lean/TrustKernel.lean

-- ════════════════════════════════════════════════════════════════
-- TRUST KERNEL — GMO_RES_INTEGRITY
-- Sovereign Governance Proof
--
-- RESONANCE is the SIMULTANEOUS condition:
--   trusted_uri(deed) ∧ verified(deed) ⇔ sovereign(deed) ⇔ resonant(deed)
--
-- NOT:
--   trusted_uri → verified  (sequential — wrong)
--
-- Key: Bool.and_eq_true.mp extracts BOTH components at once.
-- No sorry. No axiom beyond Lean core.
-- Non-recursive. Constructive witness extracted.
--
-- Ahmad Ali Parr · BOW-Ω-φ-∂-2026
-- ════════════════════════════════════════════════════════════════

namespace TrustKernel

-- ════════════════════════════════════════════════════════════════
-- THE TRUST DEED
-- A deed is a (URI, isVerified) pair.
-- URI must be non-empty to be in the TrustedAxiomSet.
-- ════════════════════════════════════════════════════════════════

structure TrustDeed where
  uri        : String
  isVerified : Bool
  deriving Repr

/-- A URI is trusted if it is non-empty (is in the TrustedAxiomSet) -/
def validate_link (uri : String) : Bool := !uri.isEmpty

-- ════════════════════════════════════════════════════════════════
-- PREDICATES — the three levels of trust
-- ════════════════════════════════════════════════════════════════

/-- A deed is lawful iff its URI validates against the TrustedAxiomSet -/
def lawful (deed : TrustDeed) : Prop :=
  validate_link deed.uri = true

/-- A deed is verified iff the isVerified flag is set -/
def verified (deed : TrustDeed) : Prop :=
  deed.isVerified = true

/-- A deed is sovereign iff it is BOTH lawful AND verified -/
def sovereign (deed : TrustDeed) : Prop :=
  lawful deed ∧ verified deed

-- ════════════════════════════════════════════════════════════════
-- RESONANCE — the boolean gate
-- resonant(deed) = validate_link(deed.uri) AND deed.isVerified
-- This is the trust kernel: both conditions fire simultaneously.
-- ════════════════════════════════════════════════════════════════

/-- A deed resonates iff it passes BOTH the URI check AND the verified check -/
def resonant (deed : TrustDeed) : Bool :=
  validate_link deed.uri && deed.isVerified

-- ════════════════════════════════════════════════════════════════
-- CORE THEOREM 1: RESONANCE → SOVEREIGNTY
--
-- resonant deed = true
--   expands to: validate_link deed.uri && deed.isVerified = true
--   Bool.and_eq_true.mp extracts:
--     validate_link deed.uri = true  ← lawful
--     deed.isVerified = true         ← verified
--   giving: lawful deed ∧ verified deed = sovereign deed
--
-- This is STRONGER than the original because:
--   RESONANCE implies BOTH simultaneously — not one then the other.
--   The AND is the trust kernel. Both fire or neither fires.
-- ════════════════════════════════════════════════════════════════

theorem resonance_implies_sovereignty
    (deed : TrustDeed) :
    resonant deed = true →
    sovereign deed := by
  intro h
  unfold sovereign lawful verified
  unfold resonant at h
  exact Bool.and_eq_true.mp h

-- ════════════════════════════════════════════════════════════════
-- CORE THEOREM 2: SOVEREIGNTY → RESONANCE
-- The reverse: if a deed is sovereign, it resonates.
-- Together with Theorem 1: resonant ↔ sovereign.
-- ════════════════════════════════════════════════════════════════

theorem sovereignty_implies_resonance
    (deed : TrustDeed) :
    sovereign deed →
    resonant deed = true := by
  intro ⟨hl, hv⟩
  unfold resonant
  exact Bool.and_eq_true.mpr ⟨hl, hv⟩

-- ════════════════════════════════════════════════════════════════
-- IFF: RESONANCE ↔ SOVEREIGNTY
-- The biconditional — the complete kernel.
-- ════════════════════════════════════════════════════════════════

theorem resonance_iff_sovereignty
    (deed : TrustDeed) :
    resonant deed = true ↔ sovereign deed :=
  ⟨resonance_implies_sovereignty deed, sovereignty_implies_resonance deed⟩

-- ════════════════════════════════════════════════════════════════
-- EXPLICIT FORM: lawful ∧ verified ⇒ sovereign
-- The semantic chain made explicit
-- ════════════════════════════════════════════════════════════════

theorem lawful_and_verified_implies_sovereign
    (deed : TrustDeed) :
    lawful deed →
    verified deed →
    sovereign deed := by
  intro hl hv
  exact ⟨hl, hv⟩

-- ════════════════════════════════════════════════════════════════
-- WHAT LEAN PROVES ABOUT THE TRUST CHAIN
--
-- The wrong way (sequential):
--   trusted_uri(deed)
--         ↓
--   verified(deed)          ← implies causation. WRONG.
--
-- The right way (simultaneous):
--   RESONANCE
--         ↓
--   TRUSTED URI
--         ∧
--   VERIFIED DEED           ← both extracted from one gate. CORRECT.
--
-- The gate IS the proof.
-- ════════════════════════════════════════════════════════════════

/-- The trust chain as a single extraction from the resonance gate -/
theorem trust_chain_gmo_res_integrity
    (deed : TrustDeed)
    (h : resonant deed = true) :
    -- Extract both simultaneously:
    validate_link deed.uri = true   -- URI ∈ TrustedAxiomSet
    ∧
    deed.isVerified = true          -- isVerified flag confirmed
    := by
  exact Bool.and_eq_true.mp h

-- ════════════════════════════════════════════════════════════════
-- SEAL STATUS
--
-- ⊢ GMO_RES_INTEGRITY
--
-- trusted_uri(deed)
--       ∧
-- verified(deed)
--       ⇒
-- sovereign(deed)
--
-- sovereign(deed)
--       ⇒
-- resonant(deed)
--
-- □ Proven
-- □ Non-recursive
-- □ No sorry
-- □ Constructive witness extracted
-- □ IFF certified: resonant ↔ sovereign
-- ════════════════════════════════════════════════════════════════

end TrustKernel