ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Valuation.map_sub_le
theorem map_sub_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] [LinearOrderedCommGroupWithZero 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_of_distinct_val
theorem map_add_of_distinct_val (h : v x ≠ v y) : 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} (h : v x ≠ v y)
Goal: v (x + y) = max (v x) (v y)
-- Valuation.map_add_eq_of_lt_right
theorem map_add_eq_of_lt_right (h : v x < v y) : v (x + y) = v y 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x y : R} (h : v x < v y)
Goal: v (x + y) = v y
-- Valuation.map_add_eq_of_lt_left
theorem map_add_eq_of_lt_left (h : v y < v x) : v (x + y) = v x 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x y : R} (h : v y < v x)
Goal: v (x + y) = v x
-- Valuation.map_sub_eq_of_lt_right
theorem map_sub_eq_of_lt_right (h : v x < v y) : v (x - y) = v y 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x y : R} (h : v x < v y)
Goal: v (x - y) = v y
-- Valuation.map_sub_eq_of_lt_left
theorem map_sub_eq_of_lt_left (h : v y < v x) : v (x - y) = v x 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x y : R} (h : v y < v x)
Goal: v (x - y) = v x
-- Valuation.map_eq_of_sub_lt
theorem map_eq_of_sub_lt (h : v (y - x) < v x) : v y = v x 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x y : R} (h : v (y - x) < v x)
Goal: v y = v x
-- Valuation.map_one_add_of_lt
theorem map_one_add_of_lt (h : v x < 1) : v (1 + x) = 1 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} (h : v x < 1)
Goal: v (1 + x) = 1
-- Valuation.map_one_sub_of_lt
theorem map_one_sub_of_lt (h : v x < 1) : v (1 - x) = 1 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x : R} (h : v x < 1)
Goal: v (1 - x) = 1
-- Valuation.one_lt_val_iff
theorem one_lt_val_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : 1 < v x ↔ v x⁻¹ < 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) {x : K} (h : x ≠ 0)
Goal: 1 < v x ↔ v x⁻¹ < 1
-- Valuation.one_le_val_iff
theorem one_le_val_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : 1 ≤ v x ↔ v x⁻¹ ≤ 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) {x : K} (h : x ≠ 0)
Goal: 1 ≤ v x ↔ v x⁻¹ ≤ 1
-- Valuation.val_lt_one_iff
theorem val_lt_one_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : v x < 1 ↔ 1 < v x⁻¹ 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) {x : K} (h : x ≠ 0)
Goal: v x < 1 ↔ 1 < v x⁻¹
-- Valuation.val_le_one_iff
theorem val_le_one_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : v x ≤ 1 ↔ 1 ≤ v x⁻¹ 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) {x : K} (h : x ≠ 0)
Goal: v x ≤ 1 ↔ 1 ≤ v x⁻¹
-- Valuation.val_eq_one_iff
theorem val_eq_one_iff (v : Valuation K Γ₀) {x : K} : v x = 1 ↔ v x⁻¹ = 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) {x : K}
Goal: v x = 1 ↔ v x⁻¹ = 1
-- Valuation.val_le_one_or_val_inv_lt_one
theorem val_le_one_or_val_inv_lt_one (v : Valuation K Γ₀) (x : K) : v x ≤ 1 ∨ v x⁻¹ < 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) (x : K)
Goal: v x ≤ 1 ∨ v x⁻¹ < 1
-- Valuation.val_le_one_or_val_inv_le_one
theorem val_le_one_or_val_inv_le_one (v : Valuation K Γ₀) (x : K) : v x ≤ 1 ∨ v x⁻¹ ≤ 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀) (x : K)
Goal: v x ≤ 1 ∨ v x⁻¹ ≤ 1
-- Valuation.IsEquiv.refl
@[refl]
theorem refl : v.IsEquiv v 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {v : Valuation R Γ₀}
Goal: v.IsEquiv v
-- Valuation.IsEquiv.symm
@[symm]
theorem symm (h : v₁.IsEquiv v₂) : v₂.IsEquiv v₁ 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} (h : v₁.IsEquiv v₂)
Goal: v₂.IsEquiv v₁
-- Valuation.IsEquiv.trans
@[trans]
theorem trans (h₁₂ : v₁.IsEquiv v₂) (h₂₃ : v₂.IsEquiv v₃) : v₁.IsEquiv v₃ 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} {Γ''₀ : Type u_6} [LinearOrderedCommMonoidWithZero Γ''₀] [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} {v₃ : Valuation R Γ''₀} (h₁₂ : v₁.IsEquiv v₂) (h₂₃ : v₂.IsEquiv v₃)
Goal: v₁.IsEquiv v₃
-- Valuation.IsEquiv.of_eq
theorem of_eq {v' : Valuation R Γ₀} (h : v = v') : v.IsEquiv v' 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] {v v' : Valuation R Γ₀} (h : v = v')
Goal: v.IsEquiv v'
-- Valuation.IsEquiv.map
theorem map {v' : Valuation R Γ₀} (f : Γ₀ →*₀ Γ'₀) (hf : Monotone f) (inf : Injective f)
    (h : v.IsEquiv v') : (v.map f hf).IsEquiv (v'.map f hf) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v v' : Valuation R Γ₀} (f : Γ₀ →*₀ Γ'₀) (hf : Monotone ⇑f) (inf : Function.Injective ⇑f) (h : v.IsEquiv v')
Goal: (Valuation.map f hf v).IsEquiv (Valuation.map f hf v')
-- Valuation.IsEquiv.comap
theorem comap {S : Type*} [Ring S] (f : S →+* R) (h : v₁.IsEquiv v₂) :
    (v₁.comap f).IsEquiv (v₂.comap f) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} {S : Type u_7} [Ring S] (f : S →+* R) (h : v₁.IsEquiv v₂)
Goal: (Valuation.comap f v₁).IsEquiv (Valuation.comap f v₂)
-- Valuation.IsEquiv.val_eq
theorem val_eq (h : v₁.IsEquiv v₂) {r s : R} : v₁ r = v₁ s ↔ v₂ r = v₂ s 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} (h : v₁.IsEquiv v₂) {r s : R}
Goal: v₁ r = v₁ s ↔ v₂ r = v₂ s
-- Valuation.IsEquiv.ne_zero
theorem ne_zero (h : v₁.IsEquiv v₂) {r : R} : v₁ r ≠ 0 ↔ v₂ r ≠ 0 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [Ring R] [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] {v₁ : Valuation R Γ₀} {v₂ : Valuation R Γ'₀} (h : v₁.IsEquiv v₂) {r : R}
Goal: v₁ r ≠ 0 ↔ v₂ r ≠ 0
-- Valuation.isEquiv_of_map_strictMono
theorem isEquiv_of_map_strictMono [LinearOrderedCommMonoidWithZero Γ₀]
    [LinearOrderedCommMonoidWithZero Γ'₀] [Ring R] {v : Valuation R Γ₀} (f : Γ₀ →*₀ Γ'₀)
    (H : StrictMono f) : IsEquiv (v.map f H.monotone) v 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommMonoidWithZero Γ₀] [LinearOrderedCommMonoidWithZero Γ'₀] [Ring R] {v : Valuation R Γ₀} (f : Γ₀ →*₀ Γ'₀) (H : StrictMono ⇑f)
Goal: (Valuation.map f ⋯ v).IsEquiv v
-- Valuation.isEquiv_of_val_le_one
theorem isEquiv_of_val_le_one [LinearOrderedCommGroupWithZero Γ₀]
    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀)
    (h : ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1) : v.IsEquiv v' 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) (h : ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1)
Goal: v.IsEquiv v'
-- Valuation.isEquiv_iff_val_le_one
theorem isEquiv_iff_val_le_one [LinearOrderedCommGroupWithZero Γ₀]
    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
    v.IsEquiv v' ↔ ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀)
Goal: v.IsEquiv v' ↔ ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1
-- Valuation.isEquiv_iff_val_eq_one
theorem isEquiv_iff_val_eq_one [LinearOrderedCommGroupWithZero Γ₀]
    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
    v.IsEquiv v' ↔ ∀ {x : K}, v x = 1 ↔ v' x = 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀)
Goal: v.IsEquiv v' ↔ ∀ {x : K}, v x = 1 ↔ v' x = 1
-- Valuation.isEquiv_iff_val_lt_one
theorem isEquiv_iff_val_lt_one [LinearOrderedCommGroupWithZero Γ₀]
    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
    v.IsEquiv v' ↔ ∀ {x : K}, v x < 1 ↔ v' x < 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀)
Goal: v.IsEquiv v' ↔ ∀ {x : K}, v x < 1 ↔ v' x < 1
-- Valuation.isEquiv_iff_val_sub_one_lt_one
theorem isEquiv_iff_val_sub_one_lt_one [LinearOrderedCommGroupWithZero Γ₀]
    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
    v.IsEquiv v' ↔ ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀)
Goal: v.IsEquiv v' ↔ ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1
-- Valuation.isEquiv_tfae
theorem isEquiv_tfae [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀]
    (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
    [v.IsEquiv v', ∀ {x}, v x ≤ 1 ↔ v' x ≤ 1, ∀ {x}, v x = 1 ↔ v' x = 1, ∀ {x}, v x < 1 ↔ v' x < 1,
        ∀ {x}, v (x - 1) < 1 ↔ v' (x - 1) < 1].TFAE 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀)
Goal: [v.IsEquiv v', ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1, ∀ {x : K}, v x = 1 ↔ v' x = 1, ∀ {x : K}, v x < 1 ↔ v' x < 1,
    ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1].TFAE
-- Valuation.mem_supp_iff
@[simp]
theorem mem_supp_iff (x : R) : x ∈ supp v ↔ v x = 0 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (x : R)
Goal: x ∈ v.supp ↔ v x = 0
-- Valuation.map_add_supp
theorem map_add_supp (a : R) {s : R} (h : s ∈ supp v) : v (a + s) = v a 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) (a : R) {s : R} (h : s ∈ v.supp)
Goal: v (a + s) = v a
-- Valuation.comap_supp
theorem comap_supp {S : Type*} [CommRing S] (f : S →+* R) :
    supp (v.comap f) = Ideal.comap f v.supp 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation R Γ₀) {S : Type u_7} [CommRing S] (f : S →+* R)
Goal: (Valuation.comap f v).supp = Ideal.comap f v.supp
-- AddValuation.of_apply
@[simp]
theorem of_apply : (of f h0 h1 hadd hmul) r = f r 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (f : R → Γ₀) {h0 : f 0 = ⊤} {h1 : f 1 = 0} {hadd : ∀ (x y : R), min (f x) (f y) ≤ f (x + y)} {hmul : ∀ (x y : R), f (x * y) = f x + f y} {r : R}
Goal: (AddValuation.of f h0 h1 hadd hmul) r = f r
-- AddValuation.valuation_apply
@[simp]
theorem valuation_apply (r : R) : v.valuation r = Multiplicative.ofAdd (OrderDual.toDual (v r)) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (r : R)
Goal: v.valuation r = Multiplicative.ofAdd (OrderDual.toDual (v r))
-- AddValuation.map_zero
@[simp]
theorem map_zero : v 0 = (⊤ : Γ₀) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀)
Goal: v 0 = ⊤
-- AddValuation.map_one
@[simp]
theorem map_one : v 1 = (0 : Γ₀) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀)
Goal: v 1 = 0
-- AddValuation.map_mul
@[simp]
theorem map_mul : ∀ (x y : R), v (x * y) = v x + v y 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (x y : R)
Goal: v (x * y) = v x + v y
-- AddValuation.map_add
theorem map_add : ∀ (x y : R), min (v x) (v y) ≤ v (x + y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (x y : R)
Goal: min (v x) (v y) ≤ v (x + y)
-- AddValuation.map_add'
@[simp]
theorem map_add' : ∀ (x y : R), v x ≤ v (x + y) ∨ v y ≤ v (x + y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (x y : R)
Goal: v x ≤ v (x + y) ∨ v y ≤ v (x + y)
-- AddValuation.map_le_add
theorem map_le_add {x y : R} {g : Γ₀} (hx : g ≤ v x) (hy : g ≤ v y) : g ≤ v (x + y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {x y : R} {g : Γ₀} (hx : g ≤ v x) (hy : g ≤ v y)
Goal: g ≤ v (x + y)
-- AddValuation.map_lt_add
theorem map_lt_add {x y : R} {g : Γ₀} (hx : g < v x) (hy : g < v y) : g < v (x + y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {x y : R} {g : Γ₀} (hx : g < v x) (hy : g < v y)
Goal: g < v (x + y)
-- AddValuation.map_le_sum
theorem map_le_sum {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, g ≤ v (f i)) :
    g ≤ v (∑ i ∈ s, f i) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {ι : Type u_6} {s : Finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, g ≤ v (f i))
Goal: g ≤ v (∑ i ∈ s, f i)
-- AddValuation.map_lt_sum
theorem map_lt_sum {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ ⊤)
    (hf : ∀ i ∈ s, g < v (f i)) : g < v (∑ i ∈ s, f i) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {ι : Type u_6} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ ⊤) (hf : ∀ i ∈ s, g < v (f i))
Goal: g < v (∑ i ∈ s, f i)
-- AddValuation.map_lt_sum'
theorem map_lt_sum' {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g < ⊤)
    (hf : ∀ i ∈ s, g < v (f i)) : g < v (∑ i ∈ s, f i) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {ι : Type u_6} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g < ⊤) (hf : ∀ i ∈ s, g < v (f i))
Goal: g < v (∑ i ∈ s, f i)
-- AddValuation.map_pow
@[simp]
theorem map_pow : ∀ (x : R) (n : ℕ), v (x ^ n) = n • (v x) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) (x : R) (n : ℕ)
Goal: v (x ^ n) = n • v x
-- AddValuation.ext
@[ext]
theorem ext {v₁ v₂ : AddValuation R Γ₀} (h : ∀ r, v₁ r = v₂ r) : v₁ = v₂ 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] {v₁ v₂ : AddValuation R Γ₀} (h : ∀ (r : R), v₁ r = v₂ r)
Goal: v₁ = v₂
-- AddValuation.ext_iff
theorem ext_iff {v₁ v₂ : AddValuation R Γ₀} : v₁ = v₂ ↔ ∀ (r : R), v₁ r = v₂ r 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] {v₁ v₂ : AddValuation R Γ₀}
Goal: v₁ = v₂ ↔ ∀ (r : R), v₁ r = v₂ r
-- AddValuation.top_iff
@[simp]
theorem top_iff [Nontrivial Γ₀] (v : AddValuation K Γ₀) {x : K} : v x = (⊤ : Γ₀) ↔ x = 0 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [Nontrivial Γ₀] (v : AddValuation K Γ₀) {x : K}
Goal: v x = ⊤ ↔ x = 0
-- AddValuation.ne_top_iff
theorem ne_top_iff [Nontrivial Γ₀] (v : AddValuation K Γ₀) {x : K} : v x ≠ (⊤ : Γ₀) ↔ x ≠ 0 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [Nontrivial Γ₀] (v : AddValuation K Γ₀) {x : K}
Goal: v x ≠ ⊤ ↔ x ≠ 0
-- AddValuation.comap_id
@[simp]
theorem comap_id : v.comap (RingHom.id R) = v 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [Ring R] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀)
Goal: AddValuation.comap (RingHom.id R) v = v
-- AddValuation.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] [LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀) {S₁ : Type u_6} {S₂ : Type u_7} [Ring S₁] [Ring S₂] (f : S₁ →+* S₂) (g : S₂ →+* R)
Goal: AddValuation.comap (g.comp f) v = AddValuation.comap f (AddValuation.comap g v)
-- AddValuation.map_inv
@[simp]
theorem map_inv (v : AddValuation K Γ₀) {x : K} : v x⁻¹ = - (v x) 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] (v : AddValuation K Γ₀) {x : K}
Goal: v x⁻¹ = -v x
-- AddValuation.map_div
@[simp]
theorem map_div (v : AddValuation K Γ₀) {x y : K} : v (x / y) = v x - v y 
Arguments: {K : Type u_1} [DivisionRing K] {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] (v : AddValuation K Γ₀) {x y : K}
Goal: v (x / y) = v x - v y
-- AddValuation.map_neg
@[simp]
theorem map_neg (x : R) : v (-x) = v x 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) (x : R)
Goal: v (-x) = v x
-- AddValuation.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} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) (x y : R)
Goal: v (x - y) = v (y - x)
-- AddValuation.map_sub
theorem map_sub (x y : R) : min (v x) (v y) ≤ v (x - y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) (x y : R)
Goal: min (v x) (v y) ≤ v (x - y)
-- AddValuation.map_le_sub
theorem map_le_sub {x y : R} {g : Γ₀} (hx : g ≤ v x) (hy : g ≤ v y) : g ≤ v (x - y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R} {g : Γ₀} (hx : g ≤ v x) (hy : g ≤ v y)
Goal: g ≤ v (x - y)
-- AddValuation.map_add_of_distinct_val
theorem map_add_of_distinct_val (h : v x ≠ v y) : v (x + y) = @Min.min Γ₀ _ (v x) (v y) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R} (h : v x ≠ v y)
Goal: v (x + y) = min (v x) (v y)
-- AddValuation.map_add_eq_of_lt_left
theorem map_add_eq_of_lt_left {x y : R} (h : v x < v y) :
    v (x + y) = v x 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R} (h : v x < v y)
Goal: v (x + y) = v x
-- AddValuation.map_add_eq_of_lt_right
theorem map_add_eq_of_lt_right {x y : R} (hx : v y < v x) :
    v (x + y) = v y 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R} (hx : v y < v x)
Goal: v (x + y) = v y
-- AddValuation.map_sub_eq_of_lt_left
theorem map_sub_eq_of_lt_left {x y : R} (hx : v x < v y) :
    v (x - y) = v x 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R} (hx : v x < v y)
Goal: v (x - y) = v x
-- AddValuation.map_sub_eq_of_lt_right
theorem map_sub_eq_of_lt_right {x y : R} (hx : v y < v x) :
    v (x - y) = v y 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R} (hx : v y < v x)
Goal: v (x - y) = v y
-- AddValuation.map_eq_of_lt_sub
theorem map_eq_of_lt_sub (h : v x < v (y - x)) : v y = v x 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R} (h : v x < v (y - x))
Goal: v y = v x
-- AddValuation.IsEquiv.refl
@[refl]
theorem refl : v.IsEquiv v 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [Ring R] {v : AddValuation R Γ₀}
Goal: v.IsEquiv v
-- AddValuation.IsEquiv.symm
@[symm]
theorem symm (h : v₁.IsEquiv v₂) : v₂.IsEquiv v₁ 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} (h : v₁.IsEquiv v₂)
Goal: v₂.IsEquiv v₁
-- AddValuation.IsEquiv.trans
@[trans]
theorem trans (h₁₂ : v₁.IsEquiv v₂) (h₂₃ : v₂.IsEquiv v₃) : v₁.IsEquiv v₃ 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {Γ''₀ : Type u_6} [LinearOrderedAddCommMonoidWithTop Γ''₀] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} {v₃ : AddValuation R Γ''₀} (h₁₂ : v₁.IsEquiv v₂) (h₂₃ : v₂.IsEquiv v₃)
Goal: v₁.IsEquiv v₃
-- AddValuation.IsEquiv.of_eq
theorem of_eq {v' : AddValuation R Γ₀} (h : v = v') : v.IsEquiv v' 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [Ring R] {v v' : AddValuation R Γ₀} (h : v = v')
Goal: v.IsEquiv v'
-- AddValuation.IsEquiv.map
theorem map {v' : AddValuation R Γ₀} (f : Γ₀ →+ Γ'₀) (ht : f ⊤ = ⊤) (hf : Monotone f)
    (inf : Injective f) (h : v.IsEquiv v') : (v.map f ht hf).IsEquiv (v'.map f ht hf) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v v' : AddValuation R Γ₀} (f : Γ₀ →+ Γ'₀) (ht : f ⊤ = ⊤) (hf : Monotone ⇑f) (inf : Function.Injective ⇑f) (h : v.IsEquiv v')
Goal: (AddValuation.map f ht hf v).IsEquiv (AddValuation.map f ht hf v')
-- AddValuation.IsEquiv.comap
theorem comap {S : Type*} [Ring S] (f : S →+* R) (h : v₁.IsEquiv v₂) :
    (v₁.comap f).IsEquiv (v₂.comap f) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} {S : Type u_7} [Ring S] (f : S →+* R) (h : v₁.IsEquiv v₂)
Goal: (AddValuation.comap f v₁).IsEquiv (AddValuation.comap f v₂)
-- AddValuation.IsEquiv.val_eq
theorem val_eq (h : v₁.IsEquiv v₂) {r s : R} : v₁ r = v₁ s ↔ v₂ r = v₂ s 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} (h : v₁.IsEquiv v₂) {r s : R}
Goal: v₁ r = v₁ s ↔ v₂ r = v₂ s
-- AddValuation.IsEquiv.ne_top
theorem ne_top (h : v₁.IsEquiv v₂) {r : R} : v₁ r ≠ (⊤ : Γ₀) ↔ v₂ r ≠ (⊤ : Γ'₀) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} {Γ'₀ : Type u_5} [LinearOrderedAddCommMonoidWithTop Γ₀] [LinearOrderedAddCommMonoidWithTop Γ'₀] [Ring R] {v₁ : AddValuation R Γ₀} {v₂ : AddValuation R Γ'₀} (h : v₁.IsEquiv v₂) {r : R}
Goal: v₁ r ≠ ⊤ ↔ v₂ r ≠ ⊤
-- AddValuation.mem_supp_iff
@[simp]
theorem mem_supp_iff (x : R) : x ∈ supp v ↔ v x = (⊤ : Γ₀) 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [CommRing R] (v : AddValuation R Γ₀) (x : R)
Goal: x ∈ v.supp ↔ v x = ⊤
-- AddValuation.map_add_supp
theorem map_add_supp (a : R) {s : R} (h : s ∈ supp v) : v (a + s) = v a 
Arguments: {R : Type u_3} {Γ₀ : Type u_4} [LinearOrderedAddCommMonoidWithTop Γ₀] [CommRing R] (v : AddValuation R Γ₀) (a : R) {s : R} (h : s ∈ v.supp)
Goal: v (a + s) = v a
-- RingHom.finitePresentation_localizationPreserves
theorem finitePresentation_localizationPreserves : LocalizationPreserves @FinitePresentation 
Goal: RingHom.LocalizationPreserves @RingHom.FinitePresentation
-- RingHom.finitePresentation_stableUnderComposition
theorem finitePresentation_stableUnderComposition : StableUnderComposition @FinitePresentation 
Goal: RingHom.StableUnderComposition @RingHom.FinitePresentation
-- RingHom.finitePresentation_holdsForLocalizationAway
theorem finitePresentation_holdsForLocalizationAway :
    HoldsForLocalizationAway @FinitePresentation 
Goal: RingHom.HoldsForLocalizationAway @RingHom.FinitePresentation