intensityAtRung_strictly_decreasing
plain-language theorem explainer
The intensity function on the phi-ladder is strictly decreasing in rung index. Strength-training designers cite the result to confirm that each higher rung yields a lower fraction of one-rep maximum. The proof is a short tactic sequence that rewrites via the successor ratio, invokes positivity, derives the inverse-phi bound from the phi-greater-than-1.5 lemma, and closes with a multiplication inequality.
Claim. Let $I(k) = I_0 phi^{-k}$ for natural number $k$, where $I_0$ is the reference intensity and $phi > 1$ is the golden ratio. Then $I(k+1) < I(k)$ holds for every $k$.
background
The Lifting Program Design module models canonical strength schemes as integer steps on the phi-ladder of relative intensity. The function intensityAtRung(k) is defined by referenceIntensity times phi to the power minus k, with rung 0 anchored at 100 percent of one-rep maximum. Upstream results establish that phi exceeds 1.5 and that intensityAtRung(k) remains positive for all natural k.
proof idea
The tactic proof rewrites the goal with the successor-ratio lemma to obtain intensityAtRung(k) times phi inverse. It invokes the positivity theorem for the left factor, derives phi inverse less than one from the phi-greater-than-1.5 lemma via linarith and inv_lt_one_of_one_lt, applies mul_lt_mul_of_pos_left, and finishes with simpa.
why it matters
The declaration supplies the strictly_decreasing field required by the liftingProgramCert definition, which packages positivity, ratio, and decrease into a single certificate. It realizes the phi-ladder structure forced by the Recognition Science self-similar fixed point, aligning intensity steps with the eight-tick octave and the three spatial dimensions of the framework.
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