awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
rw :
  ∀ (g : G.ConnectedComponent → ℝ),
    Eq.{u_1 + 1} (α :=
        ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
            ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
        (∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i))
        0 →
      ∀ (i : G.ConnectedComponent), g i = 0
⊢ LinearIndependent ℝ (M :=
    ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
        ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
    G.lapMatrix_ker_basis_aux

Tactic:
rw [Fintype.linearIndependent_iff]

Premises used: Fintype.linearIndependent_iff
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α :=
    ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
        ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
    (∑ i : G.ConnectedComponent,
      HSMul.hSMul (β :=
        ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
            ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
        (g i) (G.lapMatrix_ker_basis_aux i))
    0
⊢ Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)

Tactic:
rw [Subtype.ext_iff] at h0

Premises used: Subtype.ext_iff
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)
simp :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ x : G.ConnectedComponent,
          Subtype.mk (p := fun (x : V → ℝ) ↦
            Membership.mem (γ := Submodule ℝ (V → ℝ))
              (LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
              x)
            (HSMul.hSMul (β := V → ℝ) (g x) fun (i : V) ↦ if G.connectedComponentMk i = x then 1 else 0) ⋯) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)
⊢ Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)

Tactic:
simp only [lapMatrix_ker_basis_aux, SetLike.mk_smul_mk]

Premises used: SimpleGraph.lapMatrix_ker_basis_aux, SetLike.mk_smul_mk
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)
rw :
  Eq.{u_1 + 1} (α := V → ℝ)
    (∑ i : G.ConnectedComponent,
      (↑(Subtype.mk (p := fun (x : V → ℝ) ↦
            Membership.mem (γ := Submodule ℝ (V → ℝ))
              (LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
              x)
            (HSMul.hSMul (β := V → ℝ) (g i) fun (i_1 : V) ↦ if G.connectedComponentMk i_1 = i then 1 else 0) ⋯) :
        V → ℝ))
    fun (i : V) ↦ g (G.connectedComponentMk i)
⊢ Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ x : G.ConnectedComponent,
          Subtype.mk (p := fun (x : V → ℝ) ↦
            Membership.mem (γ := Submodule ℝ (V → ℝ))
              (LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
              x)
            (HSMul.hSMul (β := V → ℝ) (g x) fun (i : V) ↦ if G.connectedComponentMk i = x then 1 else 0) ⋯) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)

Tactic:
rw [AddSubmonoid.coe_finsetSum]

Premises used: AddSubmonoid.coe_finsetSum
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)
repeat :
  Eq.{u_1 + 1} (α := V → ℝ)
    (∑ i : G.ConnectedComponent,
      (↑(Subtype.mk (p := fun (x : V → ℝ) ↦
            Membership.mem (γ := Submodule ℝ (V → ℝ))
              (LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
              x)
            (HSMul.hSMul (β := V → ℝ) (g i) fun (i_1 : V) ↦ if G.connectedComponentMk i_1 = i then 1 else 0) ⋯) :
        V → ℝ))
    fun (i : V) ↦ g (G.connectedComponentMk i)
⊢ Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ x : G.ConnectedComponent,
          Subtype.mk (p := fun (x : V → ℝ) ↦
            Membership.mem (γ := Submodule ℝ (V → ℝ))
              (LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
              x)
            (HSMul.hSMul (β := V → ℝ) (g x) fun (i : V) ↦ if G.connectedComponentMk i = x then 1 else 0) ⋯) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)

Tactic:
repeat rw [AddSubmonoid.coe_finsetSum]

Premises used: AddSubmonoid.coe_finsetSum
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)
h :
  ∀ (i : V),
    (∑ i : G.ConnectedComponent,
          (↑(Subtype.mk (p := fun (x : V → ℝ) ↦
                Membership.mem (γ := Submodule ℝ (V → ℝ))
                  (LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                    ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
                  x)
                (HSMul.hSMul (β := V → ℝ) (g i) fun (i_1 : V) ↦ if G.connectedComponentMk i_1 = i then 1 else 0) ⋯) :
            V → ℝ))
        i =
      g (G.connectedComponentMk i)
⊢ Eq.{u_1 + 1} (α := V → ℝ)
    (∑ i : G.ConnectedComponent,
      (↑(Subtype.mk (p := fun (x : V → ℝ) ↦
            Membership.mem (γ := Submodule ℝ (V → ℝ))
              (LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
              x)
            (HSMul.hSMul (β := V → ℝ) (g i) fun (i_1 : V) ↦ if G.connectedComponentMk i_1 = i then 1 else 0) ⋯) :
        V → ℝ))
    fun (i : V) ↦ g (G.connectedComponentMk i)

Tactic:
ext i
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)
i : V
⊢ (∑ i : G.ConnectedComponent,
        (↑(Subtype.mk (p := fun (x : V → ℝ) ↦
              Membership.mem (γ := Submodule ℝ (V → ℝ))
                (LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                  ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
                x)
              (HSMul.hSMul (β := V → ℝ) (g i) fun (i_1 : V) ↦ if G.connectedComponentMk i_1 = i then 1 else 0) ⋯) :
          V → ℝ))
      i =
    g (G.connectedComponentMk i)

Tactic:
simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_ite, mul_one, mul_zero, sum_ite_eq,
      mem_univ, ↓reduceIte]

Premises used: Finset.mem_univ, Finset.sum_apply, mul_ite, MulZeroClass.mul_zero, reduceIte, smul_eq_mul, mul_one, Finset.sum_ite_eq, Pi.smul_apply
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)
⊢ Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)

Tactic:
have h : ∑ c, g c • lapMatrix_ker_basis_aux G c = fun i ↦ g (connectedComponentMk G i) := by
    simp only [lapMatrix_ker_basis_aux, SetLike.mk_smul_mk]
    repeat rw [AddSubmonoid.coe_finsetSum]
    ext i
    simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_ite, mul_one, mul_zero, sum_ite_eq,
      mem_univ, ↓reduceIte]
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)
h :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)
⊢ Eq.{u_1 + 1} (α := V → ℝ) (fun (i : V) ↦ g (G.connectedComponentMk i)) (↑0 : V → ℝ)

Tactic:
rw [h] at h0
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ i : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g i) (G.lapMatrix_ker_basis_aux i)) :
      V → ℝ)
    (↑0 : V → ℝ)
h :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)
h0₁ : Eq.{u_1 + 1} (α := V → ℝ) (fun (i : V) ↦ g (G.connectedComponentMk i)) (↑0 : V → ℝ)
⊢ Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)

Tactic:
rw [h] at h0
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 : Eq.{u_1 + 1} (α := V → ℝ) (fun (i : V) ↦ g (G.connectedComponentMk i)) (↑0 : V → ℝ)
h :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)
c : G.ConnectedComponent
⊢ ∃ (i : V), G.connectedComponentMk i = c

Tactic:
Quot.exists_rep c

Premises used: SimpleGraph.connectedComponentMk, Quot.exists_rep, SimpleGraph.ConnectedComponent
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 : Eq.{u_1 + 1} (α := V → ℝ) (fun (i : V) ↦ g (G.connectedComponentMk i)) (↑0 : V → ℝ)
h :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)
c : G.ConnectedComponent
obtain : ∀ (i : V) (h' : G.connectedComponentMk i = c), g c = 0
⊢ g c = 0

Tactic:
obtain ⟨i, h'⟩ : ∃ i : V, G.connectedComponentMk i = c := Quot.exists_rep c

Premises used: Quot.exists_rep
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 : Eq.{u_1 + 1} (α := V → ℝ) (fun (i : V) ↦ g (G.connectedComponentMk i)) (↑0 : V → ℝ)
h :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)
c : G.ConnectedComponent
i : V
h' : G.connectedComponentMk i = c
⊢ g c = 0

Tactic:
h' ▸ congrFun h0 i

Premises used: Eq.rec, congrFun
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
g : G.ConnectedComponent → ℝ
h0 : Eq.{u_1 + 1} (α := V → ℝ) (fun (i : V) ↦ g (G.connectedComponentMk i)) (↑0 : V → ℝ)
h :
  Eq.{u_1 + 1} (α := V → ℝ)
    (↑(∑ c : G.ConnectedComponent,
          HSMul.hSMul (β :=
            ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
            (g c) (G.lapMatrix_ker_basis_aux c)) :
      V → ℝ)
    fun (i : V) ↦ g (G.connectedComponentMk i)
c : G.ConnectedComponent
i : V
h' : G.connectedComponentMk i = c
⊢ g c = 0

Tactic:
exact h' ▸ congrFun h0 i
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
x :
  ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
      ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
_ :
  Membership.mem (γ :=
      Submodule ℝ
        ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
            ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
      ⊤ x →
    Membership.mem (γ :=
      Submodule ℝ
        ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
            ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
      (Submodule.span ℝ (Set.range G.lapMatrix_ker_basis_aux)) x
⊢ Membership.mem (γ :=
      Submodule ℝ
        ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
            ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
      ⊤ x →
    Membership.mem (γ :=
      Submodule ℝ
        ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
            ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
      (Submodule.span ℝ (Set.range G.lapMatrix_ker_basis_aux)) x

Tactic:
_
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
x :
  ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
      ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
h :
  Membership.mem (γ :=
    Submodule ℝ
      ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
          ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
    ⊤ x
rw :
  ∃ (c : G.ConnectedComponent → ℝ),
    ∑ i : G.ConnectedComponent,
        HSMul.hSMul (β :=
          ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
              ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
          (c i) (G.lapMatrix_ker_basis_aux i) =
      x
⊢ Membership.mem (γ :=
    Submodule ℝ
      ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
          ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
    (Submodule.span ℝ (Set.range G.lapMatrix_ker_basis_aux)) x

Tactic:
rw [Submodule.mem_span_range_iff_exists_fun]

Premises used: Submodule.mem_span_range_iff_exists_fun
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
x :
  ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
      ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
h :
  Membership.mem (γ :=
    Submodule ℝ
      ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
          ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
    ⊤ x
rw : lapMatrix ℝ G *ᵥ (↑x : V → ℝ) = 0
rw₁ :
  Eq (α := V → ℝ)
    ((((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G) : (V → ℝ) →ₗ[ℝ] V → ℝ) : (a : V → ℝ) → V → ℝ)
        (↑x : V → ℝ) :
      V → ℝ)
    (0 : V → ℝ)
rw₂ rw₃ : Eq.{u_1 + 1} (α := V → ℝ) 0 0
⊢ ∀ (a b : V) (a_1 : G.Reachable a b), (↑x : V → ℝ) a = (↑x : V → ℝ) b

Tactic:
rw [← lapMatrix_mulVec_eq_zero_iff_forall_reachable,
    ← toLin'_apply, LinearMap.map_coe_ker]

Premises used: LinearMap.map_coe_ker, SimpleGraph.lapMatrix_mulVec_eq_zero_iff_forall_reachable, Matrix.toLin'_apply
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
x :
  ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
      ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
h_1 :
  Membership.mem (γ :=
    Submodule ℝ
      ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
          ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
    ⊤ x
h :
  ∑ i : G.ConnectedComponent,
      HSMul.hSMul (β :=
        ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
            ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
        (Quot.lift (r := G.Reachable) (↑x : V → ℝ) ⋯ i) (G.lapMatrix_ker_basis_aux i) =
    x
⊢ ∃ (c : G.ConnectedComponent → ℝ),
    ∑ i : G.ConnectedComponent,
        HSMul.hSMul (β :=
          ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
              ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
          (c i) (G.lapMatrix_ker_basis_aux i) =
      x

Tactic:
use Quot.lift x.val (by rw [← lapMatrix_mulVec_eq_zero_iff_forall_reachable,
    ← toLin'_apply, LinearMap.map_coe_ker])

Premises used: Quot.lift, Subtype.val
-- Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix
Goal:
V : Type u_1
inst : Fintype V
G : SimpleGraph V
inst_1 : DecidableRel G.Adj
inst_2 : DecidableEq V
inst_3 : DecidableEq G.ConnectedComponent
x :
  ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
      ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G)))
h_1 :
  Membership.mem (γ :=
    Submodule ℝ
      ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
          ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
    ⊤ x
h :
  ∀ (j : V),
    (↑(∑ i : G.ConnectedComponent,
              HSMul.hSMul (β :=
                ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
                    ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
                (Quot.lift (r := G.Reachable) (↑x : V → ℝ) ⋯ i) (G.lapMatrix_ker_basis_aux i)) :
          V → ℝ)
        j =
      (↑x : V → ℝ) j
⊢ ∑ i : G.ConnectedComponent,
      HSMul.hSMul (β :=
        ↥(LinearMap.ker (R₂ := ℝ) (M₂ := V → ℝ) (τ₁₂ := RingHom.id ℝ)
            ((toLin' : (a : Matrix V V ℝ) → (V → ℝ) →ₗ[ℝ] V → ℝ) (lapMatrix ℝ G))))
        (Quot.lift (r := G.Reachable) (↑x : V → ℝ) ⋯ i) (G.lapMatrix_ker_basis_aux i) =
    x

Tactic:
ext j