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week-10_exercises.v
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(* week-10_exercises.v *)
(* FPP 2020 - YSC3236 2020-2021, Sem1 *)
(* ********** *)
(*
Your name:
Bernard Boey
Tristan Koh
Your e-mail address:
Your student number:
A0191234L
A0191222R
*)
(* ********* *)
(* Paraphernalia: *)
Ltac fold_unfold_tactic name := intros; unfold name; fold name; reflexivity.
Require Import Arith Bool.
(* ********** *)
Inductive binary_tree : Type :=
| Leaf : nat -> binary_tree
| Node : binary_tree -> binary_tree -> binary_tree.
(* ***** *)
Fixpoint weight (t : binary_tree) : nat :=
match t with
| Leaf n =>
n
| Node t1 t2 =>
weight t1 + weight t2
end.
Lemma fold_unfold_weight_Leaf :
forall n : nat,
weight (Leaf n) = n.
Proof.
fold_unfold_tactic weight.
Qed.
Lemma fold_unfold_weight_Node :
forall t1 t2 : binary_tree,
weight (Node t1 t2) = weight t1 + weight t2.
Proof.
fold_unfold_tactic weight.
Qed.
(* ***** *)
Fixpoint balancedp (t : binary_tree) : bool :=
match t with
| Leaf n =>
true
| Node t1 t2 =>
balancedp t1 && balancedp t2 && (weight t1 =? weight t2)
end.
Lemma fold_unfold_balancedp_Leaf :
forall n : nat,
balancedp (Leaf n) =
true.
Proof.
fold_unfold_tactic balancedp.
Qed.
Lemma fold_unfold_balancedp_Node :
forall t1 t2 : binary_tree,
balancedp (Node t1 t2) =
(balancedp t1 && balancedp t2 && (weight t1 =? weight t2)).
Proof.
fold_unfold_tactic balancedp.
Qed.
(* ********** *)
Fixpoint wb_ds_aux (t : binary_tree) : option nat :=
match t with
| Leaf n =>
Some n
| Node t1 t2 =>
match wb_ds_aux t1 with
| Some w1 =>
match wb_ds_aux t2 with
| Some w2 =>
if beq_nat w1 w2
then Some (w1 + w2)
else None
| None =>
None
end
| None =>
None
end
end.
Lemma fold_unfold_wb_ds_aux_Leaf :
forall n : nat,
wb_ds_aux (Leaf n) =
Some n.
Proof.
fold_unfold_tactic wb_ds_aux.
Qed.
Lemma fold_unfold_wb_ds_aux_Node :
forall t1 t2 : binary_tree,
wb_ds_aux (Node t1 t2) =
match wb_ds_aux t1 with
| Some w1 =>
match wb_ds_aux t2 with
| Some w2 =>
if beq_nat w1 w2
then Some (w1 + w2)
else None
| None =>
None
end
| None =>
None
end.
Proof.
fold_unfold_tactic wb_ds_aux.
Qed.
Definition wb_ds (t : binary_tree) : bool :=
match wb_ds_aux t with
| Some _ =>
true
| None =>
false
end.
(* ********** *)
Lemma soundness_of_wb_ds_aux :
forall (t : binary_tree)
(w : nat),
wb_ds_aux t = Some w ->
balancedp t = true /\ w = weight t.
Proof.
intro t.
induction t as [n | t1 IHt1 t2 IHt2]; intros w H_aux.
- rewrite -> fold_unfold_wb_ds_aux_Leaf in H_aux.
injection H_aux as H_aux.
rewrite -> fold_unfold_balancedp_Leaf.
rewrite -> fold_unfold_weight_Leaf.
symmetry in H_aux.
split; [reflexivity | exact H_aux].
- rewrite -> fold_unfold_wb_ds_aux_Node in H_aux.
case (wb_ds_aux t1) as [w1 | ] eqn:Ht1.
+ case (wb_ds_aux t2) as [w2 | ] eqn:Ht2.
* case (w1 =? w2) eqn:H_w1_w2.
** injection H_aux as H_aux.
rewrite -> fold_unfold_balancedp_Node.
rewrite -> fold_unfold_weight_Node.
Check (IHt1 w1 (eq_refl (Some w1))).
destruct (IHt1 w1 (eq_refl (Some w1))) as [Bt1 W1].
destruct (IHt2 w2 (eq_refl (Some w2))) as [Bt2 W2].
rewrite -> Bt1.
rewrite -> Bt2.
rewrite <- W1.
rewrite <- W2.
unfold andb.
symmetry in H_aux.
split; [exact H_w1_w2 | exact H_aux].
** discriminate H_aux.
* discriminate H_aux.
+ discriminate H_aux.
Qed.
Theorem soundness_of_wb_ds :
forall t : binary_tree,
wb_ds t = true ->
balancedp t = true.
Proof.
intros t H_t.
unfold wb_ds in H_t.
case (wb_ds_aux t) as [w | ] eqn:H_aux.
+ Check (soundness_of_wb_ds_aux t w H_aux).
destruct (soundness_of_wb_ds_aux t w H_aux) as [ly _].
exact ly.
+ discriminate H_t.
Qed.
(* ********** *)
Lemma completeness_of_wb_ds_aux :
forall t : binary_tree,
(exists w : nat,
wb_ds_aux t = Some w /\ w = weight t /\ balancedp t = true)
\/
(wb_ds_aux t = None /\ balancedp t = false).
Proof.
intro t.
induction t as [n | t1 [[w1 [Ht1 [W1 B1]]] | [Ht1 B1]] t2 [[w2 [Ht2 [W2 B2]]] | [Ht2 B2]]].
- rewrite -> fold_unfold_wb_ds_aux_Leaf.
rewrite -> fold_unfold_weight_Leaf.
rewrite -> fold_unfold_balancedp_Leaf.
left.
exists n.
split; [reflexivity | (split; reflexivity)].
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balancedp_Node.
rewrite -> Ht1.
rewrite -> Ht2.
rewrite <- W1.
rewrite <- W2.
rewrite -> B1.
rewrite -> B2.
case (w1 =? w2) eqn:H_w1_w2.
+ left.
exists (w1 + w2).
unfold andb.
split; [reflexivity | (split; reflexivity)].
+ right.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balancedp_Node.
rewrite -> Ht1.
rewrite -> Ht2.
right.
rewrite -> B1.
rewrite -> B2.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balancedp_Node.
rewrite -> Ht1.
right.
rewrite -> B1.
rewrite -> B2.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balancedp_Node.
rewrite -> Ht1.
right.
rewrite -> B1.
unfold andb.
split; reflexivity.
Qed.
Theorem completeness_of_wb_ds :
forall t : binary_tree,
balancedp t = true ->
wb_ds t = true.
Proof.
intros t H_t.
unfold wb_ds.
destruct (completeness_of_wb_ds_aux t) as [[w [H_aux [H_w H_b]]] | [H_aux H_b]].
- rewrite -> H_aux.
reflexivity.
- rewrite -> H_t in H_b.
discriminate H_b.
Qed.
(* ********** *)
Fixpoint balanced (t : binary_tree) : Prop :=
match t with
| Leaf n =>
True
| Node t1 t2 =>
balanced t1 /\ balanced t2 /\ weight t1 = weight t2
end.
Lemma fold_unfold_balanced_Leaf :
forall n : nat,
balanced (Leaf n) =
True.
Proof.
fold_unfold_tactic balanced.
Qed.
Lemma fold_unfold_balanced_Node :
forall t1 t2 : binary_tree,
balanced (Node t1 t2) =
(balanced t1 /\ balanced t2 /\ weight t1 = weight t2).
Proof.
fold_unfold_tactic balanced.
Qed.
(* ********** *)
(* Exercise 4a *)
Theorem soundness_of_balancedp :
forall t : binary_tree,
balancedp t = true ->
balanced t.
Proof.
intro t.
induction t as [n | t1 IHt1 t2 IHt2].
- intro H_balancedp.
rewrite -> fold_unfold_balanced_Leaf.
reflexivity.
- intro H_balancedp.
rewrite -> fold_unfold_balanced_Node.
rewrite -> fold_unfold_balancedp_Node in H_balancedp.
Check (andb_prop).
Check (andb_prop (balancedp t1 && balancedp t2) (weight t1 =? weight t2)).
assert (H_balancedp' := (andb_prop (balancedp t1 && balancedp t2) (weight t1 =? weight t2))).
assert (H_balancedp' := (H_balancedp' H_balancedp)).
destruct H_balancedp' as [H_balancedp_t1_and_t2 H_weight].
Check (andb_prop (balancedp t1) (balancedp t2)).
assert (H_balancedp_t1_and_t2' := (andb_prop (balancedp t1) (balancedp t2))).
assert (H_balancedp_t1_and_t2' := (H_balancedp_t1_and_t2' H_balancedp_t1_and_t2)).
destruct H_balancedp_t1_and_t2' as [H_balancedp_t1 H_balancedp_t2].
assert (H_balanced_t1 := (IHt1 H_balancedp_t1)).
assert (H_balanced_t2 := (IHt2 H_balancedp_t2)).
split.
+ exact H_balanced_t1.
+ split.
* exact H_balanced_t2.
* Check (beq_nat_true (weight t1) (weight t2)).
assert (H_weight' := (beq_nat_true (weight t1) (weight t2))).
assert (H_weight' := (H_weight' H_weight)).
exact H_weight'.
Qed.
Theorem completeness_of_balancedp :
forall t : binary_tree,
balanced t ->
balancedp t = true.
Proof.
intro t.
induction t as [n | t1 IHt1 t2 IHt2].
- intro H_balanced.
rewrite -> fold_unfold_balancedp_Leaf.
reflexivity.
- intro H_balanced.
rewrite -> fold_unfold_balancedp_Node.
rewrite -> fold_unfold_balanced_Node in H_balanced.
destruct H_balanced as [H_balanced_1 H_balanced_2_and_weight].
destruct H_balanced_2_and_weight as [H_balanced_2 H_weight].
assert (H_balancedp_t1 := (IHt1 H_balanced_1)).
assert (H_balancedp_t2 := (IHt2 H_balanced_2)).
rewrite -> H_balancedp_t1.
rewrite -> H_balancedp_t2.
rewrite -> H_weight.
Search (_ =? _).
rewrite -> Nat.eqb_refl.
reflexivity.
Qed.
(* Exercise 4b *)
Lemma soundness_of_wb_ds_aux_prop :
forall (t : binary_tree)
(w : nat),
wb_ds_aux t = Some w ->
balanced t /\ w = weight t.
Proof.
intro t.
induction t as [w' | t1 IHt1 t2 IHt2].
- intros w H_aux.
split.
-- Check (fold_unfold_balanced_Leaf w').
rewrite -> (fold_unfold_balanced_Leaf w').
reflexivity.
-- rewrite -> (fold_unfold_weight_Leaf).
rewrite -> (fold_unfold_wb_ds_aux_Leaf) in H_aux.
injection H_aux as H_aux.
symmetry.
exact H_aux.
- intros w H_aux.
rewrite -> fold_unfold_wb_ds_aux_Node in H_aux.
case (wb_ds_aux t1) as [w1 | ] eqn:Ht1.
+ case (wb_ds_aux t2) as [w2 | ] eqn:Ht2.
* case (w1 =? w2) eqn:H_w1_w2.
** injection H_aux as H_aux.
rewrite -> fold_unfold_balanced_Node.
rewrite -> fold_unfold_weight_Node.
Check (IHt1 w1 (eq_refl (Some w1))).
destruct (IHt1 w1 (eq_refl (Some w1))) as [Bt1 W1].
destruct (IHt2 w2 (eq_refl (Some w2))) as [Bt2 W2].
split.
*** split.
exact Bt1.
split.
**** exact Bt2.
**** rewrite <- W1.
rewrite <- W2.
Check (beq_nat_true).
assert (H_w1_w2' := (beq_nat_true w1 w2)).
assert (H_w1_w2' := (H_w1_w2' H_w1_w2)).
exact H_w1_w2'.
*** rewrite -> W1 in H_aux.
rewrite -> W2 in H_aux.
symmetry in H_aux.
exact H_aux.
** discriminate H_aux.
* discriminate H_aux.
+ discriminate H_aux.
Qed.
Theorem soundness_of_wb_ds_prop :
forall t : binary_tree,
wb_ds t = true ->
balanced t.
Proof.
intros t H_t.
unfold wb_ds in H_t.
case (wb_ds_aux t) as [w | ] eqn:H_aux.
+ Check (soundness_of_wb_ds_aux_prop t w H_aux).
destruct (soundness_of_wb_ds_aux_prop t w H_aux) as [ly _].
exact ly.
+ discriminate H_t.
Qed.
Lemma completeness_of_wb_ds_aux_prop :
forall t : binary_tree,
(exists w : nat,
wb_ds_aux t = Some w /\ w = weight t /\ balanced t)
\/
(wb_ds_aux t = None /\ ~(balanced t)).
Proof.
intro t.
induction t as [n | t1 [[w1 [Ht1 [W1 B1]]] | [Ht1 B1]] t2 [[w2 [Ht2 [W2 B2]]] | [Ht2 B2]]].
- rewrite -> fold_unfold_wb_ds_aux_Leaf.
rewrite -> fold_unfold_weight_Leaf.
rewrite -> fold_unfold_balanced_Leaf.
left.
exists n.
split; [reflexivity | (split; reflexivity)].
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balanced_Node.
rewrite -> Ht1.
rewrite -> Ht2.
rewrite <- W1.
rewrite <- W2.
case (w1 =? w2) eqn:H_w1_w2.
+ left.
exists (w1 + w2).
split.
* reflexivity.
* split.
-- reflexivity.
-- split.
++ exact B1.
++ split.
** exact B2.
** exact (beq_nat_true w1 w2 H_w1_w2).
+ right.
split.
* reflexivity.
* unfold not.
intros [_ [_ H_w1_equals_w2]].
assert (H_w1_w2' := beq_nat_false w1 w2 H_w1_w2).
contradiction.
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balanced_Node.
rewrite -> Ht1.
rewrite -> Ht2.
right.
split.
* reflexivity.
* unfold not.
intros [_ [H_B2 _]].
contradiction.
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balanced_Node.
rewrite -> Ht1.
right.
split.
* reflexivity.
* unfold not.
intros [H_B1 _].
contradiction.
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balanced_Node.
rewrite -> Ht1.
right.
split.
* reflexivity.
* unfold not.
intros [_ [H_B2 _]].
contradiction.
Qed.
Theorem completeness_of_wb_ds_prop :
forall t : binary_tree,
balanced t ->
wb_ds t = true.
Proof.
intros t H_t.
unfold wb_ds.
destruct (completeness_of_wb_ds_aux_prop t) as [[w [H_aux [H_w H_b]]] | [H_aux H_b]].
- rewrite -> H_aux.
reflexivity.
- contradiction.
Qed.
(* Exercise 5 *)
Fixpoint binary_tree_fold (V : Type) (lea : nat -> V) (nod : V -> V -> V) (t : binary_tree) : V :=
match t with
| Leaf n =>
lea n
| Node t1 t2 =>
nod (binary_tree_fold V lea nod t1)
(binary_tree_fold V lea nod t2)
end.
Lemma fold_unfold_binary_tree_fold_Leaf :
forall (V : Type)
(lea : nat -> V)
(nod : V -> V -> V)
(n : nat),
binary_tree_fold V lea nod (Leaf n) =
lea n.
Proof.
fold_unfold_tactic binary_tree_fold.
Qed.
Lemma fold_unfold_binary_tree_fold_Node :
forall (V : Type)
(lea : nat -> V)
(nod : V -> V -> V)
(t1 t2 : binary_tree),
binary_tree_fold V lea nod (Node t1 t2) =
nod (binary_tree_fold V lea nod t1)
(binary_tree_fold V lea nod t2).
Proof.
fold_unfold_tactic binary_tree_fold.
Qed.
Definition wb_ds_aux_alt (t : binary_tree) : option nat :=
binary_tree_fold
(option nat)
(fun n => Some n)
(fun ih1 ih2 => match ih1 with
| Some w1 =>
match ih2 with
| Some w2 =>
if beq_nat w1 w2
then Some (w1 + w2)
else None
| None =>
None
end
| None =>
None
end)
t.
Definition wb_ds_alt (t : binary_tree) : bool :=
match wb_ds_aux_alt t with
| Some _ =>
true
| None =>
false
end.
Lemma soundness_of_wb_ds_aux_alt :
forall (t : binary_tree)
(w : nat),
wb_ds_aux_alt t = Some w ->
balancedp t = true /\ w = weight t.
Proof.
intro t.
induction t as [n | t1 IHt1 t2 IHt2]; intros w H_aux; unfold wb_ds_aux_alt in H_aux.
- rewrite -> fold_unfold_binary_tree_fold_Leaf in H_aux.
injection H_aux as H_aux.
rewrite -> fold_unfold_balancedp_Leaf.
rewrite -> fold_unfold_weight_Leaf.
symmetry in H_aux.
split; [reflexivity | exact H_aux].
- rewrite -> fold_unfold_binary_tree_fold_Node in H_aux.
case (wb_ds_aux_alt t1) as [w1 | ] eqn:Ht1; unfold wb_ds_aux_alt in Ht1; rewrite -> Ht1 in H_aux.
+ case (wb_ds_aux_alt t2) as [w2 | ] eqn:Ht2; unfold wb_ds_aux_alt in Ht2; rewrite -> Ht2 in H_aux.
* case (w1 =? w2) eqn:H_w1_w2.
** injection H_aux as H_aux.
rewrite -> fold_unfold_balancedp_Node.
rewrite -> fold_unfold_weight_Node.
Check (IHt1 w1 (eq_refl (Some w1))).
destruct (IHt1 w1 (eq_refl (Some w1))) as [Bt1 W1].
destruct (IHt2 w2 (eq_refl (Some w2))) as [Bt2 W2].
rewrite -> Bt1.
rewrite -> Bt2.
rewrite <- W1.
rewrite <- W2.
unfold andb.
symmetry in H_aux.
split; [exact H_w1_w2 | exact H_aux].
** discriminate H_aux.
* discriminate H_aux.
+ discriminate H_aux.
Qed.
Theorem soundness_of_wb_ds_alt :
forall t : binary_tree,
wb_ds_alt t = true ->
balancedp t = true.
Proof.
intros t H_t.
unfold wb_ds_alt in H_t.
case (wb_ds_aux_alt t) as [w | ] eqn:H_aux.
+ Check (soundness_of_wb_ds_aux_alt t w H_aux).
destruct (soundness_of_wb_ds_aux_alt t w H_aux) as [ly _].
exact ly.
+ discriminate H_t.
Qed.
Lemma completeness_of_wb_ds_aux_alt :
forall t : binary_tree,
(exists w : nat,
wb_ds_aux_alt t = Some w /\ w = weight t /\ balancedp t = true)
\/
(wb_ds_aux_alt t = None /\ balancedp t = false).
Proof.
intro t.
induction t as [n | t1 [[w1 [Ht1 [W1 B1]]] | [Ht1 B1]] t2 [[w2 [Ht2 [W2 B2]]] | [Ht2 B2]]]; unfold wb_ds_aux_alt.
- rewrite -> fold_unfold_binary_tree_fold_Leaf.
rewrite -> fold_unfold_weight_Leaf.
rewrite -> fold_unfold_balancedp_Leaf.
left.
exists n.
split; [reflexivity | (split; reflexivity)].
- rewrite -> fold_unfold_binary_tree_fold_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balancedp_Node.
unfold wb_ds_aux_alt in Ht1, Ht2.
rewrite -> Ht1.
rewrite -> Ht2.
rewrite <- W1.
rewrite <- W2.
rewrite -> B1.
rewrite -> B2.
case (w1 =? w2) eqn:H_w1_w2.
+ left.
exists (w1 + w2).
unfold andb.
split; [reflexivity | (split; reflexivity)].
+ right.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_binary_tree_fold_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balancedp_Node.
unfold wb_ds_aux_alt in Ht1, Ht2.
rewrite -> Ht1.
rewrite -> Ht2.
right.
rewrite -> B1.
rewrite -> B2.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_binary_tree_fold_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balancedp_Node.
unfold wb_ds_aux_alt in Ht1.
rewrite -> Ht1.
right.
rewrite -> B1.
rewrite -> B2.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_binary_tree_fold_Node.
rewrite -> fold_unfold_weight_Node.
rewrite -> fold_unfold_balancedp_Node.
unfold wb_ds_aux_alt in Ht1.
rewrite -> Ht1.
right.
rewrite -> B1.
unfold andb.
split; reflexivity.
Qed.
Theorem completeness_of_wb_ds_alt :
forall t : binary_tree,
balancedp t = true ->
wb_ds_alt t = true.
Proof.
intros t H_t.
unfold wb_ds_alt.
destruct (completeness_of_wb_ds_aux_alt t) as [[w [H_aux [H_w H_b]]] | [H_aux H_b]].
- rewrite -> H_aux.
reflexivity.
- rewrite -> H_t in H_b.
discriminate H_b.
Qed.
Theorem equivalence_of_wb_ds_aux_and_wb_ds_aux_alt :
forall t : binary_tree,
wb_ds_aux t = wb_ds_aux_alt t.
Proof.
intro t.
induction t as [n | t1 IHt1 t2 IHt2]; unfold wb_ds_aux_alt.
- rewrite -> fold_unfold_wb_ds_aux_Leaf.
rewrite -> fold_unfold_binary_tree_fold_Leaf.
reflexivity.
- rewrite -> fold_unfold_wb_ds_aux_Node.
rewrite -> IHt1.
rewrite -> IHt2.
rewrite -> fold_unfold_binary_tree_fold_Node.
unfold wb_ds_aux_alt.
reflexivity.
Qed.
Theorem equivalence_of_wb_ds_and_wb_ds_alt :
forall t : binary_tree,
wb_ds t = wb_ds_alt t.
Proof.
intro t.
unfold wb_ds, wb_ds_alt.
rewrite -> (equivalence_of_wb_ds_aux_and_wb_ds_aux_alt t).
reflexivity.
Qed.
(* ********** *)
(* Exercise 6 *)
Inductive binary_tree' : Type :=
| Leaf' : nat -> binary_tree'
| Node' : binary_tree' -> nat -> binary_tree' -> binary_tree'.
(* ***** *)
Fixpoint weight' (t : binary_tree') : nat :=
match t with
| Leaf' n =>
n
| Node' t1 n t2 =>
weight' t1 + n + weight' t2
end.
Lemma fold_unfold_weight'_Leaf' :
forall n : nat,
weight' (Leaf' n) = n.
Proof.
fold_unfold_tactic weight'.
Qed.
Lemma fold_unfold_weight'_Node' :
forall (t1 t2 : binary_tree')
(n : nat),
weight' (Node' t1 n t2) = weight' t1 + n + weight' t2.
Proof.
fold_unfold_tactic weight'.
Qed.
(* ***** *)
Fixpoint balancedp' (t : binary_tree') : bool :=
match t with
| Leaf' n =>
true
| Node' t1 n t2 =>
balancedp' t1 && balancedp' t2 && (weight' t1 =? weight' t2)
end.
Lemma fold_unfold_balancedp'_Leaf' :
forall n : nat,
balancedp' (Leaf' n) =
true.
Proof.
fold_unfold_tactic balancedp'.
Qed.
Lemma fold_unfold_balancedp'_Node' :
forall (t1 t2 : binary_tree')
(n : nat),
balancedp' (Node' t1 n t2) =
(balancedp' t1 && balancedp' t2 && (weight' t1 =? weight' t2)).
Proof.
fold_unfold_tactic balancedp'.
Qed.
Fixpoint wb_ds_aux' (t : binary_tree') : option nat :=
match t with
| Leaf' n =>
Some n
| Node' t1 n t2 =>
match wb_ds_aux' t1 with
| Some w1 =>
match wb_ds_aux' t2 with
| Some w2 =>
if beq_nat w1 w2
then Some (w1 + n + w2)
else None
| None =>
None
end
| None =>
None
end
end.
Lemma fold_unfold_wb_ds_aux'_Leaf' :
forall n : nat,
wb_ds_aux' (Leaf' n) =
Some n.
Proof.
fold_unfold_tactic wb_ds_aux'.
Qed.
Lemma fold_unfold_wb_ds_aux'_Node' :
forall (t1 t2 : binary_tree')
(n : nat),
wb_ds_aux' (Node' t1 n t2) =
match wb_ds_aux' t1 with
| Some w1 =>
match wb_ds_aux' t2 with
| Some w2 =>
if beq_nat w1 w2
then Some (w1 + n + w2)
else None
| None =>
None
end
| None =>
None
end.
Proof.
fold_unfold_tactic wb_ds_aux'.
Qed.
Definition wb_ds' (t : binary_tree') : bool :=
match wb_ds_aux' t with
| Some _ =>
true
| None =>
false
end.
(* ********** *)
Lemma soundness_of_wb_ds_aux' :
forall (t : binary_tree')
(w : nat),
wb_ds_aux' t = Some w ->
balancedp' t = true /\ w = weight' t.
Proof.
intro t.
induction t as [n | t1 IHt1 n t2 IHt2]; intros w H_aux.
- rewrite -> fold_unfold_wb_ds_aux'_Leaf' in H_aux.
injection H_aux as H_aux.
rewrite -> fold_unfold_balancedp'_Leaf'.
rewrite -> fold_unfold_weight'_Leaf'.
symmetry in H_aux.
split; [reflexivity | exact H_aux].
- rewrite -> fold_unfold_wb_ds_aux'_Node' in H_aux.
case (wb_ds_aux' t1) as [w1 | ] eqn:Ht1.
+ case (wb_ds_aux' t2) as [w2 | ] eqn:Ht2.
* case (w1 =? w2) eqn:H_w1_w2.
** injection H_aux as H_aux.
rewrite -> fold_unfold_balancedp'_Node'.
rewrite -> fold_unfold_weight'_Node'.
Check (IHt1 w1 (eq_refl (Some w1))).
destruct (IHt1 w1 (eq_refl (Some w1))) as [Bt1 W1].
destruct (IHt2 w2 (eq_refl (Some w2))) as [Bt2 W2].
rewrite -> Bt1.
rewrite -> Bt2.
rewrite <- W1.
rewrite <- W2.
unfold andb.
symmetry in H_aux.
split; [exact H_w1_w2 | exact H_aux].
** discriminate H_aux.
* discriminate H_aux.
+ discriminate H_aux.
Qed.
Theorem soundness_of_wb_ds' :
forall t : binary_tree',
wb_ds' t = true ->
balancedp' t = true.
Proof.
intros t H_t.
unfold wb_ds' in H_t.
case (wb_ds_aux' t) as [w | ] eqn:H_aux.
+ Check (soundness_of_wb_ds_aux' t w H_aux).
destruct (soundness_of_wb_ds_aux' t w H_aux) as [ly _].
exact ly.
+ discriminate H_t.
Qed.
(* ********** *)
Lemma completeness_of_wb_ds_aux' :
forall t : binary_tree',
(exists w : nat,
wb_ds_aux' t = Some w /\ w = weight' t /\ balancedp' t = true)
\/
(wb_ds_aux' t = None /\ balancedp' t = false).
Proof.
intro t.
induction t as [n | t1 [[w1 [Ht1 [W1 B1]]] | [Ht1 B1]] n t2 [[w2 [Ht2 [W2 B2]]] | [Ht2 B2]]].
- rewrite -> fold_unfold_wb_ds_aux'_Leaf'.
rewrite -> fold_unfold_weight'_Leaf'.
rewrite -> fold_unfold_balancedp'_Leaf'.
left.
exists n.
split; [reflexivity | (split; reflexivity)].
- rewrite -> fold_unfold_wb_ds_aux'_Node'.
rewrite -> fold_unfold_weight'_Node'.
rewrite -> fold_unfold_balancedp'_Node'.
rewrite -> Ht1.
rewrite -> Ht2.
rewrite <- W1.
rewrite <- W2.
rewrite -> B1.
rewrite -> B2.
case (w1 =? w2) eqn:H_w1_w2.
+ left.
exists (w1 + n + w2).
unfold andb.
split; [reflexivity | (split; reflexivity)].
+ right.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_wb_ds_aux'_Node'.
rewrite -> fold_unfold_weight'_Node'.
rewrite -> fold_unfold_balancedp'_Node'.
rewrite -> Ht1.
rewrite -> Ht2.
right.
rewrite -> B1.
rewrite -> B2.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_wb_ds_aux'_Node'.
rewrite -> fold_unfold_weight'_Node'.
rewrite -> fold_unfold_balancedp'_Node'.
rewrite -> Ht1.
right.
rewrite -> B1.
rewrite -> B2.
unfold andb.
split; reflexivity.
- rewrite -> fold_unfold_wb_ds_aux'_Node'.
rewrite -> fold_unfold_weight'_Node'.
rewrite -> fold_unfold_balancedp'_Node'.
rewrite -> Ht1.
right.
rewrite -> B1.
unfold andb.
split; reflexivity.
Qed.
Theorem completeness_of_wb_ds' :
forall t : binary_tree',
balancedp' t = true ->
wb_ds' t = true.
Proof.
intros t H_t.
unfold wb_ds'.
destruct (completeness_of_wb_ds_aux' t) as [[w [H_aux [H_w H_b]]] | [H_aux H_b]].
- rewrite -> H_aux.
reflexivity.
- rewrite -> H_t in H_b.
discriminate H_b.
Qed.
(* ********** *)
(* end of week-10_exercises.v *)