theory IntArith imports Bin uses "int_arith.ML" begin end
theorem zless_iff_zdiff_zless_0:
x $< y <-> x $- y $< #0
theorem eq_iff_zdiff_eq_0:
[| x ∈ int; y ∈ int |] ==> x = y <-> x $- y = #0
theorem zle_iff_zdiff_zle_0:
x $≤ y <-> x $- y $≤ #0
theorem left_zadd_zmult_distrib:
i $× u $+ (j $× u $+ k) = (i $+ j) $× u $+ k
theorem eq_add_iff1:
i $× u $+ m = j $× u $+ n <-> (i $- j) $× u $+ m = intify(n)
theorem eq_add_iff2:
i $× u $+ m = j $× u $+ n <-> intify(m) = (j $- i) $× u $+ n
theorem less_add_iff1:
i $× u $+ m $< j $× u $+ n <-> (i $- j) $× u $+ m $< n
theorem less_add_iff2:
i $× u $+ m $< j $× u $+ n <-> m $< (j $- i) $× u $+ n
theorem le_add_iff1:
i $× u $+ m $≤ j $× u $+ n <-> (i $- j) $× u $+ m $≤ n
theorem le_add_iff2:
i $× u $+ m $≤ j $× u $+ n <-> m $≤ (j $- i) $× u $+ n