E5: add index monotonicity/inversion and psi0(Omega1) path#7
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…y) (#63) 2 files, SPDX-value-only suffix fix. NOT a relicence (same licence). Owner-sanctioned carve-out, standards LICENCE-POLICY A8(1). Refs LICENCE-DEBT-LEDGER-2026-05-18. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-authored-by: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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…#70) ## Summary Extends `proofs/agda/EchoApprox.agda` with the first slice of the axis-2 composition rung from the design note (drafted in `/tmp/echo-types-exploration/axis2-approximate.md`, §7 obligations 2 + 6, plus the canonical-split retract direction). Refs `docs/echo-types/roadmap.md` Axis 2 entry and `docs/echo-types/composition.md` §Q3 — does **not** close either. ## Obligations landed - **`echo-strict→approx`** (§7 #2). General strict ⇒ zero-tolerance approximate. Generalises `echo-approx-intro` from own-fibre points to arbitrary `y` via the codomain equation `p : f x ≡ y`. One extra `subst` along `p`. - **`echo-approx-comp-sound`** (§7 #6). Sound RHS-to-LHS direction of the retract shape from `composition.md` §Q3 / design-note §5. Unpacks the existential and calls `echo-approx-compose`. - **`echo-approx-comp-retract-to`**. Canonical-split LHS → RHS-Σ section of the retract: picks `b := f x`, `ε₁ := zero`, `ε₂ := ε`. Uses `echo-approx-intro` for the inner echo, original bound for the outer. - **`echo-approx-comp-retract-A`**. A-component round-trip `proj₁ ∘ sound ∘ retract-to ≡ proj₁`, proved by `refl`. The retraction direction holds definitionally on the A-component as the design note (§5) predicts. ## Obligations deferred (out of this rung) - §7 #7 separated zero-collapse — needs a separation predicate on the `PseudoMetric` record. - §7 #8 axis-1 shadow agreement — cross-axis classification. - Full retract B-component and tolerance-budget round-trip — needs a `+`-left-identity axiom on `Tolerance` (`zero + ε ≡ ε`) not in the current record. Adding it commits the carrier to a left-unital monoid; deferred as a separate decision. - Lipschitz generalisation (`L_g ≠ 1`) — needs scalar multiplication on `Tol`. ## Docs updated - `docs/echo-types/composition.md` §Q3 promoted from "entirely speculative" to landed-retract-shape with deferred items called out. - `docs/echo-types/roadmap.md` adds a "landed" entry for the composition rung first slice. ## Invariants preserved - `--safe --without-K` - No postulates - No escape pragmas - Funext untouched ## Test plan - [x] `LC_ALL=C.UTF-8 agda -i proofs/agda proofs/agda/All.agda` exits 0 - [x] `LC_ALL=C.UTF-8 agda -i proofs/agda proofs/agda/Smoke.agda` exits 0 - [x] No new `postulate`, no escape pragmas, funext not imported - [x] `Smoke.agda` already pins `module Approx` following project convention (per-lemma pins inside parameterised modules require instantiation, which the project does not do for `EchoApprox`). Co-authored-by: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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…dow (#74) ## Summary Picks up two of three §7 deferrals from PR #70 in `/tmp/echo-types-exploration/axis2-approximate.md`. Refs not Closes because §7 is multi-rung and one rung (B-component + tolerance-budget retract round-trip) stays deferred pending a `Tolerance` interface design call (see below). ## Rungs landed ### Rung A — separated zero-collapse (§7 #7) - `Separated : Set` — separation predicate on the pseudo-metric: `∀ b₁ b₂ → dist b₁ b₂ ≤ zero → b₁ ≡ b₂`. Kept out of the `PseudoMetric` record; supplied as an explicit hypothesis at the lemma site (pseudo-metrics in general only guarantee `dist y y ≡ zero`, not the converse — making them a *metric proper* is an optional refinement the record deliberately does not bake in). - `echo-approx-zero-collapses-strict : Separated → EchoR zero f y → Echo f y` formalises the §4 "Approximate → strict (only when separated, at ε = 0)" statement. ### Rung B — axis-1 shadow agreement (§7 #8) - `echo-shadow-A` — A-component projection (axis-1 shadow of an approximate echo). - `echo-shadow-iso-{to,from}` — pins the §7 #8 obligation explicitly (definitional both ways: `EchoR ε f y` IS the existential `Σ A (λ x → dist (f x) y ≤ ε)`). - `echo-strict→approx-shadow-A` — cross-axis cohere: the A-component of `echo-strict→approx` agrees with the strict A-component on the nose (`refl`). - `echo-strict→approx-collapse-shadow-A` — round-trip under separation: `zero-collapses-strict ∘ strict→approx ≡ id` on the A-component, definitionally. ## DESIGN-DECISION STOP — Rung C deferred The full B-component + tolerance-budget retract round-trip (`echo-approx-comp-retract-from-to` and friends) requires a `+`-left-identity axiom on `Tolerance` (`zero + ε ≡ ε`, plus right-identity for the symmetric variants). ### Current shape of `Tolerance` ```agda record Tolerance ℓ : Set (suc ℓ) where field Tol : Set ℓ zero : Tol _+_ : Tol → Tol → Tol _≤_ : Tol → Tol → Set ℓ ≤-refl : ∀ {ε} → ε ≤ ε ≤-trans : ∀ {ε₁ ε₂ ε₃} → ε₁ ≤ ε₂ → ε₂ ≤ ε₃ → ε₁ ≤ ε₃ +-mono-≤ : ∀ {a b c d} → a ≤ b → c ≤ d → (a + c) ≤ (b + d) ``` No identity laws on `+`. Minimal interface, lets us state `echo-approx-compose`; that's all it currently does. ### What needs to be added For the symmetric retract round-trip: ```agda +-identityˡ : ∀ {ε} → (zero + ε) ≡ ε +-identityʳ : ∀ {ε} → (ε + zero) ≡ ε ``` (`≡`, not `≤` — the round-trip is a definitional retract on the tolerance budget, so we need a propositional equality, not just mutual `≤`.) ### Two options **(a)** Extend `Tolerance` with the two identity-law fields. - Pro: minimal scaffolding; one type to thread through. - Con: ripples to every `Tolerance` instance — currently just `EchoApproxInstance.trivialTolerance` (trivial discharge), but consumers may grow. Makes `Tolerance` strictly stronger than the "ordered commutative-monoid-flavoured" framing in §3 of the axis-2 design note. **(b)** Add a separate `BalancedTolerance` record on top of `Tolerance` carrying just the identity laws. - Pro: additive; no interface breakage; layered/opt-in. Matches the design pattern of `PseudoMetric` (which is *separately* refined to a metric by adding `Separated`, not by widening the base record). Keeps the §3 "minimal moving parts" promise. - Con: an extra record name to thread through retract-round-trip lemmas. ### Recommendation **(b).** The current `Tolerance` is deliberately the minimal interface that lets us state `echo-approx-compose` (triangle + additive monotonicity, nothing else). Identity laws are needed only by the retract round-trip, which is a downstream refinement — keeping them out of the core interface preserves the "minimal moving parts" design from §3 of the axis-2 design note. The analogue is `PseudoMetric` vs. (`PseudoMetric` + `Separated`): the refinement lives outside the base record. Rung C is deferred pending owner decision. ## Rung D — also deferred The Lipschitz generalisation `L_g ≠ 1` would need scalar multiplication on `Tolerance` — a third interface call (whether to add it to the core record, or layer a `ScalableTolerance` refinement). Deferred alongside Rung C; same recommendation shape applies. ## Hygiene All new headlines pinned via the per-lemma proof-of-life pattern from PR #71: `EchoApproxInstance.agda` adds top-level identifiers at the trivial-pseudometric-on-`⊤` instance, `Smoke.agda` enumerates them in its `using` clause. ## Verification ``` $ agda -i proofs/agda proofs/agda/All.agda # exit 0 $ agda -i proofs/agda proofs/agda/Smoke.agda # exit 0 ``` `--safe --without-K` throughout. No postulates. No escape pragmas. No funext. ## Refs - PR #70 (retract direction) - `/tmp/echo-types-exploration/axis2-approximate.md` §7 Refs not Closes — §7 obligations 7 and 8 are landed, but obligation 6's "full retract round-trip" (Rung C) and the Lipschitz generalisation remain. ## Test plan - [x] `agda proofs/agda/All.agda` exits 0 - [x] `agda proofs/agda/Smoke.agda` exits 0 - [x] No postulates / escape pragmas / funext introduced - [x] All new headlines pinned in `Smoke.agda` via `EchoApproxInstance` - [ ] Owner decides Rung C `Tolerance` interface question (a vs. b) - [ ] After decision: implement chosen option, land full retract round-trip 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-authored-by: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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