree `n`, with respect to the weights `w`.
See `sum_weighted_homogeneous_component` for the statement that `φ` is equal to the sum
of all its weighted homogeneous components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_weighted_homogeneous_component [decidable_eq M] (d : σ →₀ ℕ) :
coeff d (weighted_homogeneous_component w n φ) =
if weighted_degree' w d = n then coeff d φ else 0 | finsupp.filter_apply (λ d : σ →₀ ℕ, weighted_degree' w d = n) φ d | lemma | mv_polynomial.coeff_weighted_homogeneous_component | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.filter_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weighted_homogeneous_component_apply [decidable_eq M] :
weighted_homogeneous_component w n φ =
∑ d in φ.support.filter (λ d, weighted_degree' w d = n), monomial d (coeff d φ) | finsupp.filter_eq_sum (λ d : σ →₀ ℕ, weighted_degree' w d = n) φ | lemma | mv_polynomial.weighted_homogeneous_component_apply | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsupp.filter_eq_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weighted_homogeneous_component_is_weighted_homogeneous :
(weighted_homogeneous_component w n φ).is_weighted_homogeneous w n | begin
classical,
intros d hd,
contrapose! hd,
rw [coeff_weighted_homogeneous_component, if_neg hd]
end | lemma | mv_polynomial.weighted_homogeneous_component_is_weighted_homogeneous | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | The `n` weighted homogeneous component of a polynomial is weighted homogeneous of
weighted degree `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_homogeneous_component_C_mul (n : M) (r : R) :
weighted_homogeneous_component w n (C r * φ) = C r * weighted_homogeneous_component w n φ | by simp only [C_mul', linear_map.map_smul] | lemma | mv_polynomial.weighted_homogeneous_component_C_mul | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"linear_map.map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weighted_homogeneous_component_eq_zero' (h : ∀ d : σ →₀ ℕ, d ∈ φ.support →
weighted_degree' w d ≠ n) : weighted_homogeneous_component w n φ = 0 | begin
classical,
rw [weighted_homogeneous_component_apply, sum_eq_zero],
intros d hd, rw mem_filter at hd,
exfalso, exact h _ hd.1 hd.2
end | lemma | mv_polynomial.weighted_homogeneous_component_eq_zero' | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weighted_homogeneous_component_eq_zero [semilattice_sup M] [order_bot M]
(h : weighted_total_degree w φ < n) : weighted_homogeneous_component w n φ = 0 | begin
classical,
rw [weighted_homogeneous_component_apply, sum_eq_zero],
intros d hd, rw mem_filter at hd,
exfalso,
apply lt_irrefl n,
nth_rewrite 0 ← hd.2,
exact lt_of_le_of_lt (le_weighted_total_degree w hd.1) h,
end | lemma | mv_polynomial.weighted_homogeneous_component_eq_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"order_bot",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weighted_homogeneous_component_finsupp :
(function.support (λ m, weighted_homogeneous_component w m φ)).finite | begin
suffices : function.support (λ m, weighted_homogeneous_component w m φ) ⊆
(λ d, weighted_degree' w d) '' φ.support,
{ exact finite.subset ((λ (d : σ →₀ ℕ), (weighted_degree' w) d) '' ↑(support φ)).to_finite this },
intros m hm,
by_contradiction hm', apply hm,
simp only [mem_support, ne.def] at hm,
... | lemma | mv_polynomial.weighted_homogeneous_component_finsupp | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"by_contradiction",
"finite",
"function.support",
"not_and",
"not_exists",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_weighted_homogeneous_component :
finsum (λ m, weighted_homogeneous_component w m φ) = φ | begin
classical,
rw finsum_eq_sum _ (weighted_homogeneous_component_finsupp φ),
ext1 d,
simp only [coeff_sum, coeff_weighted_homogeneous_component],
rw finset.sum_eq_single (weighted_degree' w d),
{ rw if_pos rfl, },
{ intros m hm hm', rw if_neg hm'.symm, },
{ intro hm, rw if_pos rfl,
simp only [fin... | lemma | mv_polynomial.sum_weighted_homogeneous_component | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"finsum",
"not_not"
] | Every polynomial is the sum of its weighted homogeneous components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_homogeneous_component_weighted_homogeneous_polynomial [decidable_eq M] (m n : M)
(p : mv_polynomial σ R) (h : p ∈ weighted_homogeneous_submodule R w n) :
weighted_homogeneous_component w m p = if m = n then p else 0 | begin
simp only [mem_weighted_homogeneous_submodule] at h,
ext x,
rw coeff_weighted_homogeneous_component,
by_cases zero_coeff : coeff x p = 0,
{ split_ifs,
all_goals { simp only [zero_coeff, coeff_zero], }, },
{ rw h zero_coeff,
simp only [show n = m ↔ m = n, from eq_comm],
split_ifs with h1,
... | lemma | mv_polynomial.weighted_homogeneous_component_weighted_homogeneous_polynomial | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"mv_polynomial"
] | The weighted homogeneous components of a weighted homogeneous polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
weighted_homogeneous_component_zero [no_zero_smul_divisors ℕ M]
(hw : ∀ i : σ, w i ≠ 0) : weighted_homogeneous_component w 0 φ = C (coeff 0 φ) | begin
classical,
ext1 d,
rcases em (d = 0) with (rfl|hd),
{ simp only [coeff_weighted_homogeneous_component, if_pos, map_zero, coeff_zero_C] },
{ rw [coeff_weighted_homogeneous_component, if_neg, coeff_C, if_neg (ne.symm hd)],
simp only [weighted_degree', linear_map.to_add_monoid_hom_coe, finsupp.total_ap... | lemma | mv_polynomial.weighted_homogeneous_component_zero | ring_theory.mv_polynomial | src/ring_theory/mv_polynomial/weighted_homogeneous.lean | [
"algebra.graded_monoid",
"data.mv_polynomial.variables"
] | [
"and_self_left",
"em",
"exists_prop",
"finsupp.coe_zero",
"finsupp.ext_iff",
"finsupp.mem_support_iff",
"finsupp.total_apply",
"linear_map.to_add_monoid_hom_coe",
"no_zero_smul_divisors",
"not_forall",
"not_or_distrib",
"smul_eq_zero"
] | If `M` is a `canonically_ordered_add_monoid`, then the `weighted_homogeneous_component`
of weighted degree `0` of a polynomial is its constant coefficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_subsemiring_class (S : Type*) (R : Type u)
[non_unital_non_assoc_semiring R] [set_like S R] extends add_submonoid_class S R | (mul_mem : ∀ {s : S} {a b : R}, a ∈ s → b ∈ s → a * b ∈ s) | class | non_unital_subsemiring_class | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid_class",
"non_unital_non_assoc_semiring",
"set_like"
] | `non_unital_subsemiring_class S R` states that `S` is a type of subsets `s ⊆ R` that
are both an additive submonoid and also a multiplicative subsemigroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_subsemiring_class.mul_mem_class (S : Type*) (R : Type u)
[non_unital_non_assoc_semiring R] [set_like S R] [h : non_unital_subsemiring_class S R] :
mul_mem_class S R | { .. h } | instance | non_unital_subsemiring_class.mul_mem_class | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"mul_mem_class",
"non_unital_non_assoc_semiring",
"non_unital_subsemiring_class",
"set_like"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_non_assoc_semiring : non_unital_non_assoc_semiring s | subtype.coe_injective.non_unital_non_assoc_semiring coe rfl (by simp) (λ _ _, rfl) (λ _ _, rfl) | instance | non_unital_subsemiring_class.to_non_unital_non_assoc_semiring | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_non_assoc_semiring"
] | A non-unital subsemiring of a `non_unital_non_assoc_semiring` inherits a
`non_unital_non_assoc_semiring` structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
no_zero_divisors [no_zero_divisors R] : no_zero_divisors s | subtype.coe_injective.no_zero_divisors coe rfl (λ x y, rfl) | instance | non_unital_subsemiring_class.no_zero_divisors | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype : s →ₙ+* R | { to_fun := coe, .. add_submonoid_class.subtype s, .. mul_mem_class.subtype s } | def | non_unital_subsemiring_class.subtype | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"mul_mem_class.subtype"
] | The natural non-unital ring hom from a non-unital subsemiring of a non-unital semiring `R` to
`R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_subtype : (subtype s : s → R) = coe | rfl | theorem | non_unital_subsemiring_class.coe_subtype | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_semiring {R} [non_unital_semiring R] [set_like S R]
[non_unital_subsemiring_class S R] : non_unital_semiring s | subtype.coe_injective.non_unital_semiring coe rfl (by simp) (λ _ _, rfl) (λ _ _, rfl) | instance | non_unital_subsemiring_class.to_non_unital_semiring | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_semiring",
"non_unital_subsemiring_class",
"set_like"
] | A non-unital subsemiring of a `non_unital_semiring` is a `non_unital_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_non_unital_comm_semiring {R} [non_unital_comm_semiring R] [set_like S R]
[non_unital_subsemiring_class S R] : non_unital_comm_semiring s | subtype.coe_injective.non_unital_comm_semiring coe rfl (by simp) (λ _ _, rfl) (λ _ _, rfl) | instance | non_unital_subsemiring_class.to_non_unital_comm_semiring | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_comm_semiring",
"non_unital_subsemiring_class",
"set_like"
] | A non-unital subsemiring of a `non_unital_comm_semiring` is a `non_unital_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_subsemiring (R : Type u) [non_unital_non_assoc_semiring R]
extends add_submonoid R, subsemigroup R | structure | non_unital_subsemiring | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"non_unital_non_assoc_semiring",
"subsemigroup"
] | A non-unital subsemiring of a non-unital semiring `R` is a subset `s` that is both an additive
submonoid and a semigroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier {s : non_unital_subsemiring R} {x : R} : x ∈ s.carrier ↔ x ∈ s | iff.rfl | lemma | non_unital_subsemiring.mem_carrier | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {S T : non_unital_subsemiring R} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T | set_like.ext h | theorem | non_unital_subsemiring.ext | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set_like.ext"
] | Two non-unital subsemirings are equal if they have the same elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
copy (S : non_unital_subsemiring R) (s : set R) (hs : s = ↑S) :
non_unital_subsemiring R | { carrier := s,
..S.to_add_submonoid.copy s hs,
..S.to_subsemigroup.copy s hs } | def | non_unital_subsemiring.copy | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | Copy of a non-unital subsemiring with a new `carrier` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (S : non_unital_subsemiring R) (s : set R) (hs : s = ↑S) :
(S.copy s hs : set R) = s | rfl | lemma | non_unital_subsemiring.coe_copy | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (S : non_unital_subsemiring R) (s : set R) (hs : s = ↑S) : S.copy s hs = S | set_like.coe_injective hs | lemma | non_unital_subsemiring.copy_eq | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subsemigroup_injective :
function.injective (to_subsemigroup : non_unital_subsemiring R → subsemigroup R) | | r s h := ext (set_like.ext_iff.mp h : _) | lemma | non_unital_subsemiring.to_subsemigroup_injective | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"subsemigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subsemigroup_strict_mono :
strict_mono (to_subsemigroup : non_unital_subsemiring R → subsemigroup R) | λ _ _, id | lemma | non_unital_subsemiring.to_subsemigroup_strict_mono | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"strict_mono",
"subsemigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subsemigroup_mono :
monotone (to_subsemigroup : non_unital_subsemiring R → subsemigroup R) | to_subsemigroup_strict_mono.monotone | lemma | non_unital_subsemiring.to_subsemigroup_mono | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"monotone",
"non_unital_subsemiring",
"subsemigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_injective :
function.injective (to_add_submonoid : non_unital_subsemiring R → add_submonoid R) | | r s h := ext (set_like.ext_iff.mp h : _) | lemma | non_unital_subsemiring.to_add_submonoid_injective | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_strict_mono :
strict_mono (to_add_submonoid : non_unital_subsemiring R → add_submonoid R) | λ _ _, id | lemma | non_unital_subsemiring.to_add_submonoid_strict_mono | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"non_unital_subsemiring",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_submonoid_mono :
monotone (to_add_submonoid : non_unital_subsemiring R → add_submonoid R) | to_add_submonoid_strict_mono.monotone | lemma | non_unital_subsemiring.to_add_submonoid_mono | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"monotone",
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (s : set R) (sg : subsemigroup R) (hg : ↑sg = s)
(sa : add_submonoid R) (ha : ↑sa = s) :
non_unital_subsemiring R | { carrier := s,
zero_mem' := ha ▸ sa.zero_mem,
add_mem' := λ x y, by simpa only [← ha] using sa.add_mem,
mul_mem' := λ x y, by simpa only [← hg] using sg.mul_mem } | def | non_unital_subsemiring.mk' | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"mk'",
"non_unital_subsemiring",
"subsemigroup"
] | Construct a `non_unital_subsemiring R` from a set `s`, a subsemigroup `sg`, and an additive
submonoid `sa` such that `x ∈ s ↔ x ∈ sg ↔ x ∈ sa`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk' {s : set R} {sg : subsemigroup R} (hg : ↑sg = s)
{sa : add_submonoid R} (ha : ↑sa = s) :
(non_unital_subsemiring.mk' s sg hg sa ha : set R) = s | rfl | lemma | non_unital_subsemiring.coe_mk' | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"non_unital_subsemiring.mk'",
"subsemigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk' {s : set R} {sg : subsemigroup R} (hg : ↑sg = s)
{sa : add_submonoid R} (ha : ↑sa = s) {x : R} :
x ∈ non_unital_subsemiring.mk' s sg hg sa ha ↔ x ∈ s | iff.rfl | lemma | non_unital_subsemiring.mem_mk' | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"non_unital_subsemiring.mk'",
"subsemigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_to_subsemigroup {s : set R} {sg : subsemigroup R} (hg : ↑sg = s)
{sa : add_submonoid R} (ha : ↑sa = s) :
(non_unital_subsemiring.mk' s sg hg sa ha).to_subsemigroup = sg | set_like.coe_injective hg.symm | lemma | non_unital_subsemiring.mk'_to_subsemigroup | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"non_unital_subsemiring.mk'",
"set_like.coe_injective",
"subsemigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_to_add_submonoid {s : set R} {sg : subsemigroup R} (hg : ↑sg = s)
{sa : add_submonoid R} (ha : ↑sa =s) :
(non_unital_subsemiring.mk' s sg hg sa ha).to_add_submonoid = sa | set_like.coe_injective ha.symm | lemma | non_unital_subsemiring.mk'_to_add_submonoid | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"add_submonoid",
"non_unital_subsemiring.mk'",
"set_like.coe_injective",
"subsemigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : s) : R) = (0 : R) | rfl | lemma | non_unital_subsemiring.coe_zero | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (x y : s) : ((x + y : s) : R) = (x + y : R) | rfl | lemma | non_unital_subsemiring.coe_add | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (x y : s) : ((x * y : s) : R) = (x * y : R) | rfl | lemma | non_unital_subsemiring.coe_mul | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_subsemigroup {s : non_unital_subsemiring R} {x : R} :
x ∈ s.to_subsemigroup ↔ x ∈ s | iff.rfl | lemma | non_unital_subsemiring.mem_to_subsemigroup | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_subsemigroup (s : non_unital_subsemiring R) :
(s.to_subsemigroup : set R) = s | rfl | lemma | non_unital_subsemiring.coe_to_subsemigroup | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_add_submonoid {s : non_unital_subsemiring R} {x : R} :
x ∈ s.to_add_submonoid ↔ x ∈ s | iff.rfl | lemma | non_unital_subsemiring.mem_to_add_submonoid | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.