feat: close `find` / `insert` proofs

After a complete 180 with the Order theory, we close the goals of find
and insert and we give an example of U32 order that we will upstream to
Aeneas directly.

Signed-off-by: Raito Bezarius <masterancpp@gmail.com>

1 files changed, 150 insertions, 0 deletions

diff --git a/Verification/Specifications.lean b/Verification/Specifications.lean
new file mode 100644
index 0000000..392c438
--- /dev/null
+++ b/Verification/Specifications.lean
@@ -0,0 +1,150 @@
+import «AvlVerification»
+
+namespace Primitives
+
+namespace Result
+
+def map {A B: Type} (x: Result A) (f: A -> B): Result B := match x with
+| .ok y => .ok (f y)
+| .fail e => .fail e
+| .div => .div
+
+@[inline]
+def isok {A: Type} : Result A -> Bool
+| .ok _ => true
+| .fail _ => false
+| .div => false
+
+@[inline]
+def get? {A: Type}: (r: Result A) -> isok r -> A
+| .ok x, _ => x
+
+end Result
+
+end Primitives
+
+namespace avl_verification
+
+@[simp]
+def Ordering.toLeanOrdering (o: avl_verification.Ordering): _root_.Ordering := match o with 
+| .Less => .lt
+| .Equal => .eq
+| .Greater => .gt
+
+def Ordering.ofLeanOrdering (o: _root_.Ordering): avl_verification.Ordering := match o with
+| .lt => .Less
+| .eq => .Equal
+| .gt => .Greater
+
+@[simp]
+def Ordering.toDualOrdering (o: avl_verification.Ordering): avl_verification.Ordering := match o with 
+| .Less => .Greater
+| .Equal => .Equal
+| .Greater => .Less
+
+@[simp]
+theorem Ordering.toLeanOrdering.injEq (x y: avl_verification.Ordering): (x.toLeanOrdering = y.toLeanOrdering) = (x = y) := by
+  apply propext
+  cases x <;> cases y <;> simp
+
+@[simp]
+theorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :
+    ((if c then a else b) = .Less) = if c then a = .Less else b = .Less := by
+  by_cases c <;> simp [*]
+
+@[simp]
+theorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :
+    ((if c then a else b) = .Equal) = if c then a = .Equal else b = .Equal := by
+  by_cases c <;> simp [*]
+
+@[simp]
+theorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a b : Ordering) :
+    ((if c then a else b) = .Greater) = if c then a = .Greater else b = .Greater := by
+  by_cases c <;> simp [*]
+
+end avl_verification
+
+namespace Specifications
+
+open Primitives
+open Result
+
+variable {T: Type} (H: outParam (avl_verification.Ord T))
+
+@[simp]
+def _root_.Ordering.toDualOrdering (o: _root_.Ordering): _root_.Ordering := match o with 
+| .lt => .gt
+| .eq => .eq
+| .gt => .lt
+
+
+@[simp]
+theorem toDualOrderingOfToLeanOrdering (o: avl_verification.Ordering): o.toDualOrdering.toLeanOrdering = o.toLeanOrdering.toDualOrdering := by
+  cases o <;> simp
+
+@[simp]
+theorem toDualOrderingIdempotency (o: _root_.Ordering): o.toDualOrdering.toDualOrdering = o := by 
+  cases o <;> simp
+
+-- TODO: reason about raw bundling vs. refined bundling.
+-- raw bundling: hypothesis with Rust extracted objects.
+-- refined bundling: lifted hypothesis with Lean native objects.
+class OrdSpec [Ord T] where
+  infallible: ∀ a b, ∃ (o: avl_verification.Ordering), H.cmp a b = .ok o ∧ compare a b = o.toLeanOrdering
+  
+class OrdSpecSymmetry [O: Ord T] extends OrdSpec H where
+  symmetry: ∀ a b, O.compare a b = (O.opposite.compare a b).toDualOrdering
+
+-- Must be R decidableRel and an equivalence relationship?
+class OrdSpecRel [O: Ord T] (R: outParam (T -> T -> Prop)) extends OrdSpec H where
+  equivalence: ∀ a b, H.cmp a b = .ok .Equal -> R a b
+
+class OrdSpecLinearOrderEq [O: Ord T] extends OrdSpecSymmetry H, OrdSpecRel H Eq
+
+theorem infallible [Ord T] [OrdSpec H]: ∀ a b, ∃ o, H.cmp a b = .ok o := fun a b => by 
+  obtain ⟨ o, ⟨ H, _ ⟩ ⟩ := OrdSpec.infallible a b
+  exact ⟨ o, H ⟩
+
+instance: Coe (avl_verification.Ordering) (_root_.Ordering) where
+  coe a := a.toLeanOrdering
+
+theorem rustCmpEq [Ord T] [O: OrdSpec H]: H.cmp a b = .ok o <-> compare a b = o.toLeanOrdering := by
+  apply Iff.intro
+  . intro Hcmp
+    obtain ⟨ o', ⟨ Hcmp', Hcompare ⟩ ⟩ := O.infallible a b
+    rw [Hcmp', ok.injEq] at Hcmp
+    simp [Hcompare, Hcmp', Hcmp]
+  . intro Hcompare
+    obtain ⟨ o', ⟨ Hcmp', Hcompare' ⟩ ⟩ := O.infallible a b
+    rw [Hcompare', avl_verification.Ordering.toLeanOrdering.injEq] at Hcompare
+    simp [Hcompare.symm, Hcmp']
+
+
+theorem oppositeOfOpposite {x y: _root_.Ordering}: x.toDualOrdering = y ↔ x = y.toDualOrdering := by
+  cases x <;> cases y <;> simp
+theorem oppositeRustOrder [Ord T] [Spec: OrdSpecSymmetry H] {a b}: H.cmp b a = .ok o ↔ H.cmp a b = .ok o.toDualOrdering := by
+  rw [rustCmpEq, Spec.symmetry, compare, Ord.opposite, oppositeOfOpposite, rustCmpEq, toDualOrderingOfToLeanOrdering]
+
+theorem ltOfRustOrder 
+  [LO: LinearOrder T]
+  [Spec: OrdSpec H]: 
+  ∀ a b, H.cmp a b = .ok .Less -> a < b := by
+  intros a b
+  intro Hcmp
+  -- why the typeclass search doesn't work here?
+  refine' (@compare_lt_iff_lt T LO).1 _
+  obtain ⟨ o, ⟨ Hcmp', Hcompare ⟩ ⟩ := Spec.infallible a b 
+  simp only [Hcmp', ok.injEq] at Hcmp
+  simp [Hcompare, Hcmp, avl_verification.Ordering.toLeanOrdering]
+
+theorem gtOfRustOrder 
+  [LinearOrder T]
+  [Spec: OrdSpecSymmetry H]: 
+  ∀ a b, H.cmp a b = .ok .Greater -> b < a := by
+  intros a b
+  intro Hcmp
+  refine' @ltOfRustOrder _ H _ Spec.toOrdSpec _ _ _
+  rewrite [oppositeRustOrder]
+  simp [Hcmp]
+
+end Specifications