Ассоциативность композиции функций

KompFunc.v

@@ -0,0 +1,8 @@
+ Definition cf := fun t0 t1 t2 : Type
+   => fun (f : t1 -> t2) (g : t0 -> t1) => fun x => f (g x).
+ Implicit Arguments cf [t0 t1 t2].
+ Notation "f @ g" := (cf f g) (at level 65, left associativity).
+ 
+ Definition cf_assoc := fun (t0 t1 t2 t3 : Type)
+   (f : t2 -> t3) (g : t1 -> t2) (h : t0 -> t1)
+   => (refl_equal _) : (f @ g) @ h = f @ (g @ h).

PeanoMult

@@ -1,34 +0,0 @@
- Inductive nat : Set :=
-   | O : nat
-   | S : nat -> nat.
-
- Require Import Coq.Arith.Plus.
-
- Definition mult_comm : forall n m, n * m = m * n.
- Proof.
- intros. elim n.
- auto with arith.
- intros. simpl in |- *. elim mult_n_Sm. elim H. apply plus_comm.
- Qed.
-
- Definition mult_comm := fun n:nat
- => fix rec (m : nat)
- : n * m = m * n
- := match m as m return n * m = m * n with
-   | O => sym_eq (mult_n_O _)
-   | S pm => match rec pm in _ = dep return _ = n + dep
-     with refl_equal =>
-       match mult_n_Sm _ _ in _ = dep return dep = _
-       with refl_equal => plus_comm _ _ end
-     end
-   end.
-
- Definition mult_comm := fun n:nat
- => nat_ind (fun m => n * m = m * n)
-   (sym_eq (mult_n_O _))
-   (fun _ rec =>
-     eq_ind _ (fun dep => _ = n + dep)
-     (eq_ind _ (fun dep => dep = _)
-       (plus_comm _ _) _ (mult_n_Sm _ _))
-     _ rec).
-