ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- matrixEquivTensor_apply
@[simp]
theorem matrixEquivTensor_apply (M : Matrix n n A) :
    matrixEquivTensor R A n M = ∑ p : n × n, M p.1 p.2 ⊗ₜ stdBasisMatrix p.1 p.2 1 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [Fintype n] [DecidableEq n] (M : Matrix n n A)
Goal: (matrixEquivTensor R A n) M = ∑ p : n × n, M p.1 p.2 ⊗ₜ[R] Matrix.stdBasisMatrix p.1 p.2 1
-- matrixEquivTensor_apply_std_basis
@[simp (high)]
theorem matrixEquivTensor_apply_std_basis (i j : n) (x : A) :
    matrixEquivTensor R A n (stdBasisMatrix i j x) = x ⊗ₜ stdBasisMatrix i j 1 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [Fintype n] [DecidableEq n] (i j : n) (x : A)
Goal: (matrixEquivTensor R A n) (Matrix.stdBasisMatrix i j x) = x ⊗ₜ[R] Matrix.stdBasisMatrix i j 1
-- matrixEquivTensor_apply_symm
@[simp]
theorem matrixEquivTensor_apply_symm (a : A) (M : Matrix n n R) :
    (matrixEquivTensor R A n).symm (a ⊗ₜ M) = M.map fun x => a * algebraMap R A x 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [Fintype n] [DecidableEq n] (a : A) (M : Matrix n n R)
Goal: (matrixEquivTensor R A n).symm (a ⊗ₜ[R] M) = M.map fun x => a * (algebraMap R A) x
-- Coalgebra.lTensor_counit_comp_comul
class Coalgebra (R : Type u) (A : Type v)
    [CommSemiring R] [AddCommMonoid A] [Module R A] extends CoalgebraStruct R A where
    coassoc : TensorProduct.assoc R A A A ∘ₗ comul.rTensor A ∘ₗ comul = comul.lTensor A ∘ₗ comul
    rTensor_counit_comp_comul : counit.rTensor A ∘ₗ comul = TensorProduct.mk R _ _ 1
    lTensor_counit_comp_comul : counit.lTensor A ∘ₗ comul = (TensorProduct.mk R _ _).flip 1
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [self : Coalgebra R A]
Goal: LinearMap.lTensor A Coalgebra.counit ∘ₗ Coalgebra.comul = (TensorProduct.mk R A R).flip 1
-- Coalgebra.rTensor_counit_comp_comul
class Coalgebra (R : Type u) (A : Type v)
    [CommSemiring R] [AddCommMonoid A] [Module R A] extends CoalgebraStruct R A where
    coassoc : TensorProduct.assoc R A A A ∘ₗ comul.rTensor A ∘ₗ comul = comul.lTensor A ∘ₗ comul
    rTensor_counit_comp_comul : counit.rTensor A ∘ₗ comul = TensorProduct.mk R _ _ 1
    lTensor_counit_comp_comul : counit.lTensor A ∘ₗ comul = (TensorProduct.mk R _ _).flip 1
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [self : Coalgebra R A]
Goal: LinearMap.rTensor A Coalgebra.counit ∘ₗ Coalgebra.comul = (TensorProduct.mk R R A) 1
-- Coalgebra.coassoc
class Coalgebra (R : Type u) (A : Type v)
    [CommSemiring R] [AddCommMonoid A] [Module R A] extends CoalgebraStruct R A where
    coassoc : TensorProduct.assoc R A A A ∘ₗ comul.rTensor A ∘ₗ comul = comul.lTensor A ∘ₗ comul
    rTensor_counit_comp_comul : counit.rTensor A ∘ₗ comul = TensorProduct.mk R _ _ 1
    lTensor_counit_comp_comul : counit.lTensor A ∘ₗ comul = (TensorProduct.mk R _ _).flip 1
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [self : Coalgebra R A]
Goal: ↑(TensorProduct.assoc R A A A) ∘ₗ LinearMap.rTensor A Coalgebra.comul ∘ₗ Coalgebra.comul =
  LinearMap.lTensor A Coalgebra.comul ∘ₗ Coalgebra.comul
-- Coalgebra.coassoc_apply
@[simp]
theorem coassoc_apply (a : A) :
    TensorProduct.assoc R A A A (comul.rTensor A (comul a)) = comul.lTensor A (comul a) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A)
Goal: (TensorProduct.assoc R A A A) ((LinearMap.rTensor A Coalgebra.comul) (Coalgebra.comul a)) =
  (LinearMap.lTensor A Coalgebra.comul) (Coalgebra.comul a)
-- Coalgebra.coassoc_symm_apply
@[simp]
theorem coassoc_symm_apply (a : A) :
    (TensorProduct.assoc R A A A).symm (comul.lTensor A (comul a)) = comul.rTensor A (comul a) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A)
Goal: (TensorProduct.assoc R A A A).symm ((LinearMap.lTensor A Coalgebra.comul) (Coalgebra.comul a)) =
  (LinearMap.rTensor A Coalgebra.comul) (Coalgebra.comul a)
-- Coalgebra.coassoc_symm
@[simp]
theorem coassoc_symm :
    (TensorProduct.assoc R A A A).symm ∘ₗ comul.lTensor A ∘ₗ comul =
    comul.rTensor A ∘ₗ (comul (R := R)) 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A]
Goal: ↑(TensorProduct.assoc R A A A).symm ∘ₗ LinearMap.lTensor A Coalgebra.comul ∘ₗ Coalgebra.comul =
  LinearMap.rTensor A Coalgebra.comul ∘ₗ Coalgebra.comul
-- Coalgebra.rTensor_counit_comul
@[simp]
theorem rTensor_counit_comul (a : A) : counit.rTensor A (comul a) = 1 ⊗ₜ[R] a 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A)
Goal: (LinearMap.rTensor A Coalgebra.counit) (Coalgebra.comul a) = 1 ⊗ₜ[R] a
-- Coalgebra.lTensor_counit_comul
@[simp]
theorem lTensor_counit_comul (a : A) : counit.lTensor A (comul a) = a ⊗ₜ[R] 1 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A)
Goal: (LinearMap.lTensor A Coalgebra.counit) (Coalgebra.comul a) = a ⊗ₜ[R] 1
-- CommSemiring.comul_apply
@[simp]
theorem comul_apply (r : R) : comul r = 1 ⊗ₜ[R] r 
Arguments: (R : Type u) [CommSemiring R] (r : R)
Goal: Coalgebra.comul r = 1 ⊗ₜ[R] r
-- CommSemiring.counit_apply
@[simp]
theorem counit_apply (r : R) : counit r = r 
Arguments: (R : Type u) [CommSemiring R] (r : R)
Goal: Coalgebra.counit r = r
-- Prod.comul_apply
@[simp]
theorem comul_apply (r : A × B) :
    comul r =
      TensorProduct.map (.inl R A B) (.inl R A B) (comul r.1) +
      TensorProduct.map (.inr R A B) (.inr R A B) (comul r.2) 
Arguments: (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] (r : A × B)
Goal: Coalgebra.comul r =
  (TensorProduct.map (LinearMap.inl R A B) (LinearMap.inl R A B)) (Coalgebra.comul r.1) +
    (TensorProduct.map (LinearMap.inr R A B) (LinearMap.inr R A B)) (Coalgebra.comul r.2)
-- Prod.counit_apply
@[simp]
theorem counit_apply (r : A × B) : (counit r : R) = counit r.1 + counit r.2 
Arguments: (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] (r : A × B)
Goal: Coalgebra.counit r = Coalgebra.counit r.1 + Coalgebra.counit r.2
-- Prod.comul_comp_inl
theorem comul_comp_inl :
    comul ∘ₗ inl R A B = TensorProduct.map (.inl R A B) (.inl R A B) ∘ₗ comul 
Arguments: (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B]
Goal: Coalgebra.comul ∘ₗ LinearMap.inl R A B =
  TensorProduct.map (LinearMap.inl R A B) (LinearMap.inl R A B) ∘ₗ Coalgebra.comul
-- Prod.comul_comp_inr
theorem comul_comp_inr :
    comul ∘ₗ inr R A B = TensorProduct.map (.inr R A B) (.inr R A B) ∘ₗ comul 
Arguments: (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B]
Goal: Coalgebra.comul ∘ₗ LinearMap.inr R A B =
  TensorProduct.map (LinearMap.inr R A B) (LinearMap.inr R A B) ∘ₗ Coalgebra.comul
-- Prod.comul_comp_fst
theorem comul_comp_fst :
    comul ∘ₗ .fst R A B = TensorProduct.map (.fst R A B) (.fst R A B) ∘ₗ comul 
Arguments: (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B]
Goal: Coalgebra.comul ∘ₗ LinearMap.fst R A B =
  TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ Coalgebra.comul
-- Prod.comul_comp_snd
theorem comul_comp_snd :
    comul ∘ₗ .snd R A B = TensorProduct.map (.snd R A B) (.snd R A B) ∘ₗ comul 
Arguments: (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B]
Goal: Coalgebra.comul ∘ₗ LinearMap.snd R A B =
  TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ Coalgebra.comul
-- Prod.counit_comp_inr
@[simp] theorem counit_comp_inr : counit ∘ₗ inr R A B = counit 
Arguments: (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B]
Goal: Coalgebra.counit ∘ₗ LinearMap.inr R A B = Coalgebra.counit
-- Prod.counit_comp_inl
@[simp] theorem counit_comp_inl : counit ∘ₗ inl R A B = counit 
Arguments: (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B]
Goal: Coalgebra.counit ∘ₗ LinearMap.inl R A B = Coalgebra.counit
-- Finsupp.comul_single
@[simp]
theorem comul_single (i : ι) (a : A) :
    comul (R := R) (Finsupp.single i a) =
      (TensorProduct.map (Finsupp.lsingle i) (Finsupp.lsingle i) : _ →ₗ[R] _) (comul a) 
Arguments: (R : Type u) (ι : Type v) (A : Type w) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (i : ι) (a : A)
Goal: Coalgebra.comul (Finsupp.single i a) = (TensorProduct.map (Finsupp.lsingle i) (Finsupp.lsingle i)) (Coalgebra.comul a)
-- Finsupp.counit_single
@[simp]
theorem counit_single (i : ι) (a : A) : counit (Finsupp.single i a) = counit (R := R) a 
Arguments: (R : Type u) (ι : Type v) (A : Type w) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (i : ι) (a : A)
Goal: Coalgebra.counit (Finsupp.single i a) = Coalgebra.counit a
-- Finsupp.comul_comp_lsingle
theorem comul_comp_lsingle (i : ι) :
    comul ∘ₗ (lsingle i : A →ₗ[R] _) = TensorProduct.map (lsingle i) (lsingle i) ∘ₗ comul 
Arguments: (R : Type u) (ι : Type v) (A : Type w) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (i : ι)
Goal: Coalgebra.comul ∘ₗ Finsupp.lsingle i = TensorProduct.map (Finsupp.lsingle i) (Finsupp.lsingle i) ∘ₗ Coalgebra.comul
-- Finsupp.comul_comp_lapply
theorem comul_comp_lapply (i : ι) :
    comul ∘ₗ (lapply i : _ →ₗ[R] A) = TensorProduct.map (lapply i) (lapply i) ∘ₗ comul 
Arguments: (R : Type u) (ι : Type v) (A : Type w) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (i : ι)
Goal: Coalgebra.comul ∘ₗ Finsupp.lapply i = TensorProduct.map (Finsupp.lapply i) (Finsupp.lapply i) ∘ₗ Coalgebra.comul
-- Finsupp.counit_comp_lsingle
@[simp] theorem counit_comp_lsingle (i : ι) : counit ∘ₗ (lsingle i : A →ₗ[R] _) = counit 
Arguments: (R : Type u) (ι : Type v) (A : Type w) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (i : ι)
Goal: Coalgebra.counit ∘ₗ Finsupp.lsingle i = Coalgebra.counit
-- TensorProduct.instCoalgebraStruct_comul
@[simps] instance instCoalgebraStruct :
    CoalgebraStruct R (A ⊗[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Coalgebra.comul = ↑(TensorProduct.tensorTensorTensorComm R A A B B) ∘ₗ TensorProduct.map Coalgebra.comul Coalgebra.comul
-- TensorProduct.instCoalgebraStruct_counit
@[simps] instance instCoalgebraStruct :
    CoalgebraStruct R (A ⊗[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Coalgebra.counit = LinearMap.mul' R R ∘ₗ TensorProduct.map Coalgebra.counit Coalgebra.counit
-- HahnSeries.isPWO_support'
@[ext]
structure HahnSeries (Γ : Type*) (R : Type*) [PartialOrder Γ] [Zero R] where
    coeff : Γ → R
  isPWO_support' : (Function.support coeff).IsPWO
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (self : HahnSeries Γ R)
Goal: (Function.support self.coeff).IsPWO
-- HahnSeries.coeff_injective
theorem coeff_injective : Injective (coeff : HahnSeries Γ R → Γ → R) 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R]
Goal: Function.Injective HahnSeries.coeff
-- HahnSeries.coeff_inj
@[simp]
theorem coeff_inj {x y : HahnSeries Γ R} : x.coeff = y.coeff ↔ x = y 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x y : HahnSeries Γ R}
Goal: x.coeff = y.coeff ↔ x = y
-- HahnSeries.isPWO_support
@[simp]
theorem isPWO_support (x : HahnSeries Γ R) : x.support.IsPWO 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (x : HahnSeries Γ R)
Goal: x.support.IsPWO
-- HahnSeries.isWF_support
@[simp]
theorem isWF_support (x : HahnSeries Γ R) : x.support.IsWF 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (x : HahnSeries Γ R)
Goal: x.support.IsWF
-- HahnSeries.mem_support
@[simp]
theorem mem_support (x : HahnSeries Γ R) (a : Γ) : a ∈ x.support ↔ x.coeff a ≠ 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (x : HahnSeries Γ R) (a : Γ)
Goal: a ∈ x.support ↔ x.coeff a ≠ 0
-- HahnSeries.zero_coeff
@[simp]
theorem zero_coeff {a : Γ} : (0 : HahnSeries Γ R).coeff a = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ}
Goal: HahnSeries.coeff 0 a = 0
-- HahnSeries.coeff_fun_eq_zero_iff
@[simp]
theorem coeff_fun_eq_zero_iff {x : HahnSeries Γ R} : x.coeff = 0 ↔ x = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R}
Goal: x.coeff = 0 ↔ x = 0
-- HahnSeries.ne_zero_of_coeff_ne_zero
theorem ne_zero_of_coeff_ne_zero {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g ≠ 0) : x ≠ 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g ≠ 0)
Goal: x ≠ 0
-- HahnSeries.support_zero
@[simp]
theorem support_zero : support (0 : HahnSeries Γ R) = ∅ 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R]
Goal: HahnSeries.support 0 = ∅
-- HahnSeries.support_nonempty_iff
@[simp]
nonrec theorem support_nonempty_iff {x : HahnSeries Γ R} : x.support.Nonempty ↔ x ≠ 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R}
Goal: x.support.Nonempty ↔ x ≠ 0
-- HahnSeries.support_eq_empty_iff
@[simp]
theorem support_eq_empty_iff {x : HahnSeries Γ R} : x.support = ∅ ↔ x = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R}
Goal: x.support = ∅ ↔ x = 0
-- HahnSeries.mk_eq_zero
@[simp]
lemma mk_eq_zero (f : Γ → R) (h) : HahnSeries.mk f h = 0 ↔ f = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (f : Γ → R) (h : (Function.support f).IsPWO)
Goal: { coeff := f, isPWO_support' := h } = 0 ↔ f = 0
-- HahnSeries.iterateEquiv_symm_apply
@[simps]
def iterateEquiv {Γ' : Type*} [PartialOrder Γ'] :
    HahnSeries Γ (HahnSeries Γ' R) ≃ HahnSeries (Γ ×ₗ Γ') R 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] (x : HahnSeries (Lex (Γ × Γ')) R)
Goal: HahnSeries.iterateEquiv.symm x = x.toIterate
-- HahnSeries.iterateEquiv_apply
@[simps]
def iterateEquiv {Γ' : Type*} [PartialOrder Γ'] :
    HahnSeries Γ (HahnSeries Γ' R) ≃ HahnSeries (Γ ×ₗ Γ') R 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] (x : HahnSeries Γ (HahnSeries Γ' R))
Goal: HahnSeries.iterateEquiv x = x.ofIterate
-- HahnSeries.single_coeff_same
@[simp]
theorem single_coeff_same (a : Γ) (r : R) : (single a r).coeff a = r 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (a : Γ) (r : R)
Goal: ((HahnSeries.single a) r).coeff a = r
-- HahnSeries.single_coeff_of_ne
@[simp]
theorem single_coeff_of_ne (h : b ≠ a) : (single a r).coeff b = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a b : Γ} {r : R} (h : b ≠ a)
Goal: ((HahnSeries.single a) r).coeff b = 0
-- HahnSeries.single_coeff
theorem single_coeff : (single a r).coeff b = if b = a then r else 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a b : Γ} {r : R}
Goal: ((HahnSeries.single a) r).coeff b = if b = a then r else 0
-- HahnSeries.support_single_of_ne
@[simp]
theorem support_single_of_ne (h : r ≠ 0) : support (single a r) = {a} 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} (h : r ≠ 0)
Goal: ((HahnSeries.single a) r).support = {a}
-- HahnSeries.support_single_subset
theorem support_single_subset : support (single a r) ⊆ {a} 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R}
Goal: ((HahnSeries.single a) r).support ⊆ {a}
-- HahnSeries.eq_of_mem_support_single
theorem eq_of_mem_support_single {b : Γ} (h : b ∈ support (single a r)) : b = a 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} {b : Γ} (h : b ∈ ((HahnSeries.single a) r).support)
Goal: b = a
-- HahnSeries.single_eq_zero
theorem single_eq_zero : single a (0 : R) = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ}
Goal: (HahnSeries.single a) 0 = 0
-- HahnSeries.single_injective
theorem single_injective (a : Γ) : Function.Injective (single a : R → HahnSeries Γ R) 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (a : Γ)
Goal: Function.Injective ⇑(HahnSeries.single a)
-- HahnSeries.single_ne_zero
theorem single_ne_zero (h : r ≠ 0) : single a r ≠ 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} (h : r ≠ 0)
Goal: (HahnSeries.single a) r ≠ 0
-- HahnSeries.single_eq_zero_iff
@[simp]
theorem single_eq_zero_iff {a : Γ} {r : R} : single a r = 0 ↔ r = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R}
Goal: (HahnSeries.single a) r = 0 ↔ r = 0
-- HahnSeries.orderTop_zero
@[simp]
theorem orderTop_zero : orderTop (0 : HahnSeries Γ R) = ⊤ 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R]
Goal: HahnSeries.orderTop 0 = ⊤
-- HahnSeries.orderTop_of_ne
theorem orderTop_of_ne {x : HahnSeries Γ R} (hx : x ≠ 0) :
    orderTop x = x.isWF_support.min (support_nonempty_iff.2 hx) 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} (hx : x ≠ 0)
Goal: x.orderTop = ↑(⋯.min ⋯)
-- HahnSeries.ne_zero_iff_orderTop
@[simp]
theorem ne_zero_iff_orderTop {x : HahnSeries Γ R} : x ≠ 0 ↔ orderTop x ≠ ⊤ 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R}
Goal: x ≠ 0 ↔ x.orderTop ≠ ⊤
-- HahnSeries.orderTop_eq_top_iff
theorem orderTop_eq_top_iff {x : HahnSeries Γ R} : orderTop x = ⊤ ↔ x = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R}
Goal: x.orderTop = ⊤ ↔ x = 0
-- HahnSeries.untop_orderTop_of_ne_zero
theorem untop_orderTop_of_ne_zero {x : HahnSeries Γ R} (hx : x ≠ 0) :
    WithTop.untop x.orderTop (ne_zero_iff_orderTop.mp hx) =
      x.isWF_support.min (support_nonempty_iff.2 hx) 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} (hx : x ≠ 0)
Goal: x.orderTop.untop ⋯ = ⋯.min ⋯
-- HahnSeries.coeff_orderTop_ne
theorem coeff_orderTop_ne {x : HahnSeries Γ R} {g : Γ} (hg : x.orderTop = g) :
    x.coeff g ≠ 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} {g : Γ} (hg : x.orderTop = ↑g)
Goal: x.coeff g ≠ 0
-- HahnSeries.orderTop_le_of_coeff_ne_zero
theorem orderTop_le_of_coeff_ne_zero {Γ} [LinearOrder Γ] {x : HahnSeries Γ R}
    {g : Γ} (h : x.coeff g ≠ 0) : x.orderTop ≤ g 
Arguments: {R : Type u_2} [Zero R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g ≠ 0)
Goal: x.orderTop ≤ ↑g
-- HahnSeries.orderTop_single
@[simp]
theorem orderTop_single (h : r ≠ 0) : (single a r).orderTop = a 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} (h : r ≠ 0)
Goal: ((HahnSeries.single a) r).orderTop = ↑a
-- HahnSeries.orderTop_single_le
theorem orderTop_single_le : a ≤ (single a r).orderTop 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R}
Goal: ↑a ≤ ((HahnSeries.single a) r).orderTop
-- HahnSeries.lt_orderTop_single
theorem lt_orderTop_single {g g' : Γ} (hgg' : g < g') : g < (single g' r).orderTop 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {r : R} {g g' : Γ} (hgg' : g < g')
Goal: ↑g < ((HahnSeries.single g') r).orderTop
-- HahnSeries.coeff_eq_zero_of_lt_orderTop
theorem coeff_eq_zero_of_lt_orderTop {x : HahnSeries Γ R} {i : Γ} (hi : i < x.orderTop) :
    x.coeff i = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} {i : Γ} (hi : ↑i < x.orderTop)
Goal: x.coeff i = 0
-- HahnSeries.leadingCoeff_zero
@[simp]
theorem leadingCoeff_zero : leadingCoeff (0 : HahnSeries Γ R) = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R]
Goal: HahnSeries.leadingCoeff 0 = 0
-- HahnSeries.leadingCoeff_of_ne
theorem leadingCoeff_of_ne {x : HahnSeries Γ R} (hx : x ≠ 0) :
    x.leadingCoeff = x.coeff (x.isWF_support.min (support_nonempty_iff.2 hx)) 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} (hx : x ≠ 0)
Goal: x.leadingCoeff = x.coeff (⋯.min ⋯)
-- HahnSeries.leadingCoeff_eq_iff
theorem leadingCoeff_eq_iff {x : HahnSeries Γ R} : x.leadingCoeff = 0 ↔ x = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R}
Goal: x.leadingCoeff = 0 ↔ x = 0
-- HahnSeries.leadingCoeff_ne_iff
theorem leadingCoeff_ne_iff {x : HahnSeries Γ R} : x.leadingCoeff ≠ 0 ↔ x ≠ 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R}
Goal: x.leadingCoeff ≠ 0 ↔ x ≠ 0
-- HahnSeries.leadingCoeff_of_single
theorem leadingCoeff_of_single {a : Γ} {r : R} : leadingCoeff (single a r) = r 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R}
Goal: ((HahnSeries.single a) r).leadingCoeff = r
-- HahnSeries.order_zero
@[simp]
theorem order_zero : order (0 : HahnSeries Γ R) = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ]
Goal: HahnSeries.order 0 = 0
-- HahnSeries.order_of_ne
theorem order_of_ne {x : HahnSeries Γ R} (hx : x ≠ 0) :
    order x = x.isWF_support.min (support_nonempty_iff.2 hx) 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} (hx : x ≠ 0)
Goal: x.order = ⋯.min ⋯
-- HahnSeries.order_eq_orderTop_of_ne
theorem order_eq_orderTop_of_ne {x : HahnSeries Γ R} (hx : x ≠ 0) : order x = orderTop x 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} (hx : x ≠ 0)
Goal: ↑x.order = x.orderTop
-- HahnSeries.coeff_order_ne_zero
theorem coeff_order_ne_zero {x : HahnSeries Γ R} (hx : x ≠ 0) : x.coeff x.order ≠ 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} (hx : x ≠ 0)
Goal: x.coeff x.order ≠ 0
-- HahnSeries.order_le_of_coeff_ne_zero
theorem order_le_of_coeff_ne_zero {Γ} [LinearOrderedCancelAddCommMonoid Γ] {x : HahnSeries Γ R}
    {g : Γ} (h : x.coeff g ≠ 0) : x.order ≤ g 
Arguments: {R : Type u_2} [Zero R] {Γ : Type u_3} [LinearOrderedCancelAddCommMonoid Γ] {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g ≠ 0)
Goal: x.order ≤ g
-- HahnSeries.order_single
@[simp]
theorem order_single (h : r ≠ 0) : (single a r).order = a 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} [Zero Γ] (h : r ≠ 0)
Goal: ((HahnSeries.single a) r).order = a
-- HahnSeries.coeff_eq_zero_of_lt_order
theorem coeff_eq_zero_of_lt_order {x : HahnSeries Γ R} {i : Γ} (hi : i < x.order) :
    x.coeff i = 0 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} {i : Γ} (hi : i < x.order)
Goal: x.coeff i = 0
-- HahnSeries.zero_lt_orderTop_iff
theorem zero_lt_orderTop_iff {x : HahnSeries Γ R} (hx : x ≠ 0) :
    0 < x.orderTop ↔ 0 < x.order 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} (hx : x ≠ 0)
Goal: 0 < x.orderTop ↔ 0 < x.order
-- HahnSeries.zero_lt_orderTop_of_order
theorem zero_lt_orderTop_of_order {x : HahnSeries Γ R} (hx : 0 < x.order) : 0 < x.orderTop 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} (hx : 0 < x.order)
Goal: 0 < x.orderTop
-- HahnSeries.zero_le_orderTop_iff
theorem zero_le_orderTop_iff {x : HahnSeries Γ R} : 0 ≤ x.orderTop ↔ 0 ≤ x.order 
Arguments: {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R}
Goal: 0 ≤ x.orderTop ↔ 0 ≤ x.order