# Agda → Lean 4 (mathlib) migration plan

Goal: port the static-analysis framework to Lean 4 + mathlib, preserving the
overall structure and **the same theorems/lemmas** (modulo language details),
while lifting custom machinery into mathlib wherever a standard counterpart
exists. Per discussion, the setoid equality (`_≈_`) is **dropped in favor of
propositional `=`** — it existed mainly so that unordered key-value maps could
be "equal"; representations below are chosen to be canonical so `=` works.

The Lean project lives in `lean/` (library root `Spa`). Each phase ends with a
green `lake build` and a correspondence table appended to this file, so you can
validate phase by phase.

## Design mapping

| Agda | Lean | Notes |
|---|---|---|
| `Equivalence.agda` | *lifted*: `Eq`, `Equivalence` | module disappears |
| `IsDecidable` | *lifted*: `DecidableEq` / `DecidableRel` | mathlib is classical; decidability kept only where functions compute (e.g. the fixpoint iteration) |
| `Showable.agda` | *lifted*: `ToString` | |
| `Lattice.agda` `IsSemilattice` (`⊔-assoc/comm/idemp`, `≼`, `≼-refl/trans/antisym`, `x≼x⊔y`, `⊔-Monotonicˡ/ʳ`) | *lifted*: `SemilatticeSup` (`sup_assoc`, `sup_comm`, `sup_idem`, `≤` with `sup_eq_right`, `le_refl`, `le_trans`, `le_antisymm`, `le_sup_left`, `sup_le_sup_left/right`) | `a ≼ b := a ⊔ b ≈ b` becomes `a ≤ b` with bridge lemma `sup_eq_right` |
| `IsLattice` (`absorb-⊔-⊓`, `absorb-⊓-⊔`) | *lifted*: `Lattice` (`sup_inf_self`, `inf_sup_self`) | |
| `Monotonic`, `Monotonicˡ/ʳ/₂` | *lifted*: `Monotone` (+ tiny aliases) | |
| `foldr-Mono`, `foldl-Mono`, `foldr-Mono'`, `foldl-Mono'` | custom, `Spa/Lattice.lean` | stated with `List.Forall₂` (≙ `Utils.Pairwise`) |
| `Chain.agda` (`Chain`, `concat`, `Chain-map` in `ChainMapping`) | *lifted*: `LTSeries` (`RelSeries.smash`, `LTSeries.map` + `Monotone.strictMono_of_injective`) | with `=`, the ≈-congruence steps in chains vanish |
| `Chain.Height`, `Bounded`, `Bounded-suc-n` | custom: `Spa.FixedHeight` structure (`⊥`, `⊤`, longest `LTSeries`, `bounded`) | |
| `IsFiniteHeightLattice`, `FiniteHeightLattice` | custom class `Spa.FiniteHeightLattice` | |
| `⊥≼` (chain bottom is least, given decidable eq) | custom, same proof shape (prepend `⊥⊓a ≺ ⊥` to longest chain) | decidability hypothesis dropped (classical) |
| `Fixedpoint.agda` (`doStep` with gas, `aᶠ`, `aᶠ≈faᶠ`, `aᶠ≼`) | custom, `Spa/Fixedpoint.lean`, same gas-based algorithm | **not** replaced by mathlib `lfp` (would change the proof approach and lose computability) |
| `Isomorphism.agda` (`TransportFiniteHeight`) | custom, `Spa/Isomorphism.lean` | much smaller: with `=`, f/g monotone inverse pair transports `FixedHeight` via `LTSeries.map` |
| `Lattice/Unit.agda` | *lifted*: mathlib `Lattice PUnit`; custom `FixedHeight PUnit 0` | |
| `Lattice/Nat.agda` (max/min lattice) | *lifted*: mathlib `Lattice ℕ` (`Nat.instLattice`) | kept only as a remark; file had no fixed-height content |
| `Lattice/Prod.agda` | instance *lifted* (`Prod.instLattice`); custom: `unzip` + `FixedHeight (A×B) (h₁+h₂)` | same proof: split a product chain into component chains |
| `Lattice/AboveBelow.agda` (flat lattice ⊥/[x]/⊤) | custom, same datatype; `Plain` module ⇒ `FixedHeight 2` | mathlib has no flat-lattice-on-discrete-type |
| `Lattice/ExtendBelow.agda` | *lifted*: `WithBot A` lattice instance; custom `FixedHeight (h+1)` | unused by the pipeline; ported for parity (optional) |
| `Lattice/IterProd.agda` | custom, same induction (`IterProd k = A × … × B`), lattice + height-sum by recursion | the `Everything` record trick survives as a recursive definition of bundled instances |
| `Lattice/Map.agda` (assoc list with `Unique` keys, setoid) | **deleted**: only existed to support setoid map equality | its consumers move to `Finset` / spine-fixed `FiniteMap` |
| `Lattice/MapSet.agda` (`StringSet`) | *lifted*: `Finset String` (`∪`, `{·}`, `∅`, `.toList`, `nodup_toList`) | |
| `Lattice/FiniteMap.agda` | custom: `{ l : List (A × B) // l.map Prod.fst = ks }` — key spine fixed ⇒ `=` is pointwise value equality | same API: `locate`, `_[_]`, `GeneralizedUpdate` (`f'`, `f'-Monotonic`, `f'-k∈ks-≡`, `f'-k∉ks-backward`), `m₁≼m₂⇒m₁[k]≼m₂[k]`, `Provenance-union` analog; fixed height **still via isomorphism to `IterProd`** (same approach) |
| `Lattice/Builder.agda` | **skipped** — not imported by anything in the repo | flag if you want it |
| `Utils.agda` | *lifted*: `Unique`→`List.Nodup`, `Pairwise`→`List.Forall₂`, `fins`→`List.finRange`, `∈-cartesianProduct`→`List.product`/`pair_mem_product`, `x∈xs⇒fx∈fxs`→`List.mem_map_of_mem`, `filter-++`→`List.filter_append`, `iterate`→`f^[n]`, `concat-∈`→`List.mem_join`, `All¬-¬Any` etc. → `List.All`/`Any` API | leftovers (if any) in `Spa/Utils.lean` |
| `Language/Base.agda` | custom; `Expr-vars`/`Stmt-vars : Finset String` | commented-out `∈-vars` lemmas stay omitted |
| `Language/Semantics.agda` | custom, same big-step relations; `Value`, `Env = List (String × Value)`, custom `∈` | `ℤ` → `Int` |
| `Language/Graphs.agda` | custom; `Vec` → `Vector` (mathlib `List.Vector`), `Fin._↑ˡ/_↑ʳ` → `Fin.castAdd`/`Fin.natAdd` | same `Graph` record, `∙`/`↦`/`loop`/`skipto`/`singleton`/`wrap`/`buildCfg`, `predecessors` + edge lemmas |
| `Language/Traces.agda` | custom, same `Trace`/`EndToEndTrace`/`++⟨_⟩` | |
| `Language/Properties.agda` | custom, same lemma inventory (`Trace-∙ˡ/ʳ`, `Trace-↦ˡ/ʳ`, `Trace-loop`, `EndToEndTrace-*`, `wrap-preds-∅`, `buildCfg-sufficient`) | the "ugly" `↑-≢` Fin-disjointness block should shrink via `Fin.castAdd_ne_natAdd`-style mathlib lemmas |
| `Language.agda` (`Program` record) | custom, same fields/lemmas (`trace`, `vars`, `states`, `incoming`, `initialState-pred-∅`, …) | |
| `Analysis/Forward/{Lattices,Evaluation,Adapters}.agda`, `Analysis/Forward.agda` | custom, same structure: `VariableValues`, `StateVariables`, `joinForKey`/`joinAll`, `StmtEvaluator`/`ExprEvaluator` + validity, expr→stmt adapter, `analyze`, `result`, `analyze-correct` | section variables instead of parameterized modules |
| `Analysis/Sign.agda`, `Analysis/Constant.agda` | custom, same definitions | the four monotonicity **postulates** become real proofs by `decide` (finite lattice, decidable `≤`) |
| `Main.agda` | `lake exe spa` | same test programs, same printed output |

## Phases & checkpoints

- **Phase 0 — scaffold.** `lean/` lake project, mathlib pinned to toolchain
  v4.17.0 (already installed). ✅ checkpoint: `lake build` green on empty lib.
- **Phase 1 — core order theory.** `Spa/Lattice.lean` (Monotone aliases, fold
  monotonicity, `FixedHeight`, `Bounded`, `FiniteHeightLattice`, chain-bottom-
  is-least). ✅ checkpoint: build + table below.
- **Phase 2 — fixpoint & transport.** `Spa/Fixedpoint.lean`,
  `Spa/Isomorphism.lean`. ✅ checkpoint: `fix`, `fix_eq`, `fix_le`,
  `TransportFiniteHeight`.
- **Phase 3 — basic lattice instances.** Unit, Prod (+height), AboveBelow
  (+`Plain`, height 2), ExtendBelow. ✅ checkpoint.
- **Phase 4 — map lattices.** IterProd, FiniteMap (+fixed height via IterProd
  isomorphism), MapSet→`Finset` shims. ✅ checkpoint.
- **Phase 5 — language.** Base, Semantics, Graphs, Traces, Properties,
  `Program`. ✅ checkpoint: `buildCfg_sufficient`, `Program.trace`.
- **Phase 6 — forward analysis framework.** Lattices/Evaluation/Adapters/
  Forward. ✅ checkpoint: `analyze_correct`.
- **Phase 7 — concrete analyses + executable.** Sign, Constant, Main.
  ✅ checkpoint: `lake exe spa` output vs Agda `Main` output; postulates now
  proved.

## Status

- [x] Phase 0
- [x] Phase 1
- [x] Phase 2
- [x] Phase 3
- [x] Phase 4
- [x] Phase 5
- [x] Phase 6
- [ ] Phase 7