feat(ch25): complete Johnson's algorithm — potential, triangle inequality, end-to-end correctness - #115
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…lity, end-to-end correctness - Add Bellman-Ford potential construction h(v) = δ(none, some v) via augmented graph with noNegCycle_johnsonAugmentedGraph lemma - Prove triangle inequality h(v) ≤ h(u) + w(u, v) for every edge - Prove reweighted graph edge nonnegativity (Dijkstra precondition) - Prove end-to-end Johnson correctness theorem johnsonDist_isShortestDist (CLRS Theorem 25.5) - Add generic walk-extension lemmas (append_step, walkWeight_append_step) - Add walk-projection infrastructure (chain_none_free_map, walkWeight_none_free_eq, exists_walk_in_G_of_none_free_walk) - Update chapter guide to reflect completion of Sections 25.2 and 25.3 - Update progress CSV to track completed theorems Closes #32, closes #104 Co-Authored-By: Claude <noreply@anthropic.com>
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Summary
Completes Section 25.3 (Johnson's Algorithm), filling the last remaining gaps in Chapter 25.
What's new
12 new definitions/theorems (~350 lines of Lean):
Negative-cycle equivalence for the augmented graph
adj_target_ne_none— no edge in the augmented graph targetsnonechain_no_none— chains not starting atnonenever containnonewalk_from_none_to_none_singleton— the onlynone→nonewalk is[none]noNegCycle_johnsonAugmentedGraph— G has no negative cycles iff G' has noneWalk projection (G' → G)
chain_none_free_map— lift a none-free chain from G' to GwalkWeight_none_free_eq— corresponding walk weights agreeexists_walk_in_G_of_none_free_walk— project walks back to GWalk extension utilities (reusable beyond Ch25)
IsWalkFrom.append_step— extend a walk by a single edge at the endwalkWeight_append_step— weight formula for extended walksJohnson potential
isWalkFrom_none_some— direct edgenone → some vrelaxDist_none_some_ne_top— distance finitenessjohnsonPotential—h(v) = δ(none, some v)via Bellman-FordjohnsonPotential_triangle—h(v) ≤ h(u) + w(u,v)for every edgeEnd-to-end correctness
reweightedGraph_nonneg/reweightedGraph_noNegCyclejohnsonDist— Johnson's all-pairs shortest-path distancejohnsonDist_isShortestDist— CLRS Theorem 25.5Chapter 25 status
0 sorry, full
lake buildandcheck_repository.pypass.Closes #32, closes #104
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