--- a/src/HOL/Word/Misc_Auxiliary.thy Sat Jul 04 20:45:21 2020 +0000
+++ b/src/HOL/Word/Misc_Auxiliary.thy Sat Jul 04 20:45:24 2020 +0000
@@ -8,42 +8,6 @@
imports Main
begin
-subsection \<open>Arithmetic lemmas\<close>
-
-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
-
-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 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 m1mod2k: "- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
- by (rule zmod_minus1) simp
-
-lemma sub_inc_One: "Num.sub (Num.inc n) num.One = numeral n"
- by (metis add_diff_cancel add_neg_numeral_special(3) add_uminus_conv_diff numeral_inc)
-
-lemma inc_BitM: "Num.inc (Num.BitM n) = num.Bit0 n"
- by (simp add: BitM_plus_one[symmetric] add_One)
-
-
subsection \<open>Lemmas on list operations\<close>
lemma butlast_power: "(butlast ^^ n) bl = take (length bl - n) bl"
@@ -58,122 +22,4 @@
lemma hd_butlast: "length xs > 1 \<Longrightarrow> hd (butlast xs) = hd xs"
by (cases xs) auto
-
-subsection \<open>Implicit augmentation of list prefixes\<close>
-
-primrec takefill :: "'a \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
- Z: "takefill fill 0 xs = []"
- | Suc: "takefill fill (Suc n) xs =
- (case xs of
- [] \<Rightarrow> fill # takefill fill n xs
- | y # ys \<Rightarrow> y # takefill fill n ys)"
-
-lemma nth_takefill: "m < n \<Longrightarrow> takefill fill n l ! m = (if m < length l then l ! m else fill)"
- apply (induct n arbitrary: m l)
- apply clarsimp
- apply clarsimp
- apply (case_tac m)
- apply (simp split: list.split)
- apply (simp split: list.split)
- done
-
-lemma takefill_alt: "takefill fill n l = take n l @ replicate (n - length l) fill"
- by (induct n arbitrary: l) (auto split: list.split)
-
-lemma takefill_replicate [simp]: "takefill fill n (replicate m fill) = replicate n fill"
- by (simp add: takefill_alt replicate_add [symmetric])
-
-lemma takefill_le': "n = m + k \<Longrightarrow> takefill x m (takefill x n l) = takefill x m l"
- by (induct m arbitrary: l n) (auto split: list.split)
-
-lemma length_takefill [simp]: "length (takefill fill n l) = n"
- by (simp add: takefill_alt)
-
-lemma take_takefill': "n = k + m \<Longrightarrow> take k (takefill fill n w) = takefill fill k w"
- by (induct k arbitrary: w n) (auto split: list.split)
-
-lemma drop_takefill: "drop k (takefill fill (m + k) w) = takefill fill m (drop k w)"
- by (induct k arbitrary: w) (auto split: list.split)
-
-lemma takefill_le [simp]: "m \<le> n \<Longrightarrow> takefill x m (takefill x n l) = takefill x m l"
- by (auto simp: le_iff_add takefill_le')
-
-lemma take_takefill [simp]: "m \<le> n \<Longrightarrow> take m (takefill fill n w) = takefill fill m w"
- by (auto simp: le_iff_add take_takefill')
-
-lemma takefill_append: "takefill fill (m + length xs) (xs @ w) = xs @ (takefill fill m w)"
- by (induct xs) auto
-
-lemma takefill_same': "l = length xs \<Longrightarrow> takefill fill l xs = xs"
- by (induct xs arbitrary: l) auto
-
-lemmas takefill_same [simp] = takefill_same' [OF refl]
-
-lemma tf_rev:
- "n + k = m + length bl \<Longrightarrow> takefill x m (rev (takefill y n bl)) =
- rev (takefill y m (rev (takefill x k (rev bl))))"
- apply (rule nth_equalityI)
- apply (auto simp add: nth_takefill nth_rev)
- apply (rule_tac f = "\<lambda>n. bl ! n" in arg_cong)
- apply arith
- done
-
-lemma takefill_minus: "0 < n \<Longrightarrow> takefill fill (Suc (n - 1)) w = takefill fill n w"
- by auto
-
-lemmas takefill_Suc_cases =
- list.cases [THEN takefill.Suc [THEN trans]]
-
-lemmas takefill_Suc_Nil = takefill_Suc_cases (1)
-lemmas takefill_Suc_Cons = takefill_Suc_cases (2)
-
-lemmas takefill_minus_simps = takefill_Suc_cases [THEN [2]
- takefill_minus [symmetric, THEN trans]]
-
-lemma takefill_numeral_Nil [simp]:
- "takefill fill (numeral k) [] = fill # takefill fill (pred_numeral k) []"
- by (simp add: numeral_eq_Suc)
-
-lemma takefill_numeral_Cons [simp]:
- "takefill fill (numeral k) (x # xs) = x # takefill fill (pred_numeral k) xs"
- by (simp add: numeral_eq_Suc)
-
-
-subsection \<open>Auxiliary: Range projection\<close>
-
-definition bl_of_nth :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> 'a list"
- where "bl_of_nth n f = map f (rev [0..<n])"
-
-lemma bl_of_nth_simps [simp, code]:
- "bl_of_nth 0 f = []"
- "bl_of_nth (Suc n) f = f n # bl_of_nth n f"
- by (simp_all add: bl_of_nth_def)
-
-lemma length_bl_of_nth [simp]: "length (bl_of_nth n f) = n"
- by (simp add: bl_of_nth_def)
-
-lemma nth_bl_of_nth [simp]: "m < n \<Longrightarrow> rev (bl_of_nth n f) ! m = f m"
- by (simp add: bl_of_nth_def rev_map)
-
-lemma bl_of_nth_inj: "(\<And>k. k < n \<Longrightarrow> f k = g k) \<Longrightarrow> bl_of_nth n f = bl_of_nth n g"
- by (simp add: bl_of_nth_def)
-
-lemma bl_of_nth_nth_le: "n \<le> length xs \<Longrightarrow> bl_of_nth n (nth (rev xs)) = drop (length xs - n) xs"
- apply (induct n arbitrary: xs)
- apply clarsimp
- apply clarsimp
- apply (rule trans [OF _ hd_Cons_tl])
- apply (frule Suc_le_lessD)
- apply (simp add: nth_rev trans [OF drop_Suc drop_tl, symmetric])
- apply (subst hd_drop_conv_nth)
- apply force
- apply simp_all
- apply (rule_tac f = "\<lambda>n. drop n xs" in arg_cong)
- apply simp
- done
-
-lemma bl_of_nth_nth [simp]: "bl_of_nth (length xs) ((!) (rev xs)) = xs"
- by (simp add: bl_of_nth_nth_le)
-
end