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, 57 insertions, 0 deletions

diff --git a/Verification/Order.lean b/Verification/Order.lean
new file mode 100644
index 0000000..396a524
--- /dev/null
+++ b/Verification/Order.lean
@@ -0,0 +1,57 @@
+import Verification.Specifications
+
+namespace Implementation
+
+open Primitives
+open avl_verification
+open Specifications (OrdSpecLinearOrderEq ltOfRustOrder gtOfRustOrder)
+
+instance ScalarU32DecidableLE : DecidableRel (· ≤ · : U32 -> U32 -> Prop) := by
+  simp [instLEScalar]
+  -- Lift this to the decidability of the Int version.
+  infer_instance
+
+instance : LinearOrder (Scalar .U32) where
+  le_antisymm := fun a b Hab Hba => by
+    apply (Scalar.eq_equiv a b).2; exact (Int.le_antisymm ((Scalar.le_equiv _ _).1 Hab) ((Scalar.le_equiv _ _).1 Hba))
+  le_total := fun a b => by
+    rcases (Int.le_total a b) with H | H 
+    left; exact (Scalar.le_equiv _ _).2 H 
+    right; exact (Scalar.le_equiv _ _).2 H
+  decidableLE := ScalarU32DecidableLE
+
+instance : OrdSpecLinearOrderEq OrdU32 where
+  infallible := fun a b => by
+    unfold Ord.cmp
+    unfold OrdU32
+    unfold OrdU32.cmp
+    rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]
+    if hlt : a < b then 
+      use .Less
+      simp [hlt]
+    else
+      if heq: a = b
+      then
+      use .Equal
+      simp [hlt]
+      rw [heq]
+      -- TODO: simp [hlt, heq] breaks everything???
+      else
+        use .Greater
+        simp [hlt, heq]
+  symmetry := fun a b => by
+    rw [Ordering.toDualOrdering, LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]
+    rw [compare, Ord.opposite]
+    simp [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]
+    split_ifs with hab hba hba' hab' hba'' _ hba₃ _ <;> tauto
+    exact lt_irrefl _ (lt_trans hab hba)
+    rw [hba'] at hab; exact lt_irrefl _ hab
+    rw [hab'] at hba''; exact lt_irrefl _ hba''
+    -- The order is total, therefore, we have at least one case where we are comparing something.
+    cases (lt_trichotomy a b) <;> tauto
+  equivalence := fun a b => by
+    unfold Ord.cmp
+    unfold OrdU32
+    unfold OrdU32.cmp
+    simp only []
+    split_ifs <;> simp only [Result.ok.injEq, not_false_eq_true, neq_imp, IsEmpty.forall_iff]; tauto; try assumption