ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_finset
theorem iIndepFun.indepFun_finset [IsMarkovKernel κ] (S T : Finset ι) (hST : Disjoint S T)
    (hf_Indep : iIndepFun m f κ μ) (hf_meas : ∀ i, Measurable (f i)) :
    IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : ι → Type u_8} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} [ProbabilityTheory.IsMarkovKernel κ] (S T : Finset ι) (hST : Disjoint S T) (hf_Indep : ProbabilityTheory.Kernel.iIndepFun m f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i))
Goal: ProbabilityTheory.Kernel.IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_prod_mk
theorem iIndepFun.indepFun_prod_mk [IsMarkovKernel κ] (hf_Indep : iIndepFun m f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    IndepFun (fun a => (f i a, f j a)) (f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : ι → Type u_8} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} [ProbabilityTheory.IsMarkovKernel κ] (hf_Indep : ProbabilityTheory.Kernel.iIndepFun m f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (fun a => (f i a, f j a)) (f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_prod_mk_prod_mk
lemma iIndepFun.indepFun_prod_mk_prod_mk [IsMarkovKernel κ] (hf_indep : iIndepFun m f κ μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    IndepFun (fun a ↦ (f i a, f j a)) (fun a ↦ (f k a, f l a)) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : ι → Type u_8} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun m f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.Kernel.IndepFun (fun a => (f i a, f j a)) (fun a => (f k a, f l a)) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_add_left
@[to_additive]
lemma iIndepFun.indepFun_mul_left (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    IndepFun (f i * f j) (f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Add β] [MeasurableAdd₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (f i + f j) (f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_left
@[to_additive]
lemma iIndepFun.indepFun_mul_left (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    IndepFun (f i * f j) (f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (f i * f j) (f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_right
@[to_additive]
lemma iIndepFun.indepFun_mul_right (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
    IndepFun (f i) (f j * f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (f i) (f j * f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_add_right
@[to_additive]
lemma iIndepFun.indepFun_mul_right (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
    IndepFun (f i) (f j * f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Add β] [MeasurableAdd₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (f i) (f j + f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_mul
@[to_additive]
lemma iIndepFun.indepFun_mul_mul (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    IndepFun (f i * f j) (f k * f l) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.Kernel.IndepFun (f i * f j) (f k * f l) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_add_add
@[to_additive]
lemma iIndepFun.indepFun_mul_mul (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    IndepFun (f i * f j) (f k * f l) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Add β] [MeasurableAdd₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.Kernel.IndepFun (f i + f j) (f k + f l) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_div_left
@[to_additive]
lemma iIndepFun.indepFun_div_left (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    IndepFun (f i / f j) (f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (f i / f j) (f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_left
@[to_additive]
lemma iIndepFun.indepFun_div_left (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
    IndepFun (f i / f j) (f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Sub β] [MeasurableSub₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (f i - f j) (f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_div_right
@[to_additive]
lemma iIndepFun.indepFun_div_right (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
    IndepFun (f i) (f j / f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (f i) (f j / f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_right
@[to_additive]
lemma iIndepFun.indepFun_div_right (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
    IndepFun (f i) (f j / f k) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Sub β] [MeasurableSub₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k)
Goal: ProbabilityTheory.Kernel.IndepFun (f i) (f j - f k) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_div_div
@[to_additive]
lemma iIndepFun.indepFun_div_div (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    IndepFun (f i / f j) (f k / f l) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.Kernel.IndepFun (f i / f j) (f k / f l) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_sub
@[to_additive]
lemma iIndepFun.indepFun_div_div (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i))
    (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
    IndepFun (f i / f j) (f k / f l) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [Sub β] [MeasurableSub₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l)
Goal: ProbabilityTheory.Kernel.IndepFun (f i - f j) (f k - f l) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_finset_prod_of_not_mem
@[to_additive]
theorem iIndepFun.indepFun_finset_prod_of_not_mem (hf_Indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s) :
    IndepFun (∏ j ∈ s, f j) (f i) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_Indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s)
Goal: ProbabilityTheory.Kernel.IndepFun (∏ j ∈ s, f j) (f i) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_finset_sum_of_not_mem
@[to_additive]
theorem iIndepFun.indepFun_finset_prod_of_not_mem (hf_Indep : iIndepFun (fun _ ↦ m) f κ μ)
    (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s) :
    IndepFun (∏ j ∈ s, f j) (f i) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [AddCommMonoid β] [MeasurableAdd₂ β] {f : ι → Ω → β} [ProbabilityTheory.IsMarkovKernel κ] (hf_Indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ι), Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s)
Goal: ProbabilityTheory.Kernel.IndepFun (∑ j ∈ s, f j) (f i) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_sum_range_succ
@[to_additive]
theorem iIndepFun.indepFun_prod_range_succ {f : ℕ → Ω → β}
    (hf_Indep : iIndepFun (fun _ => m) f κ μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
    IndepFun (∏ j ∈ Finset.range n, f j) (f n) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [AddCommMonoid β] [MeasurableAdd₂ β] [ProbabilityTheory.IsMarkovKernel κ] {f : ℕ → Ω → β} (hf_Indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ℕ), Measurable (f i)) (n : ℕ)
Goal: ProbabilityTheory.Kernel.IndepFun (∑ j ∈ Finset.range n, f j) (f n) κ μ
-- ProbabilityTheory.Kernel.iIndepFun.indepFun_prod_range_succ
@[to_additive]
theorem iIndepFun.indepFun_prod_range_succ {f : ℕ → Ω → β}
    (hf_Indep : iIndepFun (fun _ => m) f κ μ) (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) :
    IndepFun (∏ j ∈ Finset.range n, f j) (f n) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : Type u_8} {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] [ProbabilityTheory.IsMarkovKernel κ] {f : ℕ → Ω → β} (hf_Indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ) (hf_meas : ∀ (i : ℕ), Measurable (f i)) (n : ℕ)
Goal: ProbabilityTheory.Kernel.IndepFun (∏ j ∈ Finset.range n, f j) (f n) κ μ
-- ProbabilityTheory.Kernel.iIndepSet.iIndepFun_indicator
theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω}
    (hs : iIndepSet s κ μ) :
    iIndepFun (fun _n => m) (fun n => (s n).indicator fun _ω => 1) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {β : Type u_4} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [Zero β] [One β] {m : MeasurableSpace β} {s : ι → Set Ω} (hs : ProbabilityTheory.Kernel.iIndepSet s κ μ)
Goal: ProbabilityTheory.Kernel.iIndepFun (fun _n => m) (fun n => (s n).indicator fun _ω => 1) κ μ
-- Valuation.map_add_le_max'
structure Valuation extends R →*₀ Γ₀ where
    map_add_le_max' : ∀ x y, toFun (x + y) ≤ max (toFun x) (toFun y)
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedCommMonoidWithZero Γ₀] [Ring R] (self : Valuation R Γ₀) (x y : R)
Goal: (↑self.toMonoidWithZeroHom).toFun (x + y) ≤
  max ((↑self.toMonoidWithZeroHom).toFun x) ((↑self.toMonoidWithZeroHom).toFun y)
-- ValuationClass.map_add_le_max
class ValuationClass (F) (R Γ₀ : outParam Type*) [LinearOrderedCommMonoidWithZero Γ₀] [Ring R]
  [FunLike F R Γ₀]
  extends MonoidWithZeroHomClass F R Γ₀ : Prop where
    map_add_le_max (f : F) (x y : R) : f (x + y) ≤ max (f x) (f y)
Arguments: {F : Type u_7} {R : outParam (Type u_5)} {Γ₀ : outParam (Type u_6)} [LinearOrderedCommMonoidWithZero Γ₀] [Ring R] [FunLike F R Γ₀] [self : ValuationClass F R Γ₀] (f : F) (x y : R)
Goal: f (x + y) ≤ max (f x) (f y)
-- Valuation.coe_mk
@[simp]
theorem coe_mk (f : R →*₀ Γ₀) (h) : ⇑(Valuation.mk f h) = f 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (f : R →*₀ Γ₀) (h : ∀ (x y : R), (↑f).toFun (x + y) ≤ max ((↑f).toFun x) ((↑f).toFun y))
Goal: ⇑{ toMonoidWithZeroHom := f, map_add_le_max' := h } = ⇑f
-- Valuation.toFun_eq_coe
theorem toFun_eq_coe (v : Valuation R Γ₀) : v.toFun = v 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀)
Goal: (↑v.toMonoidWithZeroHom).toFun = ⇑v
-- Valuation.toMonoidWithZeroHom_coe_eq_coe
@[simp] -- Porting note: requested by simpNF as toFun_eq_coe LHS simplifies
theorem toMonoidWithZeroHom_coe_eq_coe (v : Valuation R Γ₀) :
    (v.toMonoidWithZeroHom : R → Γ₀) = v 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀)
Goal: ⇑v.toMonoidWithZeroHom = ⇑v
-- Valuation.ext
@[ext]
theorem ext {v₁ v₂ : Valuation R Γ₀} (h : ∀ r, v₁ r = v₂ r) : v₁ = v₂ 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {v₁ v₂ : Valuation R Γ₀} (h : ∀ (r : R), v₁ r = v₂ r)
Goal: v₁ = v₂
-- Valuation.coe_coe
@[simp, norm_cast]
theorem coe_coe : ⇑(v : R →*₀ Γ₀) = v 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀)
Goal: ⇑↑v = ⇑v
-- Valuation.map_zero
theorem map_zero : v 0 = 0 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀)
Goal: v 0 = 0
-- Valuation.map_one
theorem map_one : v 1 = 1 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀)
Goal: v 1 = 1
-- Valuation.map_mul
theorem map_mul : ∀ x y, v (x * y) = v x * v y 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x y : R)
Goal: v (x * y) = v x * v y
-- Valuation.map_add
theorem map_add : ∀ x y, v (x + y) ≤ max (v x) (v y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x y : R)
Goal: v (x + y) ≤ max (v x) (v y)
-- Valuation.map_add'
@[simp]
theorem map_add' : ∀ x y, v (x + y) ≤ v x ∨ v (x + y) ≤ v y 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x y : R)
Goal: v (x + y) ≤ v x ∨ v (x + y) ≤ v y
-- Valuation.map_add_le
theorem map_add_le {x y g} (hx : v x ≤ g) (hy : v y ≤ g) : v (x + y) ≤ g 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {x y : R} {g : Γ₀} (hx : v x ≤ g) (hy : v y ≤ g)
Goal: v (x + y) ≤ g
-- Valuation.map_add_lt
theorem map_add_lt {x y g} (hx : v x < g) (hy : v y < g) : v (x + y) < g 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {x y : R} {g : Γ₀} (hx : v x < g) (hy : v y < g)
Goal: v (x + y) < g
-- Valuation.map_sum_le
theorem map_sum_le {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, v (f i) ≤ g) :
    v (∑ i ∈ s, f i) ≤ g 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {ι : Type u_7} {s : Finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, v (f i) ≤ g)
Goal: v (∑ i ∈ s, f i) ≤ g
-- Valuation.map_sum_lt
theorem map_sum_lt {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ 0)
    (hf : ∀ i ∈ s, v (f i) < g) : v (∑ i ∈ s, f i) < g 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {ι : Type u_7} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ 0) (hf : ∀ i ∈ s, v (f i) < g)
Goal: v (∑ i ∈ s, f i) < g
-- Valuation.map_sum_lt'
theorem map_sum_lt' {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : 0 < g)
    (hf : ∀ i ∈ s, v (f i) < g) : v (∑ i ∈ s, f i) < g 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {ι : Type u_7} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : 0 < g) (hf : ∀ i ∈ s, v (f i) < g)
Goal: v (∑ i ∈ s, f i) < g
-- Valuation.map_pow
theorem map_pow : ∀ (x) (n : ℕ), v (x ^ n) = v x ^ n 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x : R) (n : ℕ)
Goal: v (x ^ n) = v x ^ n
-- Valuation.ext_iff
-- @[deprecated] Porting note: using `DFunLike.ext_iff` is not viable below for now
theorem ext_iff {v₁ v₂ : Valuation R Γ₀} : v₁ = v₂ ↔ ∀ r, v₁ r = v₂ r 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {v₁ v₂ : Valuation R Γ₀}
Goal: v₁ = v₂ ↔ ∀ (r : R), v₁ r = v₂ r
-- Valuation.zero_iff
-- @[simp] Porting note (#10618): simp can prove this
theorem zero_iff [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K} : v x = 0 ↔ x = 0 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommMonoidWithZero Γ₀] [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K}
Goal: v x = 0 ↔ x = 0
-- Valuation.ne_zero_iff
theorem ne_zero_iff [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K} : v x ≠ 0 ↔ x ≠ 0 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommMonoidWithZero Γ₀] [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K}
Goal: v x ≠ 0 ↔ x ≠ 0
-- Valuation.unit_map_eq
theorem unit_map_eq (u : Rˣ) : (Units.map (v : R →* Γ₀) u : Γ₀) = v u 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (u : Rˣ)
Goal: ↑((Units.map ↑v) u) = v ↑u
-- Valuation.comap_apply
@[simp]
theorem comap_apply {S : Type*} [Ring S] (f : S →+* R) (v : Valuation R Γ₀) (s : S) :
    v.comap f s = v (f s) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {S : Type u_7} [Ring S] (f : S →+* R) (v : Valuation R Γ₀) (s : S)
Goal: (Valuation.comap f v) s = v (f s)
-- Valuation.comap_id
@[simp]
theorem comap_id : v.comap (RingHom.id R) = v 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀)
Goal: Valuation.comap (RingHom.id R) v = v
-- Valuation.comap_comp
theorem comap_comp {S₁ : Type*} {S₂ : Type*} [Ring S₁] [Ring S₂] (f : S₁ →+* S₂) (g : S₂ →+* R) :
    v.comap (g.comp f) = (v.comap g).comap f 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {S₁ : Type u_7} {S₂ : Type u_8} [Ring S₁] [Ring S₂] (f : S₁ →+* S₂) (g : S₂ →+* R)
Goal: Valuation.comap (g.comp f) v = Valuation.comap f (Valuation.comap g v)
-- Valuation.map_neg
@[simp]
theorem map_neg (x : R) : v (-x) = v x 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) (x : R)
Goal: v (-x) = v x
-- Valuation.map_sub_swap
theorem map_sub_swap (x y : R) : v (x - y) = v (y - x) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) (x y : R)
Goal: v (x - y) = v (y - x)
-- Valuation.map_inv
theorem map_inv {R : Type*} [DivisionRing R] (v : Valuation R Γ₀) : ∀ x, v x⁻¹ = (v x)⁻¹ 
Arguments: {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] {R : Type u_7} [DivisionRing R] (v : Valuation R Γ₀) (x : R)
Goal: v x⁻¹ = (v x)⁻¹
-- Valuation.map_div
theorem map_div {R : Type*} [DivisionRing R] (v : Valuation R Γ₀) : ∀ x y, v (x / y) = v x / v y 
Arguments: {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] {R : Type u_7} [DivisionRing R] (v : Valuation R Γ₀) (x y : R)
Goal: v (x / y) = v x / v y
-- Valuation.map_sub
theorem map_sub (x y : R) : v (x - y) ≤ max (v x) (v y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) (x y : R)
Goal: v (x - y) ≤ max (v x) (v y)