Research proof index

Merkle opening uniqueness. When h is injective, two openings at the same leaf index that produce the same root must agree on both the leaf value and the entire authentication path. Proof: induction on the index. At each internal node, injectivity of h forces both hash arguments to agree , the sibling (proof head) and the recursive subtree root. Th...

Log-level binding (Collision Lemma at the log level). Log-formalized analog of getPutativeRootWithHash_binding_collision: two distinct openings (x, proof₁) ≠ (y, proof₂) of the same root at the same index, both witnessed by hash chains ChainInLog in the same log, force log to contain a hash collision (two log entries with equal responses b...

Running the monadic version of buildMerkleTree with an oracle function f is equivalent to running the functional version of buildMerkleTreeWithHash with the same oracle function.

Running the monadic getPutativeBatchRoot with an oracle function f is the same as running the functional getPutativeBatchRootWithHash with the corresponding hash function.

Under the empty env alphabet (EnvAction.empty Unit) and the trivial env scheduler (always processTick), runStepsAsync reduces to ProcessOver.runSteps with the env state pinned to () and a constant processTick trace. This is the operational core of the sync-recovery story: it factors the async engine into the synchronous ProcessOver.runSteps...

Weak monoid homomorphisms distribute over non-empty big ops.