direct-proof

name	direct-proof
description	Prove statements directly by assuming the hypothesis and deriving the conclusion through logical steps. Triggers on "prove directly", "show that", "demonstrate", "verify", "simple proof"

Direct Proof

The most straightforward proof technique: assume the hypothesis, derive the conclusion.

When to Use

• The implication P → Q has a clear logical path
• No need for contradiction or cases
• Algebraic manipulations lead directly to the result
• The statement has a constructive flavor

Step 1: Identify Hypothesis and Conclusion

For "If P then Q" (P → Q):

• Hypothesis: What are we assuming? (P)
• Conclusion: What do we need to show? (Q)
• Hidden assumptions: Types, domains, constraints

Example

"If n is even, then n² is even"

• Hypothesis: n is even (∃k, n = 2k)
• Conclusion: n² is even (∃m, n² = 2m)

Step 2: Unpack Definitions

• What does the hypothesis mean precisely?
• What does the conclusion require?
• Introduce variables from existential statements

Tools to use:

• symbolic_compute: Verify algebraic steps

Step 3: Chain of Reasoning

Build a chain of implications:

Assume P.
Then [consequence 1].
Therefore [consequence 2].
...
Thus Q. ∎

Each step must be justified by:

• Definition
• Previously proven statement
• Algebraic manipulation
• Known theorem

Step 4: Lean Formalization

theorem example (h : P) : Q := by
  -- Unfold definitions
  unfold ...
  -- Get witnesses from existentials
  obtain ⟨k, hk⟩ := h
  -- Provide witnesses for conclusion
  use ...
  -- Algebraic verification
  ring

Key tactics for direct proofs:

• intro h - assume the hypothesis
• obtain ⟨x, hx⟩ := h - unpack existential
• use x - provide witness
• exact h - conclude with exact match
• ring - algebraic simplification
• simp - simplification

Output Format

**Theorem:** If P then Q.

**Proof:**

Assume P.
[Unpack what P gives us]

Then [step 1].
[justification]

Therefore [step 2].
[justification]

...

Thus Q. ∎

**Lean proof:**
```lean
[verified code]

## Common Patterns

| Statement Type | Approach |
|----------------|----------|
| ∃x, P(x) | Construct x explicitly |
| ∀x, P(x) | Let x be arbitrary |
| P ∧ Q | Prove P and Q separately |
| P ↔ Q | Prove P → Q and Q → P |

## Example: n even ⟹ n² even

Assume n is even. By definition, ∃k ∈ ℤ such that n = 2k.

Then: n² = (2k)² = 4k² = 2(2k²)

Let m = 2k². Then n² = 2m, so n² is even. ∎