ithm

Ithm is a term encountered in discussions of mathematical logic and formal methods to denote an indexed theorem, typically written as T_i. The concept is not a universally standardized term, but it appears in contexts that emphasize structured organization of proofs and formal datasets.

Definition and notation

In a formal corpus, an ithm refers to the i-th theorem in a sequence, with T_i representing the statement and its associated proof, premises, and metadata. The indexing enables precise referencing and can be used to build a dependency graph where each theorem records which earlier theorems it relies on.

Origins and context

The idea of indexing theorems by position rather than by name arises in settings such as literate programming for mathematics, formal textbooks, and proof-dassistant datasets. It supports discussions about proof structure, induction schemes, and provenance tracking, where tracking the exact origin and dependencies of statements is important.

Usage and implications

In practice, formal work that uses ithm notation often combines T_i with metadata such as premises, proof sketches, and dependencies. Some authors prefer named lemmas or theorem identifiers, but ithm can be useful in teaching, in automated reasoning, and in documenting iterative proof development where the sequencing of results matters.

Limitations

Index-based notation can be fragile when content is inserted or reorganized, requiring rebasing of indices. Different sources may adopt different indexing schemes (starting at 0 or 1, etc.), which can reduce portability across texts and tools.

See also Theorem, Lemma, Corollary, Proof assistant, Formalization.