Numerics Interval Basic Lo Gt Implies Contains Gt

Interval arithmetic

Interval arithmetic is a way of computing with ranges of values instead of single numbers. A closed interval with rational endpoints is written as a pair (lo, hi) with lo ≤ hi, meaning the set of all real numbers x such that lo ≤ x ≤ hi. The framework's machine-checked library of formal theorems defines this structure and proves the basic facts about it.

The theorem lo_gt_implies_contains_gt states a simple consequence of that definition: if a rational number b is strictly less than the lower endpoint lo of an interval I, and x is any real number contained in I, then b is strictly less than x. In symbols, from b < I.lo and I.contains x, the theorem derives (b : ℝ) < x. The proof is a direct chain of inequalities: b < lo, and lo ≤ x because x lies in the interval, so b < x.

This result is one of four companion lemmas in the same module. Its mirror, lo_ge_implies_contains_ge, covers the non-strict case where b ≤ lo implies b ≤ x. The upper-endpoint versions, hi_lt_implies_contains_lt and hi_le_implies_contains_le, do the same work on the right side. Together they let a computation transfer a strict or non-strict bound on an interval's endpoint to every real number the interval contains.

The theorem does not claim anything about intervals whose endpoints are real numbers rather than rationals, nor does it say that every real number between lo and hi is actually attained by some computation. It only guarantees the one-way implication: a strict lower bound on the endpoint forces a strict lower bound on every member. This is a small but load-bearing step for rigorous numerical work, where a verified bound on an interval is a verified bound on all values inside it.

/-- If b < I.lo, then all x in I satisfy b < x -/
theorem lo_gt_implies_contains_gt {I : Interval} {b : ℚ} (h : b < I.lo) {x : ℝ}
    (hx : I.contains x) : (b : ℝ) < x :=
  lt_of_lt_of_le (by exact_mod_cast h) hx.1

/-- If b < I.lo, then all x in I satisfy b < x -/
theorem lo_gt_implies_contains_gt {I : Interval} {b : ℚ} (h : b < I.lo) {x : ℝ}
    (hx : I.contains x) : (b : ℝ) < x :=
  lt_of_lt_of_le (by exact_mod_cast h) hx.1

What this page does not claim

The theorem does not apply to intervals with real, non-rational endpoints. It does not assert that every real between lo and hi is produced by a computation. It says nothing about upper bounds; those are covered by separate lemmas.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Numerics/Interval/Basic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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