Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The exponential is a continuous bijection from onto
Statement
is a bijection.
Facts & Assumptions
Given: The real exponential function.
It is strictly increasing (The exponential function is strictly increasing) and continuous (The sum of a real power series is continuous at every point strictly inside its interval of convergence).
Its values are positive, and its limits at the two ends are and (The exponential tends to at and to at ).
A continuous function on an interval takes every intermediate value (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on takes every value between and ).
Proof
Strict increase gives injectivity.
Given , [L2] provides with . Applying [L3] on gives with .
Positivity gives the stated codomain, and steps 1.1 and 1.2 give bijectivity.
Depends on
The exponential function is strictly increasing
The exponential tends to at and to at
Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on takes every value between and
The sum of a real power series is continuous at every point strictly inside its interval of convergence