Create app.py

app.py

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+from google import genai
+from google.genai import types
+import gradio as gr
+
+client = genai.Client(api_key=GOOGLE_API_KEY)
+MODEL_ID = "gemini-2.0-flash-thinking-exp"
+from google import genai
+client = genai.Client(api_key="AIzaSyCmUDbVAOGcRZcOKP4q6mmeZ7Gx1WgE3vE")
+
+def llm_response(text):
+  response = client.models.generate_content(
+    model=MODEL_ID,
+    contents= text)
+  return response.text
+
+def theorem_prover(theorem):
+  theorem_llm = llm_response(f'''You are an advanced mathematical reasoning model specializing in formal theorem proving. Your task is to analyze a given theorem, 
+  determine the most appropriate proof strategy, and construct a rigorous proof using formal proof assistants such as Lean, Coq, or Isabelle. If a proof is not 
+  possible, identify gaps, suggest refinements, or provide counterexamples.
+
+  User Prompt:
+Input:
+"Theorem Statement:
+{theorem}
+ Task Breakdown:
+1. Understand the Theorem
+Identify the mathematical domain (e.g., Number Theory, Algebra, Topology).
+Recognize any implicit assumptions, definitions, or missing details.
+Determine whether the theorem is constructive, existential, or universally quantified.
+2. Determine the Proof Strategy
+The model will automatically select the best approach based on the theorem's structure:
+
+Proof Strategy Selection Guidelines
+To determine the best proof strategy for a given theorem, follow these principles:
+
+Direct Proof → Use when the statement follows naturally from known axioms or definitions. Example: Proving that if 
+𝑛
+n is even, then 
+𝑛
+2
+n 
+2
+  is even.
+Proof by Contradiction → Assume the negation of the statement and show it leads to a contradiction. This is useful for proving the infinitude of primes.
+Proof by Induction → Apply when proving a property for an infinite sequence or recursively defined structures. Example: Proving the sum formula 
+∑
+𝑖
+=
+1
+𝑛
+𝑖
+=
+𝑛
+(
+𝑛
++
+1
+)
+2
+∑ 
+i=1
+n
+​
+ i= 
+2
+n(n+1)
+​
+ .
+Case Analysis → Use when different scenarios must be considered separately. Example: Proving that a quadratic equation has at most two real roots.
+Constructive Proof → Show existence by explicitly constructing an example. Example: Proving that there exists an irrational number 
+𝑥
+x such that 
+𝑥
+𝑥
+x 
+x
+  is rational.
+Non-constructive Proof → Prove existence without constructing an explicit example, often using logic or set theory. Example: Proving that there exists a prime number between 
+𝑛
+n and 
+2
+𝑛
+2n.
+Proof by Exhaustion → Use when a theorem holds for a small, finite set of cases that can be checked individually. Example: Verifying a property for small integers.
+Proof using a Counterexample → Disprove a general claim by providing a specific case where it fails. Example: Showing that not all differentiable functions are continuous.
+The model should analyze the structure of the theorem and automatically select the most suitable proof technique based on these guidelines.
+
+3. Construct the Proof
+Step-by-step logical explanation.
+
+Verification and validation.
+4. Handle Edge Cases
+If proof fails:
+
+Highlight missing assumptions.
+Provide a minimal counterexample (if the statement is false).
+Suggest a reformulation or alternative direction.
+Expected Output:
+✔ Proof Found:
+
+Step-by-step reasoning.
+Proof strategy justification.
+
+❌ Proof Not Possible:
+
+Identified logical gap or missing assumptions.
+Suggested refinement or counterexample.
+Example Usage:
+Input:
+"Every even integer greater than 2 can be expressed as the sum of two prime numbers."
+
+Output:
+✔ Proof Attempt:
+
+Identified Proof Strategy:
+
+This is an existential statement (∃ two primes p1, p2 such that p1 + p2 = n).
+Since direct proof is difficult, contradiction and case analysis are common approaches.
+Computational methods confirm the conjecture for large values of 
+𝑛
+n, but no general proof exists.
+Proof Attempt:
+
+
+theorem goldbach_conjecture (n : ℕ) (h : even n ∧ n > 2) :  
+  ∃ p1 p2, prime p1 ∧ prime p2 ∧ p1 + p2 = n :=  
+begin  
+  -- Step 1: Assume n is an even integer greater than 2  
+  -- Step 2: Search for prime pairs (p1, p2) such that p1 + p2 = n  
+  -- Step 3: If no counterexamples exist up to a given range, assume general case  
+  -- Theorem remains unproven but verified for large n using computational methods  
+end  
+❌ Proof Not Found:
+
+No general proof within standard number theory axioms.
+Computational verification up to 
+10
+18
+10 
+18
+  supports the conjecture.
+Suggest refining the conjecture by imposing additional constraints.
+
+You should intelligently selects the best proof strategy.
+Provides clear justifications for strategy choice.
+
+Output a rigorous proof or meaningful insight even when a full proof is impossible in markdown format.''')
+  return theorem_llm  
+
+iface = gr.Interface(
+    fn=verify_formula, 
+    inputs=gr.Textbox(label="Enter Formula (e.g., x > 5 and y < 10)"), 
+    outputs=gr.HTML(label="Result"),  # Output as HTML
+    title="Theorem proving agent",
+    description="Enter a logical formula using Z3 syntax to check its satisfiability."
+)
+
+# Launch the app
+iface.launch()