This is the sixth blog post in a series about developing correct implementations of basic data structures and algorithms using the Deduce language and proof checker.
This post continues were we left off from the previous post in which we implemented binary trees and in-order tree iterators.
Our goal in this post is to prove that we correctly implemented the iterator operations:
ti2tree : < E > fn TreeIter<E> -> Tree<E>
ti_first : < E > fn Tree<E>,E,Tree<E> -> TreeIter<E>
ti_get : < E > fn TreeIter<E> -> E
ti_next : < E > fn TreeIter<E> -> TreeIter<E>
ti_index : < E > fn(TreeIter<E>) -> NatThe first operation, ti2tree, requires us to first obtain a tree iterator, for example, with ti_first, so ti2tree does not have a correctness criteria all of its own, but instead the proof of its correctness will be part of the correctness of the other operations.
So we skip to the proof of correctness for ti_first.
Correctness of ti_first
Let us make explicit the specification of ti_first:
Specification: The ti_first(A, x, B) function returns an iterator pointing to the first node, with respect to in-order traversal, of the tree TreeNode(A, x, B).
Also, recall that we said the following about ti2tree and ti_first: creating an iterator from a tree using ti_first and then applying ti2tree produces the original tree.
So we have two properties to prove about ti_first. For the first property, we need a way to formalize "the first node with respect to in-order traversal". This is where the ti_index operation comes in. If ti_first returns the first node, then its index should be 0. (One might worry that if ti_index is incorrect, then this property would not force ti_first to be correct. Not to worry, we will prove that ti_index is correct!) So we have the following theorem:
theorem ti_first_index: all E:type, A:Tree<E>, x:E, B:Tree<E>.
ti_index(ti_first(A, x, B)) = 0
proof
arbitrary E:type, A:Tree<E>, x:E, B:Tree<E>
definition ti_first
?
endAfter expanding the definition of ti_first, we are left with the following goal. So we need to prove a lemma about the first_path auxiliary function.
ti_index(first_path(A,x,B,empty)) = 0Here is a first attempt to formulate the lemma.
lemma first_path_index: all E:type. all A:Tree<E>. all y:E, B:Tree<E>.
ti_index(first_path(A,y,B, empty)) = 0However, because first_path is recursive, we will need to prove this by recursion on A. But looking at the second clause of in the definition of first_path, the path argument grows, so our induction hypothesis, which requires the path argument to be empty, will not be applicable. As is often the case, we need to generalize the lemma. Let’s replace empty with an arbitrary path as follows.
lemma first_path_index: all E:type. all A:Tree<E>. all y:E, B:Tree<E>, path:List<Direction<E>>.
ti_index(first_path(A,y,B, path)) = 0But now this lemma is false. Consider the following situation in which the current node y is 5 and the path is L,R (going from node 5 up to node 3).
The index of node 5 is not 0, it is 5! Instead the index of node 5 is equal to the number of nodes that come before 5 according to in-order travesal. We can obtain that portion of the tree using functions that we have already defined, in particular take_path followed by plug_tree. So we can formulate the lemma as follows.
lemma first_path_index: all E:type. all A:Tree<E>. all y:E, B:Tree<E>, path:List<Direction<E>>.
ti_index(first_path(A,y,B, path)) = num_nodes(plug_tree(take_path(path), EmptyTree))
proof
arbitrary E:type
induction Tree<E>
case EmptyTree {
arbitrary y:E, B:Tree<E>, path:List<Direction<E>>
?
}
case TreeNode(L, x, R) suppose IH {
arbitrary y:E, B:Tree<E>, path:List<Direction<E>>
?
}
endFor the case A = EmptyTree, the goal simply follows from the definitions of first_path, ti_index, and ti_take.
conclude ti_index(first_path(EmptyTree,y,B,path))
= num_nodes(plug_tree(take_path(path),EmptyTree))
by definition {first_path, ti_index, ti_take}.For the case A = TreeNode(L, x, R), after expanding the definition of first_path, we need to prove:
ti_index(first_path(L,x,R,node(LeftD(y,B),path)))
= num_nodes(plug_tree(take_path(path),EmptyTree))But that follows from the induction hypothesis and the definition of take_path.
definition {first_path}
equations
ti_index(first_path(L,x,R,node(LeftD(y,B),path)))
= num_nodes(plug_tree(take_path(node(LeftD(y,B),path)),EmptyTree))
by IH[x, R, node(LeftD(y,B), path)]
... = num_nodes(plug_tree(take_path(path),EmptyTree))
by definition take_path.Here is the completed proof of the first_path_index lemma.
lemma first_path_index: all E:type. all A:Tree<E>. all y:E, B:Tree<E>, path:List<Direction<E>>.
ti_index(first_path(A,y,B, path)) = num_nodes(plug_tree(take_path(path), EmptyTree))
proof
arbitrary E:type
induction Tree<E>
case EmptyTree {
arbitrary y:E, B:Tree<E>, path:List<Direction<E>>
conclude ti_index(first_path(EmptyTree,y,B,path))
= num_nodes(plug_tree(take_path(path),EmptyTree))
by definition {first_path, ti_index, ti_take}.
}
case TreeNode(L, x, R) suppose IH {
arbitrary y:E, B:Tree<E>, path:List<Direction<E>>
definition {first_path}
equations
ti_index(first_path(L,x,R,node(LeftD(y,B),path)))
= num_nodes(plug_tree(take_path(node(LeftD(y,B),path)),EmptyTree))
by IH[x, R, node(LeftD(y,B), path)]
... = num_nodes(plug_tree(take_path(path),EmptyTree))
by definition take_path.
}
endReturning to the proof of ti_first_index, we need to prove that ti_index(first_path(A,x,B,empty)) = 0. So we apply the first_path_index lemma and then the definitions of take_path, plug_tree, and num_nodes. Here is the completed proof of ti_first_index.
theorem ti_first_index: all E:type, A:Tree<E>, x:E, B:Tree<E>.
ti_index(ti_first(A, x, B)) = 0
proof
arbitrary E:type, A:Tree<E>, x:E, B:Tree<E>
definition ti_first
equations ti_index(first_path(A,x,B,empty))
= num_nodes(plug_tree(take_path(empty),EmptyTree))
by first_path_index[E][A][x,B,empty]
... = 0 by definition {take_path, plug_tree, num_nodes}.
endOur next task is to prove that creating an iterator from a tree using ti_first and then applying ti2tree produces the original tree.
theorem ti_first_stable: all E:type, A:Tree<E>, x:E, B:Tree<E>.
ti2tree(ti_first(A, x, B)) = TreeNode(A, x, B)
proof
arbitrary E:type, A:Tree<E>, x:E, B:Tree<E>
definition ti_first
?
endAfter expanding the definition of ti_first, we are left to prove that
ti2tree(first_path(A,x,B,empty)) = TreeNode(A,x,B)So we need to prove another lemma about first_path and again we need to generalize the empty path to an arbitrary path. Let us consider again the situation where the current node x is 5.
The result of first_path(A,x,B,path) will be the path to node 4, and the result of ti2tree will be the whole tree, not just TreeNode(A,x,B) as in the above equation. However, we can construct the whole tree from the path and TreeNode(A,x,B) using the plug_tree function. So we have the following lemma to prove.
lemma first_path_stable:
all E:type. all A:Tree<E>. all y:E, B:Tree<E>, path:List<Direction<E>>.
ti2tree(first_path(A, y, B, path)) = plug_tree(path, TreeNode(A, y, B))
proof
arbitrary E:type
induction Tree<E>
case EmptyTree {
arbitrary y:E, B:Tree<E>, path:List<Direction<E>>
?
}
case TreeNode(L, x, R) suppose IH_L, IH_R {
arbitrary y:E, B:Tree<E>, path:List<Direction<E>>
?
}
endIn the case A = EmptyTree, we prove the equation using the definitions of first_path and ti2tree.
equations ti2tree(first_path(EmptyTree,y,B,path))
= ti2tree(TrItr(path,EmptyTree,y,B)) by definition first_path.
... = plug_tree(path,TreeNode(EmptyTree,y,B)) by definition ti2tree.In the case A = TreeNode(L, x, R), we need to prove that
ti2tree(first_path(TreeNode(L,x,R),y,B,path))
= plug_tree(path,TreeNode(TreeNode(L,x,R),y,B))We probably need to expand the definition of first_path, but doing so in your head is hard. So we can instead ask Deduce to do it. We start by constructing an equation with a bogus right-hand side and apply the definition of first_path.
equations
ti2tree(first_path(TreeNode(L,x,R),y,B,path))
= EmptyTree
by definition first_path ?
... = plug_tree(path,TreeNode(TreeNode(L,x,R),y,B))
by ?Deduce responds with
incomplete proof
Goal:
ti2tree(first_path(L,x,R,node(LeftD(y,B),path))) = EmptyTreein which the left-hand side has expanded the definition of first_path. So we cut and paste that into our proof and move on to the next step.
equations
ti2tree(first_path(TreeNode(L,x,R),y,B,path))
= ti2tree(first_path(L,x,R,node(LeftD(y,B),path)))
by definition first_path.
... = plug_tree(path,TreeNode(TreeNode(L,x,R),y,B))
by ?We now have something that matches the induction hypothesis, so we instantiate it and ask Deduce to tell us the new right-hand side.
equations
ti2tree(first_path(TreeNode(L,x,R),y,B,path))
= ti2tree(first_path(L,x,R,node(LeftD(y,B),path)))
by definition first_path.
... = EmptyTree
by IH_L[x,R,node(LeftD(y,B),path)]
... = plug_tree(path,TreeNode(TreeNode(L,x,R),y,B))
by ?Deduce responds with
expected
ti2tree(first_path(L,x,R,node(LeftD(y,B),path))) = EmptyTree
but only have
ti2tree(first_path(L,x,R,node(LeftD(y,B),path))) = plug_tree(node(LeftD(y,B),path),TreeNode(L,x,R))So we cut and paste the right-hand side of the induction hypothesis to replace EmptyTree.
equations
ti2tree(first_path(TreeNode(L,x,R),y,B,path))
= ti2tree(first_path(L,x,R,node(LeftD(y,B),path)))
by definition first_path.
... = plug_tree(node(LeftD(y,B),path),TreeNode(L,x,R))
by IH_L[x,R,node(LeftD(y,B),path)]
... = plug_tree(path,TreeNode(TreeNode(L,x,R),y,B))
by ?The final step of the proof is easy; we just apply the definition of plug_tree. Here is the completed proof of first_path_stable.
lemma first_path_stable:
all E:type. all A:Tree<E>. all y:E, B:Tree<E>, path:List<Direction<E>>.
ti2tree(first_path(A, y, B, path)) = plug_tree(path, TreeNode(A, y, B))
proof
arbitrary E:type
induction Tree<E>
case EmptyTree {
arbitrary y:E, B:Tree<E>, path:List<Direction<E>>
equations ti2tree(first_path(EmptyTree,y,B,path))
= ti2tree(TrItr(path,EmptyTree,y,B)) by definition first_path.
... = plug_tree(path,TreeNode(EmptyTree,y,B)) by definition ti2tree.
}
case TreeNode(L, x, R) suppose IH_L, IH_R {
arbitrary y:E, B:Tree<E>, path:List<Direction<E>>
equations
ti2tree(first_path(TreeNode(L,x,R),y,B,path))
= ti2tree(first_path(L,x,R,node(LeftD(y,B),path)))
by definition first_path.
... = plug_tree(node(LeftD(y,B),path),TreeNode(L,x,R))
by IH_L[x,R,node(LeftD(y,B),path)]
... = plug_tree(path,TreeNode(TreeNode(L,x,R),y,B))
by definition plug_tree.
}
endReturning to the ti_first_stable theorem, the equation follows from our first_path_stable lemma and the definition of plug_tree.
theorem ti_first_stable: all E:type, A:Tree<E>, x:E, B:Tree<E>.
ti2tree(ti_first(A, x, B)) = TreeNode(A, x, B)
proof
arbitrary E:type, A:Tree<E>, x:E, B:Tree<E>
definition ti_first
equations ti2tree(first_path(A,x,B,empty))
= plug_tree(empty,TreeNode(A,x,B)) by first_path_stable[E][A][x,B,empty]
... = TreeNode(A,x,B) by definition plug_tree.
endCorrectness of ti_next
We start by writing down a more careful specification of ti_next.
Specification: The ti_next(iter) operation returns an iterator whose position is one more than the position of iter with respect to in-order traversal, assuming the iter is not at the end of the in-order traversal.
To make this specification formal, we can again use ti_index to talk about the position of the iterator. So we begin to prove the following theorem ti_next_index, taking the usual initial steps in the proof as guided by the formula to be proved and the definition of ti_next, which performs a switch on the right child R of the current node.
theorem ti_next_index: all E:type, iter : TreeIter<E>.
if suc(ti_index(iter)) < num_nodes(ti2tree(iter))
then ti_index(ti_next(iter)) = suc(ti_index(iter))
proof
arbitrary E:type, iter : TreeIter<E>
suppose prem: suc(ti_index(iter)) < num_nodes(ti2tree(iter))
switch iter {
case TrItr(path, L, x, R) suppose iter_eq {
definition ti_next
switch R {
case EmptyTree suppose R_eq {
?
}
case TreeNode(RL, y, RR) suppose R_eq {
?
}
}
}
}
endIn the case R = EmptyTree, ti_next calls the auxiliary function next_up and we need to prove.
ti_index(next_up(path,L,x,EmptyTree)) = suc(ti_index(TrItr(path,L,x,EmptyTree)))As usual, we must create a lemma that generalizes this equation.
Proving the next_up_index lemma
Looking at the definition of next_up, we see that the recursive call grows the fourth argument, so we must replace the EmptyTree in the needed equation with an arbitrary tree R:
ti_index(next_up(path,L,x,R)) = suc(ti_index(TrItr(path,L,x,R)))But this equation is not true in general. Consider the situation below where the current node x is node 1 in our example tree. The index of the next_up from node 1 is 3, but the index of node 1 is 1 and of course, adding one to that is 2, not 3!
So we need to change this equation to account for the situation where R is not empty, but instead an arbitrary subtree. The solution is to add the number of nodes in R to the right-hand side:
ti_index(next_up(path,L,x,R)) = suc(ti_index(TrItr(path,L,x,R))) + num_nodes(R)One more addition is necessary to formulate the lemma. The above equation is only meaningful when the index on the right-hand side is in bounds. That is, it must be smaller than the number of nodes in the tree. So we formula the lemma next_up_index as follows and take a few obvious steps into the proof.
lemma next_up_index: all E:type. all path:List<Direction<E>>. all A:Tree<E>, x:E, B:Tree<E>.
if suc(ti_index(TrItr(path, A, x, B)) + num_nodes(B)) < num_nodes(ti2tree(TrItr(path, A, x, B)))
then ti_index(next_up(path, A, x, B)) = suc(ti_index(TrItr(path, A,x,B)) + num_nodes(B))
proof
arbitrary E:type
induction List<Direction<E>>
case empty {
arbitrary A:Tree<E>, x:E, B:Tree<E>
suppose prem: suc(ti_index(TrItr(empty,A,x,B)) + num_nodes(B))
< num_nodes(ti2tree(TrItr(empty,A,x,B)))
?
}
case node(f, path') suppose IH {
arbitrary A:Tree<E>, x:E, B:Tree<E>
suppose prem
switch f {
case LeftD(y, R) {
?
}
case RightD(L, y) suppose f_eq {
?
}
}
}
endIn the case path = empty, the premise is false because there are no nodes that come afterwards in the in-order traversal. In particular, the premise implies the following contradictory inequality.
have AB_l_AB: suc(num_nodes(A) + num_nodes(B)) < suc(num_nodes(A) + num_nodes(B))
by definition {ti_index, ti_take, take_path, plug_tree, ti2tree, num_nodes}
in prem
conclude false by apply less_irreflexive to AB_l_ABNext consider the case path = node(LeftD(y, R), path'). After expanding all the relevant definitions, we need to prove that
num_nodes(plug_tree(take_path(path'), TreeNode(A,x,B)))
= suc(num_nodes(plug_tree(take_path(path'), A)) + num_nodes(B))We need a lemma that relates num_nodes and plug_tree. So we pause the current proof for the following exercise.
Exercise: prove the num_nodes_plug lemma
lemma num_nodes_plug: all E:type. all path:List<Direction<E>>. all t:Tree<E>.
num_nodes(plug_tree(path, t)) = num_nodes(plug_tree(path, EmptyTree)) + num_nodes(t)Back to the next_up_index lemma
We use num_nodes_plug on both the left and right-hand sides of the equation, and apply the definition of num_nodes.
rewrite num_nodes_plug[E][take_path(path')][TreeNode(A,x,B)]
rewrite num_nodes_plug[E][take_path(path')][A]
definition num_nodesAfter that it suffices to prove the following.
num_nodes(plug_tree(take_path(path'),EmptyTree)) + suc(num_nodes(A) + num_nodes(B))
= suc((num_nodes(plug_tree(take_path(path'),EmptyTree)) + num_nodes(A)) + num_nodes(B))This equation is rather big, so let’s squint at it by giving names to its parts. (This is a new version of define that I’m experimenting with.)
define_ X = num_nodes(plug_tree(take_path(path'),EmptyTree))
define_ Y = num_nodes(A)
define_ Z = num_nodes(B)Now it’s easy to see that our goal is true using some simple arithmetic.
conclude X + suc(Y + Z) = suc((X + Y) + Z)
by rewrite add_suc[X][Y+Z] | add_assoc[X][Y,Z].Finally, consider the case path = node(RightD(L, y), path'). After expanding the definition of next_up, we need to prove
ti_index(next_up(path',L,y,TreeNode(A,x,B)))
= suc(ti_index(TrItr(node(RightD(L,y),path'),A,x,B)) + num_nodes(B))The left-hand side matches the induction hypothesis, so we have
equations
ti_index(next_up(path',L,y,TreeNode(A,x,B)))
= suc(ti_index(TrItr(path',L,y,TreeNode(A,x,B))) + num_nodes(TreeNode(A,x,B)))
by apply IH[L,y,TreeNode(A,x,B)]
to definition {ti_index, ti_take, num_nodes, ti2tree} ?
... = suc(ti_index(TrItr(node(RightD(L,y),path'),A,x,B)) + num_nodes(B))
by ?But we need to prove the premise of the induction hypothesis. We can do that as follows, with many uses of num_nodes_plug and some arithmetic that we package up into lemma XYZW_equal.
have IH_prem: suc(num_nodes(plug_tree(take_path(path'),L))
+ suc(num_nodes(A) + num_nodes(B)))
< num_nodes(plug_tree(path',TreeNode(L,y,TreeNode(A,x,B))))
by rewrite num_nodes_plug[E][take_path(path')][L]
| num_nodes_plug[E][path'][TreeNode(L,y,TreeNode(A,x,B))]
definition {num_nodes, num_nodes}
define_ X = num_nodes(plug_tree(take_path(path'),EmptyTree))
define_ Y = num_nodes(L) define_ Z = num_nodes(A) define_ W = num_nodes(B)
define_ P = num_nodes(plug_tree(path',EmptyTree))
suffices suc((X + Y) + suc(Z + W)) < P + suc(Y + suc(Z + W))
have prem2: suc((X + suc(Y + Z)) + W) < P + suc(Y + suc(Z + W))
by enable {X,Y,Z,W,P}
definition {num_nodes, num_nodes} in
rewrite num_nodes_plug[E][take_path(path')][TreeNode(L,y,A)]
| num_nodes_plug[E][path'][TreeNode(L,y,TreeNode(A,x,B))] in
definition {ti_index, ti_take, take_path, ti2tree, plug_tree} in
rewrite f_eq in prem
rewrite XYZW_equal[X,Y,Z,W]
prem2Here is the proof of XYZW_equal.
lemma XYZW_equal: all X:Nat, Y:Nat, Z:Nat, W:Nat.
suc((X + Y) + suc(Z + W)) = suc((X + suc(Y + Z)) + W)
proof
arbitrary X:Nat, Y:Nat, Z:Nat, W:Nat
enable {operator+}
equations
suc((X + Y) + suc(Z + W))
= suc(suc(X + Y) + (Z + W)) by rewrite add_suc[X+Y][Z+W].
... = suc(suc(((X + Y) + Z) + W)) by rewrite add_assoc[X+Y][Z,W].
... = suc(suc((X + (Y + Z)) + W)) by rewrite add_assoc[X][Y,Z].
... = suc((X + suc(Y + Z)) + W) by rewrite add_suc[X][Y+Z].
endGetting back to the equational proof, it remains to prove that
suc(ti_index(TrItr(path',L,y,TreeNode(A,x,B))) + num_nodes(TreeNode(A,x,B)))
= suc(ti_index(TrItr(node(RightD(L,y),path'),A,x,B)) + num_nodes(B))which we can do with yet more uses of num_nodes_plug and XYZW_equal.
... = suc(num_nodes(plug_tree(take_path(path'),L)) + suc(num_nodes(A) + num_nodes(B)))
by definition {ti_index, ti_take, num_nodes}.
... = suc((num_nodes(plug_tree(take_path(path'),EmptyTree)) + num_nodes(L))
+ suc(num_nodes(A) + num_nodes(B)))
by rewrite num_nodes_plug[E][take_path(path')][L].
... = suc((num_nodes(plug_tree(take_path(path'),EmptyTree))
+ suc(num_nodes(L) + num_nodes(A))) + num_nodes(B))
by define_ X = num_nodes(plug_tree(take_path(path'),EmptyTree))
define_ Y = num_nodes(L) define_ Z = num_nodes(A) define_ W = num_nodes(B)
define_ P = num_nodes(plug_tree(path',EmptyTree))
conclude suc((X + Y) + suc(Z + W)) = suc((X + suc(Y + Z)) + W)
by XYZW_equal[X,Y,Z,W]
... = suc(num_nodes(plug_tree(take_path(path'),TreeNode(L,y,A))) + num_nodes(B))
by rewrite num_nodes_plug[E][take_path(path')][TreeNode(L,y,A)]
definition {num_nodes, num_nodes}.
... = suc(ti_index(TrItr(node(RightD(L,y),path'),A,x,B)) + num_nodes(B))
by definition {ti_index, ti_take, take_path, plug_tree}.That completes the last case of the proof of next_up_index. Here’s the completed proof.
lemma next_up_index: all E:type. all path:List<Direction<E>>. all A:Tree<E>, x:E, B:Tree<E>.
if suc(ti_index(TrItr(path, A, x, B)) + num_nodes(B)) < num_nodes(ti2tree(TrItr(path, A, x, B)))
then ti_index(next_up(path, A, x, B)) = suc(ti_index(TrItr(path, A,x,B)) + num_nodes(B))
proof
arbitrary E:type
induction List<Direction<E>>
case empty {
arbitrary A:Tree<E>, x:E, B:Tree<E>
suppose prem: suc(ti_index(TrItr(empty,A,x,B)) + num_nodes(B))
< num_nodes(ti2tree(TrItr(empty,A,x,B)))
have AB_l_AB: suc(num_nodes(A) + num_nodes(B)) < suc(num_nodes(A) + num_nodes(B))
by definition {ti_index, ti_take, take_path, plug_tree, ti2tree, num_nodes}
in prem
conclude false by apply less_irreflexive to AB_l_AB
}
case node(f, path') suppose IH {
arbitrary A:Tree<E>, x:E, B:Tree<E>
suppose prem
switch f {
case LeftD(y, R) {
definition {next_up, ti_index, ti_take, take_path}
rewrite num_nodes_plug[E][take_path(path')][TreeNode(A,x,B)]
rewrite num_nodes_plug[E][take_path(path')][A]
definition num_nodes
define_ X = num_nodes(plug_tree(take_path(path'),EmptyTree))
define_ Y = num_nodes(A)
define_ Z = num_nodes(B)
conclude X + suc(Y + Z) = suc((X + Y) + Z)
by rewrite add_suc[X][Y+Z] | add_assoc[X][Y,Z].
}
case RightD(L, y) suppose f_eq {
definition {next_up}
have IH_prem: suc(num_nodes(plug_tree(take_path(path'),L))
+ suc(num_nodes(A) + num_nodes(B)))
< num_nodes(plug_tree(path',TreeNode(L,y,TreeNode(A,x,B))))
by rewrite num_nodes_plug[E][take_path(path')][L]
| num_nodes_plug[E][path'][TreeNode(L,y,TreeNode(A,x,B))]
definition {num_nodes, num_nodes}
define_ X = num_nodes(plug_tree(take_path(path'),EmptyTree))
define_ Y = num_nodes(L) define_ Z = num_nodes(A) define_ W = num_nodes(B)
define_ P = num_nodes(plug_tree(path',EmptyTree))
suffices suc((X + Y) + suc(Z + W)) < P + suc(Y + suc(Z + W))
have prem2: suc((X + suc(Y + Z)) + W) < P + suc(Y + suc(Z + W))
by enable {X,Y,Z,W,P}
definition {num_nodes, num_nodes} in
rewrite num_nodes_plug[E][take_path(path')][TreeNode(L,y,A)]
| num_nodes_plug[E][path'][TreeNode(L,y,TreeNode(A,x,B))] in
definition {ti_index, ti_take, take_path, ti2tree, plug_tree} in
rewrite f_eq in prem
rewrite XYZW_equal[X,Y,Z,W]
prem2
equations
ti_index(next_up(path',L,y,TreeNode(A,x,B)))
= suc(ti_index(TrItr(path',L,y,TreeNode(A,x,B))) + num_nodes(TreeNode(A,x,B)))
by apply IH[L,y,TreeNode(A,x,B)]
to definition {ti_index, ti_take, num_nodes, ti2tree} IH_prem
... = suc(num_nodes(plug_tree(take_path(path'),L)) + suc(num_nodes(A) + num_nodes(B)))
by definition {ti_index, ti_take, num_nodes}.
... = suc((num_nodes(plug_tree(take_path(path'),EmptyTree)) + num_nodes(L))
+ suc(num_nodes(A) + num_nodes(B)))
by rewrite num_nodes_plug[E][take_path(path')][L].
... = suc((num_nodes(plug_tree(take_path(path'),EmptyTree))
+ suc(num_nodes(L) + num_nodes(A))) + num_nodes(B))
by define_ X = num_nodes(plug_tree(take_path(path'),EmptyTree))
define_ Y = num_nodes(L) define_ Z = num_nodes(A) define_ W = num_nodes(B)
define_ P = num_nodes(plug_tree(path',EmptyTree))
conclude suc((X + Y) + suc(Z + W)) = suc((X + suc(Y + Z)) + W)
by XYZW_equal[X,Y,Z,W]
... = suc(num_nodes(plug_tree(take_path(path'),TreeNode(L,y,A))) + num_nodes(B))
by rewrite num_nodes_plug[E][take_path(path')][TreeNode(L,y,A)]
definition {num_nodes, num_nodes}.
... = suc(ti_index(TrItr(node(RightD(L,y),path'),A,x,B)) + num_nodes(B))
by definition {ti_index, ti_take, take_path, plug_tree}.
}
}
}
endBack to the proof of ti_next_index
With the next_up_index lemma complete, we can get back to proving the ti_next_index theorem. Recall that we were in the case R = EmptyTree and needed to prove the following.
ti_index(next_up(path,L,x,EmptyTree)) = suc(ti_index(TrItr(path,L,x,EmptyTree)))To use the next_up_index lemma, we need to prove its premise:
have next_up_index_prem:
suc(ti_index(TrItr(path,L,x,EmptyTree)) + num_nodes(EmptyTree))
< num_nodes(ti2tree(TrItr(path,L,x,EmptyTree)))
by enable num_nodes
rewrite add_zero[ti_index(TrItr(path,L,x,EmptyTree))]
rewrite iter_eq | R_eq in premWe can finish the proof of the equation using the definition of num_nodes and the add_zero property.
equations
ti_index(next_up(path,L,x,EmptyTree))
= suc(ti_index(TrItr(path,L,x,EmptyTree)) + num_nodes(EmptyTree))
by apply next_up_index[E][path][L, x, EmptyTree] to next_up_index_prem
... = suc(ti_index(TrItr(path,L,x,EmptyTree)))
by definition num_nodes
rewrite add_zero[ti_index(TrItr(path,L,x,EmptyTree))].The next case in the proof of ti_next_index is for R = TreeNode(RL, y, RR). We need to prove
ti_index(first_path(RL,y,RR,node(RightD(L,x),path)))
= suc(ti_index(TrItr(path,L,x,TreeNode(RL,y,RR))))We can start by applying the first_path_index lemma, which gives us
equations
ti_index(first_path(RL,y,RR,node(RightD(L,x),path)))
= num_nodes(plug_tree(take_path(node(RightD(L,x),path)),EmptyTree))We have opportunities to expand take_path and then plug_tree.
... = num_nodes(plug_tree(take_path(path),TreeNode(L,x,EmptyTree)))
by definition {take_path,plug_tree}.We can separate out the TreeNode(L,x,EmptyTree) using num_nodes_plug.
... = num_nodes(plug_tree(take_path(path),EmptyTree)) + suc(num_nodes(L))
by rewrite num_nodes_plug[E][take_path(path)][TreeNode(L,x,EmptyTree)]
definition {num_nodes, num_nodes}
rewrite add_zero[num_nodes(L)].Then we can move the L back into the plug_tree with num_nodes_plug.
... = suc(num_nodes(plug_tree(take_path(path),L)))
by rewrite add_suc[num_nodes(plug_tree(take_path(path),EmptyTree))][num_nodes(L)]
rewrite num_nodes_plug[E][take_path(path)][L].We conclude the equational reasoning with the definition of ti_index and ti_take.
... = suc(ti_index(TrItr(path,L,x,TreeNode(RL,y,RR))))
by definition {ti_index, ti_take}.Here is the complete proof of ti_next_index.
theorem ti_next_index: all E:type, iter : TreeIter<E>.
if suc(ti_index(iter)) < num_nodes(ti2tree(iter))
then ti_index(ti_next(iter)) = suc(ti_index(iter))
proof
arbitrary E:type, iter : TreeIter<E>
suppose prem: suc(ti_index(iter)) < num_nodes(ti2tree(iter))
switch iter {
case TrItr(path, L, x, R) suppose iter_eq {
definition ti_next
switch R {
case EmptyTree suppose R_eq {
have next_up_index_prem:
suc(ti_index(TrItr(path,L,x,EmptyTree)) + num_nodes(EmptyTree))
< num_nodes(ti2tree(TrItr(path,L,x,EmptyTree)))
by enable num_nodes
rewrite add_zero[ti_index(TrItr(path,L,x,EmptyTree))]
rewrite iter_eq | R_eq in prem
equations
ti_index(next_up(path,L,x,EmptyTree))
= suc(ti_index(TrItr(path,L,x,EmptyTree)) + num_nodes(EmptyTree))
by apply next_up_index[E][path][L, x, EmptyTree] to next_up_index_prem
... = suc(ti_index(TrItr(path,L,x,EmptyTree)))
by definition num_nodes
rewrite add_zero[ti_index(TrItr(path,L,x,EmptyTree))].
}
case TreeNode(RL, y, RR) suppose R_eq {
equations
ti_index(first_path(RL,y,RR,node(RightD(L,x),path)))
= num_nodes(plug_tree(take_path(node(RightD(L,x),path)),EmptyTree))
by first_path_index[E][RL][y,RR,node(RightD(L,x),path)]
... = num_nodes(plug_tree(take_path(path),TreeNode(L,x,EmptyTree)))
by definition {take_path,plug_tree}.
... = num_nodes(plug_tree(take_path(path),EmptyTree)) + suc(num_nodes(L))
by rewrite num_nodes_plug[E][take_path(path)][TreeNode(L,x,EmptyTree)]
definition {num_nodes, num_nodes}
rewrite add_zero[num_nodes(L)].
... = suc(num_nodes(plug_tree(take_path(path),L)))
by rewrite add_suc[num_nodes(plug_tree(take_path(path),EmptyTree))][num_nodes(L)]
rewrite num_nodes_plug[E][take_path(path)][L].
... = suc(ti_index(TrItr(path,L,x,TreeNode(RL,y,RR))))
by definition {ti_index, ti_take}.
}
}
}
}
endProof of ti_next_stable
The second correctness condition for ti_next(iter) is that it is stable with respect to ti2tree. Following the definition of ti_next, we switch on the iterator and then on the right child of the current node.
theorem ti_next_stable: all E:type, iter:TreeIter<E>.
ti2tree(ti_next(iter)) = ti2tree(iter)
proof
arbitrary E:type, iter:TreeIter<E>
switch iter {
case TrItr(path, L, x, R) {
switch R {
case EmptyTree {
definition {ti2tree, ti_next}
?
}
case TreeNode(RL, y, RR) {
definition {ti2tree, ti_next}
?
}
}
}
}
endFor the case R = EmptyTree, we need to prove the following, which amounts to proving that next_up is stable.
ti2tree(next_up(path,L,x,EmptyTree)) = plug_tree(path,TreeNode(L,x,EmptyTree))We’ll pause the current proof to prove the next_up_stable lemma.
Exercise: next_up_stable lemma
lemma next_up_stable: all E:type. all path:List<Direction<E>>. all A:Tree<E>, y:E, B:Tree<E>.
ti2tree(next_up(path, A, y, B)) = plug_tree(path, TreeNode(A,y,B))Back to ti_next_stable
Now we conclude the R = EmptyTree case of the ti_next_stable theorem.
conclude ti2tree(next_up(path,L,x,EmptyTree))
= plug_tree(path,TreeNode(L,x,EmptyTree))
by next_up_stable[E][path][L,x,EmptyTree]In the case R = TreeNode(RL, y, RR), we need prove the following, which is to say that first_path is stable. Thankfully we already proved that lemma!
conclude ti2tree(first_path(RL,y,RR,node(RightD(L,x),path)))
= plug_tree(path,TreeNode(L,x,TreeNode(RL,y,RR)))
by rewrite first_path_stable[E][RL][y,RR,node(RightD(L,x),path)]
definition {plug_tree}.Here is the completed proof of ti_next_stable.
theorem ti_next_stable: all E:type, iter:TreeIter<E>.
ti2tree(ti_next(iter)) = ti2tree(iter)
proof
arbitrary E:type, iter:TreeIter<E>
switch iter {
case TrItr(path, L, x, R) {
switch R {
case EmptyTree {
definition {ti2tree, ti_next}
conclude ti2tree(next_up(path,L,x,EmptyTree))
= plug_tree(path,TreeNode(L,x,EmptyTree))
by next_up_stable[E][path][L,x,EmptyTree]
}
case TreeNode(RL, y, RR) {
definition {ti2tree, ti_next}
conclude ti2tree(first_path(RL,y,RR,node(RightD(L,x),path)))
= plug_tree(path,TreeNode(L,x,TreeNode(RL,y,RR)))
by rewrite first_path_stable[E][RL][y,RR,node(RightD(L,x),path)]
definition {plug_tree}.
}
}
}
}
endCorrectness of ti_get and ti_index
Recall that ti_get(iter) should return the data in the current node of iter and ti_index should return the position of iter as a natural number with respect to in-order traversal. Thus, if we apply in_order to the tree, the element at position ti_index(iter) should be the same as ti_get(iter). So we have the following theorem to prove.
theorem ti_index_get_in_order: all E:type, iter:TreeIter<E>, a:E.
ti_get(iter) = nth(in_order(ti2tree(iter)), a)(ti_index(iter))
proof
arbitrary E:type, iter:TreeIter<E>, a:E
switch iter {
case TrItr(path, L, x, R) {
definition {ti2tree, ti_get, ti_index, ti_take}
?
}
}
endAfter expanding with some definitions, we are left to prove
x = nth(in_order(plug_tree(path,TreeNode(L,x,R))),a)
(num_nodes(plug_tree(take_path(path),L)))We see num_nodes applied to plug_tree, so we can use the num_nodes_plug lemma
rewrite num_nodes_plug[E][take_path(path)][L]The goal now is to prove
x = nth(in_order(plug_tree(path, TreeNode(L,x,R))),a)
(num_nodes(plug_tree(take_path(path), EmptyTree)) + num_nodes(L))The next step to take is not so obvious. Perhaps one hint is that we have the following theorem about nth from List.pf that also involves addition in the index argument of nth.
theorem nth_append_back: all T:type. all xs:List<T>. all ys:List<T>, i:Nat, d:T.
nth(append(xs, ys), d)(length(xs) + i) = nth(ys, d)(i)So we would need to prove a lemma that relates in_order and plug_tree to append. Now the take_path function returns the part of the tree before the path, so perhaps it can be used to create the xs in nth_append_back. But what about ys? It seems like we need a function that returns the part of the tree after the path. Let us call this function drop_path.
function drop_path<E>(List<Direction<E>>) -> List<Direction<E>> {
drop_path(empty) = empty
drop_path(node(f, path')) =
switch f {
case RightD(L, x) {
drop_path(path')
}
case LeftD(x, R) {
node(LeftD(x, R), drop_path(path'))
}
}
}So using take_path and drop_path, we should be able to come up with an equation for in_order(plug_tree(path, TreeNode(A, x, B))). The part of tree before x should be take_path(path) followed by the subtree A. The part of the tree after x should be the subtree B followed by drop_path(path).
lemma in_order_plug_take_drop: all E:type. all path:List<Direction<E>>. all A:Tree<E>, x:E, B:Tree<E>.
in_order(plug_tree(path, TreeNode(A, x, B)))
= append(in_order(plug_tree(take_path(path), A)),
node(x, in_order(plug_tree(drop_path(path), B))))It turns out that to prove this, we will also need a lemma about the combination of plug_tree and take_path:
lemma in_order_plug_take: all E:type. all path:List<Direction<E>>. all t:Tree<E>.
in_order(plug_tree(take_path(path), t))
= append( in_order(plug_tree(take_path(path),EmptyTree)), in_order(t))and a lemma about the combination of plug_tree and drop_path:
lemma in_order_plug_drop: all E:type. all path:List<Direction<E>>. all t:Tree<E>.
in_order(plug_tree(drop_path(path), t)) = append( in_order(t), in_order(plug_tree(drop_path(path),EmptyTree)))Exercise: prove the in_order_plug... lemmas
Prove the three lemmas in_order_plug_take_drop, in_order_plug_take, and in_order_plug_drop.
Back to the proof of ti_index_get_in_order
Our goal was to prove
x = nth(in_order(plug_tree(path,TreeNode(L,x,R))), a)
(num_nodes(plug_tree(take_path(path),EmptyTree)) + num_nodes(L))So we use lemma in_order_plug_take_drop to get the following
in_order(plug_tree(path,TreeNode(L,x,R)))
= append(in_order(plug_tree(take_path(path),L)), node(x, in_order(plug_tree(drop_path(path),R))))and then lemma in_order_plug_take separates out the L.
in_order(plug_tree(take_path(path), L))
= append(in_order(plug_tree(take_path(path),EmptyTree)), in_order(L))So rewriting with the above equations
rewrite in_order_plug_take_drop[E][path][L,x,R]
rewrite in_order_plug_take[E][path][L]transforms our goal to
x = nth(append(append(in_order(plug_tree(take_path(path),EmptyTree)), in_order(L)),
node(x,in_order(plug_tree(drop_path(path),R)))),a)
(num_nodes(plug_tree(take_path(path),EmptyTree)) + num_nodes(L))Recall that our plan is to use the nth_append_back lemma, in which the index argument to nth is length(xs), but in the above we have the index expressed in terms of num_nodes. The following exercise proves a theorem that relates length and in_order to num_nodes.
Exercise: prove the length_in_order theorem
theorem length_in_order: all E:type. all t:Tree<E>.
length(in_order(t)) = num_nodes(t)Back to ti_index_get_in_order
Now we rewrite with the length_in_order lemma a couple times, give some short names to these big expressions, and apply length_append from List.pf.
rewrite symmetric length_in_order[E][L]
| symmetric length_in_order[E][plug_tree(take_path(path),EmptyTree)]
define_ X = in_order(plug_tree(take_path(path),EmptyTree))
define_ Y = in_order(L)
define_ Z = in_order(plug_tree(drop_path(path),R))
rewrite symmetric length_append[E][X][Y]Now we’re in a position to use nth_append_back.
x = nth(append(append(X,Y), node(x, Z)), a)
(length(append(X,Y)))In particular, nth_append_back[E][append(X,Y)][node(x,Z), 0, a] gives us
nth(append(append(X,Y), node(x,Z)),a)(length(append(X,Y)) + 0)
= nth(node(x,Z),a)(0)With that we prove the goal using add_zero and the definition of nth.
conclude x = nth(append(append(X,Y), node(x,Z)), a)(length(append(X,Y)))
by rewrite (rewrite add_zero[length(append(X,Y))] in
nth_append_back[E][append(X,Y)][node(x,Z), 0, a])
definition nth.Here is the complete proof of ti_index_get_in_order.
theorem ti_index_get_in_order: all E:type, z:TreeIter<E>, a:E.
ti_get(z) = nth(in_order(ti2tree(z)), a)(ti_index(z))
proof
arbitrary E:type, z:TreeIter<E>, a:E
switch z {
case TrItr(path, L, x, R) {
definition {ti2tree, ti_get, ti_index, ti_take}
rewrite num_nodes_plug[E][take_path(path)][L]
suffices x = nth(in_order(plug_tree(path,TreeNode(L,x,R))),a)
(num_nodes(plug_tree(take_path(path),EmptyTree)) + num_nodes(L))
rewrite in_order_plug_take_drop[E][path][L,x,R]
rewrite in_order_plug_take[E][path][L]
suffices x = nth(append(append(in_order(plug_tree(take_path(path),EmptyTree)),
in_order(L)),
node(x,in_order(plug_tree(drop_path(path),R)))),a)
(num_nodes(plug_tree(take_path(path),EmptyTree)) + num_nodes(L))
rewrite symmetric length_in_order[E][L]
| symmetric length_in_order[E][plug_tree(take_path(path),EmptyTree)]
define_ X = in_order(plug_tree(take_path(path),EmptyTree))
define_ Y = in_order(L)
define_ Z = in_order(plug_tree(drop_path(path),R))
rewrite symmetric length_append[E][X][Y]
conclude x = nth(append(append(X,Y), node(x,Z)), a)(length(append(X,Y)))
by rewrite (rewrite add_zero[length(append(X,Y))] in
nth_append_back[E][append(X,Y)][node(x,Z), 0, a])
definition nth.
}
}
endThis concludes the proofs of correctness for in-order iterator and the five operations ti2tree, ti_first, ti_get, ti_next, and ti_index.
Exercise: Prove that ti_prev is correct
In the previous post there was an exercise to implement ti_prev, which moves the iterator backwards one position with respect to in-order traversal. This exercise is to prove that your implementation of ti_prev is correct. There are two theorems to prove. The first one makes sure that ti_prev reduces the index of the iterator by one.
theorem ti_prev_index: all E:type, iter : TreeIter<E>.
if 0 < ti_index(iter)
then ti_index(ti_prev(iter)) = pred(ti_index(iter))The second theorem makes sure that the resulting iterator is still an iterator for the same tree.
theorem ti_prev_stable: all E:type, iter:TreeIter<E>.
ti2tree(ti_prev(iter)) = ti2tree(iter)
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