feat(cont): prove CONT-2 justifiable iff distance is window, square, or pronic - #37
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…or pronic Prove justifiable_iff: the executable Slot.isJustifiableAfter holds iff the slot distance from the finalized slot is at most 5, a perfect square, or a pronic number. The core is correctness of the hand-rolled isqrt (isqrt_le / isqrt_lt_succ by functional induction on the base-4 recursion, uniqueness via isqrt_eq), plus the square-expansion helpers sq_succ / odd_sq and the pronic discriminant identity 4k(k+1) + 1 = (2k+1)^2 that justifies the upstream odd-square trick. Axiom-free, no Mathlib.
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Summary
Discharges CONT-2 from
docs/lean4-proof-propositions.md:Slot.isJustifiableAfter finalized targetholds iff the slot distance δ satisfies δ ≤ 5, δ is a perfect square, or δ is pronic — i.e. the executable 3SF-mini justification-candidate check (landed with #33) matches its mathematical specification.Proof structure in
LeanSpec/Forks/Lstar/Slot.lean:isqrt_le/isqrt_lt_succ— bracketing boundsisqrt n · isqrt n ≤ n < (isqrt n + 1)²by functional induction on the base-4 recursion.isqrt_eq— uniqueness: anykwithk² ≤ n < (k+1)²equalsisqrt n.sq_succ/odd_sq— square expansions soomegacan work with products as atoms (noringwithout Mathlib).pronic_disc—4k(k+1) + 1 = (2k+1)², the identity behind upstream's odd-perfect-square pronic detection.justifiableDelta_iff(Nat-level) →justifiable_iff(catalog form).Catalog: CONT-2 checked off (CONT 1/1/0, total 13 proved),
last_updatedbumped.Note: no GitHub issue exists for CONT-2 (issue creation was blocked in-session); the catalog entry is the tracking artifact. CONT-1 follows in a stacked PR (#36).
Verification
lake buildgreen with zero warnings; nosorry, no new axioms.Part of the Containers domain (CONT-1 → #36).