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, 0 insertions, 98 deletions

diff --git a/AvlVerification/Specifications.lean b/AvlVerification/Specifications.lean
deleted file mode 100644
index ac2d056..0000000
--- a/AvlVerification/Specifications.lean
+++ /dev/null
@@ -1,98 +0,0 @@
-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
-
-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
-
-end avl_verification
-
-namespace Specifications
-
-open Primitives
-open Result
-
-variable {T: Type} (H: outParam (avl_verification.Ord T))
-
--- TODO: reason about raw bundling vs. refined bundling.
--- raw bundling: hypothesis with Rust extracted objects.
--- refined bundling: lifted hypothesis with Lean native objects.
-class OrdSpecInfaillible where
-  infallible: ∀ a b, ∃ (o: avl_verification.Ordering), H.cmp a b = .ok o
-
-class OrdSpecDuality extends OrdSpecInfaillible H where
-  duality: ∀ a b, H.cmp a b = .ok .Greater -> H.cmp b a = .ok .Less
-
-class OrdSpecRel (R: outParam (T -> T -> Prop)) extends OrdSpecInfaillible H where
-  equivalence: ∀ a b, H.cmp a b = .ok .Equal -> R a b
-
-class OrdSpecDualityEq extends OrdSpecDuality H, OrdSpecRel H Eq
-
-instance: Coe (avl_verification.Ordering) (_root_.Ordering) where
-  coe a := a.toLeanOrdering
-
-def ordOfOrdSpec (H: avl_verification.Ord T) (spec: OrdSpecInfaillible H): Ord T where
-  compare x y := (H.cmp x y).get? (by
-    cases' (spec.infallible x y) with o Hcmp
-    rewrite [isok, Hcmp]
-    simp only
-  )
-
-instance [spec: OrdSpecInfaillible H]: Ord T := ordOfOrdSpec H spec
-instance [OrdSpecInfaillible H]: LT T := ltOfOrd
-
-theorem ltOfRustOrder {Spec: OrdSpecInfaillible H}: 
-  haveI O := ordOfOrdSpec H Spec
-  haveI := @ltOfOrd _ O
-  ∀ a b, H.cmp a b = .ok .Less -> a < b := by
-  intros a b
-  intro Hcmp
-  rw [LT.lt]
-  simp [ltOfOrd]
-  rw [compare]
-  simp [ordOfOrdSpec]
-  -- https://proofassistants.stackexchange.com/questions/1062/what-does-the-motive-is-not-type-correct-error-mean-in-lean
-  simp_rw [Hcmp, get?, avl_verification.Ordering.toLeanOrdering]
-  rfl
-
-theorem gtOfRustOrder {Spec: OrdSpecDuality H}: 
-  haveI O := ordOfOrdSpec H Spec.toOrdSpecInfaillible
-  haveI := @ltOfOrd _ O
-  ∀ a b, H.cmp a b = .ok .Greater -> b < a := by
-  intros a b
-  intro Hcmp
-  apply ltOfRustOrder
-  exact (Spec.duality _ _ Hcmp)
-
-
-end Specifications