Extend source-to-TSM compiler with addition (v0.2).

Source.Expr now has intLit and add. Compile and correctness theorem
both extend.

The add case of compile_correct exercises the compositional structure:
  - IH on e1 (with extended suffix) gives the multistep for the first
    operand's evaluation.
  - IH on e2 (with extended prefix) gives the multistep for the second.
  - A single .add step at the boundary closes the trace.
  - Each intermediate state's PC is computed via array-size arithmetic
    threaded through omega.

New supporting lemmas:
  step_add               - per-instruction step for .add
  compile_add_get_op     - the instruction at the end of compile (.add e1 e2)
                           is .add. Extracted so the dependent-rewrite issue
                           with array bound proofs is contained in one place.

Engineering knowledge gained (recurring patterns when extending):
  - Array.getElem_append_left/right take the bound as an explicit positional
    arg, not via (h := ...).
  - rw on indices that appear in dependent bound proofs fails with "motive
    not type correct"; factor the lookup into a separate lemma.
  - convert tactic appears not to be available; rw + exact substitutes.
  - simp + omega closes most arithmetic on Array.size after expansion.
  - step lemmas with implicit args (a, b) need explicit (a := _) in calls
    where context doesn't determine them.

Adding a constructor still follows the v0.1 recipe — one Source
constructor, one Eval rule, one compile arm, one step_X helper, one
compile_X_get_op lemma, one case in compile_correct's induction. Each
case is ~25-40 lines of proof.

Zero sorries / axioms / admits.

TsmLean/Compile/Compile.lean

@@ -7,6 +7,7 @@ open TsmLean.Core (Instr Code)
 
 /-- Compile a source expression to TSM bytecode. -/
 def compile : Source.Expr → Code
-  | .intLit n => #[.push n]
+  | .intLit n   => #[.push n]
+  | .add e1 e2  => compile e1 ++ compile e2 ++ #[.add]
 
 end TsmLean.Compile

TsmLean/Compile/Correctness.lean

@@ -39,6 +39,27 @@ theorem step_push
     = some { code := code, pc := pc + 1, stack := .vInt n :: stack } := by
   unfold step; rw [dif_pos h_pc, h_get]
 
+theorem step_add
+    {code : Code} {pc : Nat} {a b : Int} {rest : List Value}
+    (h_pc : pc < code.size) (h_get : code[pc]'h_pc = .add) :
+    step { code := code, pc := pc, stack := .vInt b :: .vInt a :: rest }
+    = some { code := code, pc := pc + 1, stack := .vInt (a + b) :: rest } := by
+  unfold step; rw [dif_pos h_pc, h_get]
+
+/-! ## Compile-output lookup helpers. -/
+
+/-- The instruction at the end of `compile (.add e1 e2)` (right after both
+    sub-compiled segments) is `.add`. -/
+theorem compile_add_get_op
+    (e1 e2 : Source.Expr)
+    (h : (compile e1).size + (compile e2).size < (compile (Source.Expr.add e1 e2)).size) :
+    (compile (Source.Expr.add e1 e2))[(compile e1).size + (compile e2).size]'h = .add := by
+  show ((compile e1 ++ compile e2) ++ #[Instr.add])[(compile e1).size + (compile e2).size]'h = .add
+  have hle : (compile e1 ++ compile e2).size ≤ (compile e1).size + (compile e2).size :=
+    Nat.le_of_eq Array.size_append
+  rw [Array.getElem_append_right hle]
+  simp [Array.size_append, Nat.sub_self]
+
 /-! ## Code-lookup helper.
 
 Looking up an index `pre.size + i` in `pre ++ compile e ++ suf` reduces
@@ -82,6 +103,53 @@ theorem compile_correct
     have h_size : (compile (Source.Expr.intLit n)).size = 1 := by simp [compile]
     rw [h_size]
     exact step_thm
+  | @add e1 e2 a b _ _ ih1 ih2 =>
+    intros pre suf rest
+    -- Step A: compile e1
+    have stepA := ih1 pre (compile e2 ++ #[.add] ++ suf) rest
+    -- Step B: compile e2 (with prefix extended)
+    have stepB := ih2 (pre ++ compile e1) (#[.add] ++ suf) (.vInt a :: rest)
+    -- Code rearrangements
+    have h_code_A : pre ++ compile e1 ++ (compile e2 ++ #[Instr.add] ++ suf)
+                  = pre ++ compile (Source.Expr.add e1 e2) ++ suf := by
+      simp only [compile, Array.append_assoc]
+    have h_code_B : pre ++ compile e1 ++ compile e2 ++ (#[Instr.add] ++ suf)
+                  = pre ++ compile (Source.Expr.add e1 e2) ++ suf := by
+      simp only [compile, Array.append_assoc]
+    rw [h_code_A] at stepA
+    rw [show (pre ++ compile e1).size = pre.size + (compile e1).size from
+        by simp [Array.size_append]] at stepB
+    rw [h_code_B] at stepB
+    apply MultiStep.trans stepA
+    apply MultiStep.trans stepB
+    apply MultiStep.single
+    -- Final .add step
+    have h_total_size : (compile (Source.Expr.add e1 e2)).size
+                      = (compile e1).size + (compile e2).size + 1 := by
+      show (compile e1 ++ compile e2 ++ #[Instr.add]).size
+         = (compile e1).size + (compile e2).size + 1
+      simp [Array.size_append]; omega
+    have h_op_pc : pre.size + (compile e1).size + (compile e2).size
+                 < (pre ++ compile (Source.Expr.add e1 e2) ++ suf).size := by
+      simp only [Array.size_append, h_total_size]; omega
+    have h_in_comp : (compile e1).size + (compile e2).size
+                     < (compile (Source.Expr.add e1 e2)).size := by
+      simp [compile, Array.size_append]
+    have h_full_pc : pre.size + ((compile e1).size + (compile e2).size)
+                     < (pre ++ compile (Source.Expr.add e1 e2) ++ suf).size := by
+      simp only [Array.size_append, h_total_size]; omega
+    have h_op_get : (pre ++ compile (Source.Expr.add e1 e2) ++ suf)[pre.size + ((compile e1).size + (compile e2).size)]'h_full_pc = .add := by
+      rw [getElem_compile pre (Source.Expr.add e1 e2) suf
+            ((compile e1).size + (compile e2).size) h_in_comp h_full_pc]
+      exact compile_add_get_op e1 e2 h_in_comp
+    have h_step := step_add (a := a) (b := b) (rest := rest) h_full_pc h_op_get
+    have h_pre_pc : pre.size + (compile e1).size + (compile e2).size
+                  = pre.size + ((compile e1).size + (compile e2).size) := by omega
+    have h_post_pc : pre.size + ((compile e1).size + (compile e2).size) + 1
+                   = pre.size + (compile (Source.Expr.add e1 e2)).size := by
+      rw [h_total_size]; omega
+    rw [h_pre_pc, ← h_post_pc]
+    exact h_step
 
 /-! ## Demo run.
 

TsmLean/Compile/Source.lean

@@ -2,20 +2,23 @@ import TsmLean.Core.Syntax
 
 namespace TsmLean.Compile.Source
 
-/-! # Source language for compilation (v0.1).
+/-! # Source language for compilation (v0.2).
 
-The smallest meaningful source language: integer literals. The
+Integer literals + addition. The minimal "tree of operations" that
-compiler produces a one-instruction TSM program; the correctness
+exercises the compositional structure of the correctness proof. -/
-theorem demonstrates the proof framework. Subsequent versions extend
-to arithmetic, comparison, control flow, and variables. -/
 
 inductive Expr where
   | intLit : Int → Expr
+  | add    : Expr → Expr → Expr
   deriving Repr, Inhabited
 
 abbrev Value := TsmLean.Core.Value
 
 inductive Eval : Expr → Value → Prop where
   | intLit (n : Int) : Eval (.intLit n) (.vInt n)
+  | add    {e1 e2 a b}
+           (h1 : Eval e1 (.vInt a))
+           (h2 : Eval e2 (.vInt b)) :
+      Eval (.add e1 e2) (.vInt (a + b))
 
 end TsmLean.Compile.Source