Mathematics Fundamental Theorem Calculus From Rs Calculus Theorem

The five calculus theorems

The fundamental theorem of calculus is the bridge between two operations: differentiation, which measures how fast a quantity changes, and integration, which accumulates a rate over an interval. The theorem states that these operations are inverses. If you integrate a derivative, you recover the original function's change in value. The classic statement is ∫₁ʳ J'(x) dx = J(r) - J(1), where J is any well-behaved function.

This page concerns a specific function J, defined by the cost formula J(x) = (x + 1/x)/2 - 1. This function has a special property: at x = 1, its value is zero and its derivative is zero, making 1 a minimum point. The integral of J' from 1 to r equals J(r) - J(1) = J(r), since J(1) = 0. This identity links the calculus theorem directly to the cost function's structure.

The Recognition Science framework organizes the fundamental theorem of calculus together with four other standard results: the second fundamental theorem, the mean value theorem, the intermediate value theorem, and L'Hôpital's rule. The framework's machine-checked library defines an object called CalculusTheorem with exactly these five cases. A result in the library proves that the number of such cases is 5, and a certificate bundles this count with the two minimum properties of J.

In Recognition Science, the cost function J measures the price of a recognition event, and this calculus connection shows that the integral of its derivative from 1 to r equals the total cost of recognition up to r. The five theorems form a structural cluster: their count matches a dimension parameter in the framework's geometry. This is a derived fact about how the framework organizes classical calculus, not a new mathematical result about the theorems themselves.

The declaration does not prove the fundamental theorem of calculus from scratch. It assumes the theorem's validity and packages it with four other classical results into a single formal object. It also does not claim that these five theorems are the only calculus theorems, nor that the cost function J is the only function satisfying the inverse relationship. The minimum property of J at 1 is proved, but the broader claim that all calculus reduces to this structure remains a framework-internal organization.

inductive CalculusTheorem where
  | FTC1 | FTC2 | meanValue | intermediateValue | lhopital
  deriving DecidableEq, Repr, BEq, Fintype

theorem calculusTheoremCount : Fintype.card CalculusTheorem = 5 := by decide

/-- J(1) = 0 (minimum, derivative = 0 at critical point). -/
theorem jcost_minimum : Jcost 1 = 0 := Jcost_unit0

/-- J is positive off minimum (strict local minimum). -/
theorem jcost_strict_min {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

def calculusCert : CalculusCert where
  five_theorems := calculusTheoremCount
  minimum_at_1 := jcost_minimum
  strict_minimum := jcost_strict_min

What this page does not claim

This declaration does not prove the fundamental theorem of calculus from first principles. It does not claim that these five theorems are the only calculus theorems. It does not assert that the cost function J is the unique function satisfying the inverse relationship.

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/Mathematics/FundamentalTheoremCalculusFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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