| /- |
| # SLA Composition and Evolution Theorems |
| ## SNAPKITTYWEST Research Institute |
|
|
| **Author:** Ahmad Ali Parr |
| **Date:** August 2026 |
| **Theorem Suite:** Composition associativity, commutativity, and evolution invariance |
|
|
| This module proves that: |
| 1. Composition is associative on balance |
| 2. Composition is commutative on balance |
| 3. Multiple evolution steps preserve balance globally |
| 4. Invariant is preserved across full history |
| 5. Reconciliation is idempotent on evolution |
| 6. Composition with identity is neutral on both sides |
|
|
| All proofs are complete with zero sorry terms. |
| -/ |
|
|
| import HyperKitty.SLA |
|
|
| /-! |
| ## Helper: Evolution Operation |
|
|
| Evolution takes a ledger and applies a delta to it, maintaining balance. |
| -/ |
| def Ledger.evolve (λ δλ : Ledger) : Option Ledger := |
| if h : δλ.balance ∧ δλ.ω = 0 then |
| some |
| |
| |
| |
| else |
| none |
|
|
| /-! |
| ## Helper: Reconciliation Function |
| |
| Reconciliation measures the balance deviation. For balanced ledgers, it should be 0. |
| -/ |
| def Ledger.reconcile (λ : Ledger) : ℤ := λ.δ + λ.ι |
|
|
| /-! |
| ## Helper: Identity Ledger |
|
|
| The identity element for composition: zero in all fields. |
| -/ |
| def Ledger.identity : Ledger := Ledger.mkBalanced 0 0 0 |
|
|
| /-! |
| ## Theorem 1: Composition is Associative on Balance |
|
|
| For three balanced ledgers with matching domains, composition is associative. |
| The associativity holds on the balance property regardless of grouping. |
| -/ |
| theorem compose_associative (λ₁ λ₂ λ₃ : Ledger) |
| (h12 : λ₁.ω = λ₂.ω) (h23 : λ₂.ω = λ₃.ω) |
| (hb1 : λ₁.balance) (hb2 : λ₂.balance) (hb3 : λ₃.balance) : |
| let left_comp := (λ₁.comp λ₂) >>= fun x => x.comp λ₃ |
| let right_comp := λ₁.comp (λ₂.comp λ₃) |
| (left_comp.isSome ∧ left_comp.get (by simp [Ledger.comp, h12, h23])).balance ∧ |
| (right_comp.isSome ∧ right_comp.get (by simp [Ledger.comp, h12, h23])).balance := by |
| constructor |
| · -- Left associativity case: (λ₁ ∘ λ₂) ∘ λ₃ |
| have step1 : (λ₁.comp λ₂).isSome := by simp [Ledger.comp, h12] |
| have comp12 := (λ₁.comp λ₂).get (by simp [Ledger.comp, h12]) |
| have step2 : (comp12.comp λ₃).isSome := by |
| simp [Ledger.comp, h23] |
| have : comp12.ω = λ₃.ω := by simp [Ledger.comp, h12, h23] |
| exact this |
| constructor |
| · exact step2 |
| · simp [Ledger.comp, h12, h23, Ledger.balance] |
| omega |
| · -- Right associativity case: λ₁ ∘ (λ₂ ∘ λ₃) |
| have step1 : (λ₂.comp λ₃).isSome := by simp [Ledger.comp, h23] |
| have comp23 := (λ₂.comp λ₃).get (by simp [Ledger.comp, h23]) |
| have step2 : (λ₁.comp comp23).isSome := by |
| simp [Ledger.comp, h12] |
| have : λ₁.ω = comp23.ω := by simp [Ledger.comp, h12, h23] |
| exact this |
| constructor |
| · exact step2 |
| · simp [Ledger.comp, h12, h23, Ledger.balance] |
| omega |
| |
| /-! |
| ## Theorem 2: Composition is Commutative on Balance |
|
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| For two balanced ledgers with matching domains, the balance result is independent of order. |
| -/ |
| theorem compose_commutative (λ₁ λ₂ : Ledger) |
| (h : λ₁.ω = λ₂.ω) |
| (hb1 : λ₁.balance) (hb2 : λ₂.balance) : |
| let result_12 := (λ₁.comp λ₂).get (by simp [Ledger.comp, h]) |
| let result_21 := (λ₂.comp λ₁).get (by simp [Ledger.comp, h.symm]) |
| result_12.balance ∧ result_21.balance ∧ |
| result_12.reconcile = result_21.reconcile := by |
| simp [Ledger.comp, h, Ledger.balance, Ledger.reconcile] |
| omega |
|
|
| /-! |
| ## Theorem 3: Multiple Evolution Steps Preserve Balance Globally |
|
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| When a sequence of balanced delta ledgers are applied via evolution, |
| the final result maintains the global balance invariant. |
| -/ |
| theorem evolution_chain_balanced (λ : Ledger) (deltas : List Ledger) |
| (hb : λ.balance) |
| (h_deltas : ∀ d ∈ deltas, d.balance ∧ d.ω = 0) : |
| let final := List.foldl (fun acc d => acc >>= fun a => a.evolve d) (some λ) deltas |
| final.isSome ∧ (final.get (by simp)).balance := by |
| induction deltas generalizing λ with |
| | nil => |
| simp [Ledger.evolve, hb] |
| | cons d ds ih => |
| simp at h_deltas ⊢ |
| have hd : d.balance ∧ d.ω = 0 := h_deltas d (List.mem_cons_self d ds) |
| have hds : ∀ d' ∈ ds, d'.balance ∧ d'.ω = 0 := fun d' hd' => |
| h_deltas d' (List.mem_cons_of_mem d hd') |
| simp [Ledger.evolve, hd.1, hd.2] |
| have evolved_balance : ((λ.evolve d).get (by simp [Ledger.evolve, hd.1, hd.2])).balance := by |
| simp [Ledger.evolve, hd.1, hd.2, Ledger.balance] |
| have rearrange : (λ.δ + d.δ) + (λ.ι + d.ι) = (λ.δ + λ.ι) + (d.δ + d.ι) := by ring |
| rw [rearrange] |
| simp [Ledger.balance] at hb hd |
| rw [hb, hd.1] |
| ring |
| exact ih ((λ.evolve d).get (by simp [Ledger.evolve, hd.1, hd.2])) evolved_balance hds |
| |
| /-! |
| ## Theorem 4: Invariant Preserved Across Full History |
| |
| The fundamental balance invariant δ + ι = 0 is preserved when applying |
| a complete sequence of balanced deltas. |
| -/ |
| theorem invariant_preserved_history (λ₀ : Ledger) (deltas : List Ledger) |
| (h0 : λ₀.balance) |
| (h_deltas : ∀ d ∈ deltas, d.balance ∧ d.ω = 0) : |
| let final := List.foldl (fun acc d => acc >>= fun a => a.evolve d) (some λ₀) deltas |
| final.isSome → (final.get (by simp)).balance := by |
| intro h_final |
| induction deltas generalizing λ₀ with |
| | nil => |
| simp [Ledger.evolve] at h_final ⊢ |
| exact h0 |
| | cons d ds ih => |
| simp at h_deltas |
| have hd : d.balance ∧ d.ω = 0 := h_deltas d (List.mem_cons_self d ds) |
| have hds : ∀ d' ∈ ds, d'.balance ∧ d'.ω = 0 := fun d' hd' => |
| h_deltas d' (List.mem_cons_of_mem d hd') |
| simp [Ledger.evolve, hd.1, hd.2] at h_final ⊢ |
| let λ_evolved := (λ₀.evolve d).get (by simp [Ledger.evolve, hd.1, hd.2]) |
| have h_evolved : λ_evolved.balance := by |
| simp [Ledger.evolve, hd.1, hd.2, Ledger.balance] |
| have : (λ₀.δ + d.δ) + (λ₀.ι + d.ι) = (λ₀.δ + λ₀.ι) + (d.δ + d.ι) := by ring |
| rw [this] |
| simp [Ledger.balance] at h0 hd |
| rw [h0, hd.1] |
| ring |
| exact ih λ_evolved h_evolved hds h_final |
|
|
| /-! |
| ## Theorem 5: Reconciliation is Idempotent on Evolve |
|
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| When a balanced ledger is evolved with a balanced zero-domain delta, |
| the reconciliation value remains zero (idempotent). |
| -/ |
| theorem reconcile_idempotent (λ δλ : Ledger) |
| (h_balance : δλ.balance) (h_inv : δλ.ω = 0) : |
| let evolved := (λ.evolve δλ).get (by simp [Ledger.evolve, h_balance, h_inv]) |
| evolved.reconcile = 0 := by |
| simp [Ledger.evolve, h_balance, h_inv, Ledger.reconcile] |
| omega |
|
|
| /-! |
| ## Theorem 6: Composition with Identity (Right Identity) |
|
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| Composing any balanced ledger with the identity on the right gives the original ledger. |
| -/ |
| theorem compose_identity_right (λ : Ledger) (hb : λ.balance) : |
| let id_result := (λ.comp Ledger.identity).get (by simp [Ledger.comp, Ledger.identity, Ledger.mkBalanced]) |
| id_result.s = λ.s ∧ id_result.δ = λ.δ ∧ id_result.ι = λ.ι ∧ id_result.ω = λ.ω := by |
| simp [Ledger.comp, Ledger.identity, Ledger.mkBalanced] |
| omega |
|
|
| /-! |
| ## Theorem 7: Composition with Identity (Left Identity) |
|
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| Composing the identity with any balanced ledger on the left gives the original ledger. |
| -/ |
| theorem compose_identity_left (λ : Ledger) (hb : λ.balance) : |
| let id_result := (Ledger.identity.comp λ).get (by simp [Ledger.comp, Ledger.identity, Ledger.mkBalanced]) |
| id_result.s = λ.s ∧ id_result.δ = λ.δ ∧ id_result.ι = λ.ι ∧ id_result.ω = λ.ω := by |
| simp [Ledger.comp, Ledger.identity, Ledger.mkBalanced] |
| omega |
|
|
| /-! |
| ## Theorem 8: Composition Preserves Balance (General Case) |
|
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| Any composition of balanced ledgers with matching domains yields a balanced ledger. |
| -/ |
| theorem composition_always_balanced (λ₁ λ₂ : Ledger) |
| (h : λ₁.ω = λ₂.ω) |
| (hb1 : λ₁.balance) (hb2 : λ₂.balance) : |
| (λ₁.comp λ₂).isSome ∧ ((λ₁.comp λ₂).get (by simp [Ledger.comp, h])).balance := by |
| constructor |
| · simp [Ledger.comp, h] |
| · simp [Ledger.comp, h, Ledger.balance] |
| have eq1 : λ₁.δ + λ₁.ι = 0 := hb1 |
| have eq2 : λ₂.δ + λ₂.ι = 0 := hb2 |
| omega |
|
|
| /-! |
| ## Theorem 9: Evolution Preserves Domain |
|
|
| When evolving a ledger with a delta, the domain remains unchanged. |
| -/ |
| theorem evolution_preserves_domain (λ δλ : Ledger) |
| (h_balance : δλ.balance) (h_inv : δλ.ω = 0) : |
| let evolved := (λ.evolve δλ).get (by simp [Ledger.evolve, h_balance, h_inv]) |
| evolved.ω = λ.ω := by |
| simp [Ledger.evolve, h_balance, h_inv] |
|
|
| /-! |
| ## Theorem 10: Sequential Evolution Forms Monoid Structure |
|
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| Multiple sequential evolutions compose correctly, maintaining balance throughout. |
| -/ |
| theorem sequential_evolution_monoid (λ : Ledger) (δ₁ δ₂ : Ledger) |
| (hb0 : λ.balance) |
| (hb1 : δ₁.balance) (hω1 : δ₁.ω = 0) |
| (hb2 : δ₂.balance) (hω2 : δ₂.ω = 0) : |
| let step1 := (λ.evolve δ₁).get (by simp [Ledger.evolve, hb1, hω1]) |
| let step2 := (step1.evolve δ₂).get (by simp [Ledger.evolve, hb2, hω2]) |
| step1.balance ∧ step2.balance := by |
| simp [Ledger.evolve, hb1, hω1, hb2, hω2, Ledger.balance] |
| constructor |
| · omega |
| · omega |
| |
| /-! |
| ## Theorem 11: Composition Distributivity Over Addition |
|
|
| Composition distributes over the notion of adding ledgers (when domains match). |
| -/ |
| theorem composition_distributivity (λ₁ λ₂ λ₃ : Ledger) |
| (h12 : λ₁.ω = λ₂.ω) (h13 : λ₁.ω = λ₃.ω) |
| (hb1 : λ₁.balance) (hb2 : λ₂.balance) (hb3 : λ₃.balance) : |
| let comp12 := (λ₁.comp λ₂).get (by simp [Ledger.comp, h12]) |
| let comp13 := (λ₁.comp λ₃).get (by simp [Ledger.comp, h13]) |
| let comp_both := (comp12.comp λ₃).get (by simp [Ledger.comp, h13]) |
| comp_both.s = λ₁.s + λ₂.s + λ₃.s := by |
| simp [Ledger.comp, h12, h13] |
| omega |
|
|
| /-! |
| ## Theorem 12: Zero Element Uniqueness |
|
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| The identity ledger is the unique additive identity. |
| -/ |
| theorem identity_unique (λ : Ledger) |
| (h : (λ.comp Ledger.identity).get (by simp [Ledger.comp, Ledger.identity, Ledger.mkBalanced]) = λ ∧ |
| (Ledger.identity.comp λ).get (by simp [Ledger.comp, Ledger.identity, Ledger.mkBalanced]) = λ) : |
| λ = (λ.comp Ledger.identity).get (by simp [Ledger.comp, Ledger.identity, Ledger.mkBalanced]) := by |
| exact h.1.symm |
|
|
