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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>
285 lines
12 KiB
Text
285 lines
12 KiB
Text
/-
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Topolei.Cubical.Interval
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========================
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The interval I: free de Morgan algebra on dimension variables.
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Reference: CCHM §3
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Grammar: r, s ::= 0 | 1 | i | 1−r | r∧s | r∨s
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Strategy: prove the de Morgan laws *semantically* — they hold for every
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Boolean assignment of the dimension variables. Sound because Bool is itself
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a de Morgan algebra and DimExpr is freely generated over it.
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-/
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-- ── Dimension variables ───────────────────────────────────────────────────────
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structure DimVar where
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name : String
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deriving DecidableEq, Repr
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-- ── The Interval ──────────────────────────────────────────────────────────────
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inductive DimExpr where
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| zero : DimExpr
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| one : DimExpr
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| var (i : DimVar) : DimExpr
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| inv (r : DimExpr) : DimExpr -- 1 − r
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| meet (r s : DimExpr) : DimExpr -- r ∧ s
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| join (r s : DimExpr) : DimExpr -- r ∨ s
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deriving Repr, Inhabited, DecidableEq
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-- ── Semantic evaluation ───────────────────────────────────────────────────────
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def DimExpr.eval (env : DimVar → Bool) : DimExpr → Bool
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| .zero => false
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| .one => true
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| .var i => env i
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| .inv r => !(r.eval env)
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| .meet r s => r.eval env && s.eval env
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| .join r s => r.eval env || s.eval env
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-- ── de Morgan algebra laws ────────────────────────────────────────────────────
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-- All proved by simp-unfolding eval and then Bool case analysis.
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-- We use `simp only [eval]` (not plain `simp`) to avoid over-simplification
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-- that would leave `<;> rfl` with no goals.
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namespace DimExpr
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theorem inv_zero (env : DimVar → Bool) :
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(inv zero).eval env = (one).eval env := by
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simp only [eval]; rfl
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theorem inv_one (env : DimVar → Bool) :
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(inv one).eval env = (zero).eval env := by
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simp only [eval]; rfl
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theorem inv_inv (r : DimExpr) (env : DimVar → Bool) :
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(inv (inv r)).eval env = r.eval env := by
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simp only [eval]; cases r.eval env <;> rfl
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theorem inv_meet (r s : DimExpr) (env : DimVar → Bool) :
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(inv (meet r s)).eval env = (join (inv r) (inv s)).eval env := by
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simp only [eval]; cases r.eval env <;> cases s.eval env <;> rfl
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theorem inv_join (r s : DimExpr) (env : DimVar → Bool) :
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(inv (join r s)).eval env = (meet (inv r) (inv s)).eval env := by
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simp only [eval]; cases r.eval env <;> cases s.eval env <;> rfl
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theorem meet_comm (r s : DimExpr) (env : DimVar → Bool) :
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(meet r s).eval env = (meet s r).eval env := by
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simp only [eval]; cases r.eval env <;> cases s.eval env <;> rfl
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theorem join_comm (r s : DimExpr) (env : DimVar → Bool) :
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(join r s).eval env = (join s r).eval env := by
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simp only [eval]; cases r.eval env <;> cases s.eval env <;> rfl
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theorem meet_assoc (r s t : DimExpr) (env : DimVar → Bool) :
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(meet r (meet s t)).eval env = (meet (meet r s) t).eval env := by
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simp only [eval]
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cases r.eval env <;> cases s.eval env <;> cases t.eval env <;> rfl
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theorem join_assoc (r s t : DimExpr) (env : DimVar → Bool) :
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(join r (join s t)).eval env = (join (join r s) t).eval env := by
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simp only [eval]
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cases r.eval env <;> cases s.eval env <;> cases t.eval env <;> rfl
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theorem meet_one (r : DimExpr) (env : DimVar → Bool) :
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(meet r one).eval env = r.eval env := by
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simp only [eval]; cases r.eval env <;> rfl
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theorem one_meet (r : DimExpr) (env : DimVar → Bool) :
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(meet one r).eval env = r.eval env := by
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simp only [eval]; cases r.eval env <;> rfl
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theorem join_zero (r : DimExpr) (env : DimVar → Bool) :
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(join r zero).eval env = r.eval env := by
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simp only [eval]; cases r.eval env <;> rfl
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theorem zero_join (r : DimExpr) (env : DimVar → Bool) :
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(join zero r).eval env = r.eval env := by
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simp only [eval]; cases r.eval env <;> rfl
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theorem meet_zero (r : DimExpr) (env : DimVar → Bool) :
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(meet r zero).eval env = false := by
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simp only [eval]; cases r.eval env <;> rfl
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theorem join_one (r : DimExpr) (env : DimVar → Bool) :
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(join r one).eval env = true := by
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simp only [eval]; cases r.eval env <;> rfl
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theorem meet_distrib_join (r s t : DimExpr) (env : DimVar → Bool) :
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(meet r (join s t)).eval env = (join (meet r s) (meet r t)).eval env := by
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simp only [eval]
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cases r.eval env <;> cases s.eval env <;> cases t.eval env <;> rfl
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theorem join_distrib_meet (r s t : DimExpr) (env : DimVar → Bool) :
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(join r (meet s t)).eval env = (meet (join r s) (join r t)).eval env := by
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simp only [eval]
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cases r.eval env <;> cases s.eval env <;> cases t.eval env <;> rfl
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-- r ∧ (1−r) = 0 and r ∨ (1−r) = 1 hold semantically (not in the free algebra)
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theorem meet_inv_self (r : DimExpr) (env : DimVar → Bool) :
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(meet r (inv r)).eval env = false := by
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simp only [eval]; cases r.eval env <;> rfl
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theorem join_inv_self (r : DimExpr) (env : DimVar → Bool) :
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(join r (inv r)).eval env = true := by
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simp only [eval]; cases r.eval env <;> rfl
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-- ── Substitution ──────────────────────────────────────────────────────────────
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def subst (i : DimVar) (r : DimExpr) : DimExpr → DimExpr
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| .zero => .zero
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| .one => .one
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| .var j => if j = i then r else .var j
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| .inv s => .inv (subst i r s)
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| .meet s t => .meet (subst i r s) (subst i r t)
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| .join s t => .join (subst i r s) (subst i r t)
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theorem eval_subst (i : DimVar) (r s : DimExpr) (env : DimVar → Bool) :
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(subst i r s).eval env =
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s.eval (fun j => if j = i then r.eval env else env j) := by
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induction s with
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| zero => rfl
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| one => rfl
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| var j =>
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simp only [subst, eval]
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by_cases h : j = i <;> simp [h, eval]
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| inv s ih => simp only [subst, eval, ih]
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| meet s t ih_s ih_t => simp only [subst, eval, ih_s, ih_t]
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| join s t ih_s ih_t => simp only [subst, eval, ih_s, ih_t]
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theorem eval_subst_zero (i : DimVar) (s : DimExpr) (env : DimVar → Bool) :
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(subst i zero s).eval env =
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s.eval (fun j => if j = i then false else env j) := by
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simp [eval_subst, eval]
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theorem eval_subst_one (i : DimVar) (s : DimExpr) (env : DimVar → Bool) :
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(subst i one s).eval env =
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s.eval (fun j => if j = i then true else env j) := by
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simp [eval_subst, eval]
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theorem eval_subst_inv (i : DimVar) (s : DimExpr) (env : DimVar → Bool) :
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(subst i (inv (var i)) s).eval env =
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s.eval (fun j => if j = i then !(env i) else env j) := by
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simp [eval_subst, eval]
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-- ── Dimension absence ────────────────────────────────────────────────────────
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/-- Syntactic check: `i` does not appear in the dim expression. -/
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def dimAbsent (i : DimVar) : DimExpr → Bool
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| .zero => true
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| .one => true
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| .var j => j != i
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| .inv r => r.dimAbsent i
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| .meet r s => r.dimAbsent i && s.dimAbsent i
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| .join r s => r.dimAbsent i && s.dimAbsent i
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/-- Bool endpoints have `dimAbsent = true` for every dimension. -/
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theorem dimAbsent_endpoint (i : DimVar) (b : Bool) :
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(if b then DimExpr.one else DimExpr.zero).dimAbsent i = true := by
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cases b <;> rfl
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-- ── Normalisation (Stage 4.1, 2026-04-23) ────────────────────────────────────
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--
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-- Reduce DimExprs to a canonical form by pushing `inv` reductions to
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-- the literals. Rust's cubical evaluator needs this to canonicalise
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-- face formulas (otherwise `.inv .zero ≠ .one` structurally, and face
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-- dispatch becomes unreliable).
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--
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-- Reductions implemented:
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-- · `.inv .zero → .one`
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-- · `.inv .one → .zero`
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-- · `.inv (.inv r) → r` (double-negation cancellation)
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--
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-- The `.meet` / `.join` sub-terms are themselves normalised recursively
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-- so that iterated inversion reduces through layers of conjunction /
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-- disjunction. Unit / absorption laws (e.g. `.meet .zero _ → .zero`)
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-- are semantically valid but deliberately *not* implemented here — they
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-- don't discharge any Lean-side axiom, and adding them would complicate
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-- structural termination. A future `DimExpr.normalize_full` can add
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-- them when a well-founded measure is introduced.
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--
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-- Semantic correctness: `normalize_preserves_eval` shows the normal
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-- form has the same Boolean evaluation under every environment.
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@[extern "topolei_cubical_dimexpr_normalize"]
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private opaque normalizeRust : DimExpr → DimExpr
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/-- Canonical form under `inv`-reduction. Idempotent:
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`normalize (normalize r) = normalize r`. -/
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@[implemented_by normalizeRust]
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def normalize : DimExpr → DimExpr
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| .zero => .zero
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| .one => .one
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| .var i => .var i
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| .inv r =>
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match normalize r with
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| .zero => .one -- `.inv .zero` → `.one`
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| .one => .zero -- `.inv .one` → `.zero`
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| .inv s => s -- double-negation cancel
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| nr => .inv nr -- opaque; keep wrapped
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| .meet r s => .meet (normalize r) (normalize s)
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| .join r s => .join (normalize r) (normalize s)
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-- ── Preservation of evaluation ──────────────────────────────────────────────
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/-- Auxiliary lemma: the four-way match inside `normalize (.inv r)`
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produces a DimExpr whose `eval` equals the logical negation of
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`(normalize r).eval env`. Each literal reduction (`.inv .zero → .one`,
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`.inv .one → .zero`, double-negation cancel) preserves this equation
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under Boolean semantics. -/
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private theorem normalize_inv_aux (r : DimExpr) (env : DimVar → Bool) :
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(normalize (.inv r)).eval env = !((normalize r).eval env) := by
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show (match normalize r with
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| .zero => DimExpr.one | .one => DimExpr.zero
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| .inv s => s | nr => DimExpr.inv nr).eval env =
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!((normalize r).eval env)
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cases hnr : normalize r with
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| zero => rfl
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| one => rfl
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| var j => rfl
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| inv s =>
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-- match returns s; RHS is !((.inv s).eval env) = !(!(s.eval env)).
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show s.eval env = !((DimExpr.inv s).eval env)
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simp [eval]
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| meet a b => rfl
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| join a b => rfl
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/-- Normalisation is semantically transparent: the normalised form has
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the same `eval` as the original, under every Bool environment. -/
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theorem normalize_preserves_eval (r : DimExpr) (env : DimVar → Bool) :
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(normalize r).eval env = r.eval env := by
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induction r with
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| zero => rfl
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| one => rfl
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| var i => rfl
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| inv r ih =>
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rw [normalize_inv_aux, ih]
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rfl
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| meet a b ih_a ih_b =>
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show (DimExpr.meet (normalize a) (normalize b)).eval env =
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(DimExpr.meet a b).eval env
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simp only [eval, ih_a, ih_b]
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| join a b ih_a ih_b =>
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show (DimExpr.join (normalize a) (normalize b)).eval env =
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(DimExpr.join a b).eval env
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simp only [eval, ih_a, ih_b]
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-- ── Canonical-form literals ─────────────────────────────────────────────────
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/-- `.inv .zero` normalises to `.one` — the fundamental literal rule. -/
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theorem normalize_inv_zero : normalize (.inv .zero) = .one := rfl
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/-- `.inv .one` normalises to `.zero`. -/
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theorem normalize_inv_one : normalize (.inv .one) = .zero := rfl
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/-- `.inv (.inv r)` normalises to the same value as `r` (semantically).
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Stated via eval rather than syntactically because structural double-
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negation cancellation depends on `r`'s post-normalisation shape. -/
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theorem normalize_inv_inv_eval (r : DimExpr) (env : DimVar → Bool) :
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(normalize (.inv (.inv r))).eval env = r.eval env := by
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rw [normalize_preserves_eval]
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simp [eval]
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end DimExpr
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