## Diagram: Proof Tactics with and without Backtracking ### Overview The image presents two diagrams illustrating proof tactics, one "Without Backtracking" and the other "With Backtracking." Each diagram shows a sequence of states and the tactics applied to transition between them, demonstrating how a proof can be completed. The diagram on the right shows a backtracking step. ### Components/Axes * **Titles:** "Without Backtracking" (left), "With Backtracking" (right) * **States:** "Initial State", "State 1", "State 2", "State 3", "State 4" * **Tactics:** "Intro: h₁", "Apply Or.inr", "Apply Or.inl", "Exact h₁" * **Logic Statements:** * ⊢ p₂ → p₃ ∨ p₂ * h₁ : p₂ ⊢ p₃ ∨ p₂ * h₁ : p₂ ⊢ p₂ * h₁ : p₂ ⊢ p₃ * **Arrows:** Indicate the flow of applying tactics. * **Backtrack Arrow:** A dashed blue arrow labeled "backtrack" indicates a return to a previous state. * **Tactics Bracket:** A bracket on the left side of the "Without Backtracking" diagram is labeled "Tactics". ### Detailed Analysis **Left Diagram: Without Backtracking** * **Initial State:** ⊢ p₂ → p₃ ∨ p₂ * **Intro: h₁:** Applied to the Initial State. * **State 1:** h₁ : p₂ ⊢ p₃ ∨ p₂ * **Apply Or.inr:** Applied to State 1. * **State 2:** h₁ : p₂ ⊢ p₂ * **Exact h₁:** Applied to State 2. * **State 3:** Complete **Right Diagram: With Backtracking** * **Initial State:** ⊢ p₂ → p₃ ∨ p₂ * **Intro h₁:** Applied to the Initial State. * **State 1:** h₁ : p₂ ⊢ p₃ ∨ p₂ * **Apply Or.inl:** Applied to State 1. * **State 2:** h₁ : p₂ ⊢ p₃ * **Apply Or.inr:** Applied to State 2, resulting in a loop back to State 1. * **State 3:** h₁ : p₂ ⊢ p₂ * **Exact h₁:** Applied to State 3. * **State 4:** Complete * **Backtrack:** A blue dashed arrow points from "Apply Or.inl" back to "State 1", labeled "backtrack". ### Key Observations * The "Without Backtracking" diagram shows a direct path to completing the proof. * The "With Backtracking" diagram demonstrates a scenario where applying a tactic ("Apply Or.inl") leads to a state from which the proof cannot be directly completed, necessitating a return to a previous state ("State 1") and the application of a different tactic ("Apply Or.inr"). ### Interpretation The diagrams illustrate the concept of backtracking in proof tactics. In the "Without Backtracking" scenario, the correct sequence of tactics is applied directly, leading to a completed proof. However, in the "With Backtracking" scenario, an initial tactic choice leads to a dead end, requiring the system to "backtrack" to a previous state and try a different tactic. This highlights the search process involved in automated theorem proving, where multiple paths may be explored before a successful proof is found. The "backtrack" arrow visually represents this process of undoing a step and trying an alternative.