(*
Author: Jeremy Dawson, NICTA
*)
header {* Useful Numerical Lemmas *}
theory Misc_Numeric
imports Main Parity
begin
lemma mod_2_neq_1_eq_eq_0:
fixes k :: int
shows "k mod 2 \<noteq> 1 \<longleftrightarrow> k mod 2 = 0"
by (fact not_mod_2_eq_1_eq_0)
lemma z1pmod2:
fixes b :: int
shows "(2 * b + 1) mod 2 = (1::int)"
by arith
lemma diff_le_eq':
"a - b \<le> c \<longleftrightarrow> a \<le> b + (c::int)"
by arith
lemma emep1:
fixes n d :: int
shows "even n \<Longrightarrow> even d \<Longrightarrow> 0 \<le> d \<Longrightarrow> (n + 1) mod d = (n mod d) + 1"
by (auto simp add: pos_zmod_mult_2 add.commute even_equiv_def)
lemma int_mod_ge:
"a < n \<Longrightarrow> 0 < (n :: int) \<Longrightarrow> a \<le> a mod n"
by (metis dual_order.trans le_cases mod_pos_pos_trivial pos_mod_conj)
lemma int_mod_ge':
"b < 0 \<Longrightarrow> 0 < (n :: int) \<Longrightarrow> b + n \<le> b mod n"
by (metis add_less_same_cancel2 int_mod_ge mod_add_self2)
lemma int_mod_le':
"(0::int) \<le> b - n \<Longrightarrow> b mod n \<le> b - n"
by (metis minus_mod_self2 zmod_le_nonneg_dividend)
lemma zless2:
"0 < (2 :: int)"
by (fact zero_less_numeral)
lemma zless2p:
"0 < (2 ^ n :: int)"
by arith
lemma zle2p:
"0 \<le> (2 ^ n :: int)"
by arith
lemma m1mod2k:
"- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
using zless2p by (rule zmod_minus1)
lemma p1mod22k':
fixes b :: int
shows "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
using zle2p by (rule pos_zmod_mult_2)
lemma p1mod22k:
fixes b :: int
shows "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
by (simp add: p1mod22k' add.commute)
lemma int_mod_lem:
"(0 :: int) < n ==> (0 <= b & b < n) = (b mod n = b)"
apply safe
apply (erule (1) mod_pos_pos_trivial)
apply (erule_tac [!] subst)
apply auto
done
end