ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HahnModule.single_smul_coeff_add
theorem single_smul_coeff_add [MulZeroClass R] [SMulWithZero R V] {r : R} {x : HahnModule Γ' R V}
    {a : Γ'} {b : Γ} :
    ((of R).symm (HahnSeries.single b r • x)).coeff (b +ᵥ a) = r • ((of R).symm x).coeff a 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [AddCommMonoid V] [MulZeroClass R] [SMulWithZero R V] {r : R} {x : HahnModule Γ' R V} {a : Γ'} {b : Γ}
Goal: ((HahnModule.of R).symm ((HahnSeries.single b) r • x)).coeff (b +ᵥ a) = r • ((HahnModule.of R).symm x).coeff a
-- HahnModule.single_zero_smul_coeff
theorem single_zero_smul_coeff {Γ} [OrderedAddCommMonoid Γ] [AddAction Γ Γ']
    [IsOrderedCancelVAdd Γ Γ'] [MulZeroClass R] [SMulWithZero R V] {r : R}
    {x : HahnModule Γ' R V} {a : Γ'} :
    ((of R).symm ((HahnSeries.single 0 r : HahnSeries Γ R) • x)).coeff a =
    r • ((of R).symm x).coeff a 
Arguments: {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ'] [AddCommMonoid V] {Γ : Type u_5} [OrderedAddCommMonoid Γ] [AddAction Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [MulZeroClass R] [SMulWithZero R V] {r : R} {x : HahnModule Γ' R V} {a : Γ'}
Goal: ((HahnModule.of R).symm ((HahnSeries.single 0) r • x)).coeff a = r • ((HahnModule.of R).symm x).coeff a
-- HahnModule.single_zero_smul_eq_smul
@[simp]
theorem single_zero_smul_eq_smul (Γ) [OrderedAddCommMonoid Γ] [AddAction Γ Γ']
    [IsOrderedCancelVAdd Γ Γ'] [MulZeroClass R] [SMulWithZero R V] {r : R}
    {x : HahnModule Γ' R V} :
    (HahnSeries.single (0 : Γ) r) • x = r • x 
Arguments: {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ'] [AddCommMonoid V] (Γ : Type u_5) [OrderedAddCommMonoid Γ] [AddAction Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [MulZeroClass R] [SMulWithZero R V] {r : R} {x : HahnModule Γ' R V}
Goal: (HahnSeries.single 0) r • x = r • x
-- HahnModule.zero_smul'
@[simp]
theorem zero_smul' [Zero R] [SMulWithZero R V] {x : HahnModule Γ' R V} :
    (0 : HahnSeries Γ R) • x = 0 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [AddCommMonoid V] [Zero R] [SMulWithZero R V] {x : HahnModule Γ' R V}
Goal: 0 • x = 0
-- HahnModule.one_smul'
@[simp]
theorem one_smul' {Γ} [OrderedAddCommMonoid Γ] [AddAction Γ Γ'] [IsOrderedCancelVAdd Γ Γ']
    [MonoidWithZero R] [MulActionWithZero R V] {x : HahnModule Γ' R V} :
    (1 : HahnSeries Γ R) • x = x 
Arguments: {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ'] [AddCommMonoid V] {Γ : Type u_5} [OrderedAddCommMonoid Γ] [AddAction Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [MonoidWithZero R] [MulActionWithZero R V] {x : HahnModule Γ' R V}
Goal: 1 • x = x
-- HahnModule.support_smul_subset_vadd_support'
theorem support_smul_subset_vadd_support' [MulZeroClass R] [SMulWithZero R V] {x : HahnSeries Γ R}
    {y : HahnModule Γ' R V} :
    ((of R).symm (x • y)).support ⊆ x.support +ᵥ ((of R).symm y).support 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [AddCommMonoid V] [MulZeroClass R] [SMulWithZero R V] {x : HahnSeries Γ R} {y : HahnModule Γ' R V}
Goal: ((HahnModule.of R).symm (x • y)).support ⊆ x.support +ᵥ ((HahnModule.of R).symm y).support
-- HahnModule.support_smul_subset_vadd_support
theorem support_smul_subset_vadd_support [MulZeroClass R] [SMulWithZero R V] {x : HahnSeries Γ R}
    {y : HahnModule Γ' R V} :
    ((of R).symm (x • y)).support ⊆ x.support +ᵥ ((of R).symm y).support 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [PartialOrder Γ] [PartialOrder Γ'] [VAdd Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [AddCommMonoid V] [MulZeroClass R] [SMulWithZero R V] {x : HahnSeries Γ R} {y : HahnModule Γ' R V}
Goal: ((HahnModule.of R).symm (x • y)).support ⊆ x.support +ᵥ ((HahnModule.of R).symm y).support
-- HahnModule.smul_coeff_order_add_order
theorem smul_coeff_order_add_order {Γ} [LinearOrderedCancelAddCommMonoid Γ] [Zero R]
    [SMulWithZero R V] (x : HahnSeries Γ R) (y : HahnModule Γ R V) :
    ((of R).symm (x • y)).coeff (x.order + ((of R).symm y).order) =
    x.leadingCoeff • ((of R).symm y).leadingCoeff 
Arguments: {R : Type u_3} {V : Type u_4} [AddCommMonoid V] {Γ : Type u_5} [LinearOrderedCancelAddCommMonoid Γ] [Zero R] [SMulWithZero R V] (x : HahnSeries Γ R) (y : HahnModule Γ R V)
Goal: ((HahnModule.of R).symm (x • y)).coeff (x.order + ((HahnModule.of R).symm y).order) =
  x.leadingCoeff • ((HahnModule.of R).symm y).leadingCoeff
-- HahnSeries.of_symm_smul_of_eq_mul
theorem of_symm_smul_of_eq_mul [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    (HahnModule.of R).symm (x • HahnModule.of R y) = x * y 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R}
Goal: (HahnModule.of R).symm (x • (HahnModule.of R) y) = x * y
-- HahnSeries.mul_coeff
theorem mul_coeff [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} {a : Γ} :
    (x * y).coeff a =
      ∑ ij ∈ addAntidiagonal x.isPWO_support y.isPWO_support a, x.coeff ij.fst * y.coeff ij.snd 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} {a : Γ}
Goal: (x * y).coeff a = ∑ ij ∈ Finset.addAntidiagonal ⋯ ⋯ a, x.coeff ij.1 * y.coeff ij.2
-- HahnSeries.mul_coeff_left'
theorem mul_coeff_left' [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} {a : Γ} {s : Set Γ}
    (hs : s.IsPWO) (hxs : x.support ⊆ s) :
    (x * y).coeff a =
      ∑ ij ∈ addAntidiagonal hs y.isPWO_support a, x.coeff ij.fst * y.coeff ij.snd 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} {a : Γ} {s : Set Γ} (hs : s.IsPWO) (hxs : x.support ⊆ s)
Goal: (x * y).coeff a = ∑ ij ∈ Finset.addAntidiagonal hs ⋯ a, x.coeff ij.1 * y.coeff ij.2
-- HahnSeries.mul_coeff_right'
theorem mul_coeff_right' [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} {a : Γ} {s : Set Γ}
    (hs : s.IsPWO) (hys : y.support ⊆ s) :
    (x * y).coeff a =
      ∑ ij ∈ addAntidiagonal x.isPWO_support hs a, x.coeff ij.fst * y.coeff ij.snd 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} {a : Γ} {s : Set Γ} (hs : s.IsPWO) (hys : y.support ⊆ s)
Goal: (x * y).coeff a = ∑ ij ∈ Finset.addAntidiagonal ⋯ hs a, x.coeff ij.1 * y.coeff ij.2
-- HahnSeries.single_mul_coeff_add
theorem single_mul_coeff_add [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ}
    {b : Γ} : (single b r * x).coeff (a + b) = r * x.coeff a 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a b : Γ}
Goal: ((HahnSeries.single b) r * x).coeff (a + b) = r * x.coeff a
-- HahnSeries.mul_single_coeff_add
theorem mul_single_coeff_add [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ}
    {b : Γ} : (x * single b r).coeff (a + b) = x.coeff a * r 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a b : Γ}
Goal: (x * (HahnSeries.single b) r).coeff (a + b) = x.coeff a * r
-- HahnSeries.mul_single_zero_coeff
@[simp]
theorem mul_single_zero_coeff [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ} :
    (x * single 0 r).coeff a = x.coeff a * r 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ}
Goal: (x * (HahnSeries.single 0) r).coeff a = x.coeff a * r
-- HahnSeries.single_zero_mul_coeff
theorem single_zero_mul_coeff [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ} :
    ((single 0 r : HahnSeries Γ R) * x).coeff a = r * x.coeff a 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ}
Goal: ((HahnSeries.single 0) r * x).coeff a = r * x.coeff a
-- HahnSeries.single_zero_mul_eq_smul
@[simp]
theorem single_zero_mul_eq_smul [Semiring R] {r : R} {x : HahnSeries Γ R} :
    single 0 r * x = r • x 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [Semiring R] {r : R} {x : HahnSeries Γ R}
Goal: (HahnSeries.single 0) r * x = r • x
-- HahnSeries.support_mul_subset_add_support
theorem support_mul_subset_add_support [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    support (x * y) ⊆ support x + support y 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R}
Goal: (x * y).support ⊆ x.support + y.support
-- HahnSeries.mul_coeff_order_add_order
theorem mul_coeff_order_add_order {Γ} [LinearOrderedCancelAddCommMonoid Γ]
    [NonUnitalNonAssocSemiring R] (x y : HahnSeries Γ R) :
    (x * y).coeff (x.order + y.order) = x.leadingCoeff * y.leadingCoeff 
Arguments: {R : Type u_3} {Γ : Type u_5} [LinearOrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] (x y : HahnSeries Γ R)
Goal: (x * y).coeff (x.order + y.order) = x.leadingCoeff * y.leadingCoeff
-- _private.Mathlib.RingTheory.HahnSeries.Multiplication.0.HahnSeries.mul_assoc'
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalSemiring R] (x y z : HahnSeries Γ R)
Goal: x * y * z = x * (y * z)
-- _private.Mathlib.RingTheory.HahnSeries.Multiplication.0.HahnModule.mul_smul'
private theorem mul_smul' [Semiring R] [Module R V] (x y : HahnSeries Γ R)
    (z : HahnModule Γ' R V) : (x * y) • z = x • (y • z) 
Arguments: {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} [OrderedCancelAddCommMonoid Γ] [PartialOrder Γ'] [AddAction Γ Γ'] [IsOrderedCancelVAdd Γ Γ'] [AddCommMonoid V] [Semiring R] [Module R V] (x y : HahnSeries Γ R) (z : HahnModule Γ' R V)
Goal: (x * y) • z = x • y • z
-- HahnSeries.orderTop_add_orderTop_le_orderTop_mul
theorem orderTop_add_orderTop_le_orderTop_mul {Γ} [LinearOrderedCancelAddCommMonoid Γ]
    [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    x.orderTop + y.orderTop ≤ (x * y).orderTop 
Arguments: {R : Type u_3} {Γ : Type u_5} [LinearOrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R}
Goal: x.orderTop + y.orderTop ≤ (x * y).orderTop
-- HahnSeries.order_mul
@[simp]
theorem order_mul {Γ} [LinearOrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R]
    [NoZeroDivisors R] {x y : HahnSeries Γ R} (hx : x ≠ 0) (hy : y ≠ 0) :
    (x * y).order = x.order + y.order 
Arguments: {R : Type u_3} {Γ : Type u_5} [LinearOrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] [NoZeroDivisors R] {x y : HahnSeries Γ R} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: (x * y).order = x.order + y.order
-- HahnSeries.order_pow
@[simp]
theorem order_pow {Γ} [LinearOrderedCancelAddCommMonoid Γ] [Semiring R] [NoZeroDivisors R]
    (x : HahnSeries Γ R) (n : ℕ) : (x ^ n).order = n • x.order 
Arguments: {R : Type u_3} {Γ : Type u_5} [LinearOrderedCancelAddCommMonoid Γ] [Semiring R] [NoZeroDivisors R] (x : HahnSeries Γ R) (n : ℕ)
Goal: (x ^ n).order = n • x.order
-- HahnSeries.single_mul_single
@[simp]
theorem single_mul_single {a b : Γ} {r s : R} :
    single a r * single b s = single (a + b) (r * s) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R] {a b : Γ} {r s : R}
Goal: (HahnSeries.single a) r * (HahnSeries.single b) s = (HahnSeries.single (a + b)) (r * s)
-- HahnSeries.single_pow
@[simp]
theorem single_pow (a : Γ) (n : ℕ) (r : R) : single a r ^ n = single (n • a) (r ^ n) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [Semiring R] (a : Γ) (n : ℕ) (r : R)
Goal: (HahnSeries.single a) r ^ n = (HahnSeries.single (n • a)) (r ^ n)
-- HahnSeries.C_apply
@[simps]
def C : R →+* HahnSeries Γ R 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonAssocSemiring R] (a : R)
Goal: HahnSeries.C a = (HahnSeries.single 0) a
-- HahnSeries.C_zero
theorem C_zero : C (0 : R) = (0 : HahnSeries Γ R) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonAssocSemiring R]
Goal: HahnSeries.C 0 = 0
-- HahnSeries.C_one
theorem C_one : C (1 : R) = (1 : HahnSeries Γ R) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonAssocSemiring R]
Goal: HahnSeries.C 1 = 1
-- HahnSeries.C_injective
theorem C_injective : Function.Injective (C : R → HahnSeries Γ R) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonAssocSemiring R]
Goal: Function.Injective ⇑HahnSeries.C
-- HahnSeries.C_ne_zero
theorem C_ne_zero {r : R} (h : r ≠ 0) : (C r : HahnSeries Γ R) ≠ 0 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonAssocSemiring R] {r : R} (h : r ≠ 0)
Goal: HahnSeries.C r ≠ 0
-- HahnSeries.order_C
theorem order_C {r : R} : order (C r : HahnSeries Γ R) = 0 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [NonAssocSemiring R] {r : R}
Goal: (HahnSeries.C r).order = 0
-- HahnSeries.C_mul_eq_smul
theorem C_mul_eq_smul {r : R} {x : HahnSeries Γ R} : C r * x = r • x 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [Semiring R] {r : R} {x : HahnSeries Γ R}
Goal: HahnSeries.C r * x = r • x
-- HahnSeries.embDomain_mul
theorem embDomain_mul [NonUnitalNonAssocSemiring R] (f : Γ ↪o Γ')
    (hf : ∀ x y, f (x + y) = f x + f y) (x y : HahnSeries Γ R) :
    embDomain f (x * y) = embDomain f x * embDomain f y 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] {Γ' : Type u_5} [OrderedCancelAddCommMonoid Γ'] [NonUnitalNonAssocSemiring R] (f : Γ ↪o Γ') (hf : ∀ (x y : Γ), f (x + y) = f x + f y) (x y : HahnSeries Γ R)
Goal: HahnSeries.embDomain f (x * y) = HahnSeries.embDomain f x * HahnSeries.embDomain f y
-- HahnSeries.embDomain_one
theorem embDomain_one [NonAssocSemiring R] (f : Γ ↪o Γ') (hf : f 0 = 0) :
    embDomain f (1 : HahnSeries Γ R) = (1 : HahnSeries Γ' R) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] {Γ' : Type u_5} [OrderedCancelAddCommMonoid Γ'] [NonAssocSemiring R] (f : Γ ↪o Γ') (hf : f 0 = 0)
Goal: HahnSeries.embDomain f 1 = 1
-- HahnSeries.embDomainRingHom_apply
@[simps]
def embDomainRingHom [NonAssocSemiring R] (f : Γ →+ Γ') (hfi : Function.Injective f)
    (hf : ∀ g g' : Γ, f g ≤ f g' ↔ g ≤ g') : HahnSeries Γ R →+* HahnSeries Γ' R 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] {Γ' : Type u_5} [OrderedCancelAddCommMonoid Γ'] [NonAssocSemiring R] (f : Γ →+ Γ') (hfi : Function.Injective ⇑f) (hf : ∀ (g g' : Γ), f g ≤ f g' ↔ g ≤ g')
Goal: ∀ (a : HahnSeries Γ R),
  (HahnSeries.embDomainRingHom f hfi hf) a = HahnSeries.embDomain { toFun := ⇑f, inj' := hfi, map_rel_iff' := ⋯ } a
-- HahnSeries.embDomainRingHom_C
theorem embDomainRingHom_C [NonAssocSemiring R] {f : Γ →+ Γ'} {hfi : Function.Injective f}
    {hf : ∀ g g' : Γ, f g ≤ f g' ↔ g ≤ g'} {r : R} : embDomainRingHom f hfi hf (C r) = C r 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] {Γ' : Type u_5} [OrderedCancelAddCommMonoid Γ'] [NonAssocSemiring R] {f : Γ →+ Γ'} {hfi : Function.Injective ⇑f} {hf : ∀ (g g' : Γ), f g ≤ f g' ↔ g ≤ g'} {r : R}
Goal: (HahnSeries.embDomainRingHom f hfi hf) (HahnSeries.C r) = HahnSeries.C r
-- HahnSeries.C_eq_algebraMap
theorem C_eq_algebraMap : C = algebraMap R (HahnSeries Γ R) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [CommSemiring R]
Goal: HahnSeries.C = algebraMap R (HahnSeries Γ R)
-- HahnSeries.algebraMap_apply
theorem algebraMap_apply {r : R} : algebraMap R (HahnSeries Γ A) r = C (algebraMap R A r) 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [CommSemiring R] {A : Type u_5} [Semiring A] [Algebra R A] {r : R}
Goal: (algebraMap R (HahnSeries Γ A)) r = HahnSeries.C ((algebraMap R A) r)
-- HahnSeries.embDomainAlgHom_apply_coeff
@[simps!]
def embDomainAlgHom (f : Γ →+ Γ') (hfi : Function.Injective f)
    (hf : ∀ g g' : Γ, f g ≤ f g' ↔ g ≤ g') : HahnSeries Γ A →ₐ[R] HahnSeries Γ' A 
Arguments: {Γ : Type u_1} {R : Type u_3} [OrderedCancelAddCommMonoid Γ] [CommSemiring R] {A : Type u_5} [Semiring A] [Algebra R A] {Γ' : Type u_6} [OrderedCancelAddCommMonoid Γ'] (f : Γ →+ Γ') (hfi : Function.Injective ⇑f) (hf : ∀ (g g' : Γ), f g ≤ f g' ↔ g ≤ g')
Goal: ∀ (a : HahnSeries Γ A) (b : Γ'),
  ((HahnSeries.embDomainAlgHom f hfi hf) a).coeff b =
    if h : ∃ x, ¬a.coeff x = 0 ∧ f x = b then a.coeff (Classical.choose ⋯) else 0
-- Algebra.IsIntegral.isIntegral
protected class Algebra.IsIntegral : Prop :=
  isIntegral : ∀ x : A, IsIntegral R x
Arguments: {R : Type u_1} {A : Type u_3} [CommRing R] [Ring A] [Algebra R A] [self : Algebra.IsIntegral R A] (x : A)
Goal: IsIntegral R x
-- Algebra.isIntegral_def
lemma Algebra.isIntegral_def : Algebra.IsIntegral R A ↔ ∀ x : A, IsIntegral R x 
Arguments: {R : Type u_1} {A : Type u_3} [CommRing R] [Ring A] [Algebra R A]
Goal: Algebra.IsIntegral R A ↔ ∀ (x : A), IsIntegral R x
-- Ideal.coe_comap
@[simp]
theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I : Ideal S)
Goal: ↑(Ideal.comap f I) = ⇑f ⁻¹' ↑I
-- Ideal.map_mono
theorem map_mono (h : I ≤ J) : map f I ≤ map f J 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {I J : Ideal R} (h : I ≤ J)
Goal: Ideal.map f I ≤ Ideal.map f J
-- Ideal.mem_map_of_mem
theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {I : Ideal R} {x : R} (h : x ∈ I)
Goal: f x ∈ Ideal.map f I
-- Ideal.apply_coe_mem_map
theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (I : Ideal R) (x : ↥I)
Goal: f ↑x ∈ Ideal.map f I
-- Ideal.map_le_iff_le_comap
theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} {I : Ideal R} {K : Ideal S}
Goal: Ideal.map f I ≤ K ↔ I ≤ Ideal.comap f K
-- Ideal.mem_comap
@[simp]
theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} {K : Ideal S} {x : R}
Goal: x ∈ Ideal.comap f K ↔ f x ∈ K
-- Ideal.comap_mono
theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} {K L : Ideal S} (h : K ≤ L)
Goal: Ideal.comap f K ≤ Ideal.comap f L
-- Ideal.comap_ne_top
theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} (hK : K ≠ ⊤)
Goal: Ideal.comap f K ≠ ⊤
-- Ideal.map_le_comap_of_inv_on
theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) :
    I.map f ≤ I.comap g 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {G : Type u_2} [FunLike G S R] [rcg : RingHomClass G S R] (g : G) (I : Ideal R) (hf : Set.LeftInvOn ⇑g ⇑f ↑I)
Goal: Ideal.map f I ≤ Ideal.comap g I
-- Ideal.comap_le_map_of_inv_on
theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) :
    I.comap f ≤ I.map g 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {G : Type u_2} [FunLike G S R] (g : G) (I : Ideal S) (hf : Set.LeftInvOn (⇑g) (⇑f) (⇑f ⁻¹' ↑I))
Goal: Ideal.comap f I ≤ Ideal.map g I
-- Ideal.map_le_comap_of_inverse
theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) :
    I.map f ≤ I.comap g 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {G : Type u_2} [FunLike G S R] [rcg : RingHomClass G S R] (g : G) (I : Ideal R) (h : Function.LeftInverse ⇑g ⇑f)
Goal: Ideal.map f I ≤ Ideal.comap g I
-- Ideal.comap_le_map_of_inverse
theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) :
    I.comap f ≤ I.map g 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {G : Type u_2} [FunLike G S R] (g : G) (I : Ideal S) (h : Function.LeftInverse ⇑g ⇑f)
Goal: Ideal.comap f I ≤ Ideal.map g I
-- Ideal.map_top
theorem map_top : map f ⊤ = ⊤ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F)
Goal: Ideal.map f ⊤ = ⊤
-- Ideal.gc_map_comap
theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F)
Goal: GaloisConnection (Ideal.map f) (Ideal.comap f)
-- Ideal.comap_id
@[simp]
theorem comap_id : I.comap (RingHom.id R) = I 
Arguments: {R : Type u} [Semiring R] (I : Ideal R)
Goal: Ideal.comap (RingHom.id R) I = I
-- Ideal.map_id
@[simp]
theorem map_id : I.map (RingHom.id R) = I 
Arguments: {R : Type u} [Semiring R] (I : Ideal R)
Goal: Ideal.map (RingHom.id R) I = I
-- Ideal.comap_comap
theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) :
    (I.comap g).comap f = I.comap (g.comp f) 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] {T : Type u_3} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T)
Goal: Ideal.comap f (Ideal.comap g I) = Ideal.comap (g.comp f) I
-- Ideal.map_map
theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) :
    (I.map f).map g = I.map (g.comp f) 
Arguments: {R : Type u} {S : Type v} [Semiring R] [Semiring S] {T : Type u_3} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T)
Goal: Ideal.map g (Ideal.map f I) = Ideal.map (g.comp f) I
-- Ideal.map_span
theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (s : Set R)
Goal: Ideal.map f (Ideal.span s) = Ideal.span (⇑f '' s)
-- Ideal.map_le_of_le_comap
theorem map_le_of_le_comap : I ≤ K.comap f → I.map f ≤ K 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} {I : Ideal R} {K : Ideal S}
Goal: I ≤ Ideal.comap f K → Ideal.map f I ≤ K
-- Ideal.le_comap_of_map_le
theorem le_comap_of_map_le : I.map f ≤ K → I ≤ K.comap f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} {I : Ideal R} {K : Ideal S}
Goal: Ideal.map f I ≤ K → I ≤ Ideal.comap f K
-- Ideal.le_comap_map
theorem le_comap_map : I ≤ (I.map f).comap f 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} {I : Ideal R}
Goal: I ≤ Ideal.comap f (Ideal.map f I)
-- Ideal.map_comap_le
theorem map_comap_le : (K.comap f).map f ≤ K 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} {K : Ideal S}
Goal: Ideal.map f (Ideal.comap f K) ≤ K
-- Ideal.comap_top
@[simp]
theorem comap_top : (⊤ : Ideal S).comap f = ⊤ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F}
Goal: Ideal.comap f ⊤ = ⊤
-- Ideal.comap_eq_top_iff
@[simp]
theorem comap_eq_top_iff {I : Ideal S} : I.comap f = ⊤ ↔ I = ⊤ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} {I : Ideal S}
Goal: Ideal.comap f I = ⊤ ↔ I = ⊤
-- Ideal.map_bot
@[simp]
theorem map_bot : (⊥ : Ideal R).map f = ⊥ 
Arguments: {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F}
Goal: Ideal.map f ⊥ = ⊥