-variant le_inv_S_S: ∀m,n. S m ≤ S n → m ≤ n
-≝ le_S_S_to_le.
-
-theorem plus_inv_S_S_S: ∀x,y,z. S x = S y + S z → S y ≤ x ∧ S z ≤ x.
- simplify; intros; destruct;autobatch.
-qed.
-
-theorem times_inv_S_m_SS: ∀k,n,m. S n = m * (S (S k)) → m ≤ n.
- intros 3; elim m names 0; clear m; simplify; intros; destruct;
- clear H; autobatch by le_S_S, transitive_le, le_plus_n, le_plus_n_r.
-qed.
-
-theorem plus_3_S3n: ∀n. S (S n * 3) = 3 + S (n * 3).
- intros; autobatch depth = 1.
-qed.
-
-theorem times_exp_x_y_Sz: ∀x,y,z. x * y \sup (S z) = (x * y \sup z) * y.
- intros; autobatch depth = 1.
-qed.
-
-definition acc_nat: (nat → Prop) → nat →Prop ≝
- λP:nat→Prop. λn. ∀m. m ≤ n → P m.
-
-theorem wf_le: ∀P. P 0 → (∀n. acc_nat P n → P (S n)) → ∀n. acc_nat P n.
- unfold acc_nat; intros 4; elim n names 0; clear n;
- [ intros; autobatch by (eq_ind ? ? P), H, H2, le_n_O_to_eq.
- (* lapply linear le_n_O_to_eq to H2; destruct; autobatch *)
- | intros 3; elim m; clear m; intros; clear H3;
- [ clear H H1; autobatch depth = 2
- | clear H; lapply linear le_inv_S_S to H4;
- apply H1; clear H1; intros;
- apply H2; clear H2; autobatch depth = 2
- ]
- ].
-qed.
-
-theorem wf_nat_ind:
- ∀P:nat→Prop. P O → (∀n. (∀m. m ≤ n → P m) → P (S n)) → ∀n. P n.
- intros; lapply linear depth=2 wf_le to H, H1 as H0;
- autobatch.
-qed.
+lemma lt_plus_nmn_false: ∀m,n. n + m < n → False.
+#m #n elim n -n
+[ #H elim (lt_zero_false … H)
+| /3 width=1/
+]
+qed-.
+
+lemma not_b_divides_nbr: ∀b,r. 0 < r → r < b →
+ ∀n,m. n * b + r = m * b → False.
+#b #r #Hr #Hrb #n elim n -n
+[ * normalize
+ [ -Hrb #H destruct elim (lt_refl_false … Hr)
+ | -Hr #m #H destruct
+ elim (lt_plus_nmn_false … Hrb)
+ ]
+| #n #IHn * normalize
+ [ -IHn -Hrb #H destruct
+ elim (plus_inv_O3 … H) -H #_ #H destruct
+ elim (lt_refl_false … Hr)
+ | -Hr -Hrb /3 width=3/
+ ]
+]
+qed-.