IndisputableMonolith.Numerics.Interval.Tactic
This module supplies tactics for proving bounds using interval containment in real arithmetic. Researchers verifying numerical inequalities for phi and related constants in Recognition Science would cite it. It organizes reusable tactics that compose interval lemmas from the Basic, Exp, Log, and Pow modules into automated bound proofs.
claimTactics to establish bounds such as $l_1 < f(x) < u_1$ or $l_2 < g(x) < u_2$ via containment in verified intervals with rational endpoints.
background
The module operates within the verified interval arithmetic setting of the Numerics.Interval hierarchy. Basic supplies rigorous interval arithmetic using rational endpoints to bound real values exactly. Exp, Log, and Pow extend this to the exponential, natural logarithm, and power functions with domain-specific strategies for error control on intervals in [0,1), (0,∞), and positive reals respectively.
proof idea
This is a tactic module that defines helper tactics applying interval containment lemmas. It wraps the imported operations from Basic, Exp, Log, and Pow into reusable steps for lower and upper bound goals, plus specialized checks such as phi interval containment.
why it matters in Recognition Science
The module enables numerical verification steps that support Recognition Science claims on constants and fixed points. It underpins sibling results such as phi interval contains and log phi bounds, aiding closure of numerical aspects in the phi-ladder and self-similar fixed point without floating-point reliance.
scope and limits
- Does not implement general interval arithmetic solvers.
- Does not extend bounds to complex numbers or other fields.
- Does not cover arbitrary functions beyond exp, log, and pow.
- Does not replace manual proofs for non-interval goals.