59 lines
1.6 KiB
Plaintext
59 lines
1.6 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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section "More Lemmas on Division"
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theory More_Divides
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imports Main
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begin
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lemma div_mult_le:
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"(a::nat) div b * b \<le> a"
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by (fact div_times_less_eq_dividend)
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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 minus_mod_eq_mult_div)
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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
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nonneg1_imp_zdiv_pos_iff zabs_less_one_iff)
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declare div_eq_dividend_iff [simp]
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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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