ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Holor.cprankMax_1
theorem cprankMax_1 [Monoid α] [AddMonoid α] {x : Holor α ds} (h : CPRankMax1 x) :
    CPRankMax 1 x 
Arguments: {α : Type} {ds : List ℕ} [Monoid α] [AddMonoid α] {x : Holor α ds} (h : x.CPRankMax1)
Goal: Holor.CPRankMax 1 x
-- Holor.cprankMax_add
theorem cprankMax_add [Monoid α] [AddMonoid α] :
    ∀ {m : ℕ} {n : ℕ} {x : Holor α ds} {y : Holor α ds},
      CPRankMax m x → CPRankMax n y → CPRankMax (m + n) (x + y)
  
Arguments: {α : Type} {ds : List ℕ} [Monoid α] [AddMonoid α] {m n : ℕ} {x y : Holor α ds}
Goal: Holor.CPRankMax m x → Holor.CPRankMax n y → Holor.CPRankMax (m + n) (x + y)
-- Holor.cprankMax_mul
theorem cprankMax_mul [Ring α] :
    ∀ (n : ℕ) (x : Holor α [d]) (y : Holor α ds), CPRankMax n y → CPRankMax n (x ⊗ y)
  
Arguments: {α : Type} {d : ℕ} {ds : List ℕ} [Ring α] (n : ℕ) (x : Holor α [d]) (y : Holor α ds)
Goal: Holor.CPRankMax n y → Holor.CPRankMax n (x.mul y)
-- Holor.cprankMax_sum
theorem cprankMax_sum [Ring α] {β} {n : ℕ} (s : Finset β) (f : β → Holor α ds) :
    (∀ x ∈ s, CPRankMax n (f x)) → CPRankMax (s.card * n) (∑ x ∈ s, f x) 
Arguments: {α : Type} {ds : List ℕ} [Ring α] {β : Type u_1} {n : ℕ} (s : Finset β) (f : β → Holor α ds)
Goal: (∀ x ∈ s, Holor.CPRankMax n (f x)) → Holor.CPRankMax (s.card * n) (∑ x ∈ s, f x)
-- Holor.cprankMax_upper_bound
theorem cprankMax_upper_bound [Ring α] : ∀ {ds}, ∀ x : Holor α ds, CPRankMax ds.prod x
  
Arguments: {α : Type} [Ring α] {ds : List ℕ} (x : Holor α ds)
Goal: Holor.CPRankMax ds.prod x
-- Holor.cprank_upper_bound
theorem cprank_upper_bound [Ring α] : ∀ {ds}, ∀ x : Holor α ds, cprank x ≤ ds.prod 
Arguments: {α : Type} [Ring α] {ds : List ℕ} (x : Holor α ds)
Goal: x.cprank ≤ ds.prod
-- Finsupp.rangeSingleton_support
@[simps]
def rangeSingleton (f : ι →₀ α) : ι →₀ Finset α 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] (f : ι →₀ α)
Goal: f.rangeSingleton.support = f.support
-- Finsupp.rangeSingleton_toFun
@[simps]
def rangeSingleton (f : ι →₀ α) : ι →₀ Finset α 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] (f : ι →₀ α) (i : ι)
Goal: f.rangeSingleton i = {f i}
-- Finsupp.mem_rangeSingleton_apply_iff
theorem mem_rangeSingleton_apply_iff : a ∈ f.rangeSingleton i ↔ a = f i 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] {f : ι →₀ α} {i : ι} {a : α}
Goal: a ∈ f.rangeSingleton i ↔ a = f i
-- Finsupp.rangeIcc_toFun
@[simps toFun]
def rangeIcc (f g : ι →₀ α) : ι →₀ Finset α 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [PartialOrder α] [LocallyFiniteOrder α] (f g : ι →₀ α) (i : ι)
Goal: (f.rangeIcc g) i = Finset.Icc (f i) (g i)
-- Finsupp.coe_rangeIcc
lemma coe_rangeIcc (f g : ι →₀ α) : rangeIcc f g i = Icc (f i) (g i) 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [PartialOrder α] [LocallyFiniteOrder α] {i : ι} (f g : ι →₀ α)
Goal: (f.rangeIcc g) i = Finset.Icc (f i) (g i)
-- Finsupp.rangeIcc_support
@[simp]
theorem rangeIcc_support (f g : ι →₀ α) :
    (rangeIcc f g).support = f.support ∪ g.support 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [PartialOrder α] [LocallyFiniteOrder α] (f g : ι →₀ α)
Goal: (f.rangeIcc g).support = f.support ∪ g.support
-- Finsupp.mem_rangeIcc_apply_iff
theorem mem_rangeIcc_apply_iff : a ∈ f.rangeIcc g i ↔ f i ≤ a ∧ a ≤ g i 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [PartialOrder α] [LocallyFiniteOrder α] {f g : ι →₀ α} {i : ι} {a : α}
Goal: a ∈ (f.rangeIcc g) i ↔ f i ≤ a ∧ a ≤ g i
-- Finsupp.Icc_eq
theorem Icc_eq : Icc f g = (f.support ∪ g.support).finsupp (f.rangeIcc g) 
Arguments: {ι : Type u_1} {α : Type u_2} [PartialOrder α] [Zero α] [LocallyFiniteOrder α] (f g : ι →₀ α)
Goal: Finset.Icc f g = (f.support ∪ g.support).finsupp ⇑(f.rangeIcc g)
-- Finsupp.card_Icc
theorem card_Icc : (Icc f g).card = ∏ i ∈ f.support ∪ g.support, (Icc (f i) (g i)).card 
Arguments: {ι : Type u_1} {α : Type u_2} [PartialOrder α] [Zero α] [LocallyFiniteOrder α] (f g : ι →₀ α)
Goal: (Finset.Icc f g).card = ∏ i ∈ f.support ∪ g.support, (Finset.Icc (f i) (g i)).card
-- Finsupp.card_Ico
theorem card_Ico : (Ico f g).card = (∏ i ∈ f.support ∪ g.support, (Icc (f i) (g i)).card) - 1 
Arguments: {ι : Type u_1} {α : Type u_2} [PartialOrder α] [Zero α] [LocallyFiniteOrder α] (f g : ι →₀ α)
Goal: (Finset.Ico f g).card = ∏ i ∈ f.support ∪ g.support, (Finset.Icc (f i) (g i)).card - 1
-- Finsupp.card_Ioc
theorem card_Ioc : (Ioc f g).card = (∏ i ∈ f.support ∪ g.support, (Icc (f i) (g i)).card) - 1 
Arguments: {ι : Type u_1} {α : Type u_2} [PartialOrder α] [Zero α] [LocallyFiniteOrder α] (f g : ι →₀ α)
Goal: (Finset.Ioc f g).card = ∏ i ∈ f.support ∪ g.support, (Finset.Icc (f i) (g i)).card - 1
-- Finsupp.card_Ioo
theorem card_Ioo : (Ioo f g).card = (∏ i ∈ f.support ∪ g.support, (Icc (f i) (g i)).card) - 2 
Arguments: {ι : Type u_1} {α : Type u_2} [PartialOrder α] [Zero α] [LocallyFiniteOrder α] (f g : ι →₀ α)
Goal: (Finset.Ioo f g).card = ∏ i ∈ f.support ∪ g.support, (Finset.Icc (f i) (g i)).card - 2
-- Finsupp.card_uIcc
theorem card_uIcc :
    (uIcc f g).card = ∏ i ∈ f.support ∪ g.support, (uIcc (f i) (g i)).card 
Arguments: {ι : Type u_1} {α : Type u_2} [Lattice α] [Zero α] [LocallyFiniteOrder α] (f g : ι →₀ α)
Goal: (Finset.uIcc f g).card = ∏ i ∈ f.support ∪ g.support, (Finset.uIcc (f i) (g i)).card
-- Finsupp.card_Iic
theorem card_Iic : (Iic f).card = ∏ i ∈ f.support, (Iic (f i)).card 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] [LocallyFiniteOrder α] (f : ι →₀ α)
Goal: (Finset.Iic f).card = ∏ i ∈ f.support, (Finset.Iic (f i)).card
-- Finsupp.card_Iio
theorem card_Iio : (Iio f).card = (∏ i ∈ f.support, (Iic (f i)).card) - 1 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] [LocallyFiniteOrder α] (f : ι →₀ α)
Goal: (Finset.Iio f).card = ∏ i ∈ f.support, (Finset.Iic (f i)).card - 1
-- Finsupp.indicator_of_mem
theorem indicator_of_mem (hi : i ∈ s) (f : ∀ i ∈ s, α) : indicator s f i = f i hi 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] {s : Finset ι} {i : ι} (hi : i ∈ s) (f : (i : ι) → i ∈ s → α)
Goal: (Finsupp.indicator s f) i = f i hi
-- Finsupp.indicator_of_not_mem
theorem indicator_of_not_mem (hi : i ∉ s) (f : ∀ i ∈ s, α) : indicator s f i = 0 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] {s : Finset ι} {i : ι} (hi : i ∉ s) (f : (i : ι) → i ∈ s → α)
Goal: (Finsupp.indicator s f) i = 0
-- Finsupp.indicator_apply
@[simp]
theorem indicator_apply [DecidableEq ι] : indicator s f i = if hi : i ∈ s then f i hi else 0 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] (s : Finset ι) (f : (i : ι) → i ∈ s → α) (i : ι) [DecidableEq ι]
Goal: (Finsupp.indicator s f) i = if hi : i ∈ s then f i hi else 0
-- Finsupp.indicator_injective
theorem indicator_injective : Injective fun f : ∀ i ∈ s, α => indicator s f 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] (s : Finset ι)
Goal: Function.Injective fun f => Finsupp.indicator s f
-- Finsupp.support_indicator_subset
theorem support_indicator_subset : ((indicator s f).support : Set ι) ⊆ s 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] (s : Finset ι) (f : (i : ι) → i ∈ s → α)
Goal: ↑(Finsupp.indicator s f).support ⊆ ↑s
-- Finsupp.single_eq_indicator
lemma single_eq_indicator (b : α) : single i b = indicator {i} (fun _ _ => b) 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] (i : ι) (b : α)
Goal: Finsupp.single i b = Finsupp.indicator {i} fun x x => b
-- Finsupp.toDFinsupp_coe
@[simp]
theorem Finsupp.toDFinsupp_coe [Zero M] (f : ι →₀ M) : ⇑f.toDFinsupp = f 
Arguments: {ι : Type u_1} {M : Type u_3} [Zero M] (f : ι →₀ M)
Goal: ⇑f.toDFinsupp = ⇑f
-- Finsupp.toDFinsupp_single
@[simp]
theorem Finsupp.toDFinsupp_single (i : ι) (m : M) :
    (Finsupp.single i m).toDFinsupp = DFinsupp.single i m 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] (i : ι) (m : M)
Goal: (Finsupp.single i m).toDFinsupp = DFinsupp.single i m
-- toDFinsupp_support
@[simp]
theorem toDFinsupp_support (f : ι →₀ M) : f.toDFinsupp.support = f.support 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)] (f : ι →₀ M)
Goal: f.toDFinsupp.support = f.support
-- DFinsupp.toFinsupp_coe
@[simp]
theorem DFinsupp.toFinsupp_coe (f : Π₀ _ : ι, M) : ⇑f.toFinsupp = f 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)] (f : Π₀ (x : ι), M)
Goal: ⇑f.toFinsupp = ⇑f
-- DFinsupp.toFinsupp_support
@[simp]
theorem DFinsupp.toFinsupp_support (f : Π₀ _ : ι, M) : f.toFinsupp.support = f.support 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)] (f : Π₀ (x : ι), M)
Goal: f.toFinsupp.support = f.support
-- DFinsupp.toFinsupp_single
@[simp]
theorem DFinsupp.toFinsupp_single (i : ι) (m : M) :
    (DFinsupp.single i m : Π₀ _ : ι, M).toFinsupp = Finsupp.single i m 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)] (i : ι) (m : M)
Goal: (DFinsupp.single i m).toFinsupp = Finsupp.single i m
-- Finsupp.toDFinsupp_toFinsupp
@[simp]
theorem Finsupp.toDFinsupp_toFinsupp (f : ι →₀ M) : f.toDFinsupp.toFinsupp = f 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)] (f : ι →₀ M)
Goal: f.toDFinsupp.toFinsupp = f
-- DFinsupp.toFinsupp_toDFinsupp
@[simp]
theorem DFinsupp.toFinsupp_toDFinsupp (f : Π₀ _ : ι, M) : f.toFinsupp.toDFinsupp = f 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)] (f : Π₀ (x : ι), M)
Goal: f.toFinsupp.toDFinsupp = f
-- Finsupp.toDFinsupp_zero
@[simp]
theorem toDFinsupp_zero [Zero M] : (0 : ι →₀ M).toDFinsupp = 0 
Arguments: {ι : Type u_1} {M : Type u_3} [Zero M]
Goal: Finsupp.toDFinsupp 0 = 0
-- Finsupp.toDFinsupp_add
@[simp]
theorem toDFinsupp_add [AddZeroClass M] (f g : ι →₀ M) :
    (f + g).toDFinsupp = f.toDFinsupp + g.toDFinsupp 
Arguments: {ι : Type u_1} {M : Type u_3} [AddZeroClass M] (f g : ι →₀ M)
Goal: (f + g).toDFinsupp = f.toDFinsupp + g.toDFinsupp
-- Finsupp.toDFinsupp_neg
@[simp]
theorem toDFinsupp_neg [AddGroup M] (f : ι →₀ M) : (-f).toDFinsupp = -f.toDFinsupp 
Arguments: {ι : Type u_1} {M : Type u_3} [AddGroup M] (f : ι →₀ M)
Goal: (-f).toDFinsupp = -f.toDFinsupp
-- Finsupp.toDFinsupp_sub
@[simp]
theorem toDFinsupp_sub [AddGroup M] (f g : ι →₀ M) :
    (f - g).toDFinsupp = f.toDFinsupp - g.toDFinsupp 
Arguments: {ι : Type u_1} {M : Type u_3} [AddGroup M] (f g : ι →₀ M)
Goal: (f - g).toDFinsupp = f.toDFinsupp - g.toDFinsupp
-- Finsupp.toDFinsupp_smul
@[simp]
theorem toDFinsupp_smul [Monoid R] [AddMonoid M] [DistribMulAction R M] (r : R) (f : ι →₀ M) :
    (r • f).toDFinsupp = r • f.toDFinsupp 
Arguments: {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Monoid R] [AddMonoid M] [DistribMulAction R M] (r : R) (f : ι →₀ M)
Goal: (r • f).toDFinsupp = r • f.toDFinsupp
-- DFinsupp.toFinsupp_zero
@[simp]
theorem toFinsupp_zero [Zero M] [∀ m : M, Decidable (m ≠ 0)] : toFinsupp 0 = (0 : ι →₀ M) 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)]
Goal: DFinsupp.toFinsupp 0 = 0
-- DFinsupp.toFinsupp_add
@[simp]
theorem toFinsupp_add [AddZeroClass M] [∀ m : M, Decidable (m ≠ 0)] (f g : Π₀ _ : ι, M) :
    (toFinsupp (f + g) : ι →₀ M) = toFinsupp f + toFinsupp g 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddZeroClass M] [(m : M) → Decidable (m ≠ 0)] (f g : Π₀ (x : ι), M)
Goal: (f + g).toFinsupp = f.toFinsupp + g.toFinsupp
-- DFinsupp.toFinsupp_neg
@[simp]
theorem toFinsupp_neg [AddGroup M] [∀ m : M, Decidable (m ≠ 0)] (f : Π₀ _ : ι, M) :
    (toFinsupp (-f) : ι →₀ M) = -toFinsupp f 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddGroup M] [(m : M) → Decidable (m ≠ 0)] (f : Π₀ (x : ι), M)
Goal: (-f).toFinsupp = -f.toFinsupp
-- DFinsupp.toFinsupp_sub
@[simp]
theorem toFinsupp_sub [AddGroup M] [∀ m : M, Decidable (m ≠ 0)] (f g : Π₀ _ : ι, M) :
    (toFinsupp (f - g) : ι →₀ M) = toFinsupp f - toFinsupp g 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddGroup M] [(m : M) → Decidable (m ≠ 0)] (f g : Π₀ (x : ι), M)
Goal: (f - g).toFinsupp = f.toFinsupp - g.toFinsupp
-- DFinsupp.toFinsupp_smul
@[simp]
theorem toFinsupp_smul [Monoid R] [AddMonoid M] [DistribMulAction R M] [∀ m : M, Decidable (m ≠ 0)]
    (r : R) (f : Π₀ _ : ι, M) : (toFinsupp (r • f) : ι →₀ M) = r • toFinsupp f 
Arguments: {ι : Type u_1} {R : Type u_2} {M : Type u_3} [DecidableEq ι] [Monoid R] [AddMonoid M] [DistribMulAction R M] [(m : M) → Decidable (m ≠ 0)] (r : R) (f : Π₀ (x : ι), M)
Goal: (r • f).toFinsupp = r • f.toFinsupp
-- finsuppEquivDFinsupp_symm_apply
@[simps (config := .asFn)]
def finsuppEquivDFinsupp [DecidableEq ι] [Zero M] [∀ m : M, Decidable (m ≠ 0)] :
    (ι →₀ M) ≃ Π₀ _ : ι, M 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)]
Goal: ⇑finsuppEquivDFinsupp.symm = DFinsupp.toFinsupp
-- finsuppEquivDFinsupp_apply
@[simps (config := .asFn)]
def finsuppEquivDFinsupp [DecidableEq ι] [Zero M] [∀ m : M, Decidable (m ≠ 0)] :
    (ι →₀ M) ≃ Π₀ _ : ι, M 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [Zero M] [(m : M) → Decidable (m ≠ 0)]
Goal: ⇑finsuppEquivDFinsupp = Finsupp.toDFinsupp
-- finsuppAddEquivDFinsupp_apply
@[simps (config := .asFn)]
def finsuppAddEquivDFinsupp [DecidableEq ι] [AddZeroClass M] [∀ m : M, Decidable (m ≠ 0)] :
    (ι →₀ M) ≃+ Π₀ _ : ι, M 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddZeroClass M] [(m : M) → Decidable (m ≠ 0)]
Goal: ⇑finsuppAddEquivDFinsupp = Finsupp.toDFinsupp
-- finsuppAddEquivDFinsupp_symm_apply
@[simps (config := .asFn)]
def finsuppAddEquivDFinsupp [DecidableEq ι] [AddZeroClass M] [∀ m : M, Decidable (m ≠ 0)] :
    (ι →₀ M) ≃+ Π₀ _ : ι, M 
Arguments: {ι : Type u_1} {M : Type u_3} [DecidableEq ι] [AddZeroClass M] [(m : M) → Decidable (m ≠ 0)]
Goal: ⇑finsuppAddEquivDFinsupp.symm = DFinsupp.toFinsupp
-- finsuppLequivDFinsupp_apply_apply
@[simp]
theorem finsuppLequivDFinsupp_apply_apply [DecidableEq ι] [Semiring R] [AddCommMonoid M]
    [∀ m : M, Decidable (m ≠ 0)] [Module R M] :
    (↑(finsuppLequivDFinsupp (M := M) R) : (ι →₀ M) → _) = Finsupp.toDFinsupp 
Arguments: {ι : Type u_1} (R : Type u_2) {M : Type u_3} [DecidableEq ι] [Semiring R] [AddCommMonoid M] [(m : M) → Decidable (m ≠ 0)] [Module R M]
Goal: ⇑(finsuppLequivDFinsupp R) = Finsupp.toDFinsupp
-- finsuppLequivDFinsupp_symm_apply
@[simp]
theorem finsuppLequivDFinsupp_symm_apply [DecidableEq ι] [Semiring R] [AddCommMonoid M]
    [∀ m : M, Decidable (m ≠ 0)] [Module R M] :
    ↑(LinearEquiv.symm (finsuppLequivDFinsupp (ι := ι) (M := M) R)) = DFinsupp.toFinsupp 
Arguments: {ι : Type u_1} (R : Type u_2) {M : Type u_3} [DecidableEq ι] [Semiring R] [AddCommMonoid M] [(m : M) → Decidable (m ≠ 0)] [Module R M]
Goal: ⇑(finsuppLequivDFinsupp R).symm = DFinsupp.toFinsupp
-- sigmaFinsuppEquivDFinsupp_apply
@[simp]
theorem sigmaFinsuppEquivDFinsupp_apply [Zero N] (f : (Σi, η i) →₀ N) :
    (sigmaFinsuppEquivDFinsupp f : ∀ i, η i →₀ N) = Finsupp.split f 
Arguments: {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} [Zero N] (f : (i : ι) × η i →₀ N)
Goal: ⇑(sigmaFinsuppEquivDFinsupp f) = f.split
-- sigmaFinsuppEquivDFinsupp_symm_apply
@[simp]
theorem sigmaFinsuppEquivDFinsupp_symm_apply [Zero N] (f : Π₀ i, η i →₀ N) (s : Σi, η i) :
    (sigmaFinsuppEquivDFinsupp.symm f : (Σi, η i) →₀ N) s = f s.1 s.2 
Arguments: {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} [Zero N] (f : Π₀ (i : ι), η i →₀ N) (s : (i : ι) × η i)
Goal: (sigmaFinsuppEquivDFinsupp.symm f) s = (f s.fst) s.snd
-- sigmaFinsuppEquivDFinsupp_support
@[simp]
theorem sigmaFinsuppEquivDFinsupp_support [DecidableEq ι] [Zero N]
    [∀ (i : ι) (x : η i →₀ N), Decidable (x ≠ 0)] (f : (Σi, η i) →₀ N) :
    (sigmaFinsuppEquivDFinsupp f).support = Finsupp.splitSupport f 
Arguments: {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} [DecidableEq ι] [Zero N] [(i : ι) → (x : η i →₀ N) → Decidable (x ≠ 0)] (f : (i : ι) × η i →₀ N)
Goal: (sigmaFinsuppEquivDFinsupp f).support = f.splitSupport
-- sigmaFinsuppEquivDFinsupp_single
@[simp]
theorem sigmaFinsuppEquivDFinsupp_single [DecidableEq ι] [Zero N] (a : Σi, η i) (n : N) :
    sigmaFinsuppEquivDFinsupp (Finsupp.single a n) =
      @DFinsupp.single _ (fun i => η i →₀ N) _ _ a.1 (Finsupp.single a.2 n) 
Arguments: {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} [DecidableEq ι] [Zero N] (a : (i : ι) × η i) (n : N)
Goal: sigmaFinsuppEquivDFinsupp (Finsupp.single a n) = DFinsupp.single a.fst (Finsupp.single a.snd n)
-- sigmaFinsuppEquivDFinsupp_add
@[simp]
theorem sigmaFinsuppEquivDFinsupp_add [AddZeroClass N] (f g : (Σi, η i) →₀ N) :
    sigmaFinsuppEquivDFinsupp (f + g) =
      (sigmaFinsuppEquivDFinsupp f + sigmaFinsuppEquivDFinsupp g : Π₀ i : ι, η i →₀ N) 
Arguments: {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} [AddZeroClass N] (f g : (i : ι) × η i →₀ N)
Goal: sigmaFinsuppEquivDFinsupp (f + g) = sigmaFinsuppEquivDFinsupp f + sigmaFinsuppEquivDFinsupp g
-- sigmaFinsuppAddEquivDFinsupp_apply
@[simps]
def sigmaFinsuppAddEquivDFinsupp [AddZeroClass N] : ((Σi, η i) →₀ N) ≃+ Π₀ i, η i →₀ N 
Arguments: {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} [AddZeroClass N] (a : (i : ι) × η i →₀ N)
Goal: sigmaFinsuppAddEquivDFinsupp a = sigmaFinsuppEquivDFinsupp a
-- sigmaFinsuppAddEquivDFinsupp_symm_apply
@[simps]
def sigmaFinsuppAddEquivDFinsupp [AddZeroClass N] : ((Σi, η i) →₀ N) ≃+ Π₀ i, η i →₀ N 
Arguments: {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} [AddZeroClass N] (a : Π₀ (i : ι), η i →₀ N)
Goal: sigmaFinsuppAddEquivDFinsupp.symm a = sigmaFinsuppEquivDFinsupp.symm a
-- sigmaFinsuppEquivDFinsupp_smul
@[simp]
theorem sigmaFinsuppEquivDFinsupp_smul {R} [Monoid R] [AddMonoid N] [DistribMulAction R N] (r : R)
    (f : (Σ i, η i) →₀ N) :
    sigmaFinsuppEquivDFinsupp (r • f) = r • sigmaFinsuppEquivDFinsupp f 
Arguments: {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} {R : Type u_6} [Monoid R] [AddMonoid N] [DistribMulAction R N] (r : R) (f : (i : ι) × η i →₀ N)
Goal: sigmaFinsuppEquivDFinsupp (r • f) = r • sigmaFinsuppEquivDFinsupp f
-- sigmaFinsuppLequivDFinsupp_symm_apply
@[simps]
def sigmaFinsuppLequivDFinsupp [AddCommMonoid N] [Module R N] :
    ((Σi, η i) →₀ N) ≃ₗ[R] Π₀ i, η i →₀ N 
Arguments: {ι : Type u_1} (R : Type u_2) {η : ι → Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] [Module R N] (a : Π₀ (i : ι), η i →₀ N)
Goal: (sigmaFinsuppLequivDFinsupp R).symm a = sigmaFinsuppEquivDFinsupp.symm a
-- sigmaFinsuppLequivDFinsupp_apply
@[simps]
def sigmaFinsuppLequivDFinsupp [AddCommMonoid N] [Module R N] :
    ((Σi, η i) →₀ N) ≃ₗ[R] Π₀ i, η i →₀ N 
Arguments: {ι : Type u_1} (R : Type u_2) {η : ι → Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] [Module R N] (a : (i : ι) × η i →₀ N)
Goal: (sigmaFinsuppLequivDFinsupp R) a = sigmaFinsuppEquivDFinsupp a
-- Finsupp.le_def
lemma le_def : f ≤ g ↔ ∀ i, f i ≤ g i 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [LE α] {f g : ι →₀ α}
Goal: f ≤ g ↔ ∀ (i : ι), f i ≤ g i
-- Finsupp.coe_le_coe
@[simp, norm_cast] lemma coe_le_coe : ⇑f ≤ g ↔ f ≤ g 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [LE α] {f g : ι →₀ α}
Goal: ⇑f ≤ ⇑g ↔ f ≤ g
-- Finsupp.orderEmbeddingToFun_apply
@[simp]
theorem orderEmbeddingToFun_apply {f : ι →₀ α} {i : ι} : orderEmbeddingToFun f i = f i 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [LE α] {f : ι →₀ α} {i : ι}
Goal: Finsupp.orderEmbeddingToFun f i = f i
-- Finsupp.lt_def
lemma lt_def : f < g ↔ f ≤ g ∧ ∃ i, f i < g i 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [Preorder α] {f g : ι →₀ α}
Goal: f < g ↔ f ≤ g ∧ ∃ i, f i < g i
-- Finsupp.coe_lt_coe
@[simp, norm_cast] lemma coe_lt_coe : ⇑f < g ↔ f < g 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [Preorder α] {f g : ι →₀ α}
Goal: ⇑f < ⇑g ↔ f < g
-- Finsupp.coe_mono
lemma coe_mono : Monotone (Finsupp.toFun : (ι →₀ α) → ι → α) 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [Preorder α]
Goal: Monotone Finsupp.toFun
-- Finsupp.coe_strictMono
lemma coe_strictMono : Monotone (Finsupp.toFun : (ι →₀ α) → ι → α) 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [Preorder α]
Goal: Monotone Finsupp.toFun
-- Finsupp.inf_apply
@[simp]
theorem inf_apply [SemilatticeInf α] {i : ι} {f g : ι →₀ α} : (f ⊓ g) i = f i ⊓ g i 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [SemilatticeInf α] {i : ι} {f g : ι →₀ α}
Goal: (f ⊓ g) i = f i ⊓ g i
-- Finsupp.sup_apply
@[simp]
theorem sup_apply [SemilatticeSup α] {i : ι} {f g : ι →₀ α} : (f ⊔ g) i = f i ⊔ g i 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [SemilatticeSup α] {i : ι} {f g : ι →₀ α}
Goal: (f ⊔ g) i = f i ⊔ g i
-- Finsupp.support_inf_union_support_sup
theorem support_inf_union_support_sup : (f ⊓ g).support ∪ (f ⊔ g).support = f.support ∪ g.support 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [DecidableEq ι] [Lattice α] (f g : ι →₀ α)
Goal: (f ⊓ g).support ∪ (f ⊔ g).support = f.support ∪ g.support
-- Finsupp.support_sup_union_support_inf
theorem support_sup_union_support_inf : (f ⊔ g).support ∪ (f ⊓ g).support = f.support ∪ g.support 
Arguments: {ι : Type u_1} {α : Type u_2} [Zero α] [DecidableEq ι] [Lattice α] (f g : ι →₀ α)
Goal: (f ⊔ g).support ∪ (f ⊓ g).support = f.support ∪ g.support
-- Finsupp.bot_eq_zero
protected theorem bot_eq_zero : (⊥ : ι →₀ α) = 0 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α]
Goal: ⊥ = 0
-- Finsupp.add_eq_zero_iff
@[simp]
theorem add_eq_zero_iff (f g : ι →₀ α) : f + g = 0 ↔ f = 0 ∧ g = 0 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] (f g : ι →₀ α)
Goal: f + g = 0 ↔ f = 0 ∧ g = 0
-- Finsupp.le_iff'
theorem le_iff' (f g : ι →₀ α) {s : Finset ι} (hf : f.support ⊆ s) : f ≤ g ↔ ∀ i ∈ s, f i ≤ g i 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] (f g : ι →₀ α) {s : Finset ι} (hf : f.support ⊆ s)
Goal: f ≤ g ↔ ∀ i ∈ s, f i ≤ g i
-- Finsupp.le_iff
theorem le_iff (f g : ι →₀ α) : f ≤ g ↔ ∀ i ∈ f.support, f i ≤ g i 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] (f g : ι →₀ α)
Goal: f ≤ g ↔ ∀ i ∈ f.support, f i ≤ g i
-- Finsupp.support_monotone
lemma support_monotone : Monotone (support (α := ι) (M := α)) 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α]
Goal: Monotone Finsupp.support
-- Finsupp.support_mono
lemma support_mono (hfg : f ≤ g) : f.support ⊆ g.support 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] {f g : ι →₀ α} (hfg : f ≤ g)
Goal: f.support ⊆ g.support
-- Finsupp.single_le_iff
@[simp]
theorem single_le_iff {i : ι} {x : α} {f : ι →₀ α} : single i x ≤ f ↔ x ≤ f i 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] {i : ι} {x : α} {f : ι →₀ α}
Goal: Finsupp.single i x ≤ f ↔ x ≤ f i
-- Finsupp.coe_tsub
@[simp, norm_cast] lemma coe_tsub (f g : ι →₀ α) : ⇑(f - g) = f - g 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] (f g : ι →₀ α)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- Finsupp.tsub_apply
theorem tsub_apply (f g : ι →₀ α) (a : ι) : (f - g) a = f a - g a 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] (f g : ι →₀ α) (a : ι)
Goal: (f - g) a = f a - g a
-- Finsupp.single_tsub
@[simp]
theorem single_tsub : single i (a - b) = single i a - single i b 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] {i : ι} {a b : α}
Goal: Finsupp.single i (a - b) = Finsupp.single i a - Finsupp.single i b
-- Finsupp.support_tsub
theorem support_tsub {f1 f2 : ι →₀ α} : (f1 - f2).support ⊆ f1.support 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] {f1 f2 : ι →₀ α}
Goal: (f1 - f2).support ⊆ f1.support
-- Finsupp.subset_support_tsub
theorem subset_support_tsub [DecidableEq ι] {f1 f2 : ι →₀ α} :
    f1.support \ f2.support ⊆ (f1 - f2).support 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] [DecidableEq ι] {f1 f2 : ι →₀ α}
Goal: f1.support \ f2.support ⊆ (f1 - f2).support
-- Finsupp.support_inf
@[simp]
theorem support_inf [DecidableEq ι] (f g : ι →₀ α) : (f ⊓ g).support = f.support ∩ g.support 
Arguments: {ι : Type u_1} {α : Type u_2} [CanonicallyLinearOrderedAddCommMonoid α] [DecidableEq ι] (f g : ι →₀ α)
Goal: (f ⊓ g).support = f.support ∩ g.support