70 lines
1.9 KiB
Plaintext
70 lines
1.9 KiB
Plaintext
(*
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* Copyright 2014, NICTA
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*
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* This software may be distributed and modified according to the terms of
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* the BSD 2-Clause license. Note that NO WARRANTY is provided.
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* See "LICENSE_BSD2.txt" for details.
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*
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* @TAG(NICTA_BSD)
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*)
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theory MoreDivides
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imports "~~/src/HOL/Main"
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begin
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(* FIXME: to Isabelle lib *)
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lemma div_mult_le:
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"(a::nat) div b * b \<le> a"
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apply(subgoal_tac "a = a div b * b + a mod b")
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apply arith
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apply simp
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done
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(* FIXME: to Isabelle lib *)
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lemma mod_le_dividend:
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"m mod n \<le> (m::nat)"
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by (induct m) (auto simp: mod_Suc)
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(* diff_mod_le now also in Num_lemmas.ML *)
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(* FIXME: to Isabelle lib? *)
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lemma diff_mod_le:
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"\<lbrakk> (a::nat) < d; b dvd d \<rbrakk> \<Longrightarrow> a - a mod b \<le> d - b"
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apply(subst mult_div_cancel [symmetric])
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apply(clarsimp simp: dvd_def)
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apply(case_tac "b = 0")
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apply simp
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apply(subgoal_tac "a div b \<le> k - 1")
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prefer 2
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apply(subgoal_tac "a div b < k")
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apply(simp add: less_Suc_eq_le [symmetric])
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apply(subgoal_tac "b * (a div b) < b * ((b * k) div b)")
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apply clarsimp
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apply(subst div_mult_self1_is_m)
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apply arith
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apply(rule le_less_trans)
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apply simp
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apply(subst mult.commute)
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apply(rule div_mult_le)
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apply assumption
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apply clarsimp
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apply(subgoal_tac "b * (a div b) \<le> b * (k - 1)")
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apply(erule le_trans)
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apply(simp add: diff_mult_distrib2)
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apply simp
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done
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lemma int_div_same_is_1 [simp]:
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"0 < a \<Longrightarrow> ((a :: int) div b = a) = (b = 1)"
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by (metis div_by_1 abs_ge_zero abs_of_pos int_div_less_self neq_iff nonneg1_imp_zdiv_pos_iff zabs_less_one_iff)
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lemma nat_div_same_is_1 [simp]:
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"a \<noteq> 0 \<Longrightarrow> ((a :: nat) div b = a) = (b = 1)"
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by (metis div_by_0 div_by_1 div_le_dividend div_less_dividend div_self nat_less_le)
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lemma int_div_minus_is_minus1 [simp]:
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"a < 0 \<Longrightarrow> ((a :: int) div b = -a) = (b = -1)"
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by (metis div_minus_right equation_minus_iff int_div_same_is_1 neg_0_less_iff_less)
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end
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