Коммутативность умножения в арифметике Пеано

PeanoMult.v

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+ 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).
+