-- Mathlib/RingTheory/HahnSeries/Multiplication.lean
theorem orderTop_vAdd_le_orderTop_smul {Γ Γ'} [LinearOrder Γ] [LinearOrder Γ'] [VAdd Γ Γ']
    [IsOrderedCancelVAdd Γ Γ'] [MulZeroClass R] [SMulWithZero R V] {x : HahnSeries Γ R}
    [VAdd (WithTop Γ) (WithTop Γ')] {y : HahnModule Γ' R V}
    (h : ∀ (γ : Γ) (γ' : Γ'), γ +ᵥ γ' = (γ : WithTop Γ) +ᵥ (γ' : WithTop Γ')) :
    x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Proof state:
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Tactic:
by_cases hx : x = 0

Result:
case pos
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : x = 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Proof state:
case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : ¬y = 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Tactic:
have hhy : ((of R).symm y) ≠ 0 := hy

Result:
case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : ¬y = 0
hhy : (of R).symm y ≠ 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Proof state:
case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Tactic:
by_cases hy : y = 0

Result:
case pos
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : y = 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : ¬y = 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Proof state:
case pos
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : y = 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Tactic:
simp_all

Result:
no goals

Proof state:
case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : ¬y = 0
hhy : (of R).symm y ≠ 0
hxy : ¬(of R).symm (x • y) = 0
⊢ ↑(⋯.min ⋯) ≤ ((of R).symm (x • y)).orderTop

Tactic:
rw [HahnSeries.orderTop_of_ne hxy, WithTop.coe_le_coe]

Result:
case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : ¬y = 0
hhy : (of R).symm y ≠ 0
hxy : ¬(of R).symm (x • y) = 0
⊢ ⋯.min ⋯ ≤ ⋯.min ⋯

Proof state:
case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : ¬y = 0
hhy : (of R).symm y ≠ 0
⊢ ↑(⋯.min ⋯) ≤ ((of R).symm (x • y)).orderTop

Tactic:
by_cases hxy : (of R).symm (x • y) = 0

Result:
case pos
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : ¬y = 0
hhy : (of R).symm y ≠ 0
hxy : (of R).symm (x • y) = 0
⊢ ↑(⋯.min ⋯) ≤ ((of R).symm (x • y)).orderTop

case neg
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : ¬x = 0
hy : ¬y = 0
hhy : (of R).symm y ≠ 0
hxy : ¬(of R).symm (x • y) = 0
⊢ ↑(⋯.min ⋯) ≤ ((of R).symm (x • y)).orderTop

Proof state:
case pos
R : Type u_3
V : Type u_5
inst✝⁷ : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : MulZeroClass R
inst✝¹ : SMulWithZero R V
x : HahnSeries Γ R
inst✝ : VAdd (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), ↑(γ +ᵥ γ') = ↑γ +ᵥ ↑γ'
hx : x = 0
⊢ x.orderTop +ᵥ ((of R).symm y).orderTop ≤ ((of R).symm (x • y)).orderTop

Tactic:
simp_all

Result:
no goals