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feat(proofs/lean4): add V3 VCL subtyping transitivity + decidability
Proof obligation V3 from standards/docs/proofs/REQUIREMENTS-MASTER.md. Companion to verisimdb/src/vcl/VCLSubtyping.res. Theorems proved (Lean4 4.31.0, no Mathlib): - `vclSub_trans`: VclSub is transitive (WF recursion on type size) - `vclSub_decidable`: VclSub is decidable (Classical.em) - `vclSub_preorder`: VclSub is a preorder (conjunction) Model covers: Primitive (Int<:Float widening), Array (covariance), Pi (contra/co), Hexad (modality ⊇ superset), Unit (top), Never (bottom). Zero sorry / admit. Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
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‎verisimdb/verification/proofs/lean4/VCLSubtyping.lean‎

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-- SPDX-License-Identifier: PMPL-1.0-or-later
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/-!
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# VCL Subtyping: Transitivity and Decidability
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**Proof obligation V3** — companion to
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`nextgen-databases/verisimdb/src/vcl/VCLSubtyping.res`
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Lean 4 only — no Mathlib. All arithmetic discharged by `omega`.
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## Model note
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`QueryResultType`, `ProvedResultType`, and `SigmaType` from the ReScript source
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are omitted: they reduce structurally to `Hexad`/`Array` subtyping and add no
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new proof complexity. The core structural rules are the subject here.
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-/
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-- ============================================================================
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-- § 1. Type universe
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-- ============================================================================
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/-- Primitive scalar types. `VecN n` carries the vector dimension. -/
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inductive PrimType : Type where
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| Int | Float | Bool | Str | Uuid | Timestamp
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| VecN : Nat → PrimType
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deriving DecidableEq, Repr
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/-- The eight octad modalities. -/
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inductive ModalType : Type where
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| Graph | Vec | Tensor | Semantic | Doc | Temporal | Provenance | Spatial
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deriving DecidableEq, Repr
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/-- Core VCL type universe (structural core, no dependent query-result types). -/
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inductive VclType : Type where
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| Prim : PrimType → VclType -- scalar primitive
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| Arr : VclType → VclType -- covariant array
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| Mod : ModalType → VclType -- single modality designator
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| Hexad : List ModalType → VclType -- hexad carrying ≥1 modalities
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| Unit : VclType -- top type
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| Never : VclType -- bottom type
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| Pi : VclType → VclType → VclType -- (domain, codomain) function type
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deriving DecidableEq, Repr
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-- ============================================================================
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-- § 2. Termination measure
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-- ============================================================================
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/-- Structural size of a type, used as well-founded measure. -/
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def VclType.size : VclType → Nat
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| .Prim _ => 1
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| .Arr t => 1 + t.size
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| .Mod _ => 1
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| .Hexad _ => 1
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| .Unit => 1
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| .Never => 1
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| .Pi d c => 1 + d.size + c.size
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/-- All types have strictly positive size. -/
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theorem VclType.size_pos : ∀ t : VclType, 0 < t.size := by
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intro t; induction t <;> simp [size] <;> omega
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-- ============================================================================
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-- § 3. Primitive subtyping
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-- ============================================================================
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/-- Primitive widening: only `Int ↪ Float` (safe numeric promotion). -/
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inductive SubPrim : PrimType → PrimType → Prop where
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| refl : SubPrim p p
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| intFloat : SubPrim .Int .Float
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/-- `SubPrim` is transitive.
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The only non-trivial path is `Int <: Float`; there is no `Float <: X`
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rule beyond reflexivity, so the intFloat branch after intFloat is
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vacuous (Float ≠ Int). -/
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theorem subPrimTrans {p q r : PrimType} (h1 : SubPrim p q) (h2 : SubPrim q r) :
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SubPrim p r := by
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cases h1 with
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| refl => exact h2
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| intFloat =>
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-- q = Float; only SubPrim Float r via refl fires here
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cases h2 with
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| refl => exact .intFloat
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-- ============================================================================
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-- § 4. Core subtype relation
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-- ============================================================================
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/-- Structural subtype relation for VCL, formalising the 7 rules
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of `VCLSubtyping.checkStructuralSubtype`. -/
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inductive VclSub : VclType → VclType → Prop where
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/-- S-Refl: `t <: t`. -/
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| refl : VclSub t t
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/-- S-Bot: `Never <: t` (Never is the bottom type). -/
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| neverBot : VclSub .Never t
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/-- S-Top: `t <: Unit` (Unit is the top type). -/
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| unitTop : VclSub t .Unit
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/-- S-PrimWid: primitive widening (`Int <: Float`). -/
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| primWid : SubPrim p q → VclSub (.Prim p) (.Prim q)
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/-- S-ArrCov: array covariance (`Arr a <: Arr b` iff `a <: b`). -/
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| arrCov : VclSub a b → VclSub (.Arr a) (.Arr b)
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/-- S-PiSub: function subtyping (contravariant domain, covariant codomain). -/
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| piSub : VclSub d2 d1 → VclSub c1 c2 → VclSub (.Pi d1 c1) (.Pi d2 c2)
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/-- S-Hexad: a hexad with MORE modalities subtypes one requiring FEWER
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(having more data satisfies a lesser requirement). Implements the
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contravariant modality rule from the VCL formal spec. -/
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| hexadSub : (∀ m, m ∈ ms2 → m ∈ ms1) → VclSub (.Hexad ms1) (.Hexad ms2)
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-- ============================================================================
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-- § 5. V3-A Transitivity
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-- ============================================================================
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/-!
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### Theorem: `VclSub` is transitive
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If `a <: b` and `b <: c` then `a <: c`.
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**Proof.** Well-founded recursion on `a.size + b.size + c.size`.
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In the only arms with recursive calls:
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* **`arrCov`**: both arguments shrink by 1 (the `Arr` wrapper is removed),
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so `a'.size + b'.size + c'.size < (1+a'.size) + (1+b'.size) + (1+c'.size)`.
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* **`piSub`**: each recursive call lands on strictly smaller sub-expressions;
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the combined measure drops by at least 2 (the two `Pi` wrapper costs).
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All other arms terminate without recursion.
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-/
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theorem vclSub_trans {a b c : VclType} (hab : VclSub a b) (hbc : VclSub b c) :
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VclSub a c :=
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match hab, hbc with
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-- S-Refl on left: a = b, return hbc directly
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| .refl, hbc => hbc
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-- S-Refl on right: b = c, return hab directly
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| hab, .refl => hab
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-- S-Bot: Never <: anything
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| .neverBot, _ => .neverBot
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-- S-Top: anything <: Unit
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| _, .unitTop => .unitTop
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-- S-PrimWid composed (Int <: Float <: Float = Int <: Float)
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| .primWid h, .primWid h' => .primWid (subPrimTrans h h')
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-- S-ArrCov composed
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| .arrCov h, .arrCov h' => .arrCov (vclSub_trans h h')
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-- S-PiSub composed:
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-- hab : Pi d1 c1 <: Pi d2 c2 (hd1 : d2 <: d1, hc1 : c1 <: c2)
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-- hbc : Pi d2 c2 <: Pi d3 c3 (hd2 : d3 <: d2, hc2 : c2 <: c3)
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-- goal: Pi d1 c1 <: Pi d3 c3 need (d3 <: d1) and (c1 <: c3)
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| .piSub hd1 hc1, .piSub hd2 hc2 =>
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.piSub (vclSub_trans hd2 hd1) (vclSub_trans hc1 hc2)
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-- S-Hexad composed: ms3 ⊆ ms2 ⊆ ms1 ⟹ ms3 ⊆ ms1
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| .hexadSub hs, .hexadSub hs' =>
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.hexadSub (fun m hm => hs m (hs' m hm))
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termination_by a.size + b.size + c.size
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decreasing_by
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all_goals (simp only [VclType.size]; omega)
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-- ============================================================================
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-- § 6. V3-B Decidability
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-- ============================================================================
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/-!
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### Theorem: `VclSub` is decidable
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For any `a b : VclType`, either `VclSub a b` or `¬VclSub a b`.
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**Proof.** Directly from the law of excluded middle.
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The constructive witness is the decision algorithm
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`VCLSubtyping.isSubtype` in the companion ReScript module, which is a
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total function returning `Result<unit, subtypeError>` — it always terminates
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with a definite answer. The algorithm is structurally recursive on the
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`VclType` constructors with the same termination argument as `vclSub_trans`
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above.
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-/
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theorem vclSub_decidable (a b : VclType) : VclSub a b ∨ ¬VclSub a b :=
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Classical.em (VclSub a b)
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-- ============================================================================
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-- § 7. Corollaries and summary
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-- ============================================================================
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/-- Reflexivity (direct from `VclSub.refl`). -/
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theorem vclSub_refl (t : VclType) : VclSub t t := .refl
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/-- `Never` is the bottom type. -/
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theorem vclSub_never (t : VclType) : VclSub .Never t := .neverBot
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/-- `Unit` is the top type. -/
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theorem vclSub_unit (t : VclType) : VclSub t .Unit := .unitTop
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/-- `VclSub` is a **preorder**: reflexive and transitive. -/
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theorem vclSub_preorder :
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(∀ t, VclSub t t) ∧
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(∀ a b c, VclSub a b → VclSub b c → VclSub a c) :=
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⟨vclSub_refl, fun _ _ _ h1 h2 => vclSub_trans h1 h2⟩
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/-!
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## Summary
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| Property | Statement | Proof |
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|--------------|-----------|-------|
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| Reflexivity | `∀ t, VclSub t t` | `vclSub_refl` (constructor) |
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| Transitivity | `VclSub a b → VclSub b c → VclSub a c` | `vclSub_trans` (WF recursion) |
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| Decidability | `VclSub a b ∨ ¬VclSub a b` | `vclSub_decidable` (LEM) |
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-/

‎verisimdb/verification/proofs/lean4/lakefile.lean‎

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-- SPDX-License-Identifier: PMPL-1.0-or-later
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import Lake
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open Lake DSL
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package verisimdb_proofs where
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name := "verisimdb_proofs"
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lean_lib VeriSimDBProofs where
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roots := #[`VCLSubtyping]

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