diff --git a/Game/Levels/VectorSpaceWorld/Level04.lean b/Game/Levels/VectorSpaceWorld/Level04.lean index 66e7e0c..66ffd1d 100644 --- a/Game/Levels/VectorSpaceWorld/Level04.lean +++ b/Game/Levels/VectorSpaceWorld/Level04.lean @@ -82,14 +82,14 @@ Statement subspace_contains_zero {W : Set V} (hW : isSubspace (K := K) (V := V) Hint "Start by expanding the subspace definition using obtain. This will give you the three properties: nonempty, closed under addition, and closed under scalar multiplication." Hint (hidden := true) "Try `obtain ⟨h1, h2, h3⟩ := hW`" obtain ⟨h1, _h2, h3⟩ := hW - Hint "The nonempty property h1 means there exists some element in W. Use obtain to extract this element." + Hint "The nonempty property {h1} means there exists some element in W. Use obtain to extract this element." Hint (hidden := true) "Try `obtain ⟨w, hw⟩ := h1`" obtain ⟨w, hw⟩ := h1 Hint "We want to show 0 ∈ W, but we know that 0 = 0 • w (from Level 1). Rewrite the goal using this fact." Hint (hidden := true) "Try `rw [(zero_smul_v K V w).symm]`" rw [(zero_smul_v K V w).symm] Hint "Now apply the scalar multiplication closure property h3. Since w ∈ W and subspaces are closed under scalar multiplication, 0 • w ∈ W." - Hint (hidden := true) "Try `apply h3`" + Hint (hidden := true) "Try `apply {h3}`" apply h3 Hint "Finally, provide the proof that w ∈ W, which we have from our obtain step." Hint (hidden := true) "Try `exact hw`"