\n ## Diagram: Formal Proof of Addition Associativity in a Proof Assistant ### Overview The image is a technical diagram illustrating a formal proof of the associativity of addition for natural numbers within a proof assistant environment (likely Coq or a similar system). It is divided into two primary panels: a left panel containing the formal definitions, theorem statement, and proof script, and a right panel visualizing the corresponding proof tree with tactics, goals, and local contexts. ### Components/Axes The diagram is structured into two main regions: **Left Panel (Code & Proof Script):** * **Environment Section:** Contains foundational definitions. * `Inductive nat :=` with constructors `0 : nat` and `S : nat -> nat`. Comments define these as natural numbers and the successor function. * `Fixpoint add (x y : nat) :=` defines addition recursively using pattern matching on `x`. Comments explain the base case (`x = 0, x + y = y`) and the recursive case (`x = 1 + x', x + y = 1 + (x' + y)`). * `Notation "x + y" := (add x y).` defines the infix notation. * **Theorem Section:** States the goal. * `Theorem add_assoc:` with the statement `forall a b c : nat, (a + b) + c = a + (b + c).` A comment notes this is to "prove that addition is associative." * **Proof Section:** Contains the proof script. * `Proof.` and `Qed.` delimit the proof. * Tactics listed: `intros.`, `induction a as [ | a' IHa'].`, `+ trivial.`, `+ simpl; rewrite IHa'. trivial.` **Right Panel (Proof Tree):** * **Title:** "Proof tree" at the top left. * **Structure:** A tree diagram showing the proof state evolution. * **Nodes:** Each node contains a "Local context" (hypotheses) and a "Goal" (the formula to prove). * **Edges:** Labeled with the tactic applied to transform the parent node into the child node(s). * **Key Labels:** * `Tactic intros` at the root. * `induction a as [ | a']` at the first branching point. * `trivial` at the leaf nodes for the base case and the final step. * `simpl; rewrite IHa'` at the inductive step node. ### Detailed Analysis **Proof Tree Flow and Data Points:** 1. **Root Node (Top Center):** * **Local Context:** `a b c : nat` * **Goal:** `(a + b) + c = a + (b + c)` * **Action:** The tactic `intros` is applied. This corresponds to introducing the universally quantified variables `a`, `b`, and `c` from the theorem statement into the local context. 2. **First Branching (Induction on `a`):** * The tactic `induction a as [ | a']` is applied. This performs structural induction on the natural number `a`, creating two subgoals: one for the base case (`a = 0`) and one for the inductive case (`a = S a'`). * **Left Branch (Base Case):** * **Local Context:** `b c : nat` * **Goal:** `(0 + b) + c = 0 + (b + c)` * **Action:** The tactic `trivial` is applied. This solves the goal automatically because, by the definition of `add`, both sides of the equation reduce to `b + c`. * **Right Branch (Inductive Case):** * **Local Context:** `a' b c : nat` and the **Inductive Hypothesis (IHa'):** `(a' + b) + c = a' + (b + c)` * **Goal:** `(S a' + b) + c = S a' + (b + c)` * **Action:** The tactic sequence `simpl; rewrite IHa'` is applied. * `simpl` likely simplifies the goal using the definition of `add`. The goal becomes `S (a' + b) + c = S (a' + (b + c))` (or a similar intermediate form). * `rewrite IHa'` uses the inductive hypothesis to replace `(a' + b) + c` with `a' + (b + c)` in the goal. * **Resulting Node:** * **Local Context:** `a' b c : nat`, `IHa' : (a' + b) + c = a' + (b + c)` * **Goal:** `S (a' + (b + c)) = S (a' + (b + c))` * **Action:** The tactic `trivial` is applied, solving this reflexive equality. ### Key Observations * **Spatial Grounding:** The proof tree is positioned on the right, with the code on the left. The tree flows top-down. The legend/labels ("Local context", "Goal") are placed at the top-right of the tree area. The tactic names are placed directly on the connecting arrows between nodes. * **Trend Verification:** The proof follows a standard inductive proof structure: introduce variables, perform induction, solve the base case trivially, and use the inductive hypothesis to solve the inductive step. The visual trend is a single root splitting into two branches, with the right branch having an additional step before reaching a trivial conclusion. * **Component Isolation:** The left panel is purely declarative (definitions and script). The right panel is operational, showing the state transitions. The connection is that the tactics in the left panel's `Proof` section directly generate the tree on the right. * **Text Transcription:** All text, including comments (e.g., `(* define natural numbers *)`), has been captured. The mathematical notation (`:=`, `->`, `forall`, `+`, `=`, `S`) is integral to the content. ### Interpretation This diagram serves as a pedagogical or documentation tool to explain how a proof assistant verifies a fundamental mathematical property. It demonstrates the **Curry-Howard correspondence**, where the proof script (left) is a program that constructs a proof object, and the proof tree (right) is the visual representation of that construction. * **What the data suggests:** The proof is constructive and relies on the principle of mathematical induction. The `trivial` tactic highlights where the proof follows directly from definitions (base case) or from reflexivity (after rewriting). The `rewrite` tactic is the crucial step that leverages the inductive hypothesis (`IHa'`). * **Relationship between elements:** The `Fixpoint add` definition on the left is the computational content that the `simpl` tactic uses. The `Inductive nat` definition provides the structure for the `induction` tactic. The proof tree is a direct, step-by-step visualization of the logical steps dictated by the proof script. * **Notable patterns:** The proof is minimal and efficient. The base case is solved in one step. The inductive case requires simplification and one rewrite step. This pattern is characteristic of proofs for properties of recursively defined functions over inductive types. The diagram effectively bridges the gap between the textual proof script and the underlying logical proof state, making the abstract process concrete.