Generate a tactic that can transform one step from the current tactic state to the 'no goals' tactic state.
Current tactic state:
```
R : Type u
ι : Type v
inst✝⁶ : CommRing R
L : Type w₁
M : ι → Type w
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : (i : ι) → LieRingModule L (M i)
inst✝ : ∀ (i : ι), LieModule R L (M i)
x y : L
m : ⨁ (i : ι), M i
⊢ ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆
```

refine
  DFinsupp.ext fun _ =>
    ?_
      -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): Originally `ext`

Given a natural language math problem and the initial solution state, please generate a Lean 4 solution sketch.
You can use Lean 4 comments to conduct natural language reasoning.
Please only use forward reasoning; do not use tactics that modify the final goal.
For new hypotheses, please do not prove them and use `sorry` to close them.
Please assume the following header code has already been executed, and do not add any imports or openings.
```lean4
import Mathlib
```

# Problem
"""
Given $2\sin (π-x)+1=0$, find the value of $\cos 2x$.
"""

# Initial Solution State
```
case h.mp
answer x : ℝ
h_eq : 2 * Real.sin (Real.pi - x) + 1 = 0
h_answer : Real.cos (2 * x) = answer
⊢ ?w
```

# Formal Solution Draft
```lean4
-- The given equation can be simplified as follows: $2\sin (π-x)+1=0$
have h_given_eq : 2 * Real.sin (Real.pi - x) + 1 = 0 := h_eq

-- Using the cofunction identity $\sin(π-x)=\sin(x)$, we have: $2\sin(x)+1=0$
have h_cofunction : Real.sin (Real.pi - x) = Real.sin x := sorry
have h_simplified_eq : 2 * Real.sin x + 1 = 0 := sorry

-- Now, solve for $\sin(x)$: $\sin(x)=-\frac{1}{2}$
have h_sin_x : Real.sin x = -1/2 := sorry

-- Now, we need to find $\cos 2x$. We can use the double angle identity for cosine: $\cos 2x = 1 - 2\sin^2 x$
have h_double_angle : Real.cos (2 * x) = 1 - 2 * (Real.sin x)^2 := sorry

-- Substitute the value of $\sin x$: $\cos 2x = 1 - 2(-\frac{1}{2})^2$
have h_substituted : Real.cos (2 * x) = 1 - 2 * (-1/2)^2 := sorry

-- Simplify: $\cos 2x = 1 - 2(\frac{1}{4})$
have h_simplified : Real.cos (2 * x) = 1 - 2 * (1/4) := sorry

-- $\cos 2x = 1 - \frac{1}{2}$
have h_final_simplification : Real.cos (2 * x) = 1 - 1/2 := sorry

-- $\cos 2x = \frac{1}{2}$
have h_cos_2x : Real.cos (2 * x) = 1/2 := sorry

-- Thus, the value of $\cos 2x$ is $\boxed{\frac{1}{2}}$
have h_answer : answer = 1/2 := sorry

exact h_answer -- submit

```