Some examples from Coq, and equivalents in Cur

* Examples of Coq code, and what one wishes one could do.
* Examples of similar model in Cur, and what is currently required.

examples/try.rkt

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+#lang cur
+
+(require
+ cur/stdlib/sugar
+ cur/stdlib/nat
+ cur/stdlib/prop)
+
+
+;; TODO: Axioms
+;; TODO: implicits
+;; No way to define axioms in Cur, yet, although it is a simple change.
+;; No implicits in Cur, yet
+(define (plus_0_n (f_equal : (-> (A : Type) (a1 : A) (a2 : A) (B : Type) (f : (-> A B))
+    (== A a1 a2)
+    (== B (f a1) (f a2)))) (n : Nat))
+  (match n
+    ;; TODO: This should work
+    #:return (== Nat n (plus z n))
+    [z (refl Nat z)]
+    [(s (n : Nat))
+     (f_equal Nat n (plus z n) Nat s (recur n))]))
+(:: plus_n_0
+    (forall
+     (f_equal : (-> (A : Type) (a1 : A) (a2 : A) (B : Type) (f : (-> A B))
+                    (== A a1 a2) (== B (f a1) (f a2))))
+     (n : Nat)
+     (== Nat n (plus z n))))
+
+#;(define (plus_n_Sm (n : Nat) (m : Nat))
+  (elim Nat Type
+        (lambda (n : Nat) (== Nat (plus n m) (plus m n)))
+        (plus_n_0 m)
+        (lambda (y : Nat) (ih : ))
+        (elim == Type)))
+#;(:: plus_n_Sm (forall (n : Nat) (m : Nat) (== (s (plus n m)) (plus n (s m)))))

examples/try.v

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+Definition plus_n_O : forall n : nat, n = n + 0 :=
+fun n : nat =>
+nat_ind (fun n0 : nat => n0 = n0 + 0) eq_refl
+  (fun (n0 : nat) (IHn : n0 = n0 + 0) => f_equal S IHn) n.
+
+Definition plus_n_Sm : forall n m : nat, S (n + m) = n + S m :=
+fun n m : nat =>
+nat_ind (fun n0 : nat => S (n0 + m) = n0 + S m) eq_refl
+  (fun (n0 : nat) (IHn : S (n0 + m) = n0 + S m) => f_equal S IHn) n.
+    
+Definition plus_comm : forall n m : nat, n + m = m + n := 
+fun n m : nat =>
+nat_ind (fun n0 : nat => n0 + m = m + n0) (plus_n_O m)
+  (fun (y : nat) (H : y + m = m + y) =>
+   eq_ind (S (m + y)) (fun n0 : nat => S (y + m) = n0) 
+     (f_equal S H) (m + S y) (plus_n_Sm m y)) n.
+
+Fixpoint ft (l: nat): Type :=
+  match l with
+  | O => unit
+  | S n => (bool * ft n) % type
+  end.
+
+Fixpoint concat {n1 n2} : ft n1 -> ft n2 -> ft (n1 + n2) :=
+  match n1 as n1_PAT return ft n1_PAT -> ft n2 -> ft (n1_PAT + n2) with
+  | O => fun _ l2 => l2
+  | S n => fun l1 =>
+    match l1 with
+    | (b, l1') => fun l2 => (b, concat l1' l2)
+    end
+  end.
+
+(* Question 1: Ideally, I could have the following syntactic sugar, can you do that?
+
+Fixpoint concat {n1 n2} (l1: ft n1) (l2: ft n2) : ft (n1 + n2) :=
+  match n1 with
+  | O => l2
+  | S n => 
+    match l1 with
+    | (b, l1') => (b, concat l1' l2)
+    end
+  end.
+
+*)
+
+Definition concat' {n1 n2} (l1: ft n1) (l2: ft n2): ft (n1 + n2) :=
+  eq_rect_r ft (concat l2 l1) (plus_comm _ _).
+
+Goal (@concat' 1 1 (true, tt) (false, tt)) = (false, (true, tt)).
+Proof.
+  unfold concat'.
+  simpl (@concat 1 1 (false, tt) (true, tt)).
+  cbv delta [plus_comm plus_n_Sm plus_n_O nat_ind eq_ind].
+  simpl.
+  (* Question 2: In this step, Coq does a large number of computation to compute a proof term to eq_refl. This is super inefficient for any function application of eq_rect_r. Is there a possible way to customize the method of reducing a term in your system? For example, when you see a eq_rect_r, directly computes two types (which should be equivalent) and check if they are convertible. If yes, tag new type on the term directly. *)
+  reflexivity.
+Qed.