Theorem: HahnSeries.SummableFamily.hsum_smul
File: Mathlib/RingTheory/HahnSeries/Summable.lean

tactic: ext g
state_before: Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
⊢ (x • s).hsum = x * s.hsum
state_after: case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ (x • s).hsum.coeff g = (x * s.hsum).coeff g

tactic: simp only [mul_coeff, hsum_coeff, smul_apply]
state_before: case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ (x • s).hsum.coeff g = (x * s.hsum).coeff g
state_after: case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∑ᶠ (i : α), ∑ ij ∈ addAntidiagonal ⋯ ⋯ g, x.coeff ij.1 * (s i).coeff ij.2 =
    ∑ x_1 ∈ addAntidiagonal ⋯ ⋯ g, x.coeff x_1.1 * ∑ᶠ (i : α), (s i).coeff x_1.2

tactic: refine
  (Eq.trans (finsum_congr fun a => ?_)
        (finsum_sum_comm (addAntidiagonal x.isPWO_support s.isPWO_iUnion_support g)
          (fun i ij => x.coeff (Prod.fst ij) * (s i).coeff ij.snd) ?_)).trans
    ?_
state_before: case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∑ᶠ (i : α), ∑ ij ∈ addAntidiagonal ⋯ ⋯ g, x.coeff ij.1 * (s i).coeff ij.2 =
    ∑ x_1 ∈ addAntidiagonal ⋯ ⋯ g, x.coeff x_1.1 * ∑ᶠ (i : α), (s i).coeff x_1.2
state_after: case coeff.h.refine_1
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
a : α
⊢ ∑ ij ∈ addAntidiagonal ⋯ ⋯ g, x.coeff ij.1 * (s a).coeff ij.2 =
    ∑ b ∈ addAntidiagonal ⋯ ⋯ g, x.coeff b.1 * (s a).coeff b.2

case coeff.h.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∀ b ∈ addAntidiagonal ⋯ ⋯ g, (Function.support fun a => (fun i ij => x.coeff ij.1 * (s i).coeff ij.2) a b).Finite

case coeff.h.refine_3
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∑ b ∈ addAntidiagonal ⋯ ⋯ g, ∑ᶠ (a : α), x.coeff b.1 * (s a).coeff b.2 =
    ∑ x_1 ∈ addAntidiagonal ⋯ ⋯ g, x.coeff x_1.1 * ∑ᶠ (i : α), (s i).coeff x_1.2

tactic: refine sum_subset (addAntidiagonal_mono_right
  (Set.subset_iUnion (fun j => support (toFun s j)) a)) ?_
state_before: case coeff.h.refine_1
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
a : α
⊢ ∑ ij ∈ addAntidiagonal ⋯ ⋯ g, x.coeff ij.1 * (s a).coeff ij.2 =
    ∑ b ∈ addAntidiagonal ⋯ ⋯ g, x.coeff b.1 * (s a).coeff b.2
state_after: case coeff.h.refine_1
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
a : α
⊢ ∀ x_1 ∈ addAntidiagonal ⋯ ⋯ g, x_1 ∉ addAntidiagonal ⋯ ⋯ g → x.coeff x_1.1 * (s a).coeff x_1.2 = 0

tactic: rintro ⟨i, j⟩ hU ha
state_before: case coeff.h.refine_1
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
a : α
⊢ ∀ x_1 ∈ addAntidiagonal ⋯ ⋯ g, x_1 ∉ addAntidiagonal ⋯ ⋯ g → x.coeff x_1.1 * (s a).coeff x_1.2 = 0
state_after: case coeff.h.refine_1.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
a : α
i j : Γ
hU : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
ha : (i, j) ∉ addAntidiagonal ⋯ ⋯ g
⊢ x.coeff (i, j).1 * (s a).coeff (i, j).2 = 0

tactic: rw [mem_addAntidiagonal] at *
state_before: case coeff.h.refine_1.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
a : α
i j : Γ
hU : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
ha : (i, j) ∉ addAntidiagonal ⋯ ⋯ g
⊢ x.coeff (i, j).1 * (s a).coeff (i, j).2 = 0
state_after: case coeff.h.refine_1.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
a : α
i j : Γ
hU : (i, j).1 ∈ x.support ∧ (i, j).2 ∈ ⋃ a, (s a).support ∧ (i, j).1 + (i, j).2 = g
ha : ¬((i, j).1 ∈ x.support ∧ (i, j).2 ∈ (s a).support ∧ (i, j).1 + (i, j).2 = g)
⊢ x.coeff (i, j).1 * (s a).coeff (i, j).2 = 0

tactic: rw [Classical.not_not.1 fun con => ha ⟨hU.1, con, hU.2.2⟩, mul_zero]
state_before: case coeff.h.refine_1.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
a : α
i j : Γ
hU : (i, j).1 ∈ x.support ∧ (i, j).2 ∈ ⋃ a, (s a).support ∧ (i, j).1 + (i, j).2 = g
ha : ¬((i, j).1 ∈ x.support ∧ (i, j).2 ∈ (s a).support ∧ (i, j).1 + (i, j).2 = g)
⊢ x.coeff (i, j).1 * (s a).coeff (i, j).2 = 0
state_after: no goals

tactic: rintro ⟨i, j⟩ _
state_before: case coeff.h.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∀ b ∈ addAntidiagonal ⋯ ⋯ g, (Function.support fun a => (fun i ij => x.coeff ij.1 * (s i).coeff ij.2) a b).Finite
state_after: case coeff.h.refine_2.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ (Function.support fun a => (fun i ij => x.coeff ij.1 * (s i).coeff ij.2) a (i, j)).Finite

tactic: refine (s.finite_co_support j).subset ?_
state_before: case coeff.h.refine_2.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ (Function.support fun a => (fun i ij => x.coeff ij.1 * (s i).coeff ij.2) a (i, j)).Finite
state_after: case coeff.h.refine_2.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ (Function.support fun a => (fun i ij => x.coeff ij.1 * (s i).coeff ij.2) a (i, j)) ⊆
    Function.support fun a => (s a).coeff j

tactic: simp_rw [Function.support_subset_iff', Function.mem_support, Classical.not_not]
state_before: case coeff.h.refine_2.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ (Function.support fun a => (fun i ij => x.coeff ij.1 * (s i).coeff ij.2) a (i, j)) ⊆
    Function.support fun a => (s a).coeff j
state_after: case coeff.h.refine_2.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ ∀ (x_1 : α), (s x_1).coeff j = 0 → x.coeff i * (s x_1).coeff j = 0

tactic: intro a ha
state_before: case coeff.h.refine_2.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ ∀ (x_1 : α), (s x_1).coeff j = 0 → x.coeff i * (s x_1).coeff j = 0
state_after: case coeff.h.refine_2.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
a : α
ha : (s a).coeff j = 0
⊢ x.coeff i * (s a).coeff j = 0

tactic: rw [ha, mul_zero]
state_before: case coeff.h.refine_2.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
a : α
ha : (s a).coeff j = 0
⊢ x.coeff i * (s a).coeff j = 0
state_after: no goals

tactic: refine (sum_congr rfl ?_).trans (sum_subset (addAntidiagonal_mono_right ?_) ?_).symm
state_before: case coeff.h.refine_3
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∑ b ∈ addAntidiagonal ⋯ ⋯ g, ∑ᶠ (a : α), x.coeff b.1 * (s a).coeff b.2 =
    ∑ x_1 ∈ addAntidiagonal ⋯ ⋯ g, x.coeff x_1.1 * ∑ᶠ (i : α), (s i).coeff x_1.2
state_after: case coeff.h.refine_3.refine_1
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∀ x_1 ∈ addAntidiagonal ⋯ ⋯ g,
    ∑ᶠ (a : α), x.coeff x_1.1 * (s a).coeff x_1.2 = x.coeff x_1.1 * ∑ᶠ (i : α), (s i).coeff x_1.2

case coeff.h.refine_3.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ s.hsum.support ⊆ ⋃ a, (s a).support

case coeff.h.refine_3.refine_3
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∀ x_1 ∈ addAntidiagonal ⋯ ⋯ g, x_1 ∉ addAntidiagonal ⋯ ⋯ g → x.coeff x_1.1 * ∑ᶠ (i : α), (s i).coeff x_1.2 = 0

tactic: rintro ⟨i, j⟩ _
state_before: case coeff.h.refine_3.refine_1
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∀ x_1 ∈ addAntidiagonal ⋯ ⋯ g,
    ∑ᶠ (a : α), x.coeff x_1.1 * (s a).coeff x_1.2 = x.coeff x_1.1 * ∑ᶠ (i : α), (s i).coeff x_1.2
state_after: case coeff.h.refine_3.refine_1.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ ∑ᶠ (a : α), x.coeff (i, j).1 * (s a).coeff (i, j).2 = x.coeff (i, j).1 * ∑ᶠ (i_1 : α), (s i_1).coeff (i, j).2

tactic: rw [mul_finsum]
state_before: case coeff.h.refine_3.refine_1.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ ∑ᶠ (a : α), x.coeff (i, j).1 * (s a).coeff (i, j).2 = x.coeff (i, j).1 * ∑ᶠ (i_1 : α), (s i_1).coeff (i, j).2
state_after: case coeff.h.refine_3.refine_1.mk.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ (Function.support fun i_1 => (s i_1).coeff (i, j).2).Finite

tactic: apply s.finite_co_support
state_before: case coeff.h.refine_3.refine_1.mk.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
a✝ : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
⊢ (Function.support fun i_1 => (s i_1).coeff (i, j).2).Finite
state_after: no goals

tactic: intro x hx
state_before: case coeff.h.refine_3.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ s.hsum.support ⊆ ⋃ a, (s a).support
state_after: case coeff.h.refine_3.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x✝ : HahnSeries Γ R
s : SummableFamily Γ R α
g x : Γ
hx : x ∈ s.hsum.support
⊢ x ∈ ⋃ a, (s a).support

tactic: simp only [Set.mem_iUnion, Ne, mem_support]
state_before: case coeff.h.refine_3.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x✝ : HahnSeries Γ R
s : SummableFamily Γ R α
g x : Γ
hx : x ∈ s.hsum.support
⊢ x ∈ ⋃ a, (s a).support
state_after: case coeff.h.refine_3.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x✝ : HahnSeries Γ R
s : SummableFamily Γ R α
g x : Γ
hx : x ∈ s.hsum.support
⊢ ∃ i, ¬(s i).coeff x = 0

tactic: contrapose! hx
state_before: case coeff.h.refine_3.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x✝ : HahnSeries Γ R
s : SummableFamily Γ R α
g x : Γ
hx : x ∈ s.hsum.support
⊢ ∃ i, ¬(s i).coeff x = 0
state_after: case coeff.h.refine_3.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x✝ : HahnSeries Γ R
s : SummableFamily Γ R α
g x : Γ
hx : ∀ (i : α), (s i).coeff x = 0
⊢ x ∉ s.hsum.support

tactic: simp [hx]
state_before: case coeff.h.refine_3.refine_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x✝ : HahnSeries Γ R
s : SummableFamily Γ R α
g x : Γ
hx : ∀ (i : α), (s i).coeff x = 0
⊢ x ∉ s.hsum.support
state_after: no goals

tactic: rintro ⟨i, j⟩ hU ha
state_before: case coeff.h.refine_3.refine_3
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g : Γ
⊢ ∀ x_1 ∈ addAntidiagonal ⋯ ⋯ g, x_1 ∉ addAntidiagonal ⋯ ⋯ g → x.coeff x_1.1 * ∑ᶠ (i : α), (s i).coeff x_1.2 = 0
state_after: case coeff.h.refine_3.refine_3.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
hU : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
ha : (i, j) ∉ addAntidiagonal ⋯ ⋯ g
⊢ x.coeff (i, j).1 * ∑ᶠ (i_1 : α), (s i_1).coeff (i, j).2 = 0

tactic: rw [mem_addAntidiagonal] at *
state_before: case coeff.h.refine_3.refine_3.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
hU : (i, j) ∈ addAntidiagonal ⋯ ⋯ g
ha : (i, j) ∉ addAntidiagonal ⋯ ⋯ g
⊢ x.coeff (i, j).1 * ∑ᶠ (i_1 : α), (s i_1).coeff (i, j).2 = 0
state_after: case coeff.h.refine_3.refine_3.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
hU : (i, j).1 ∈ x.support ∧ (i, j).2 ∈ ⋃ a, (s a).support ∧ (i, j).1 + (i, j).2 = g
ha : ¬((i, j).1 ∈ x.support ∧ (i, j).2 ∈ s.hsum.support ∧ (i, j).1 + (i, j).2 = g)
⊢ x.coeff (i, j).1 * ∑ᶠ (i_1 : α), (s i_1).coeff (i, j).2 = 0

tactic: rw [← hsum_coeff, Classical.not_not.1 fun con => ha ⟨hU.1, con, hU.2.2⟩,
  mul_zero]
state_before: case coeff.h.refine_3.refine_3.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
α : Type u_3
x : HahnSeries Γ R
s : SummableFamily Γ R α
g i j : Γ
hU : (i, j).1 ∈ x.support ∧ (i, j).2 ∈ ⋃ a, (s a).support ∧ (i, j).1 + (i, j).2 = g
ha : ¬((i, j).1 ∈ x.support ∧ (i, j).2 ∈ s.hsum.support ∧ (i, j).1 + (i, j).2 = g)
⊢ x.coeff (i, j).1 * ∑ᶠ (i_1 : α), (s i_1).coeff (i, j).2 = 0
state_after: no goals