lean_proposal.lean

namespace Cicsc.Evolution.FFIntegration

/-- Axiomatic bridge: we trust git's --ff-only as the ground truth for descent.
    In a full formalization, we'd prove this equivalent to our graph definition. -/
axiom git_ff_implies_descent : 
  ∀ (main worktree : Commit) (g : GitGraph),
    git_merge_ff_succeeds main worktree → IsFFMorph main worktree g

/-- The integration trace: executable witness extraction.
    When git succeeds, we get a proof object.
    When git fails, we get none. -/
def integrate_sh_trace (main : Commit) (worktree : Commit) (g : GitGraph) : 
  IO (Option (IsFFMorph main worktree g)) := do
  let result ← IO.Process.output { 
    cmd := "git", 
    args := #["merge", "--ff-only", worktree] 
  }
  if result.exitCode = 0 then 
    -- Git succeeded: we have computational evidence of FF property
    -- The proof is axiomatic (we trust git), but the witness is real
    return some (git_ff_implies_descent main worktree g (by native_decide))
  else 
    -- Git failed: no FF morphism exists
    return none

/-- Alternative: Use the execution as an oracle directly -/
axiom GitOracle : Type
axiom query_ff (oracle : GitOracle) (main worktree : Commit) : Bool
axiom oracle_sound : 
  ∀ o main worktree, 
    query_ff o main worktree = true → 
    ∃ g, IsFFMorph main worktree g

/-- Practical pattern: Extract witness, carry proof -/
structure IntegrationResult where
  newMain : Commit
  proof : IsFFMorph main newMain graph  -- Proof-carrying code

def integrate_with_evidence (oracle : GitOracle) (main : Commit) (worktree : Commit) :
  IO (Option IntegrationResult) := do
  if query_ff oracle main worktree then
    -- We have proof-carrying evidence
    let newGraph := graph.addEdge main worktree
    return some {
      newMain := worktree,
      proof := by { 
        -- Proof constructed from oracle soundness
        apply oracle_sound oracle main worktree
        simp [query_ff]
      }
    }
  else
    return none

/-- The key theorem: Traced integration implies categorical validity -/
theorem traced_integration_valid (main worktree : Commit) (g : GitGraph)
  (h : integrate_sh_trace main worktree g = IO.pure (some proof)) :
  IsFFMorph main worktree g := by
  -- The proof object is the witness
  exact proof

end Cicsc.Evolution.FFIntegration