Numerics Interval Basic Hi Lt Implies Contains Lt

The upper-bound guarantee

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 of rational numbers, a lower bound and an upper bound, with the guarantee that the lower bound is no larger than the upper bound. The framework's machine-checked library of formal theorems defines such an interval as a structure with two rational fields, lo and hi, together with a proof that lo ≤ hi. The library also defines what it means for a real number x to be contained in an interval: x is contained exactly when lo ≤ x and x ≤ hi, where the rational endpoints are read as real numbers.

The theorem hi_lt_implies_contains_lt states a simple consequence of that definition. If the interval's upper endpoint hi is strictly less than some rational number b, and a real number x is contained in the interval, then x is strictly less than b. The proof is a direct chain of inequalities: x ≤ hi follows from containment, hi < b is the assumption, and the two combine to give x < b. The same library also proves the mirror-image statement for lower endpoints: if b is strictly less than the interval's lower endpoint lo, then every contained real x satisfies b < x.

These lemmas matter because they let a computation conclude a strict bound on a real number without ever touching the real number directly. The interval's endpoints are rational, so they can be computed exactly. If the computed interval's upper endpoint is below a target threshold, the lemma certifies that every real value the interval represents is also below that threshold. This is the rigorous backbone for bounding transcendental functions: the library's docstring says the key insight is using rational endpoints, which the machine can compute exactly, to bound real values.

In Recognition Science, this theorem is a small piece of the framework's numerical infrastructure. It does not derive any physical constant, nor does it touch the forcing chain that produces the golden ratio or the number of spatial dimensions. It is a routine fact about ordered fields, stated and proved inside the machine-checked library so that larger proofs can rely on it. What it establishes is narrow and precise: a strict upper bound on an interval's rational endpoint transfers to a strict upper bound on every real number the interval contains.

The practical consequence for a reader is that interval computations carry certificates. When a computation returns an interval whose upper endpoint is below a bound, the certificate says the true value is below that bound, with no floating-point rounding and no unstated assumption. The theorem does not claim that the interval is tight, that it contains all relevant values, or that any particular real number lies inside it; it only says what follows when containment is already known.

/-- If I.hi < b, then all x in I satisfy x < b -/
theorem hi_lt_implies_contains_lt {I : Interval} {b : ℚ} (h : I.hi < b) {x : ℝ}
    (hx : I.contains x) : x < (b : ℝ) :=
  lt_of_le_of_lt hx.2 (by exact_mod_cast h)

/-- 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

/-- A closed interval with rational endpoints. -/
structure Interval where
  lo : ℚ
  hi : ℚ
  valid : lo ≤ hi
  deriving DecidableEq

/-- Containment: a real number x is in interval I if lo ≤ x ≤ hi -/
def contains (I : Interval) (x : ℝ) : Prop :=
  (I.lo : ℝ) ≤ x ∧ x ≤ (I.hi : ℝ)

What this page does not claim

The theorem does not claim that the interval is the smallest interval containing the value, nor that it contains any particular real number. It does not derive any physical constant or any result from the framework's forcing chain. It does not address floating-point arithmetic or any approximate computation method.

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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Derived articles

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