--- a/src/HOL/Word/Misc_Numeric.thy Tue Apr 16 19:50:03 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,71 +0,0 @@
-(* Title: HOL/Word/Misc_Numeric.thy
- Author: Jeremy Dawson, NICTA
-*)
-
-section \<open>Useful Numerical Lemmas\<close>
-
-theory Misc_Numeric
- imports Main
-begin
-
-lemma one_mod_exp_eq_one [simp]:
- "1 mod (2 * 2 ^ n) = (1::int)"
- using power_gt1 [of 2 n] by (auto intro: mod_pos_pos_trivial)
-
-lemma mod_2_neq_1_eq_eq_0: "k mod 2 \<noteq> 1 \<longleftrightarrow> k mod 2 = 0"
- for k :: int
- by (fact not_mod_2_eq_1_eq_0)
-
-lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)"
- for b :: int
- by arith
-
-lemma diff_le_eq': "a - b \<le> c \<longleftrightarrow> a \<le> b + c"
- for a b c :: int
- by arith
-
-lemma emep1: "even n \<Longrightarrow> even d \<Longrightarrow> 0 \<le> d \<Longrightarrow> (n + 1) mod d = (n mod d) + 1"
- for n d :: int
- by (auto simp add: pos_zmod_mult_2 add.commute dvd_def)
-
-lemma int_mod_ge: "a < n \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a mod n"
- for a n :: int
- by (metis dual_order.trans le_cases mod_pos_pos_trivial pos_mod_conj)
-
-lemma int_mod_ge': "b < 0 \<Longrightarrow> 0 < n \<Longrightarrow> b + n \<le> b mod n"
- for b n :: int
- by (metis add_less_same_cancel2 int_mod_ge mod_add_self2)
-
-lemma int_mod_le': "0 \<le> b - n \<Longrightarrow> b mod n \<le> b - n"
- for b n :: int
- 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': "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
- for b :: int
- using zle2p by (rule pos_zmod_mult_2)
-
-lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
- for b :: int
- by (simp add: p1mod22k' add.commute)
-
-lemma int_mod_lem: "0 < n \<Longrightarrow> 0 \<le> b \<and> b < n \<longleftrightarrow> b mod n = b"
- for b n :: int
- apply safe
- apply (erule (1) mod_pos_pos_trivial)
- apply (erule_tac [!] subst)
- apply auto
- done
-
-end