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
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import Bookspec.Statements
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/-!
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# Closing a period, and splitting the net
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Mirrors `kernel::closing_line`, `kernel::credit_line`, `kernel::plug_last`,
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`periods::allocate`, and `Ledger::close_period_to` (income closing, net to
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one equity account or split among members).
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-/
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namespace Bookspec
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/-- The line that zeroes an account with movement `m`: the opposite side,
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same amount; nothing for zero (`kernel::closing_line`). It is also the
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line that credits `m` to an account (`kernel::credit_line`). -/
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def closingLine (a : Nat) (m : Int) : List Line :=
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if 0 < m then [⟨a, .credit, m⟩] else if m < 0 then [⟨a, .debit, -m⟩] else []
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theorem net_closingLine (a : Nat) (m : Int) : net (closingLine a m) = -m := by
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unfold closingLine
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split
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· simp [Line.signed]
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· split <;> simp [Line.signed] <;> omega
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theorem movement_closingLine (b a : Nat) (m : Int) :
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movement b (closingLine a m) = if b = a then -m else 0 := by
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by_cases e : b = a
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· subst e
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rcases (by omega : 0 < m ∨ m < 0 ∨ m = 0) with h | h | h
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· simp [closingLine, h, Line.signed]
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· have : ¬ 0 < m := by omega
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simp [closingLine, h, this, Line.signed]
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· simp [closingLine, h]
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· have e' : ¬ a = b := fun x => e x.symm
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unfold closingLine
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split <;> (try split) <;> simp [e']
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theorem closingLine_pos (a : Nat) (m : Int) : ∀ l ∈ closingLine a m, 0 < l.amount := by
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unfold closingLine
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split
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· simp; omega
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· split
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· simp; omega
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· simp
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theorem closingLine_account (a : Nat) (m : Int) : ∀ l ∈ closingLine a m, l.account = a := by
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unfold closingLine
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split
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· simp
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· split <;> simp
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theorem closingLine_eq_nil (a : Nat) (m : Int) : closingLine a m = [] ↔ m = 0 := by
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unfold closingLine
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split
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· simp; omega
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· split
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· simp; omega
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· simp; omega
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/-- The closing lines for some accounts: one per account with a nonzero
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movement. -/
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def closings (as : List Nat) (ls : List Line) : List Line :=
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match as with
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| [] => []
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| a :: as => closingLine a (movement a ls) ++ closings as ls
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theorem net_closings (as : List Nat) (ls : List Line) :
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net (closings as ls) = -sumOver (fun a => movement a ls) as := by
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induction as with
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| nil => rfl
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| cons a as ih => simp only [closings, net_append, net_closingLine, ih, sumOver_cons]; omega
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theorem movement_closings (b : Nat) (as : List Nat) (ls : List Line) (hd : Distinct as) :
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movement b (closings as ls) = if b ∈ as then -movement b ls else 0 := by
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induction as with
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| nil => simp [closings]
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| cons a as ih =>
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obtain ⟨ha, hd⟩ := hd
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simp only [closings, movement_append, movement_closingLine, ih hd]
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by_cases e : b = a
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· subst e; simp [ha]
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· simp [e]
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theorem closings_pos (as : List Nat) (ls : List Line) : ∀ l ∈ closings as ls, 0 < l.amount := by
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induction as with
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| nil => simp [closings]
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| cons a as ih =>
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intro l hl
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simp only [closings, List.mem_append] at hl
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cases hl with
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| inl h => exact closingLine_pos _ _ l h
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| inr h => exact ih l h
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theorem closings_account (as : List Nat) (ls : List Line) : ∀ l ∈ closings as ls, l.account ∈ as := by
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induction as with
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| nil => simp [closings]
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| cons a as ih =>
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intro l hl
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simp only [closings, List.mem_append] at hl
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cases hl with
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| inl h => rw [closingLine_account _ _ l h]; simp
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| inr h => exact List.mem_cons_of_mem _ (ih l h)
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/-- Nothing to close means every closed account was already at zero. -/
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theorem closings_eq_nil (as : List Nat) (ls : List Line) (h : closings as ls = []) :
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∀ a ∈ as, movement a ls = 0 := by
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induction as with
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| nil => simp
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| cons a as ih =>
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simp only [closings, List.append_eq_nil_iff] at h
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intro b hb
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cases hb with
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| head => exact (closingLine_eq_nil _ _).1 h.1
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| tail _ hb => exact ih h.2 b hb
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/-! ## The net to equity -/
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/-- The sum of some amounts. -/
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def total : List Int → Int
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| [] => 0
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| x :: xs => x + total xs
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/-- Lines crediting each target its part (`kernel::credit_line` per
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member). -/
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def equityLines : List (Nat × Int) → List Line
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| [] => []
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| (t, c) :: ts => closingLine t c ++ equityLines ts
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theorem net_equityLines (ts : List (Nat × Int)) :
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net (equityLines ts) = -total (ts.map Prod.snd) := by
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induction ts with
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| nil => rfl
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| cons x ts ih =>
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obtain ⟨t, c⟩ := x
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simp only [equityLines, net_append, net_closingLine, ih, List.map_cons, total]; omega
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theorem movement_equityLines_absent (b : Nat) (ts : List (Nat × Int))
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(h : ∀ x ∈ ts, x.1 ≠ b) : movement b (equityLines ts) = 0 := by
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induction ts with
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| nil => rfl
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| cons x ts ih =>
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obtain ⟨t, c⟩ := x
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have ht : b ≠ t := fun e => h (t, c) (by simp) e.symm
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simp only [equityLines, movement_append, movement_closingLine, ht, ite_false,
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ih (fun y hy => h y (List.mem_cons_of_mem _ hy))]
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rfl
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theorem equityLines_pos (ts : List (Nat × Int)) : ∀ l ∈ equityLines ts, 0 < l.amount := by
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induction ts with
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| nil => simp [equityLines]
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| cons x ts ih =>
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obtain ⟨t, c⟩ := x
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intro l hl
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simp only [equityLines, List.mem_append] at hl
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cases hl with
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| inl h => exact closingLine_pos _ _ l h
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| inr h => exact ih l h
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theorem equityLines_account (ts : List (Nat × Int)) :
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∀ l ∈ equityLines ts, ∃ x ∈ ts, l.account = x.1 := by
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induction ts with
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| nil => simp [equityLines]
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| cons x ts ih =>
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obtain ⟨t, c⟩ := x
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intro l hl
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simp only [equityLines, List.mem_append] at hl
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cases hl with
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| inl h => exact ⟨(t, c), by simp, closingLine_account _ _ l h⟩
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| inr h =>
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obtain ⟨y, hy, e⟩ := ih l h
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exact ⟨y, List.mem_cons_of_mem _ hy, e⟩
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theorem equityLines_eq_nil (ts : List (Nat × Int)) (h : equityLines ts = []) :
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total (ts.map Prod.snd) = 0 := by
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induction ts with
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| nil => rfl
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| cons x ts ih =>
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obtain ⟨t, c⟩ := x
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simp only [equityLines, List.append_eq_nil_iff] at h
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simp only [List.map_cons, total, (closingLine_eq_nil _ _).1 h.1, ih h.2]; rfl
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/-- The closing entry: zero the closed accounts, credit the net to the
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targets. -/
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def closingEntry (as : List Nat) (ts : List (Nat × Int)) (ls : List Line) : List Line :=
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closings as ls ++ equityLines ts
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/-- **A closing entry balances** when the targets' parts sum to the net
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of the closing lines (the net income), however that net was split. -/
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theorem closingEntry_net (as : List Nat) (ts : List (Nat × Int)) (ls : List Line)
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(hsum : total (ts.map Prod.snd) = net (closings as ls)) :
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net (closingEntry as ts ls) = 0 := by
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rw [closingEntry, net_append, net_equityLines, hsum]; omega
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/-- **A closing entry zeroes every account it closes.** -/
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theorem closingEntry_zeroes (as : List Nat) (ts : List (Nat × Int)) (ls : List Line)
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(hd : Distinct as) (b : Nat) (hb : b ∈ as) (ht : ∀ x ∈ ts, x.1 ≠ b) :
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movement b (ls ++ closingEntry as ts ls) = 0 := by
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rw [movement_append, closingEntry, movement_append, movement_closings b as ls hd,
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movement_equityLines_absent b ts ht]
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simp only [hb, ite_true]; omega
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/-- **A non-empty closing entry is acceptable**: two or more lines, every
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amount positive, debits equal to credits. -/
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theorem closingEntry_accepted (as : List Nat) (ts : List (Nat × Int)) (ls : List Line)
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(hsum : total (ts.map Prod.snd) = net (closings as ls)) (hne : closings as ls ≠ []) :
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Accepted (closingEntry as ts ls) := by
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refine ⟨?_, ?_, ?_⟩
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· rw [closingEntry, List.length_append]
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match hc : closings as ls with
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| [] => exact absurd hc hne
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| [l] =>
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have hl : 0 < l.amount := closings_pos as ls l (by rw [hc]; simp)
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have hnet : net (closings as ls) ≠ 0 := by
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rw [hc]; obtain ⟨a, s, m⟩ := l
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cases s <;> simp [Line.signed] at hl ⊢ <;> omega
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have : equityLines ts ≠ [] := fun h => hnet (hsum ▸ equityLines_eq_nil ts h)
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have : 1 ≤ (equityLines ts).length := by
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cases h : equityLines ts with
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| nil => exact absurd h this
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| cons _ _ => simp
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simp; omega
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| _ :: _ :: _ => simp; omega
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· intro l hl
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simp only [closingEntry, List.mem_append] at hl
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cases hl with
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| inl h => exact closings_pos as ls l h
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| inr h => exact equityLines_pos ts l h
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· have := closingEntry_net as ts ls hsum
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rw [net_eq_debits_sub_credits] at this; omega
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/-! ## Net income -/
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/-- The chart's revenue and expense codes: what a close zeroes. -/
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def nominalCodes : Chart → List Nat
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| [] => []
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| (a, t) :: c => if t = .revenue ∨ t = .expense then a :: nominalCodes c else nominalCodes c
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theorem nominalCodes_sub (c : Chart) : ∀ a ∈ nominalCodes c, a ∈ codes c := by
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induction c with
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| nil => simp [nominalCodes]
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| cons x c ih =>
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obtain ⟨a, t⟩ := x
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intro b hb
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simp only [nominalCodes] at hb
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split at hb
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· cases hb with
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| head => simp
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| tail _ h => exact List.mem_cons_of_mem _ (ih b h)
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· exact List.mem_cons_of_mem _ (ih b hb)
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theorem nominalCodes_distinct (c : Chart) (hd : Distinct (codes c)) : Distinct (nominalCodes c) := by
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induction c with
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| nil => trivial
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| cons x c ih =>
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obtain ⟨a, t⟩ := x
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obtain ⟨ha, hd⟩ := hd
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simp only [nominalCodes]
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split
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· exact ⟨fun h => ha (nominalCodes_sub c a h), ih hd⟩
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· exact ih hd
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/-- **The net a close credits to equity is the period's net income**:
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revenue balances minus expense balances. -/
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theorem net_closings_is_net_income (c : Chart) (ls : List Line) :
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net (closings (nominalCodes c) ls) = typeTotal .revenue ls c - typeTotal .expense ls c := by
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rw [net_closings]
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induction c with
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| nil => rfl
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| cons x c ih =>
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obtain ⟨a, t⟩ := x
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cases t <;> simp [nominalCodes, typeTotal, balance, AccountType.normal, signed] at ih ⊢ <;> omega
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/-! ## Splitting among members -/
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/-- Make rounded parts sum to the whole: the last becomes the whole minus
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the others (`kernel::plug_last`). -/
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def plugLast (whole : Int) : List Int → List Int
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| [] => []
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| [_] => [whole]
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| p :: q :: rest => p :: plugLast (whole - p) (q :: rest)
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/-- **An allocation's parts sum to the whole, exactly**, however the
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other parts were rounded. -/
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theorem plugLast_total (whole : Int) (ps : List Int) (h : ps ≠ []) :
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total (plugLast whole ps) = whole := by
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induction ps generalizing whole with
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| nil => exact absurd rfl h
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| cons p rest ih =>
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match rest, ih with
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| [], _ => show total [whole] = whole; simp [total]
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| q :: rest, ih =>
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simp only [plugLast, total]
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have := ih (whole - p) (by simp)
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omega
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theorem plugLast_length (whole : Int) (ps : List Int) : (plugLast whole ps).length = ps.length := by
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induction ps generalizing whole with
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| nil => rfl
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| cons p rest ih =>
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match rest, ih with
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| [], _ => rfl
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| q :: rest, ih => simp only [plugLast, List.length_cons]; rw [ih]; rfl
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/-- Only the last part changes. -/
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theorem plugLast_keeps (whole : Int) (ps : List Int) :
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(plugLast whole ps).dropLast = ps.dropLast := by
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induction ps generalizing whole with
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| nil => rfl
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| cons p rest ih =>
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match rest, ih with
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| [], _ => rfl
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| q :: rest, ih =>
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have hne : plugLast (whole - p) (q :: rest) ≠ [] := by
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intro h
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have := plugLast_length (whole - p) (q :: rest)
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rw [h] at this; simp at this
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rw [show plugLast whole (p :: q :: rest) = p :: plugLast (whole - p) (q :: rest) from rfl,
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List.dropLast_cons_of_ne_nil hne, ih, List.dropLast_cons_cons]
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/-! ## Cumulative member closes
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A run of members closes under the run rule of 2026-10-13 to P3
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(`CloseTo::MembersRun`, journaled `"cumulative"`; rules.md B-13) gives each
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member, at each close, their allocation of the run's total so far minus
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their allocation of the total before it. `alloc` is any allocation of an
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amount to members (member `i` gets `alloc x i`), such as
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`periods::allocate`. -/
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/-- What one close gives member `i`: `before` is the run's total before
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it, `net` what it closes. -/
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def closePart (alloc : Int → Nat → Int) (before net : Int) (i : Nat) : Int :=
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alloc (before + net) i - alloc before i
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/-- What a run of closes with these nets, starting after `before`, gives
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member `i` in all. -/
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def runPart (alloc : Int → Nat → Int) : Int → List Int → Nat → Int
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| _, [], _ => 0
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| before, net :: rest, i => closePart alloc before net i + runPart alloc (before + net) rest i
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/-- **A run telescopes**: member `i`'s total from a run is their allocation
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of everything the run closed, less their allocation of where it started.
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From an empty start (`alloc 0 = 0`), it is exactly their allocation of
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the run's total, so its rounding is one allocation's rounding, never a
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sum of them. -/
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theorem runPart_telescopes (alloc : Int → Nat → Int) (before : Int) (nets : List Int) (i : Nat) :
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runPart alloc before nets i = alloc (before + nets.sum) i - alloc before i := by
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induction nets generalizing before with
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| nil => simp [runPart]
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| cons n rest ih =>
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simp only [runPart, closePart, ih, List.sum_cons]
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rw [show before + n + rest.sum = before + (n + rest.sum) by omega]
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omega
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/-- A sum of differences is the difference of the sums. -/
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theorem sum_map_sub (f g : Nat → Int) (ms : List Nat) :
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(ms.map (fun i => f i - g i)).sum = (ms.map f).sum - (ms.map g).sum := by
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induction ms with
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| nil => simp
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| cons m ms ih => simp only [List.map_cons, List.sum_cons, ih]; omega
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/-- **Each close hands out exactly its net** when every allocation sums
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to its amount (`periods::allocate` does: `kernel::apportion`). -/
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theorem closePart_total (alloc : Int → Nat → Int) (members : List Nat)
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(sums : ∀ x, (members.map (alloc x)).sum = x) (before net : Int) :
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(members.map (closePart alloc before net)).sum = net := by
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have h := sum_map_sub (alloc (before + net)) (alloc before) members
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unfold closePart
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rw [h, sums, sums]
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omega
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/-! ## Members closes counted from a count's start (P3)
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The rule since P3 (`CloseTo::Members`, rules.md B-13; D-10): a close gives
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member `i` their allocation of everything the count has closed (`closed`
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before it, plus its `net`), less what the count's earlier closes gave them
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(`given i`), whatever rule those closes were made under
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(`periods::allocate_cumulative`, `kernel::apportion_cumulative`). The run
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rule of 2026-10-13 (`CloseTo::MembersRun`, `closePart`) is the special case
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where the earlier closes gave exactly the allocation. -/
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||||
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/-- What a cumulative close gives member `i`. -/
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def trueUp (alloc : Int → Nat → Int) (closed net : Int) (given : Nat → Int) (i : Nat) : Int :=
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alloc (closed + net) i - given i
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|
||||
/-- **A cumulative close hands out exactly its net**, whatever rule the
|
||||
earlier closes used, as long as they handed out what they closed. -/
|
||||
theorem trueUp_total (alloc : Int → Nat → Int) (members : List Nat)
|
||||
(sums : ∀ x, (members.map (alloc x)).sum = x) (closed net : Int) (given : Nat → Int)
|
||||
(hg : (members.map given).sum = closed) :
|
||||
(members.map (trueUp alloc closed net given)).sum = net := by
|
||||
have h := sum_map_sub (alloc (closed + net)) given members
|
||||
unfold trueUp
|
||||
rw [h, sums, hg]
|
||||
omega
|
||||
|
||||
/-- **A running total is one allocation**: after the close, what the count
|
||||
has given member `i` is their allocation of everything it closed, so its
|
||||
rounding is one allocation's, never a sum of them. -/
|
||||
theorem trueUp_running (alloc : Int → Nat → Int) (closed net : Int) (given : Nat → Int)
|
||||
(i : Nat) : given i + trueUp alloc closed net given i = alloc (closed + net) i := by
|
||||
unfold trueUp
|
||||
omega
|
||||
|
||||
/-- The run rule is the special case `given = alloc closed`. -/
|
||||
theorem closePart_eq_trueUp (alloc : Int → Nat → Int) (before net : Int) (i : Nat) :
|
||||
closePart alloc before net i = trueUp alloc before net (alloc before) i := rfl
|
||||
|
||||
/-- **Bounded drift, for any history**: when every allocation is under one
|
||||
unit from exact (scaled by `W`, the shares being `w i / W`), so is every
|
||||
member's running total after a cumulative close, whatever the earlier
|
||||
closes gave (`Apportion.allocW_drift` supplies the hypothesis). -/
|
||||
theorem cumulative_drift (alloc : Int → Nat → Int) (w : Nat → Int) (W : Int)
|
||||
(hd : ∀ x i, -W < alloc x i * W - x * w i ∧ alloc x i * W - x * w i < W)
|
||||
(closed net : Int) (given : Nat → Int) (i : Nat) :
|
||||
-W < (given i + trueUp alloc closed net given i) * W - (closed + net) * w i ∧
|
||||
(given i + trueUp alloc closed net given i) * W - (closed + net) * w i < W := by
|
||||
rw [trueUp_running]
|
||||
exact hd _ _
|
||||
|
||||
/-- The statement of members' equity's share of the earnings `U` not yet
|
||||
closed, as of a day: what the count's next close would give of them, its
|
||||
allocation of `closed + U` less its allocation of `closed` (D-38; P3 review
|
||||
R-1: never the true-up a close hasn't posted). -/
|
||||
def unclosedShare (alloc : Int → Nat → Int) (closed U : Int) (i : Nat) : Int :=
|
||||
alloc (closed + U) i - alloc closed i
|
||||
|
||||
/-- **Closing a period moves a member's capital on a day by exactly the
|
||||
true-up the close posts** (D-38, R-1): before, their capital `cap` plus
|
||||
their share of the unclosed `U`; after closing `net` of it, `cap` plus the
|
||||
close's part plus their share of the rest. The difference is
|
||||
`alloc closed i - given i`. -/
|
||||
theorem statement_after_close (alloc : Int → Nat → Int) (cap closed net U : Int)
|
||||
(given : Nat → Int) (i : Nat) :
|
||||
cap + trueUp alloc closed net given i + unclosedShare alloc (closed + net) (U - net) i
|
||||
= cap + unclosedShare alloc closed U i + (alloc closed i - given i) := by
|
||||
unfold trueUp unclosedShare
|
||||
rw [show closed + net + (U - net) = closed + U by omega]
|
||||
omega
|
||||
|
||||
/-- **A member's capital on a day never moves when its period is closed**
|
||||
(D-38), once the count's closes gave what the cumulative rule gives
|
||||
(`given = alloc closed`: every count the rule began, and every count after
|
||||
its first close that posts). -/
|
||||
theorem statement_unmoved_by_close (alloc : Int → Nat → Int) (cap closed net U : Int)
|
||||
(i : Nat) :
|
||||
cap + trueUp alloc closed net (alloc closed) i + unclosedShare alloc (closed + net) (U - net) i
|
||||
= cap + unclosedShare alloc closed U i := by
|
||||
rw [statement_after_close]
|
||||
omega
|
||||
|
||||
end Bookspec
|
||||
Reference in new issue
Block a user