feat: 3234 week 2

cs3234/labs/week-01_polymorphism.v

@@ -0,0 +1,180 @@
+(* week-01_polymorphism.v *)
+(* LPP 2024 - CS3234 2023-2024, Sem2 *)
+(* Olivier Danvy <danvy@yale-nus.edu.sg> *)
+(* Version of 18 Jan 2024 *)
+
+(* ********** *)
+
+Require Import Arith Bool.
+
+(* ********** *)
+
+Definition make_pair_v0 : forall (A : Type) (B : Type), A -> B -> A * B :=
+  fun A B a b => (a, b).
+
+Definition make_pair_v1 (A : Type) : forall B : Type, A -> B -> A * B :=
+  fun B a b => (a, b).
+
+Definition make_pair_v2 (A : Type) (B : Type) : A -> B -> A * B :=
+  fun a b => (a, b).
+
+Definition make_pair_v3 (A : Type) (B : Type) (a : A) : B -> A * B :=
+  fun b => (a, b).
+
+Definition make_pair_v4 (A : Type) (B : Type) (a : A) (b : B) : A * B :=
+  (a, b).
+
+(* ***** *)
+Compute make_pair_v0 nat bool 1 true.
+
+Compute (make_pair_v4 nat bool 1 true).
+(* = (1, true) : nat * bool *)
+
+Definition dupl (V : Type) (v : V) : V * V :=
+  make_pair_v4 V V v v.
+
+(* ********** *)
+
+Inductive polymorphic_binary_tree (V : Type) : Type :=
+| PLeaf : V -> polymorphic_binary_tree V
+| PNode : polymorphic_binary_tree V -> polymorphic_binary_tree V -> polymorphic_binary_tree V.
+
+Definition test_number_of_leaves (candidate : forall V : Type, polymorphic_binary_tree V -> nat) :=
+  (candidate bool (PLeaf bool true) =? 1) &&
+  (candidate nat (PNode nat (PLeaf nat 0) (PLeaf nat 1)) =? 2).
+
+Fixpoint number_of_leaves (V : Type) (t : polymorphic_binary_tree V) : nat :=
+  match t with
+  | PLeaf _ v =>
+    1
+  | PNode _ t1 t2 =>
+    number_of_leaves V t1 + number_of_leaves V t2
+  end.
+
+Compute (test_number_of_leaves number_of_leaves).
+
+(* ********** *)
+
+Fixpoint eqb_polymorphic_binary_tree (V : Type) (eqb_V : V -> V -> bool) (t1 t2 : polymorphic_binary_tree V) : bool :=
+  match t1 with
+  | PLeaf _ v1 =>
+    match t2 with
+    | PLeaf _ v2 =>
+      eqb_V v1 v2
+    | PNode _ t11 t12 =>
+      false
+    end
+  | PNode _ t11 t12 =>
+    match t2 with
+    | PLeaf _ v2 =>
+      false
+    | PNode _ t21 t22 =>
+      eqb_polymorphic_binary_tree V eqb_V t11 t21
+      &&
+      eqb_polymorphic_binary_tree V eqb_V t12 t22
+    end
+  end.
+
+(* ********** *)
+
+Definition eqb_nat := Nat.eqb.
+
+Definition eqb_binary_tree_of_nats (t1 t2 : polymorphic_binary_tree nat) : bool :=
+  eqb_polymorphic_binary_tree nat eqb_nat t1 t2.
+
+(* ********** *)
+
+(* Exercise 05 *)
+Definition e5a : polymorphic_binary_tree (nat * bool) :=
+  PNode (nat * bool)
+    (PLeaf (nat * bool) (0, true))
+    (PLeaf (nat * bool) (1, false))
+.
+
+Check e5a.
+Compute e5a.
+
+Definition nTree := (polymorphic_binary_tree nat).
+
+Definition e5b : polymorphic_binary_tree (polymorphic_binary_tree nat) :=
+  PNode (nTree)
+    (PLeaf (nTree) (PLeaf nat 1))
+    (PLeaf (nTree) (PLeaf nat 2))
+.
+
+Check e5b.
+Compute e5b.
+(* Exercise 07 *)
+
+Definition eqb_option (V : Type) (eqb_V : V -> V -> bool) (ov1 ov2 : option V) : bool :=
+  match ov1 with
+  | Some v1 =>
+    match ov2 with
+    | Some v2 =>
+      eqb_V v1 v2
+    | None =>
+      false
+    end
+  | None =>
+    match ov2 with
+    | Some v =>
+      false
+    | None =>
+      true
+    end
+  end.
+
+Check (eqb_polymorphic_binary_tree nat).
+Check (eqb_polymorphic_binary_tree (option nat)).
+Check (eqb_option (polymorphic_binary_tree nat)).
+
+(*
+Definition eqb_binary_tree_of_optional_nats (t1 t2 : polymorphic_binary_tree (option nat)) : bool :=
+*)
+
+(*
+Definition eqb_optional_binary_tree_of_nats (t1 t2 : option (polymorphic_binary_tree nat)) : bool :=
+*)
+
+(*
+Definition eqb_optional_binary_tree_of_optional_nats (t1 t2 : option (polymorphic_binary_tree (option nat))) : bool :=
+*)
+
+(*
+Definition eqb_binary_tree_of_optional_binary_tree_of_nats (t1 t2 : polymorphic_binary_tree (option (polymorphic_binary_tree nat))) : bool :=
+*)
+
+(*
+Definition eqb_binary_tree_of_optional_binary_tree_of_optional_nats (t1 t2 : polymorphic_binary_tree (option (polymorphic_binary_tree (option nat)))) : bool :=
+*)
+
+(* ********** *)
+
+Inductive option' (V : Type) : Type :=
+| Some' : V -> option' V
+| None' : option' V.
+
+Print option'.
+
+Set Implicit Arguments.
+
+Inductive option'' (V : Type) : Type :=
+| Some'' : V -> option'' V
+| None'' : option'' V.
+
+Print option''.
+
+Check (Some 10, Some true).
+Check (Some' nat 10, Some' bool true).
+Check (Some'' 10, Some'' true).
+
+Check None.
+Check None'.
+Check None''.
+Check (@None nat).
+Check (@None' nat).
+Check (@None'' nat).
+
+(* ********** *)
+
+(* end of week-01_polymorphism.v *)