mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.HahnSeries.Addition

Goal state before:
pp: R : Type u_3
inst✝¹ : AddGroup R
Γ : Type u_8
inst✝ : LinearOrder Γ
x y : R⟦Γ⟧
hxy : x.orderTop < (-y).orderTop
⊢ (x + -y).leadingCoeff = x.leadingCoeff

Tactic:
]

Goal state after:
R : Type u_3
inst✝¹ : AddGroup R
Γ : Type u_8
inst✝ : LinearOrder Γ
x y : R⟦Γ⟧
hxy : x.orderTop < (-y).orderTop
⊢ (x + -y).leadingCoeff = x.leadingCoeff
Module: Mathlib.RingTheory.HahnSeries.Addition

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddGroup R
x y : R⟦Γ⟧
g : Γ
hyg : y.orderTop = ↑g
hxyc : x.leadingCoeff = y.leadingCoeff
hx : x ≠ 0
hxg : ⋯.min ⋯ = g
⊢ (x - y).coeff g = 0

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddGroup R
x y : R⟦Γ⟧
g : Γ
hyg : y.orderTop = ↑g
hxyc : x.leadingCoeff = y.leadingCoeff
hx : x ≠ 0
hxg : ⋯.min ⋯ = g
⊢ (x - y).coeff g = 0
Module: Mathlib.RingTheory.HahnSeries.Addition

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddGroup R
x y : R⟦Γ⟧
g : Γ
hxyc : x.leadingCoeff = y.leadingCoeff
hx : x ≠ 0
hxg : ⋯.min ⋯ = g
hy : y ≠ 0
hyg : ⋯.min ⋯ = g
⊢ (x - y).coeff g = 0

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddGroup R
x y : R⟦Γ⟧
g : Γ
hxyc : x.leadingCoeff = y.leadingCoeff
hx : x ≠ 0
hxg : ⋯.min ⋯ = g
hy : y ≠ 0
hyg : ⋯.min ⋯ = g
⊢ (x - y).coeff g = 0
Module: Mathlib.RingTheory.HahnSeries.Addition

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddGroup R
x y : R⟦Γ⟧
g : Γ
hx : x ≠ 0
hxg : ⋯.min ⋯ = g
hy : y ≠ 0
hyg : ⋯.min ⋯ = g
hxyc : x.coeff g = y.coeff g
⊢ x.coeff g = y.coeff g

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddGroup R
x y : R⟦Γ⟧
g : Γ
hx : x ≠ 0
hxg : ⋯.min ⋯ = g
hy : y ≠ 0
hyg : ⋯.min ⋯ = g
hxyc : x.coeff g = y.coeff g
⊢ x.coeff g = y.coeff g
Module: Mathlib.RingTheory.HahnSeries.Addition

Goal state before:
pp: R : Type u_3
inst✝¹ : AddGroup R
Γ : Type u_8
inst✝ : LinearOrder Γ
x y : R⟦Γ⟧
g : Γ
hxg : x.orderTop = ↑g
hyg : y.orderTop = ↑g
hxyc : x.leadingCoeff = y.leadingCoeff
⊢ ↑g = ↑g

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Module.PositiveLinearMap

Goal state before:
pp: R : Type u_1
E₁ : Type u_2
E₂ : Type u_3
inst✝⁸ : Semiring R
inst✝⁷ : AddCommGroup E₁
inst✝⁶ : PartialOrder E₁
inst✝⁵ : IsOrderedAddMonoid E₁
inst✝⁴ : AddCommGroup E₂
inst✝³ : PartialOrder E₂
inst✝² : IsOrderedAddMonoid E₂
inst✝¹ : Module R E₁
inst✝ : Module R E₂
f : E₁ →ₗ[R] E₂
hf : ∀ (x : E₁), 0 ≤ x → 0 ≤ f x
a b : E₁
hab : 0 ≤ b - a
⊢ 0 ≤ f.toFun b - f.toFun a

Tactic:
]

Goal state after:
R : Type u_1
E₁ : Type u_2
E₂ : Type u_3
inst✝⁸ : Semiring R
inst✝⁷ : AddCommGroup E₁
inst✝⁶ : PartialOrder E₁
inst✝⁵ : IsOrderedAddMonoid E₁
inst✝⁴ : AddCommGroup E₂
inst✝³ : PartialOrder E₂
inst✝² : IsOrderedAddMonoid E₂
inst✝¹ : Module R E₁
inst✝ : Module R E₂
f : E₁ →ₗ[R] E₂
hf : ∀ (x : E₁), 0 ≤ x → 0 ≤ f x
a b : E₁
hab : 0 ≤ b - a
⊢ 0 ≤ f.toFun b - f.toFun a
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: Γ : Type u_1
R : Type u_2
inst✝² : LinearOrder Γ
inst✝¹ : Zero R
inst✝ : LinearOrder R
a b : Lex R⟦Γ⟧
hab✝ : a ≠ b
hab : ∃ a_1, a.coeff a_1 ≠ b.coeff a_1
u : Set Γ := {i | (ofLex a).coeff i ≠ 0} ∪ {i | (ofLex b).coeff i ≠ 0}
v : Set Γ := {i | (ofLex a).coeff i ≠ (ofLex b).coeff i}
i : Γ
h : i ∈ v
⊢ (ofLex a).coeff i ≠ 0 ∨ (ofLex b).coeff i ≠ 0

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_2
inst✝² : LinearOrder Γ
inst✝¹ : Zero R
inst✝ : LinearOrder R
a b : Lex R⟦Γ⟧
hab✝ : a ≠ b
hab : ∃ a_1, a.coeff a_1 ≠ b.coeff a_1
u : Set Γ := {i | (ofLex a).coeff i ≠ 0} ∪ {i | (ofLex b).coeff i ≠ 0}
v : Set Γ := {i | (ofLex a).coeff i ≠ (ofLex b).coeff i}
i : Γ
h : i ∈ v
⊢ (ofLex a).coeff i ≠ 0 ∨ (ofLex b).coeff i ≠ 0
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: Γ : Type u_1
R : Type u_2
inst✝² : LinearOrder Γ
inst✝¹ : Zero R
inst✝ : LinearOrder R
a b : Lex R⟦Γ⟧
hab✝ : a ≠ b
hab : ∃ a_1, a.coeff a_1 ≠ b.coeff a_1
u : Set Γ := {i | (ofLex a).coeff i ≠ 0} ∪ {i | (ofLex b).coeff i ≠ 0}
v : Set Γ := {i | (ofLex a).coeff i ≠ (ofLex b).coeff i}
i : Γ
h : (ofLex a).coeff i = 0 ∧ (ofLex b).coeff i = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: Γ : Type u_1
R : Type u_2
inst✝² : LinearOrder Γ
inst✝¹ : Zero R
inst✝ : LinearOrder R
x : Lex R⟦Γ⟧
⊢ 0 < (ofLex x).leadingCoeff ↔
    ∃ i, (∀ j < i, (ofLex 0).coeff j = (ofLex x).coeff j) ∧ (ofLex 0).coeff i < (ofLex x).coeff i

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_2
inst✝² : LinearOrder Γ
inst✝¹ : Zero R
inst✝ : LinearOrder R
x : Lex R⟦Γ⟧
⊢ 0 < (ofLex x).leadingCoeff ↔
    ∃ i, (∀ j < i, (ofLex 0).coeff j = (ofLex x).coeff j) ∧ (ofLex 0).coeff i < (ofLex x).coeff i
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case mpr
Γ : Type u_1
R : Type u_2
inst✝² : LinearOrder Γ
inst✝¹ : Zero R
inst✝ : LinearOrder R
x : Lex R⟦Γ⟧
i : Γ
hj : ∀ j < i, (ofLex 0).coeff j = (ofLex x).coeff j
hi : (ofLex 0).coeff i < (ofLex x).coeff i
horder : (ofLex x).orderTop = ↑i
htop : (ofLex x).orderTop ≠ ⊤
hne : ofLex x ≠ 0
horder' : (ofLex x).orderTop.untop htop = i
⊢ 0 < (ofLex x).coeff i

Tactic:
]

Goal state after:
case mpr
Γ : Type u_1
R : Type u_2
inst✝² : LinearOrder Γ
inst✝¹ : Zero R
inst✝ : LinearOrder R
x : Lex R⟦Γ⟧
i : Γ
hj : ∀ j < i, (ofLex 0).coeff j = (ofLex x).coeff j
hi : (ofLex 0).coeff i < (ofLex x).coeff i
horder : (ofLex x).orderTop = ↑i
htop : (ofLex x).orderTop ≠ ⊤
hne : ofLex x ≠ 0
horder' : (ofLex x).orderTop.untop htop = i
⊢ 0 < (ofLex x).coeff i
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr
Γ : Type u_1
R : Type u_2
inst✝⁴ : LinearOrder Γ
inst✝³ : PartialOrder R
inst✝² : AddCommMonoid R
inst✝¹ : AddLeftStrictMono R
inst✝ : IsOrderedAddMonoid R
a b : Lex R⟦Γ⟧
hab : a ≤ b
c : Lex R⟦Γ⟧
hlt : ∃ i, (∀ j < i, (ofLex a).coeff j = (ofLex b).coeff j) ∧ (ofLex a).coeff i < (ofLex b).coeff i
⊢ ∃ i, (∀ j < i, (ofLex (a + c)).coeff j = (ofLex (b + c)).coeff j) ∧ (ofLex (a + c)).coeff i < (ofLex (b + c)).coeff i

Tactic:
]

Goal state after:
case inr
Γ : Type u_1
R : Type u_2
inst✝⁴ : LinearOrder Γ
inst✝³ : PartialOrder R
inst✝² : AddCommMonoid R
inst✝¹ : AddLeftStrictMono R
inst✝ : IsOrderedAddMonoid R
a b : Lex R⟦Γ⟧
hab : a ≤ b
c : Lex R⟦Γ⟧
hlt : ∃ i, (∀ j < i, (ofLex a).coeff j = (ofLex b).coeff j) ∧ (ofLex a).coeff i < (ofLex b).coeff i
⊢ ∃ i, (∀ j < i, (ofLex (a + c)).coeff j = (ofLex (b + c)).coeff j) ∧ (ofLex (a + c)).coeff i < (ofLex (b + c)).coeff i
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inl
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hle : x ≤ 0
⊢ (ofLex (-x)).support = (ofLex x).support

Tactic:
]

Goal state after:
case inl
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hle : x ≤ 0
⊢ (ofLex (-x)).support = (ofLex x).support
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hge : 0 ≤ x
⊢ (ofLex x).support = (ofLex x).support

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inl
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hle : x ≤ 0
⊢ (ofLex x).orderTop = (ofLex x).orderTop

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hge : 0 ≤ x
⊢ (ofLex x).orderTop = (ofLex x).orderTop

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr
Γ : Type u_1
R : Type u_2
inst✝⁴ : LinearOrder Γ
inst✝³ : LinearOrder R
inst✝² : AddCommGroup R
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Zero Γ
x : Lex R⟦Γ⟧
hne : x ≠ 0
hne' : ofLex x ≠ 0
habs : ofLex |x| ≠ 0
⊢ (ofLex |x|).orderTop = (ofLex x).orderTop

Tactic:
]

Goal state after:
case inr
Γ : Type u_1
R : Type u_2
inst✝⁴ : LinearOrder Γ
inst✝³ : LinearOrder R
inst✝² : AddCommGroup R
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Zero Γ
x : Lex R⟦Γ⟧
hne : x ≠ 0
hne' : ofLex x ≠ 0
habs : ofLex |x| ≠ 0
⊢ (ofLex |x|).orderTop = (ofLex x).orderTop
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inl
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hlt : x < 0
hlt' : (ofLex x).leadingCoeff < 0
⊢ -(ofLex x).leadingCoeff = -(ofLex x).leadingCoeff

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr.inr
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hgt : 0 < x
hgt' : 0 < (ofLex x).leadingCoeff
⊢ (ofLex x).leadingCoeff = (ofLex x).leadingCoeff

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex |y|).orderTop < (ofLex |x|).orderTop
⊢ |x| < |y|

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex |y|).orderTop < (ofLex |x|).orderTop
⊢ |x| < |y|
Module: Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder

Goal state before:
pp: M : Type u_1
inst✝⁴ : CancelCommMonoid M
inst✝³ : LinearOrder M
inst✝² : IsOrderedMonoid M
inst✝¹ : LocallyFiniteOrder M
inst✝ : ExistsMulOfLE M
a b : M
ha : 1 ≤ a
hb : 1 ≤ b
this : Ico 1 b ∪ Ico (1 * b) (a * b) = Ico 1 (a * b)
⊢ #(Ico 1 b) + #(Ico 1 a) - #(Ico 1 b ∩ Ico (1 * b) (a * b)) = #(Ico 1 a) + #(Ico 1 b)

Tactic:
]

Goal state after:
M : Type u_1
inst✝⁴ : CancelCommMonoid M
inst✝³ : LinearOrder M
inst✝² : IsOrderedMonoid M
inst✝¹ : LocallyFiniteOrder M
inst✝ : ExistsMulOfLE M
a b : M
ha : 1 ≤ a
hb : 1 ≤ b
this : Ico 1 b ∪ Ico (1 * b) (a * b) = Ico 1 (a * b)
⊢ #(Ico 1 b) + #(Ico 1 a) - #(Ico 1 b ∩ Ico (1 * b) (a * b)) = #(Ico 1 a) + #(Ico 1 b)
Module: Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder

Goal state before:
pp: G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : LocallyFiniteOrder G
⊢ ∀ (a : G), 0 < a → 0 < (orderAddMonoidHom G) a

Tactic:
]

Goal state after:
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : LocallyFiniteOrder G
⊢ ∀ (a : G), 0 < a → 0 < (orderAddMonoidHom G) a
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr.mp
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |y| ≤ n • |x|
hn' : |y| < (n + 1) • |x|
j : Γ
hj : ∀ j_1 < j, (ofLex |y|).coeff j_1 = (n + 1) • (ofLex |x|).coeff j_1
hi : (ofLex |y|).coeff j < (n + 1) • (ofLex |x|).coeff j
⊢ (ofLex |y|).coeff ((ofLex |y|).orderTop.untop ⋯) ≤ (n + 1) • (ofLex |x|).coeff ((ofLex |x|).orderTop.untop ⋯)

Tactic:
]

Goal state after:
case inr.mp
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |y| ≤ n • |x|
hn' : |y| < (n + 1) • |x|
j : Γ
hj : ∀ j_1 < j, (ofLex |y|).coeff j_1 = (n + 1) • (ofLex |x|).coeff j_1
hi : (ofLex |y|).coeff j < (n + 1) • (ofLex |x|).coeff j
⊢ (ofLex |y|).coeff ((ofLex |y|).orderTop.untop ⋯) ≤ (n + 1) • (ofLex |x|).coeff ((ofLex |x|).orderTop.untop ⋯)
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulLeftMono M
u : Mˣ
a b : M
⊢ a ≤ ↑u * b ↔ a ≤ ↑u * b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulLeftMono M
u : Mˣ
a b : M
⊢ ↑u * a ≤ b ↔ ↑u * a ≤ b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulLeftMono M
u : Mˣ
⊢ ↑u ≤ 1 ↔ ↑u ≤ 1

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulLeftMono M
u : Mˣ
⊢ 1 ≤ ↑u ↔ 1 ≤ ↑u

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulLeftMono M
u : Mˣ
a : M
⊢ ↑u ≤ a ↔ ↑u ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulLeftMono M
u : Mˣ
a : M
⊢ a ≤ ↑u ↔ a ≤ ↑u

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulRightMono M
a b : M
u : Mˣ
⊢ a ≤ b * ↑u ↔ a ≤ b * ↑u

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulRightMono M
a b : M
u : Mˣ
⊢ a * ↑u ≤ b ↔ a * ↑u ≤ b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulRightMono M
a : M
u : Mˣ
⊢ ↑u ≤ a ↔ ↑u ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Units

Goal state before:
pp: M : Type u_1
inst✝² : Monoid M
inst✝¹ : LE M
inst✝ : MulRightMono M
a : M
u : Mˣ
⊢ a ≤ ↑u ↔ a ≤ ↑u

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr.mpr.refine_2
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|
⊢ (ofLex |y|).coeff ((ofLex x).orderTop.untop ⋯) < (n + 1) • (ofLex |x|).coeff ((ofLex x).orderTop.untop ⋯)

Tactic:
]

Goal state after:
case inr.mpr.refine_2
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|
⊢ (ofLex |y|).coeff ((ofLex x).orderTop.untop ⋯) < (n + 1) • (ofLex |x|).coeff ((ofLex x).orderTop.untop ⋯)
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr.mpr.refine_2
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|
⊢ (ofLex |y|).coeff ((ofLex x).orderTop.untop ⋯) < (ofLex ((n + 1) • |x|)).coeff ((ofLex x).orderTop.untop ⋯)

Tactic:
rw [ofLex_smul, coeff_smul]

Goal state after:
case inr.mpr.refine_2
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|
⊢ (ofLex |y|).coeff ((ofLex x).orderTop.untop ⋯) < (n + 1) • (ofLex |x|).coeff ((ofLex x).orderTop.untop ⋯)
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|
this : (ofLex |y|).coeff ((ofLex |y|).orderTop.untop ⋯) < (n + 1) • (ofLex |x|).coeff ((ofLex |x|).orderTop.untop ⋯)
⊢ (ofLex |y|).coeff ((ofLex x).orderTop.untop ⋯) < (n + 1) • (ofLex |x|).coeff ((ofLex x).orderTop.untop ⋯)

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|
this : (ofLex |y|).coeff ((ofLex |y|).orderTop.untop ⋯) < (n + 1) • (ofLex |x|).coeff ((ofLex |x|).orderTop.untop ⋯)
⊢ (ofLex |y|).coeff ((ofLex x).orderTop.untop ⋯) < (n + 1) • (ofLex |x|).coeff ((ofLex x).orderTop.untop ⋯)
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝⁸ : DivisionRing K
inst✝⁷ : LinearOrder K
inst✝⁶ : IsOrderedRing K
inst✝⁵ : Archimedean K
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : LinearOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : Module K M
inst✝ : IsOrderedModule K M
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : M
ha : a ∈ u.stratum c
h0 : a ≠ 0
⊢ a ∈ ball K c ⊔ u.stratum c

Tactic:
]

Goal state after:
K : Type u_1
inst✝⁸ : DivisionRing K
inst✝⁷ : LinearOrder K
inst✝⁶ : IsOrderedRing K
inst✝⁵ : Archimedean K
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : LinearOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : Module K M
inst✝ : IsOrderedModule K M
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : M
ha : a ∈ u.stratum c
h0 : a ≠ 0
⊢ a ∈ ball K c ⊔ u.stratum c
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : ↥(u.stratum c)
ha : a ≠ 0
b : ↥(u.stratum c)
hb : b ≠ 0
this : (∃ m, |↑b| ≤ m • |↑a|) ∧ ∃ n, |↑a| ≤ n • |↑b|
⊢ (∃ m, |b| ≤ m • |a|) ∧ ∃ n, |a| ≤ n • |b|

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : ↥(u.stratum c)
ha : a ≠ 0
b : ↥(u.stratum c)
hb : b ≠ 0
this : (∃ m, |↑b| ≤ m • |↑a|) ∧ ∃ n, |↑a| ≤ n • |↑b|
⊢ (∃ m, |b| ≤ m • |a|) ∧ ∃ n, |a| ≤ n • |b|
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inr.mpr.refine_2
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|
⊢ ofLex x ≠ 0

Tactic:
]

Goal state after:
case inr.mpr.refine_2
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex x).orderTop = (ofLex y).orderTop
hy : y ≠ 0
hx : x ≠ 0
h' : (ofLex |x|).orderTop = (ofLex |y|).orderTop
n : ℕ
hn : |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|
⊢ ofLex x ≠ 0
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : ↥(u.stratum c)
ha : a ≠ 0
b : ↥(u.stratum c)
hb : b ≠ 0
⊢ ↑c = ArchimedeanClass.mk ↑b

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : ↥(u.stratum c)
ha : a ≠ 0
b : ↥(u.stratum c)
hb : b ≠ 0
⊢ ↑c = ArchimedeanClass.mk ↑b
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : ↥(u.stratum c)
ha : a ≠ 0
b : ↥(u.stratum c)
hb : b ≠ 0
⊢ ↑c = ↑c

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝⁸ : DivisionRing K
inst✝⁷ : LinearOrder K
inst✝⁶ : IsOrderedRing K
inst✝⁵ : Archimedean K
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : LinearOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : Module K M
inst✝ : IsOrderedModule K M
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
⊢ ∀ x ∈ u.stratum c, ∀ y ∈ ⨆ j, ⨆ (_ : j ≠ c), u.stratum j, x = y → x = 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝⁸ : DivisionRing K
inst✝⁷ : LinearOrder K
inst✝⁶ : IsOrderedRing K
inst✝⁵ : Archimedean K
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : LinearOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : Module K M
inst✝ : IsOrderedModule K M
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
⊢ ∀ x ∈ u.stratum c, ∀ y ∈ ⨆ j, ⨆ (_ : j ≠ c), u.stratum j, x = y → x = 0
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝⁸ : DivisionRing K
inst✝⁷ : LinearOrder K
inst✝⁶ : IsOrderedRing K
inst✝⁵ : Archimedean K
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : LinearOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : Module K M
inst✝ : IsOrderedModule K M
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : M
ha : a ∈ u.stratum c
b : M
hb : b ∈ ⨆ j, ⨆ (_ : j ≠ c), u.stratum j
hab : a = b
f : Π₀ (i : FiniteArchimedeanClass M), ↥(⨆ (_ : i ≠ c), u.stratum i)
hf : (f.sum fun x xi ↦ ↑xi) = b
h0 : a ≠ 0
⊢ ArchimedeanClass.mk (∑ i ∈ f.support, ↑(f i)) ≠ ArchimedeanClass.mk a

Tactic:
]

Goal state after:
K : Type u_1
inst✝⁸ : DivisionRing K
inst✝⁷ : LinearOrder K
inst✝⁶ : IsOrderedRing K
inst✝⁵ : Archimedean K
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : LinearOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : Module K M
inst✝ : IsOrderedModule K M
u : ArchimedeanStrata K M
c : FiniteArchimedeanClass M
a : M
ha : a ∈ u.stratum c
b : M
hb : b ∈ ⨆ j, ⨆ (_ : j ≠ c), u.stratum j
hab : a = b
f : Π₀ (i : FiniteArchimedeanClass M), ↥(⨆ (_ : i ≠ c), u.stratum i)
hf : (f.sum fun x xi ↦ ↑xi) = b
h0 : a ≠ 0
⊢ ArchimedeanClass.mk (∑ i ∈ f.support, ↑(f i)) ≠ ArchimedeanClass.mk a