(* int-div-examp.ec *) require import AllCore. require import IntDiv StdOrder. (* lemmas for integer mod and div *) import IntOrder. (* lemmas about <, <= on int *) (* n %/ x is the integer division of n by x, discarding any remainder n %% x is the remainder of integer division of n by x x %| n tests whether x divides n, i.e., n %% x = 0 %/ and %% are actually abbreviations, defined in terms of edivz; consequently, when using `search` to look for lemmas involving these abbreviations, one must search for `edivz` instead. *) lemma div_le (x n : int) : 1 <= x => 1 <= n => x %| n => x <= n. proof. move => ge1_x ge1_n x_div_n. have n_eq : n = (n %/ x) * x. by rewrite eq_sym -dvdz_eq. rewrite n_eq -{1}mul1z ler_pmul // (lez_trans 1) //. case (1 <= n %/ x) => //. rewrite -ltrNge => lt1_n_div_x. have le0_n : n <= 0. by rewrite n_eq mulr_le0_ge0 1:-ltzS // (lez_trans 1). have // : 1 <= 0. by apply (lez_trans n). qed.