Antiecho partition dec#98
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… (f x ≡ y) Lands obligation 5 of `coecho.md` §5 (deferred since the original AntiEcho thin slice in PR #69 / commit 428b4b8): the per-element partition of A into the Echo-side and the AntiEcho-side, given decidability of `f x ≡ y`. antiecho-partition-dec : ∀ {A B} {f : A → B} {y : B} (x : A) → Dec (f x ≡ y) → Echo f y ⊎ AntiEcho f y antiecho-partition-codomain-dec : ∀ {A B} {f : A → B} {y : B} → (∀ b → Dec (b ≡ y)) → (x : A) → Echo f y ⊎ AntiEcho f y Companion to the existing `antiecho-disjoint` (per-element). Together they exhibit A as the disjoint union of the Echo-side and AntiEcho- side parameterised by y: every x classifies to one side, and no x appears on both. Also corrects a misleading line in the prior preamble. The earlier comment claimed the asymmetric joint statement `Echo f y → AntiEcho f y → ⊥` (with possibly distinct domain witnesses) "requires decidability on the codomain to collapse the two witnesses". That joint statement is genuinely *not* a theorem at any decidability strength — Bool counterexample with `f = id` simultaneously inhabits both sides at `y = true`. The deferred-but-now-landed obligation 5 is the per-element classification above, not joint disjointness; the preamble is rewritten to say so. Build: --safe --without-K, zero postulates, no funext. Both `antiecho-partition-dec` and `antiecho-partition-codomain-dec` pinned in `Smoke.agda`. `agda proofs/agda/All.agda` and `agda proofs/agda/Smoke.agda` exit 0. Refs: project_echo_types_swarm_2026_05_20 (small deferrals) Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
# Conflicts: # proofs/agda/AntiEcho.agda
🔍 Hypatia Security ScanFindings: 10 issues detected
View findings[
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