271b47102e
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Lands the foundational DecidableEq machinery and the @[simp]-
based question-form routing in one go (per project discipline:
no shortcuts, no compat shims).
CubicalTransport/DecEq.lean (new, ~290 lines):
- Mutual decEq for the 5-way AST block (CType, CTerm, CTypeArg,
CtorSpec, CTypeSchema) plus list/clause/branch helpers
(decEqListCType, decEqListCTerm, decEqListCTypeArg,
decEqListCtorSpec, decEqClauses, decEqBranches).
- Returns Decidable (a = b) directly; uses OR-patterns for
cross-constructor mismatches, discharged uniformly via cases.
- Five DecidableEq instances declared post-block.
- Lean 4 deriving doesn't support mutual inductives; manual
decEq is the canonical approach.
CubicalTransport/Interval.lean: deriving DecidableEq on DimExpr.
CubicalTransport/Face.lean: deriving DecidableEq on FaceFormula.
CubicalTransport/Question.lean:
- All 11 classifier Decidable instances now land:
IsConstLine, IsFullFace, IsEmptyFace, IsTransport,
IsIntervalLine, IsUnivLine — direct from DecidableEq.
IsPathLine, IsPiLine, IsSigmaLine, IsGlueLine, IsIndLine —
via match h : q.body with on the head constructor +
existential-witness reconstruction in the isTrue arm.
- @[simp] tags on ask_of_full_face / ask_of_empty_face /
ask_of_const_line / ask_of_transport_full_face — the Level 2
routing through CompQ.Equiv.
- Three example proofs at end of file demonstrating that the
simp-set composes (full-face C1, empty-face C2, transport-
shaped interval-line reduction).
CubicalTransport/FFITest.lean: 6 new classifier-decidability
smoke tests (IsFullFace, IsTransport×2, IsPiLine, IsIntervalLine).
Test count: 84 → 89 passing.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
347 lines
19 KiB
Text
347 lines
19 KiB
Text
/-
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Topolei.Cubical.Syntax
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======================
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Deep embedding of the cubical term language (CCHM §2–3).
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Grammar:
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A, B ::= U | Π A B | Path A a b
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t, u ::= x | λx.t | t u | ⟨i⟩t | t@r | transpⁱ A φ t | compⁱ A φ u t
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CType and CTerm are mutually inductive because path types carry endpoint
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terms, and terms carry path applications over DimExprs.
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transp and comp store (i : DimVar) (A : CType) inline rather than DimLine,
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because DimLine is defined later in the import chain (Subst → DimLine).
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The typing rules in Typing.lean use DimLine as a convenient wrapper.
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The path β-rule (⟨i⟩ t) @ r ↝ t[i := r]
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and the four "fully-reducing" transport/comp cases (T1, T2, C1, C2)
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are NbE theorems in `Cubical/Readback.lean`. The residual step-level
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axioms — T3, T5, C4 (subject reduction + face congruence) and T4
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(path-line shape preservation) — live in `TransportLaws.lean` /
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`CompLaws.lean`.
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-/
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import CubicalTransport.Face
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-- ── Syntax ────────────────────────────────────────────────────────────────────
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mutual
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/-- Types in the cubical calculus. -/
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inductive CType where
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| univ : CType -- U
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| pi (A : CType) (B : CType) : CType -- Π A B
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| path (A : CType) (a b : CTerm) : CType -- Path A a b
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/-- Non-dependent Σ type (cells-spec §6.2, §8.1, §9.2).
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`Sigma A B` — pairs whose first component has type `A` and whose
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second has type `B`. Non-dependent: `B` does not refer to the
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first component. Dependent Σ (where `B : A → CType`) is deferred
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for the same reason as dependent Π — requires a term evaluator
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to apply `B` to a term. -/
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| sigma (A B : CType) : CType -- Σ A B
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/-- Glue type (CCHM §6).
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`Glue [φ ↦ (T, e)] A` — on face `φ` the type is `T` with `e : T ≃ A`
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as witness; off face `φ` the type is `A`. The equivalence `e` is
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inlined as five `CTerm` fields (f, fInv, sec, ret, coh) rather than
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a nested `EquivData` so that `CType` / `CTerm` remain a closed
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mutual inductive block — `EquivData` lives in `Equiv.lean` and is
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downstream in the import chain. Use `EquivData.toGlueType` in
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`Equiv.lean` for an ergonomic wrapper. -/
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| glue (φ : FaceFormula) (T : CType)
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(f fInv sec ret coh : CTerm)
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(A : CType) : CType -- Glue [φ ↦ (T, e)] A
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/-- Schema-defined inductive type (REL1, INDUCTIVE_TYPES.md).
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`ind S params` is an instance of the schema `S` at the given
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type parameters. `params.length = S.numParams` is a typing-side
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condition (not enforced syntactically — see `HasType.ind` in
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`Typing.lean`).
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The schema `S` carries the constructor list (point + path
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ctors) and their boundary partial-element systems. Both plain
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inductives (`Nat`, `List A`, `Bool`) and HITs (`S¹`, `‖A‖₋₁`,
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suspensions, pushouts) are encoded uniformly via
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`CTypeSchema`. -/
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| ind (S : CTypeSchema) (params : List CType) : CType -- ind S params
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/-- The cubical interval `𝕀` as a first-class type (REL2).
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Inhabited by `CTerm.dimExpr r` for `r : DimExpr`. No
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non-trivial transport on `.interval` (it is dim-absent in
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itself); transport on `.interval` is always the identity by
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`eval_transp_const` / `eval_transp_top`.
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Pre-REL2, `CTerm.dimExpr` was typed at the placeholder
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`.univ`; promoting `.interval` to a CType primitive lets
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path-constructor dim arguments (`loop @ r`, `seg @ r`,
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`squash _ _ @ r`, etc.) carry real semantic ground.
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See `docs/REL2_PLAN.md` Phase 1. Tag 6. -/
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| interval : CType -- 𝕀
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/-- Terms in the cubical calculus. -/
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inductive CTerm where
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| var (x : String) : CTerm -- x
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| lam (x : String) (t : CTerm) : CTerm -- λx. t
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| app (f a : CTerm) : CTerm -- f a
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| plam (i : DimVar) (t : CTerm) : CTerm -- ⟨i⟩ t
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| papp (t : CTerm) (r : DimExpr) : CTerm -- t @ r
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| transp (i : DimVar) (A : CType) (φ : FaceFormula)
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(t : CTerm) : CTerm -- transpⁱ A φ t
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| comp (i : DimVar) (A : CType) (φ : FaceFormula)
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(u t : CTerm) : CTerm -- compⁱ A φ u t
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/-- Multi-clause heterogeneous composition.
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`compⁱ A [φ₁ ↦ u₁, …, φₙ ↦ uₙ] t` — a partial element defined
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over the union of the clause faces. Coherence (clauses agree
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on overlaps and with `t` at `i = 0`) is a typing-level side
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condition, not enforced syntactically. Used by CCHM path
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transport and heterogeneous Π composition to express systems
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that a single-clause `.comp` cannot represent. -/
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| compN (i : DimVar) (A : CType)
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(clauses : List (FaceFormula × CTerm))
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(t : CTerm) : CTerm -- compⁱ A [φ₁↦u₁,…] t
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/-- Glue introduction (CCHM §6.1).
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`glueIn [φ ↦ t] a` — on face `φ`, equals `t : T`; off face `φ`,
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equals `a : A`. Well-typedness requires `e.f t = a` on the
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overlap (a typing-side condition, not enforced syntactically).
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The equivalence `e` is carried by the type, not the term. -/
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| glueIn (φ : FaceFormula) (t a : CTerm) : CTerm -- glue [φ↦t] a
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/-- Glue elimination (CCHM §6.1).
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`unglue [φ ↦ f] g` — extract the underlying `A`-value from a
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glued term. On face `φ`, equals `f g` (apply the forward map);
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off face `φ`, equals `g` (already an `A`-value). `f` is the
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`f`-field of the equivalence witnessing `T ≃ A` — carried
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explicitly at the term level because we don't have type
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annotations on terms. -/
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| unglue (φ : FaceFormula) (f g : CTerm) : CTerm -- unglue [φ↦f] g
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/-- Σ introduction (pair). -/
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| pair (a b : CTerm) : CTerm -- (a, b)
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/-- Σ elimination (first projection). -/
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| fst (t : CTerm) : CTerm -- t.1
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/-- Σ elimination (second projection). -/
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| snd (t : CTerm) : CTerm -- t.2
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/-- A dimension expression lifted into the term language (REL1).
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Used to fill `.dim`-typed argument positions of schema
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constructors (path ctors) and to hand a `DimExpr` to any
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future term-level operation that needs one. `DimExpr` is
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de Morgan algebra over `DimVar` and 0/1; lifting it lets path
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constructors take general dim arguments like `.meet i j`,
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`.inv r`, etc., not just bare `.var`-shaped dim names. -/
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| dimExpr (r : DimExpr) : CTerm -- ⟦ r ⟧
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/-- Schema constructor application (REL1, INDUCTIVE_TYPES.md §2.4).
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`ctor S c params args` applies the constructor named `c` of
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schema `S` at parameters `params`, with `args.length =
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(S.ctors.find c).args.length`.
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Args are positional and follow `CtorSpec.args` order. `.dim`-
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typed arg positions carry `CTerm.dimExpr r` (or any other
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CTerm that evaluates to a dim) — eval consults the
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constructor's boundary system on these to decide whether to
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produce the canonical `vctor` or fire a boundary clause. -/
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| ctor (S : CTypeSchema) (ctorName : String)
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(params : List CType) (args : List CTerm) : CTerm
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/-- Inductive eliminator (REL1, INDUCTIVE_TYPES.md §5).
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`indElim S params motive branches target` eliminates a value
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of type `.ind S params`.
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- `motive` is a CTerm of (function-CType) shape `.pi (.ind S
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params) .univ` — i.e. `λx : .ind S params. SomeType x`.
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- `branches` is one entry per ctor in *schema-declaration
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order*. Entry `(c, body)` says: when target reduces to the
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ctor `c`, evaluate `body` applied to the ctor's args plus
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(for each `.self` arg) the recursive elimination result.
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- `target : .ind S params` is the term being eliminated.
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For path constructors, the branch must respect boundary
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coherence: at every face in the ctor's boundary system, the
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branch agrees with the corresponding eliminated body. This
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is a typing-level side condition (see `HasType.indElim`). -/
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| indElim (S : CTypeSchema) (params : List CType)
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(motive : CTerm)
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(branches : List (String × CTerm))
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(target : CTerm) : CTerm
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/-- Argument shape for a schema constructor (REL1, §2.1).
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Distinguishes ordinary CType-typed args (which may reference
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schema parameters via `.param`) from recursive `.self` args
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(the inductive being defined) and `.dim` args (used by path
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constructors). Boundary clauses inside `CtorSpec` reference
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args positionally; `.dim` args are addressed in face formulae
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via `DimVar.mk "$d_k"` where `k` is the dim-arg index counted
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among `.dim` arg positions only. -/
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inductive CTypeArg where
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/-- A non-recursive arg whose type is a closed CType. May
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reference schema parameters via embedded `.param i` inside `A`
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(parameter substitution happens at instantiation time). -/
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| type (A : CType) : CTypeArg
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/-- The `i`th schema parameter (zero-indexed against
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`CTypeSchema.numParams`). -/
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| param (i : Nat) : CTypeArg
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/-- Recursive reference to the inductive type being defined. -/
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| self : CTypeArg
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/-- A dimension binder, used by path constructors. -/
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| dim : CTypeArg
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/-- Constructor specification (REL1, §2.2).
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`name` is unique within the schema. `args` is the positional
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arg list. `boundary` is the partial-element system for path
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constructors (empty list ≡ point ctor). Each clause `(φ, body)`
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says: on the face `φ` of the dim args, the constructor reduces
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to `body`. Coherence obligations on the clauses are enforced
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at typing time (see `HasType.ctor` in `Typing.lean`).
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Boundary clause bodies are CTerms in a scope where the ctor's
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args are bound positionally as `.var "$arg_k"`. Face formulae
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reference dim args as `DimVar.mk "$d_k"`. -/
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inductive CtorSpec where
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| mk (name : String) (args : List CTypeArg)
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(boundary : List (FaceFormula × CTerm)) : CtorSpec
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/-- Schema for an inductive (or higher-inductive) type (REL1, §2.3).
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`name` is the schema's symbolic name (used for diagnostics, not
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identity — schemas are compared structurally). `numParams` is
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the count of type parameters; `ctors` is the constructor list
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in declaration order. Equality is structural (no interning).
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The schema mutually depends on `CType` (via `.type`-typed args
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and via `params : List CType` in `CType.ind`) and `CTerm` (via
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boundary clauses), so all five types live in this `mutual`
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block. -/
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inductive CTypeSchema where
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| mk (name : String) (numParams : Nat)
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(ctors : List CtorSpec) : CTypeSchema
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end
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-- ── Repr derivations ─────────────────────────────────────────────────────────
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-- All five mutual inductives get `Repr` instances post-hoc (Lean's
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-- `deriving Repr` clause inside a mutual block of five doesn't always
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-- compose; explicit `deriving instance` is the robust form).
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deriving instance Repr for CType
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deriving instance Repr for CTerm
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deriving instance Repr for CTypeArg
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deriving instance Repr for CtorSpec
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deriving instance Repr for CTypeSchema
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-- DecidableEq for the 5-way mutual block lives in `CubicalTransport.DecEq`
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-- (Lean's `deriving instance DecidableEq` doesn't currently support mutual
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-- inductives — has to be written manually).
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-- ── Dimension substitution ────────────────────────────────────────────────────
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-- Substitute dimension variable i with DimExpr r throughout a term.
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--
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-- Scope inside transp/comp:
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-- · j is the binder of the transport line, bound in A and in φ.
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-- · The base term t (and system u) are in outer scope — we substitute there.
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--
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-- Approximation: `substDim` does NOT descend into A or φ — even when j ≠ i
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-- and i would be free under the binder. Consequence: this substitution is
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-- only faithful for *endpoint* calls (`substDimBool`), where downstream
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-- uses the dimension-absent predicate to justify correctness. Full
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-- DimExpr-in-FaceFormula substitution is deferred (see cells-spec §5.5).
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mutual
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def CTerm.substDim (i : DimVar) (r : DimExpr) : CTerm → CTerm
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| .var x => .var x
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| .lam x t => .lam x (t.substDim i r)
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| .app f a => .app (f.substDim i r) (a.substDim i r)
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| .plam j t => if j = i then .plam j t -- i bound; stop
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else .plam j (t.substDim i r)
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| .papp t s => .papp (t.substDim i r) (DimExpr.subst i r s)
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-- transp/comp: leave A alone (approximation); descend into t, u and
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-- substitute in φ via the general DimExpr face-formula substitution.
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| .transp j A φ t => .transp j A (φ.substDim i r) (t.substDim i r)
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| .comp j A φ u t => .comp j A (φ.substDim i r) (u.substDim i r) (t.substDim i r)
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-- Multi-clause comp: substitute in each clause's face and body, and in t.
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-- Uses an explicit recursive helper `substDim.clauses` so Lean can see
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-- structural termination through the clause list.
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| .compN j A clauses t =>
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.compN j A (CTerm.substDim.clauses i r clauses) (t.substDim i r)
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-- Glue introduction / elimination: descend into all sub-terms and
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-- substitute into the face formula. Same approximation for types
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-- as transp/comp (A not touched — the `φ` face formula carries the
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-- whole dim-dependency story; in our approximation we still don't
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-- recurse into CType sub-trees here, matching `.transp`/`.comp`).
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| .glueIn φ t a => .glueIn (φ.substDim i r) (t.substDim i r) (a.substDim i r)
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| .unglue φ f g => .unglue (φ.substDim i r) (f.substDim i r) (g.substDim i r)
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-- Σ constructors: structural recursion into sub-terms.
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| .pair a b => .pair (a.substDim i r) (b.substDim i r)
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| .fst t => .fst (t.substDim i r)
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| .snd t => .snd (t.substDim i r)
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-- REL1 inductive-type constructors. Same CType-approximation as
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-- transp/comp/glue: schemas and `params : List CType` are *not*
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-- recursed into. The schemas are static and the params are
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-- supplied externally to any binder we'd be substituting into.
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| .dimExpr s => .dimExpr (DimExpr.subst i r s)
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| .ctor S c params args =>
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.ctor S c params (CTerm.substDim.list i r args)
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| .indElim S params motive branches target =>
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.indElim S params
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(motive.substDim i r)
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(CTerm.substDim.branches i r branches)
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(target.substDim i r)
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/-- Helper: apply `CTerm.substDim i r` to each clause body (and
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`FaceFormula.substDim` to each face) in a system's clause list.
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Defined mutually with `substDim` so Lean can verify termination. -/
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def CTerm.substDim.clauses (i : DimVar) (r : DimExpr) :
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List (FaceFormula × CTerm) → List (FaceFormula × CTerm)
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| [] => []
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| (φ, u) :: rest =>
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(φ.substDim i r, u.substDim i r) :: CTerm.substDim.clauses i r rest
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/-- Helper: apply `CTerm.substDim i r` to each element of a CTerm
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list (ctor argument lists). Mutually recursive for termination. -/
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def CTerm.substDim.list (i : DimVar) (r : DimExpr) :
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List CTerm → List CTerm
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| [] => []
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| t :: rest => t.substDim i r :: CTerm.substDim.list i r rest
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/-- Helper: apply `CTerm.substDim i r` to the body of each branch in
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an `indElim`. Branch *names* are not affected; only bodies. -/
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def CTerm.substDim.branches (i : DimVar) (r : DimExpr) :
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List (String × CTerm) → List (String × CTerm)
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| [] => []
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| (n, b) :: rest =>
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(n, b.substDim i r) :: CTerm.substDim.branches i r rest
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end
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-- ── One-step reduction ────────────────────────────────────────────────────────
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-- `CTerm.step` is left *opaque*. Semantically it is what a CCHM-style
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-- evaluator will compute. Keeping `step` opaque is required for
|
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-- soundness: if `step` were a concrete `def` with a wildcard identity
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-- arm, any axiom of the shape `step (.transp …) = t` would rfl-collapse
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-- to `.transp … = t`, contradicting `CTerm.noConfusion`.
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--
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-- **Stage 4.4 decision (2026-04-23).** After the Phase 1 step↔eval
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-- bridge + Stream B #2d + Stage 2.3, only one step-level axiom remains:
|
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-- `transp_plam_is_plam_path` (T4, path-restricted form; Stage 4.2).
|
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-- T1/T2/C1/C2/T5/`step_papp_plam` are NbE theorems in `Readback.lean`;
|
||
-- T3/C4 are theorems via `CTerm.step_preserves_type` (ValueTyping.lean);
|
||
-- glue β/η are theorems via `eval_unglue_of_glueIn`/`eval_glueIn_of_unglue`.
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--
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-- **Rust discharge flexibility.** The Rust backend has two valid
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||
-- implementation strategies for `CTerm.step`:
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--
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-- · *Option A — native step*: implement `step` directly as a C ABI
|
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-- entry point. The single T4 axiom is discharged by emitting the
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-- CCHM comp-shaped body for path-typed transp-of-plam inputs, plus
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-- identity behaviour on other shapes.
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--
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-- · *Option B — derived step*: define `step := readback ∘ eval .nil`
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-- entirely in Rust (no separate `step` FFI entry). This matches
|
||
-- the Lean-side "bridge" intuition and keeps the Rust FFI surface
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-- smaller by one function. Satisfies T4 on path-typed lines via
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-- the `vPathTransp` → `.compN` readback arm.
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--
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-- The Lean-side spec is agnostic to which strategy Rust picks — both
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-- satisfy the same axiom set. See FFI_DESIGN.md (Stage 4.5) for the
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-- concrete recommendation.
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opaque CTerm.step : CTerm → CTerm := id
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