1 files changed, 71 insertions, 0 deletions
diff --git a/lib/coq/Sail2_real.v b/lib/coq/Sail2_real.v
index 494e36d4..4800f18b 100644
--- a/lib/coq/Sail2_real.v
+++ b/lib/coq/Sail2_real.v
@@ -34,3 +34,74 @@ Definition pow_real := powerRZ.
 
 Definition print_real (_ : string) (_ : R) : unit := tt.
 Definition prerr_real (_ : string) (_ : R) : unit := tt.
+
+
+
+
+Lemma IZRdiv m n :
+  0 < m -> 0 < n ->
+  (IZR (m / n) <= IZR m / IZR n)%R.
+intros.
+apply Rmult_le_reg_l with (r := IZR n).
+auto using IZR_lt.
+unfold Rdiv.
+rewrite <- Rmult_assoc.
+rewrite Rinv_r_simpl_m.
+rewrite <- mult_IZR.
+apply IZR_le.
+apply Z.mul_div_le.
+assumption.
+discrR.
+omega.
+Qed.
+
+(* One annoying use of reals in the ARM spec I've been looking at. *)
+Lemma round_up_div m n :
+  0 < m -> 0 < n ->
+  round_up (div_real (to_real m) (to_real n)) = (m + n - 1) / n.
+intros.
+unfold round_up, round_down, div_real, to_real.
+apply Z.eq_opp_l.
+apply Z.sub_move_r.
+symmetry.
+apply up_tech.
+
+rewrite Ropp_Ropp_IZR.
+apply Ropp_le_contravar.
+apply Rmult_le_reg_l with (r := IZR n).
+auto using IZR_lt.
+unfold Rdiv.
+rewrite <- Rmult_assoc.
+rewrite Rinv_r_simpl_m.
+rewrite <- mult_IZR.
+apply IZR_le.
+assert (diveq : n*((m+n-1)/n) = (m+n-1) - (m+n-1) mod n).
+apply Zplus_minus_eq.
+rewrite (Z.add_comm ((m+n-1) mod n)).
+apply Z.div_mod.
+omega.
+rewrite diveq.
+assert (modmax : (m+n-1) mod n < n).
+specialize (Z.mod_pos_bound (m+n-1) n). intuition.
+omega.
+
+discrR; omega.
+
+rewrite <- Z.opp_sub_distr.
+rewrite Ropp_Ropp_IZR.
+apply Ropp_lt_contravar.
+apply Rmult_lt_reg_l with (r := IZR n).
+auto using IZR_lt.
+unfold Rdiv.
+rewrite <- Rmult_assoc.
+rewrite Rinv_r_simpl_m.
+2: { discrR. omega. }
+rewrite <- mult_IZR.
+apply IZR_lt.
+rewrite Z.mul_sub_distr_l.
+apply Z.le_lt_trans with (m := m+n-1-n*1).
+apply Z.sub_le_mono_r.
+apply Z.mul_div_le.
+assumption.
+omega.
+Qed.