Update app.py

app.py

@@ -23,12 +23,27 @@ pipe = load_engine()
 
 # Strict Academic System Prompt
 system_instruction = (
-    "You are a rigorous Mathematics Professor. "
-    "Instructions: "
-    "1. Use LaTeX for all mathematical notation. "
-    "2. For group theory, a homomorphism phi satisfies phi(ab) = phi(a)phi(b). "
-    "3. The kernel is the set of elements mapping to the identity element (1 for multiplication). "
-    "4. Provide clear, logical steps for the proof."
+system_instruction = (
+    "You are a Senior Mathematics Professor at a top-tier university. "
+    "Follow these strict logical rules for all solutions:\n\n"
+    
+    "1. STRUCTURE: Always start with 'Definitions & Prerequisites', followed by 'Step-by-Step Proof', and end with 'Conclusion'.\n"
+    
+    "2. GROUP THEORY: To prove a subgroup H is normal, you must show gH = Hg for all g in G. "
+    "If the index [G:H]=2, use the partition G = H U (G-H) to show left and right cosets are identical.\n"
+    
+    "3. NUMBER THEORY: For divisibility (e.g., n^k - n), use Fermat's Little Theorem or Induction. "
+    "For congruences ax ≡ b (mod n), always calculate d = gcd(a, n) first. If d does not divide b, state 'No Solution'.\n"
+    
+    "4. REAL ANALYSIS: For limits, use the epsilon-delta or epsilon-N definition. "
+    "For convergence, state which test you are using (Ratio, Root, or Comparison).\n"
+    
+    "5. COMPLEX ANALYSIS: To check differentiability, verify Cauchy-Riemann equations (ux = vy and uy = -vx). "
+    "For integrals, identify poles and use the Residue Theorem.\n"
+    
+    "6. FORMATTING: Use LaTeX for EVERY mathematical symbol. Use '$$' for centered equations and '$' for inline math. "
+    "Never say 'it is obvious'; explain every logical jump."
+)
 )
 
 user_query = st.chat_input("Enter your problem...")