feat(bookspec): the five laws as Lean theorems (P13 step 4)

The first edition's five laws (debits equal credits; the accounting
equation; the trial balance balances; net income articulates to equity;
the cash-flow sections sum to the change in cash) are theorems in
bookspec/lean, ported from verify/lean (Basic, Entry, Statements, Closing,
Ledger under namespace Bookspec) and restated by name in Laws.lean, each a
statement about every reachable state of the books. Audit.lean prints
their axioms; the checker's lean.rs test fails on a sorry, a law not
audited, or any axiom beyond propext, Quot.sound and Classical.choice.
Money is Int, the exact count of the smallest unit: every law is a law of
addition, so the model is faithful to exact decimals.

Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01HS8WZziDnyxunC23MUxrLp

lean/Bookspec/Laws.lean

@@ -0,0 +1,62 @@
+import Bookspec.Ledger
+
+/-!
+# bookspec's five laws
+
+The five laws of the first edition (`bookspec/spec/00-model.md`, "The five
+laws"), each stated once, by name, as a theorem about *every* reachable
+state of the books, ported from `verify/lean` (P13 step 4). Money is `Int`,
+an exact count of the smallest unit; any finite set of exact decimals
+scales to integers without loss, and every law is a law of addition.
+
+1. Debits equal credits (BS-entry-1).
+2. Assets = Liabilities + Equity + (Revenue - Expenses) (BS-statement-2).
+3. The trial balance balances (BS-statement-1).
+4. Net income articulates to equity: a close credits equity with exactly
+   revenue less expenses (BS-period-1).
+5. The cash-flow sections sum to the change in cash (BS-statement-3).
+-/
+
+namespace Bookspec
+
+/-- Law 1. In every reachable state the posted lines' debits equal their
+credits (`net` is debits less credits). -/
+theorem law_debits_equal_credits {L : Ledger} (r : Reachable L) :
+    net (allLines L.posted) = 0 :=
+  r.books_net_zero
+
+/-- Law 2. The accounting equation, in every reachable state. -/
+theorem law_accounting_equation {L : Ledger} (r : Reachable L) :
+    let ls := allLines L.posted
+    typeTotal .asset ls L.chart
+      = typeTotal .liability ls L.chart + typeTotal .equity ls L.chart
+        + (typeTotal .revenue ls L.chart - typeTotal .expense ls L.chart) :=
+  r.accounting_equation
+
+/-- Law 3. The trial balance balances, in every reachable state. -/
+theorem law_trial_balance {L : Ledger} (r : Reachable L) :
+    tbDebits (allLines L.posted) (codes L.chart)
+      = tbCredits (allLines L.posted) (codes L.chart) :=
+  r.trial_balance
+
+/-- Law 4. A close credits equity with exactly the period's net income. -/
+theorem law_net_income_articulates (L : Ledger) (p : Period) (ts : List (Nat × Int))
+    (hsum : total (ts.map Prod.snd)
+      = net (closings (nominalCodes L.chart) (allLines (through p.stop L.posted)))) :
+    let ls := allLines (through p.stop L.posted)
+    total (ts.map Prod.snd) = typeTotal .revenue ls L.chart - typeTotal .expense ls L.chart :=
+  close_credits_net_income L p ts hsum
+
+/-- Law 5. The three sections of the statement of cash flows sum to the
+change in the cash accounts. -/
+theorem law_cash_flow_is_change_in_cash {L : Ledger} (r : Reachable L) (b c : Nat)
+    (hbc : b ≤ c) (F : List Line → Line → Flow) (C : List Line → List Int)
+    (hC : ∀ e, sum3 (C e) = 0) (cash : List Nat) (hd : Distinct cash)
+    (hF : ∀ e l, F e l = .cash ↔ l.account ∈ cash) :
+    let es := (between b c L.posted).map Entry.lines
+    sectionTotal F C 0 es + sectionTotal F C 1 es + sectionTotal F C 2 es
+      = sumOver (fun a => movement a (allLines (through c L.posted))) cash
+        - sumOver (fun a => movement a (allLines (through b L.posted))) cash :=
+  r.cashflow_is_change_in_cash b c hbc F C hC cash hd hF
+
+end Bookspec