Add theorem prover docs page with Phase 2 constraint solving

- prove.sx Phase 2: bounded model checking with 34 algebraic properties
  (commutativity, associativity, distributivity, inverses, bounds, transitivity)
- prove.sx generates SMT-LIB for unbounded Z3 verification via z3-expr
- New docs page /plans/theorem-prover with live results (91/91 sat, 34/34 verified)
- Page helper runs both proof phases and returns structured data
- Parser: re-add ' quote syntax (removed by prior edit)
- Load prove.sx alongside z3.sx at startup

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

sx/sxc/pages/docs.sx

@@ -626,10 +626,14 @@
     :sub-nav (~section-nav :items plans-nav-items
                :current (find-current plans-nav-items slug))
     :selected (or (find-current plans-nav-items slug) ""))
+  :data (case slug
+    "theorem-prover" (prove-data)
+    :else nil)
   :content (case slug
     "status" (~plan-status-content)
     "reader-macros" (~plan-reader-macros-content)
     "reader-macro-demo" (~plan-reader-macro-demo-content)
+    "theorem-prover" (~plan-theorem-prover-content)
     "sx-activity" (~plan-sx-activity-content)
     "predictive-prefetch" (~plan-predictive-prefetch-content)
     "content-addressed-components" (~plan-content-addressed-components-content)

sx/sxc/pages/helpers.py

@@ -31,6 +31,7 @@ def _register_sx_helpers() -> None:
         "optimistic-demo-data": _optimistic_demo_data,
         "action:add-demo-item": _add_demo_item,
         "offline-demo-data": _offline_demo_data,
+        "prove-data": _prove_data,
     })
 
 
@@ -1000,6 +1001,109 @@ def _add_demo_item(**kwargs) -> dict:
     }
 
 
+def _prove_data() -> dict:
+    """Run prove.sx against the SX spec — both phases.
+
+    Phase 1: Translate all define-* forms via z3.sx, verify satisfiability.
+    Phase 2: Evaluate algebraic properties via bounded model checking.
+    Returns results for the docs page to render.
+    """
+    import time
+    from shared.sx.parser import parse_all
+    from shared.sx.evaluator import evaluate
+    from shared.sx.primitives import all_primitives
+    from shared.sx.evaluator import _trampoline, _call_lambda
+
+    env = all_primitives()
+
+    ref_dir = _ref_dir()
+    for lib in ("z3.sx", "prove.sx"):
+        path = __import__("os").path.join(ref_dir, lib)
+        with open(path, encoding="utf-8") as f:
+            for expr in parse_all(f.read()):
+                evaluate(expr, env)
+
+    # Phase 1: definitional satisfiability
+    with open(__import__("os").path.join(ref_dir, "primitives.sx"), encoding="utf-8") as f:
+        prim_exprs = parse_all(f.read())
+
+    t0 = time.monotonic()
+    phase1 = _trampoline(_call_lambda(env["prove-file"], [prim_exprs], env))
+    phase1_ms = round((time.monotonic() - t0) * 1000)
+
+    # Phase 2: property-based constraint solving
+    t1 = time.monotonic()
+    phase2 = _trampoline(_call_lambda(env["prove-all-properties"], [], env))
+    phase2_ms = round((time.monotonic() - t1) * 1000)
+
+    # Flatten Phase 1 results for rendering
+    phase1_results = []
+    for r in phase1.get("results", []):
+        phase1_results.append({
+            "name": r.get("name", "?"),
+            "status": r.get("status", "?"),
+        })
+
+    # Flatten Phase 2 results for rendering
+    phase2_results = []
+    total_tested = 0
+    for r in phase2.get("results", []):
+        tested = r.get("tested", 0)
+        skipped = r.get("skipped", 0)
+        total_tested += tested
+        entry = {
+            "name": r.get("name", "?"),
+            "status": r.get("status", "?"),
+            "tested": tested,
+            "skipped": skipped,
+        }
+        ce = r.get("counterexample")
+        if ce:
+            entry["counterexample"] = str(ce)
+        phase2_results.append(entry)
+
+    # Generate SMT-LIB sample for a few properties
+    props = env["sx-properties"]
+    smtlib_samples = []
+    for p in props[:3]:
+        smt = _trampoline(_call_lambda(env["prove-property-smtlib"], [p], env))
+        smtlib_samples.append(smt)
+    # One with precondition
+    for p in props:
+        if p.get("given-expr"):
+            smt = _trampoline(_call_lambda(env["prove-property-smtlib"], [p], env))
+            smtlib_samples.append(smt)
+            break
+
+    return {
+        "phase1-total": phase1.get("total", 0),
+        "phase1-sat": phase1.get("sat", 0),
+        "phase1-all-sat": phase1.get("all-sat", False),
+        "phase1-ms": phase1_ms,
+        "phase1-results": phase1_results,
+        "phase2-total": phase2.get("total", 0),
+        "phase2-verified": phase2.get("verified", 0),
+        "phase2-falsified": phase2.get("falsified", 0),
+        "phase2-all-verified": phase2.get("all-verified", False),
+        "phase2-ms": phase2_ms,
+        "phase2-results": phase2_results,
+        "phase2-total-tested": total_tested,
+        "smtlib-sample": "\n".join(smtlib_samples),
+        "prove-source": _read_spec_file("prove.sx"),
+        "z3-source": _read_spec_file("z3.sx"),
+    }
+
+
+def _ref_dir() -> str:
+    """Return the path to the SX ref directory."""
+    import os
+    # Same resolution as _read_spec_file
+    ref_dir = os.path.join(os.path.dirname(__file__), "..", "..", "shared", "sx", "ref")
+    if not os.path.isdir(ref_dir):
+        ref_dir = "/app/shared/sx/ref"
+    return ref_dir
+
+
 def _offline_demo_data() -> dict:
     """Return demo data for the offline data layer test page."""
     from datetime import datetime, timezone