feat: upgrade and close all proofs except Preorder on U32

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

5 files changed, 138 insertions, 81 deletions

diff --git a/AvlVerification.lean b/AvlVerification.lean
index e0516b4..828b2aa 100644
--- a/AvlVerification.lean
+++ b/AvlVerification.lean
@@ -30,7 +30,7 @@ structure AVLTreeSet (T : Type) where
 /- [avl_verification::{avl_verification::AVLTreeSet<T>}::new]:
    Source: 'src/main.rs', lines 24:4-24:24 -/
 def AVLTreeSet.new (T : Type) (OrdInst : Ord T) : Result (AVLTreeSet T) :=
-  Result.ret { root := none }
+  Result.ok { root := none }
 
 /- [avl_verification::{avl_verification::AVLTreeSet<T>}::insert]: loop 0:
    Source: 'src/main.rs', lines 28:4-46:5 -/
@@ -41,7 +41,7 @@ divergent def AVLTreeSet.insert_loop
   :=
   match current_tree with
   | none => let a := AVLNode.mk value none none
-            Result.ret (true, some a)
+            Result.ok (true, some a)
   | some current_node =>
     do
     let ⟨ t, current_tree1, current_tree2 ⟩ := current_node
@@ -50,13 +50,13 @@ divergent def AVLTreeSet.insert_loop
     | Ordering.Less =>
       do
       let (b, back) ← AVLTreeSet.insert_loop T OrdInst value current_tree2
-      Result.ret (b, some (AVLNode.mk t current_tree1 back))
+      Result.ok (b, some (AVLNode.mk t current_tree1 back))
     | Ordering.Equal =>
-      Result.ret (false, some (AVLNode.mk t current_tree1 current_tree2))
+      Result.ok (false, some (AVLNode.mk t current_tree1 current_tree2))
     | Ordering.Greater =>
       do
       let (b, back) ← AVLTreeSet.insert_loop T OrdInst value current_tree1
-      Result.ret (b, some (AVLNode.mk t back current_tree2))
+      Result.ok (b, some (AVLNode.mk t back current_tree2))
 
 /- [avl_verification::{avl_verification::AVLTreeSet<T>}::insert]:
    Source: 'src/main.rs', lines 28:4-28:46 -/
@@ -66,17 +66,17 @@ def AVLTreeSet.insert
   :=
   do
   let (b, back) ← AVLTreeSet.insert_loop T OrdInst value self.root
-  Result.ret (b, { root := back })
+  Result.ok (b, { root := back })
 
 /- [avl_verification::{(avl_verification::Ord for u32)#1}::cmp]:
    Source: 'src/main.rs', lines 50:4-50:43 -/
 def OrdU32.cmp (self : U32) (other : U32) : Result Ordering :=
   if self < other
-  then Result.ret Ordering.Less
+  then Result.ok Ordering.Less
   else
     if self = other
-    then Result.ret Ordering.Equal
-    else Result.ret Ordering.Greater
+    then Result.ok Ordering.Equal
+    else Result.ok Ordering.Greater
 
 /- Trait implementation: [avl_verification::{(avl_verification::Ord for u32)#1}]
    Source: 'src/main.rs', lines 49:0-49:16 -/
diff --git a/AvlVerification/BinarySearchTree.lean b/AvlVerification/BinarySearchTree.lean
index 954f21f..2b17d52 100644
--- a/AvlVerification/BinarySearchTree.lean
+++ b/AvlVerification/BinarySearchTree.lean
@@ -10,34 +10,45 @@ inductive ForallNode (p: T -> Prop): AVLTree T -> Prop
 | none : ForallNode p none
 | some (a: T) (left: AVLTree T) (right: AVLTree T) : ForallNode p left -> p a -> ForallNode p right -> ForallNode p (some (AVLNode.mk a left right))
 
-theorem ForallNode.left {p: T -> Prop} {t: AVLTree T}: ForallNode p t -> ForallNode p t.left := sorry
-
-theorem ForallNode.right {p: T -> Prop} {t: AVLTree T}: ForallNode p t -> ForallNode p t.right := sorry
-
-theorem ForallNode.label {a: T} {p: T -> Prop} {left right: AVLTree T}: ForallNode p (AVLNode.mk a left right) -> p a := sorry
+theorem ForallNode.left {p: T -> Prop} {t: AVLTree T}: ForallNode p t -> ForallNode p t.left := by
+  intro Hpt
+  cases Hpt with 
+  | none => simp [AVLTree.left, ForallNode.none]
+  | some a left right f_pleft f_pa f_pright => simp [AVLTree.left, f_pleft]
+
+theorem ForallNode.right {p: T -> Prop} {t: AVLTree T}: ForallNode p t -> ForallNode p t.right := by
+  intro Hpt
+  cases Hpt with 
+  | none => simp [AVLTree.right, ForallNode.none]
+  | some a left right f_pleft f_pa f_pright => simp [AVLTree.right, f_pright]
+
+theorem ForallNode.label {a: T} {p: T -> Prop} {left right: AVLTree T}: ForallNode p (AVLNode.mk a left right) -> p a := by
+  intro Hpt
+  cases Hpt with 
+  | some a left right f_pleft f_pa f_pright => exact f_pa
 
 -- This is the binary search invariant.
-inductive BST [LT T]: AVLTree T -> Prop
-| none : BST none
+inductive Invariant [LT T]: AVLTree T -> Prop
+| none : Invariant none
 | some (a: T) (left: AVLTree T) (right: AVLTree T) : 
   ForallNode (fun v => v < a) left -> ForallNode (fun v => a < v) right
-  -> BST left -> BST right -> BST (some (AVLNode.mk a left right))
+  -> Invariant left -> Invariant right -> Invariant (some (AVLNode.mk a left right))
 
 @[simp]
-theorem singleton_bst [LT T] {a: T}: BST (some (AVLNode.mk a none none)) := by
-  apply BST.some
-  all_goals simp [ForallNode.none, BST.none] 
+theorem singleton_bst [LT T] {a: T}: Invariant (some (AVLNode.mk a none none)) := by
+  apply Invariant.some
+  all_goals simp [ForallNode.none, Invariant.none] 
 
-theorem left [LT T] {t: AVLTree T}: BST t -> BST t.left := by
+theorem left [LT T] {t: AVLTree T}: Invariant t -> Invariant t.left := by
   intro H
   induction H with 
-  | none => exact BST.none
+  | none => exact Invariant.none
   | some _ _ _ _ _ _ _ _ _ => simp [AVLTree.left]; assumption
 
-theorem right [LT T] {t: AVLTree T}: BST t -> BST t.right := by
+theorem right [LT T] {t: AVLTree T}: Invariant t -> Invariant t.right := by
   intro H
   induction H with 
-  | none => exact BST.none
+  | none => exact Invariant.none
   | some _ _ _ _ _ _ _ _ _ => simp [AVLTree.right]; assumption
 
 end BST
diff --git a/AvlVerification/Insert.lean b/AvlVerification/Insert.lean
index 3c2b8b3..baf3441 100644
--- a/AvlVerification/Insert.lean
+++ b/AvlVerification/Insert.lean
@@ -7,9 +7,47 @@ namespace Implementation
 open Primitives
 open avl_verification
 open Tree (AVLTree)
-open Specifications (OrdSpec ordSpecOfInfaillible ordOfOrdSpec ltOfRustOrder)
+open Specifications (OrdSpec ordSpecOfTotalityAndDuality ordOfOrdSpec ltOfRustOrder gtOfRustOrder)
+
+example: OrdSpec OrdU32 := ordSpecOfTotalityAndDuality _ 
+  (by 
+  -- Totality
+  intro a b
+  unfold Ord.cmp
+  unfold OrdU32
+  unfold OrdU32.cmp
+  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]
+  ) (by 
+  -- Duality
+  intro a b Hgt
+  if hlt : b < a then
+    unfold Ord.cmp
+    unfold OrdU32
+    unfold OrdU32.cmp
+    simp [hlt]
+  else
+    unfold Ord.cmp at Hgt
+    unfold OrdU32 at Hgt
+    unfold OrdU32.cmp at Hgt
+    have hnlt : ¬ (a < b) := sorry
+    have hneq : ¬ (a = b) := sorry
+    exfalso
+    apply hlt
+    -- I need a Preorder on U32 now.
+    sorry)
 
--- TODO: OSpec should be proven.
 variable (T: Type) (H: avl_verification.Ord T) (Ospec: @OrdSpec T H)
 
 instance rustOrder {U: Type} {O: avl_verification.Ord U} (OSpec: OrdSpec O): _root_.Ord U := ordOfOrdSpec O OSpec
@@ -20,9 +58,9 @@ instance rustLt' : LT T := rustLt Ospec
 
 @[pspec]
 theorem AVLTreeSet.insert_loop_spec_local (p: T -> Prop)
-  (Hcmp_spec: ∀ a b, ∃ o, H.cmp a b = Result.ret o)
+  (Hcmp_spec: ∀ a b, ∃ o, H.cmp a b = Result.ok o)
   (a: T) (t: Option (AVLNode T)):
-  ∃ added t_new, AVLTreeSet.insert_loop T H a t = Result.ret ⟨ added, t_new ⟩
+  ∃ added t_new, AVLTreeSet.insert_loop T H a t = Result.ok ⟨ added, t_new ⟩
 --  ∧ AVLTree.set t'.2 = insert a (AVLTree.set t)
   ∧ (BST.ForallNode p t -> p a -> BST.ForallNode p t_new)
   := by match t with
@@ -68,8 +106,8 @@ lemma AVLTreeSet.insert_loop_spec_global
   (a: T) (t: Option (AVLNode T))
   :
   haveI : LT T := (rustLt Ospec)
-  BST.BST t -> ∃ added t_new, AVLTreeSet.insert_loop T H a t = Result.ret ⟨ added, t_new ⟩
-  ∧ BST.BST t_new := by 
+  BST.Invariant t -> ∃ added t_new, AVLTreeSet.insert_loop T H a t = Result.ok ⟨ added, t_new ⟩
+  ∧ BST.Invariant t_new := by 
   intro Hbst
   letI instOrd : _root_.Ord T := (rustOrder Ospec)
   letI instLt : LT T := (rustLt Ospec)
@@ -78,9 +116,7 @@ lemma AVLTreeSet.insert_loop_spec_global
   | some (AVLNode.mk b left right) =>
     rw [AVLTreeSet.insert_loop]
     simp only []
-    have := Ospec.infallible; simp at this
-    -- TODO: unbundle `this`...
-    have : ∀ a b, ∃ o, H.cmp a b = .ret o := sorry
+    have : ∀ a b, ∃ o, H.cmp a b = .ok o := Ospec.infallible
     progress keep Hordering as ⟨ ordering ⟩
     cases ordering
     all_goals simp only []
@@ -104,34 +140,33 @@ lemma AVLTreeSet.insert_loop_spec_global
       simp; tauto
     }
     {
-      sorry
+      have ⟨ added₂, left₂, ⟨ H_result, H_bst ⟩ ⟩ := AVLTreeSet.insert_loop_spec_global a left (BST.left Hbst)
+      progress keep Htree with AVLTreeSet.insert_loop_spec_local as ⟨ added₁, left₁, Hnode ⟩
+      exact (fun x => x < b)
+      rewrite [Htree] at H_result; simp at H_result
+      refine' ⟨ added₁, ⟨ some (AVLNode.mk b left₁ right), _ ⟩ ⟩
+      simp only [true_and]
+      cases' Hbst with _ _ _ H_forall_left H_forall_right H_bst_left H_bst_right
+      constructor
+      apply Hnode; exact H_forall_left
+      exact (gtOfRustOrder H b a Hordering)
+      exact H_forall_right
+      convert H_bst
+      exact H_result.2
+      exact H_bst_right
     }
 
 @[pspec]
 def AVLTreeSet.insert_spec 
-  {H: avl_verification.Ord T} 
-  -- TODO: this can be generalized into `H.cmp` must be an equivalence relation.
-  -- and insert works no longer on Sets but on set quotiented by this equivalence relation.
-  (Hcmp_eq: ∀ a b, H.cmp a b = Result.ret Ordering.Equal -> a = b)
-  (Hcmp_spec: ∀ a b, ∃ o, H.cmp a b = Result.ret o)
-  (Hbs_search: @BSTree.searchInvariant _ H t)
   (a: T) (t: AVLTreeSet T):
-  ∃ t', t.insert _ H a = Result.ret t' 
-  -- set of values *POST* insertion is {a} \cup set of values of the *PRE* tree.
-  ∧ AVLTree.setOf t'.2.root = insert a (AVLTree.setOf t.root) 
+  haveI : LT T := (rustLt Ospec)
+  BST.Invariant t.root -> (∃ t' added,t.insert _ H a = Result.ok (added, t')
   -- it's still a binary search tree.
-  ∧ @BSTree.searchInvariant _ H t'.2.root
-  -- TODO(reinforcement): (t'.1 is false <=> a \in AVLTree.setOf t.root)
+  ∧ BST.Invariant t'.root)
   := by
-  rw [AVLTreeSet.insert]
-  progress keep h as ⟨ t', Hset ⟩; simp
-  rw [Hset]
-  sorry
-
-
--- insert is infaillible, right?
--- instance [LT T] (o: avl_verification.Ord T): Insert T (AVLTreeSet T) where
---   insert x tree := (AVLTreeSet.insert T o tree x).map (fun (added, tree) => tree)
+  rw [AVLTreeSet.insert]; intro Hbst
+  progress keep h as ⟨ t', Hset ⟩; 
+  simp; assumption
 
 end Implementation
 
diff --git a/AvlVerification/Specifications.lean b/AvlVerification/Specifications.lean
index 247fd00..958a3e7 100644
--- a/AvlVerification/Specifications.lean
+++ b/AvlVerification/Specifications.lean
@@ -5,19 +5,19 @@ namespace Primitives
 namespace Result
 
 def map {A B: Type} (x: Result A) (f: A -> B): Result B := match x with
-| .ret y => .ret (f y)
+| .ok y => .ok (f y)
 | .fail e => .fail e
 | .div => .div
 
 @[inline]
-def isRet {A: Type} : Result A -> Bool
-| .ret _ => true
+def isok {A: Type} : Result A -> Bool
+| .ok _ => true
 | .fail _ => false
 | .div => false
 
 @[inline]
-def get? {A: Type}: (r: Result A) -> isRet r -> A
-| .ret x, _ => x
+def get? {A: Type}: (r: Result A) -> isok r -> A
+| .ok x, _ => x
 
 end Result
 
@@ -30,6 +30,11 @@ def Ordering.toLeanOrdering (o: avl_verification.Ordering): _root_.Ordering := m
 | .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
@@ -39,42 +44,51 @@ open Result
 
 variable {T: Type} (H: avl_verification.Ord T)
 
+-- TODO: reason about raw bundling vs. refined bundling.
 class OrdSpec where
-  cmp: T -> T -> Result Ordering := fun x y => (H.cmp x y).map (fun z => z.toLeanOrdering)
-  infallible: ∀ a b, ∃ (o: Ordering), cmp a b = .ret o
-
-@[simp]
-theorem OrdSpec.cmpEq {Spec: OrdSpec H}: ∀ x y, Spec.cmp x y = (H.cmp x y).map (fun z => z.toLeanOrdering) := sorry
-
-@[simp]
-theorem OrdSpec.infallibleEq {Spec: OrdSpec H}: ∀ a b, ∃ (o: Ordering), Spec.cmp a b = .ret o <-> ∀ a b, ∃ (o: Ordering), (H.cmp a b).map (fun z => z.toLeanOrdering) = .ret o := sorry
+  infallible: ∀ a b, ∃ (o: avl_verification.Ordering), H.cmp a b = .ok o
+  duality: ∀ a b, H.cmp a b = .ok .Greater -> H.cmp b a = .ok .Less
 
 instance: Coe (avl_verification.Ordering) (_root_.Ordering) where
   coe a := a.toLeanOrdering
 
-def ordSpecOfInfaillible
+def ordSpecOfTotalityAndDuality
   (H: avl_verification.Ord T)
-  (Hresult: ∀ a b, ∃ o, H.cmp a b = Primitives.Result.ret o)
+  (Hresult: ∀ a b, ∃ o, H.cmp a b = Primitives.Result.ok o)
+  (Hduality: ∀ a b, H.cmp a b = .ok .Greater -> H.cmp b a = .ok .Less)
   : OrdSpec H where
-  infallible := by
-    intros a b; cases' (Hresult a b) with o Hcmp; refine' ⟨ o.toLeanOrdering, _ ⟩
-    simp [Result.map, Hcmp]
+  infallible := Hresult
+  duality := Hduality
 
 def ordOfOrdSpec (H: avl_verification.Ord T) (spec: OrdSpec H): Ord T where
-  compare x y := (spec.cmp x y).get? (by
+  compare x y := (H.cmp x y).get? (by
     cases' (spec.infallible x y) with o Hcmp
-    rewrite [isRet, Hcmp]
+    rewrite [isok, Hcmp]
     simp only
   )
 
 theorem ltOfRustOrder {Spec: OrdSpec H}: 
   haveI O := ordOfOrdSpec H Spec
   haveI := @ltOfOrd _ O
-  ∀ a b, H.cmp a b = .ret .Less -> a < b := by
+  ∀ a b, H.cmp a b = .ok .Less -> a < b := by
   intros a b
   intro Hcmp
-  change ((Spec.cmp a b).get? _ == .lt)
-  simp [Hcmp, map, avl_verification.Ordering.toLeanOrdering]
-  simp [get?]
+  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: OrdSpec H}: 
+  haveI O := ordOfOrdSpec H Spec
+  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
diff --git a/AvlVerification/Tree.lean b/AvlVerification/Tree.lean
index 3316e9e..fdd4b78 100644
--- a/AvlVerification/Tree.lean
+++ b/AvlVerification/Tree.lean
@@ -75,7 +75,4 @@ def AVLTree.mem_some {x: T} {left right: AVLTree T}: AVLTree.mem (some (AVLNode.
 
 def AVLTree.set (t: AVLTree T): Set T := _root_.setOf (AVLTree.mem t)
 
-def AVLTree.setOf_left_incl (t: AVLTree T): t.left.set ⊂ t.set := sorry
-def AVLTree.setOf_right_incl (t: AVLTree T): t.right.set ⊂ t.set := sorry
-
 end Tree