src/HOL/Divides.thy
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(*  Title:      HOL/Divides.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div, mod and the divides relation "dvd" *}
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theory Divides
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imports Datatype Power
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uses "~~/src/Provers/Arith/cancel_div_mod.ML"
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begin
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(*We use the same class for div and mod;
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  moreover, dvd is defined whenever multiplication is*)
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class div = type +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>div" 70)
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  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>mod" 70)
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instance nat :: Divides.div
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  div_def: "m div n == wfrec (pred_nat^+)
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                          (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m"
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  mod_def: "m mod n == wfrec (pred_nat^+)
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                          (%f j. if j<n | n=0 then j else f (j-n)) m" ..
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definition (in times)
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  dvd  :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<^loc>dvd" 50)
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where
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  "m \<^loc>dvd n \<longleftrightarrow> (\<exists>k. n = m \<^loc>* k)"
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lemmas dvd_def = dvd_def [folded times_class.dvd]
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class dvd_mod = times + div + zero + -- {* for code generation *}
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  assumes dvd_def_mod: "times.dvd (op \<^loc>*) x y \<longleftrightarrow> y \<^loc>mod x = \<^loc>0"
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lemmas dvd_def_mod [code func] = dvd_def_mod [folded times_class.dvd]
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definition
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  quorem :: "(nat*nat) * (nat*nat) => bool" where
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  (*This definition helps prove the harder properties of div and mod.
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    It is copied from IntDiv.thy; should it be overloaded?*)
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  "quorem = (%((a,b), (q,r)).
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                    a = b*q + r &
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                    (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"
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subsection{*Initial Lemmas*}
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lemmas wf_less_trans =
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       def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],
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                  standard]
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lemma mod_eq: "(%m. m mod n) =
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              wfrec (pred_nat^+) (%f j. if j<n | n=0 then j else f (j-n))"
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by (simp add: mod_def)
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lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+)
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               (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
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by (simp add: div_def)
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(** Aribtrary definitions for division by zero.  Useful to simplify
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    certain equations **)
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lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"
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  by (rule div_eq [THEN wf_less_trans], simp)
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lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"
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  by (rule mod_eq [THEN wf_less_trans], simp)
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subsection{*Remainder*}
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lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"
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  by (rule mod_eq [THEN wf_less_trans]) simp
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lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
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  apply (cases "n=0")
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   apply simp
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  apply (rule mod_eq [THEN wf_less_trans])
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  apply (simp add: cut_apply less_eq)
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  done
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(*Avoids the ugly ~m<n above*)
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lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n"
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  by (simp add: mod_geq linorder_not_less)
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lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)"
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  by (simp add: mod_geq)
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lemma mod_1 [simp]: "m mod Suc 0 = 0"
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  by (induct m) (simp_all add: mod_geq)
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lemma mod_self [simp]: "n mod n = (0::nat)"
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  by (cases "n = 0") (simp_all add: mod_geq)
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lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
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  apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
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   apply (simp add: add_commute)
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  apply (subst mod_geq [symmetric], simp_all)
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  done
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lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
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  by (simp add: add_commute mod_add_self2)
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lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
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  by (induct k) (simp_all add: add_left_commute [of _ n])
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lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
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  by (simp add: mult_commute mod_mult_self1)
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lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
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  apply (cases "n = 0", simp)
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  apply (cases "k = 0", simp)
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  apply (induct m rule: nat_less_induct)
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  apply (subst mod_if, simp)
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  apply (simp add: mod_geq diff_mult_distrib)
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  done
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lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
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  by (simp add: mult_commute [of k] mod_mult_distrib)
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lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
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  apply (cases "n = 0", simp)
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  apply (induct m, simp)
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  apply (rename_tac k)
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  apply (cut_tac m = "k * n" and n = n in mod_add_self2)
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  apply (simp add: add_commute)
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  done
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lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
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  by (simp add: mult_commute mod_mult_self_is_0)
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subsection{*Quotient*}
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lemma div_less [simp]: "m<n ==> m div n = (0::nat)"
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  by (rule div_eq [THEN wf_less_trans], simp)
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lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
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  apply (rule div_eq [THEN wf_less_trans])
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  apply (simp add: cut_apply less_eq)
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  done
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(*Avoids the ugly ~m<n above*)
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lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"
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  by (simp add: div_geq linorder_not_less)
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lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
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  by (simp add: div_geq)
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(*Main Result about quotient and remainder.*)
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lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"
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  apply (cases "n = 0", simp)
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  apply (induct m rule: nat_less_induct)
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  apply (subst mod_if)
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  apply (simp_all add: add_assoc div_geq add_diff_inverse)
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  done
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lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
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  apply (cut_tac m = m and n = n in mod_div_equality)
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  apply (simp add: mult_commute)
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  done
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subsection{*Simproc for Cancelling Div and Mod*}
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lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
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  by (simp add: mod_div_equality2)
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ML
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{*
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structure CancelDivModData =
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struct
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val div_name = @{const_name Divides.div};
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val mod_name = @{const_name Divides.mod};
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val mk_binop = HOLogic.mk_binop;
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val mk_sum = NatArithUtils.mk_sum;
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val dest_sum = NatArithUtils.dest_sum;
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(*logic*)
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val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
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val trans = trans
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val prove_eq_sums =
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  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
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  in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
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end;
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structure CancelDivMod = CancelDivModFun(CancelDivModData);
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val cancel_div_mod_proc = NatArithUtils.prep_simproc
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      ("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc);
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Addsimprocs[cancel_div_mod_proc];
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*}
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(* a simple rearrangement of mod_div_equality: *)
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lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
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  by (cut_tac m = m and n = n in mod_div_equality2, arith)
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lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
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  apply (induct m rule: nat_less_induct)
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  apply (rename_tac m)
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  apply (case_tac "m<n", simp)
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  txt{*case @{term "n \<le> m"}*}
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  apply (simp add: mod_geq)
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  done
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lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
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  apply (drule mod_less_divisor [where m = m])
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  apply simp
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  done
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lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
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  by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)
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lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
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  by (simp add: mult_commute div_mult_self_is_m)
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(*mod_mult_distrib2 above is the counterpart for remainder*)
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subsection{*Proving facts about Quotient and Remainder*}
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lemma unique_quotient_lemma:
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     "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]
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      ==> q' \<le> (q::nat)"
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  apply (rule leI)
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  apply (subst less_iff_Suc_add)
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  apply (auto simp add: add_mult_distrib2)
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  done
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lemma unique_quotient:
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     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
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      ==> q = q'"
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  apply (simp add: split_ifs quorem_def)
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  apply (blast intro: order_antisym
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    dest: order_eq_refl [THEN unique_quotient_lemma] sym)
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  done
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lemma unique_remainder:
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     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
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      ==> r = r'"
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  apply (subgoal_tac "q = q'")
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   prefer 2 apply (blast intro: unique_quotient)
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  apply (simp add: quorem_def)
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  done
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lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"
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  unfolding quorem_def by simp
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lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"
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  by (simp add: quorem_div_mod [THEN unique_quotient])
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lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"
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  by (simp add: quorem_div_mod [THEN unique_remainder])
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(** A dividend of zero **)
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lemma div_0 [simp]: "0 div m = (0::nat)"
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  by (cases "m = 0") simp_all
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lemma mod_0 [simp]: "0 mod m = (0::nat)"
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  by (cases "m = 0") simp_all
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(** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
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lemma quorem_mult1_eq:
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     "[| quorem((b,c),(q,r));  0 < c |]
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      ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
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parents: 22473
diff changeset
   279
  by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
14267
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   280
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   281
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
22718
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wenzelm
parents: 22473
diff changeset
   282
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   283
  apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   284
  done
14267
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diff changeset
   285
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   286
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   287
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   288
  apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   289
  done
14267
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   290
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diff changeset
   291
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   292
  apply (rule trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   293
   apply (rule_tac s = "b*a mod c" in trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   294
    apply (rule_tac [2] mod_mult1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   295
   apply (simp_all add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   296
  done
14267
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diff changeset
   297
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diff changeset
   298
lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   299
  apply (rule mod_mult1_eq' [THEN trans])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   300
  apply (rule mod_mult1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   301
  done
14267
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diff changeset
   302
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diff changeset
   303
(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
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diff changeset
   304
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   305
lemma quorem_add1_eq:
22718
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wenzelm
parents: 22473
diff changeset
   306
     "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]
14267
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parents: 14208
diff changeset
   307
      ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   308
  by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
14267
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paulson
parents: 14208
diff changeset
   309
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parents: 14208
diff changeset
   310
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
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diff changeset
   311
lemma div_add1_eq:
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diff changeset
   312
     "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   313
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   314
  apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   315
  done
14267
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parents: 14208
diff changeset
   316
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   317
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   318
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   319
  apply (blast intro: quorem_div_mod quorem_div_mod quorem_add1_eq [THEN quorem_mod])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   320
  done
14267
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parents: 14208
diff changeset
   321
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   322
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   323
subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
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parents: 14208
diff changeset
   324
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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diff changeset
   325
(** first, a lemma to bound the remainder **)
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diff changeset
   326
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   327
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   328
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   329
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   330
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   331
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   332
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   333
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   334
lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
14267
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paulson
parents: 14208
diff changeset
   335
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   336
  by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
14267
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parents: 14208
diff changeset
   337
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   338
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   339
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   340
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   341
  apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   342
  done
14267
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parents: 14208
diff changeset
   343
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   344
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   345
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   346
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   347
  apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   348
  done
14267
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diff changeset
   349
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parents: 14208
diff changeset
   350
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   351
subsection{*Cancellation of Common Factors in Division*}
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parents: 14208
diff changeset
   352
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   353
lemma div_mult_mult_lemma:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   354
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   355
  by (auto simp add: div_mult2_eq)
14267
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paulson
parents: 14208
diff changeset
   356
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   357
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   358
  apply (cases "b = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   359
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   360
  done
14267
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paulson
parents: 14208
diff changeset
   361
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   362
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   363
  apply (drule div_mult_mult1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   364
  apply (auto simp add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   365
  done
14267
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paulson
parents: 14208
diff changeset
   366
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   367
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   368
subsection{*Further Facts about Quotient and Remainder*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   369
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   370
lemma div_1 [simp]: "m div Suc 0 = m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   371
  by (induct m) (simp_all add: div_geq)
14267
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paulson
parents: 14208
diff changeset
   372
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   373
lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   374
  by (simp add: div_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   375
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   376
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   377
  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   378
   apply (simp add: add_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   379
  apply (subst div_geq [symmetric], simp_all)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   380
  done
14267
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paulson
parents: 14208
diff changeset
   381
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   382
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   383
  by (simp add: add_commute div_add_self2)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   384
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   385
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   386
  apply (subst div_add1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   387
  apply (subst div_mult1_eq, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   388
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   389
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   390
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   391
  by (simp add: mult_commute div_mult_self1)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   392
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   393
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   394
(* Monotonicity of div in first argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   395
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   396
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   397
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   398
apply (induct "n" rule: nat_less_induct, clarify)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   399
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   400
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   401
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   402
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   403
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   404
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   405
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   406
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   407
apply (simp add: div_geq diff_le_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   408
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   409
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   410
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   411
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   412
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   413
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   414
apply (induct_tac k rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   415
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   416
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   417
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   418
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   419
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   420
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   421
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   422
 apply (blast intro: div_le_mono diff_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   423
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   424
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   425
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   426
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   427
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   428
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   429
apply (subgoal_tac "m div n \<le> m div 1", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   430
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   431
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   432
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   433
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   434
(* Similar for "less than" *)
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   435
lemma div_less_dividend [rule_format]:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   436
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   437
apply (induct_tac m rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   438
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   439
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   440
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   441
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   442
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   443
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   444
 apply (subgoal_tac "(m-n) div n < (m-n) ")
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   445
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   446
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   447
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   448
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   449
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   450
declare div_less_dividend [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   451
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   452
text{*A fact for the mutilated chess board*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   453
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   454
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   455
apply (induct "m" rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   456
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   457
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   458
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   459
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   460
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   461
apply (auto simp add: Suc_diff_le le_mod_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   462
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   463
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   464
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   465
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   466
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   467
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   468
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   469
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   470
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   471
subsection{*The Divides Relation*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   472
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   473
lemma dvdI [intro?]: "n = m * k ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   474
  unfolding dvd_def by blast
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   475
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   476
lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   477
  unfolding dvd_def by blast
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   478
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   479
lemma dvd_0_right [iff]: "m dvd (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   480
  unfolding dvd_def by (blast intro: mult_0_right [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   481
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   482
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   483
  by (force simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   484
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   485
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   486
  by (blast intro: dvd_0_left)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   487
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   488
lemma dvd_1_left [iff]: "Suc 0 dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   489
  unfolding dvd_def by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   490
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   491
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   492
  by (simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   493
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   494
lemma dvd_refl [simp]: "m dvd (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   495
  unfolding dvd_def by (blast intro: mult_1_right [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   496
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   497
lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   498
  unfolding dvd_def by (blast intro: mult_assoc)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   499
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   500
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   501
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   502
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   503
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   504
text {* @{term "op dvd"} is a partial order *}
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   505
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   506
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> m \<noteq> n"]
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   507
  by unfold_locales (auto intro: dvd_trans dvd_anti_sym)
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   508
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   509
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   510
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   511
  by (blast intro: add_mult_distrib2 [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   512
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   513
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   514
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   515
  by (blast intro: diff_mult_distrib2 [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   516
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   517
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   518
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   519
  apply (blast intro: dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   520
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   521
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   522
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   523
  by (drule_tac m = m in dvd_diff, auto)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   524
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   525
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   526
  unfolding dvd_def by (blast intro: mult_left_commute)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   527
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   528
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   529
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   530
  apply (erule dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   531
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   532
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   533
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   534
  by (rule dvd_refl [THEN dvd_mult])
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   535
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   536
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   537
  by (rule dvd_refl [THEN dvd_mult2])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   538
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   539
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   540
  apply (rule iffI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   541
   apply (erule_tac [2] dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   542
   apply (rule_tac [2] dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   543
  apply (subgoal_tac "n = (n+k) -k")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   544
   prefer 2 apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   545
  apply (erule ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   546
  apply (erule dvd_diff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   547
  apply (rule dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   548
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   549
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   550
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   551
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   552
  apply (case_tac "n = 0", auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   553
  apply (blast intro: mod_mult_distrib2 [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   554
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   555
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   556
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   557
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   558
   apply (simp add: mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   559
  apply (simp only: dvd_add dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   560
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   561
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   562
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   563
  by (blast intro: dvd_mod_imp_dvd dvd_mod)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   564
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   565
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   566
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   567
  apply (erule exE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   568
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   569
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   570
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   571
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   572
  apply auto
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   573
   apply (subgoal_tac "m*n dvd m*1")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   574
   apply (drule dvd_mult_cancel, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   575
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   576
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   577
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   578
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   579
  apply (erule dvd_mult_cancel1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   580
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   581
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   582
lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   583
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   584
  apply (rule_tac x = "k*ka" in exI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   585
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   586
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   587
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   588
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   589
  by (simp add: dvd_def mult_assoc, blast)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   590
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   591
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   592
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   593
  apply (rule_tac x = "i*k" in exI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   594
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   595
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   596
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   597
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   598
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   599
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   600
  apply (erule conjE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   601
  apply (rule le_trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   602
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   603
   apply (erule_tac [2] Suc_leI, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   604
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   605
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   606
lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   607
  apply (unfold dvd_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   608
  apply (case_tac "k=0", simp, safe)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   609
   apply (simp add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   610
  apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   611
  apply (subst mult_commute, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   612
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   613
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   614
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   615
  apply (subgoal_tac "m mod n = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   616
   apply (simp add: mult_div_cancel)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   617
  apply (simp only: dvd_eq_mod_eq_0)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   618
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   619
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   620
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   621
  apply (unfold dvd_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   622
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   623
  apply (simp add: power_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   624
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   625
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   626
lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   627
  by (induct n) auto
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   628
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   629
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   630
  apply (induct j)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   631
   apply (simp_all add: le_Suc_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   632
  apply (blast dest!: dvd_mult_right)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   633
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   634
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   635
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   636
  apply (rule power_le_imp_le_exp, assumption)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   637
  apply (erule dvd_imp_le, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   638
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   639
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   640
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   641
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   642
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   643
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   644
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   645
(*Loses information, namely we also have r<d provided d is nonzero*)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   646
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   647
  apply (cut_tac m = m in mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   648
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   649
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   650
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   651
14131
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13882
diff changeset
   652
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   653
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   654
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   655
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   656
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   657
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   658
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   659
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   660
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   661
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   662
    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   663
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   664
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   665
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   666
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   667
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   668
      assume n: "n = k*i + j" and j: "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   669
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   670
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   671
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   672
        with n j P show "P i" by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   673
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   674
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   675
        with not0 n j P show "P i" by(simp add:add_ac)
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   676
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   677
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   678
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   679
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   680
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   681
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   682
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   683
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   684
    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   685
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   686
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   687
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   688
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
   689
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   690
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   691
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   692
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   693
lemma split_div_lemma:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   694
  "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   695
  apply (rule iffI)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   696
  apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 15439
diff changeset
   697
prefer 3; apply assumption
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20380
diff changeset
   698
  apply (simp_all add: quorem_def) apply arith
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   699
  apply (rule conjI)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   700
  apply (rule_tac P="%x. n * (m div n) \<le> x" in
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   701
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   702
  apply (simp only: add: mult_ac)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   703
  apply (rule_tac P="%x. x < n + n * (m div n)" in
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   704
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   705
  apply (simp only: add: mult_ac add_ac)
14208
144f45277d5a misc tidying
paulson
parents: 14131
diff changeset
   706
  apply (rule add_less_mono1, simp)
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   707
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   708
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   709
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   710
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   711
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   712
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   713
  apply (simp only: add: split_div_lemma)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   714
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   715
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   716
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   717
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   718
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   719
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   720
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   721
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   722
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   723
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   724
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   725
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   726
    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   727
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   728
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   729
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   730
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   731
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   732
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   733
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   734
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   735
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   736
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   737
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   738
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   739
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   740
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   741
    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   742
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   743
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   744
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   745
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
   746
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   747
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   748
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   749
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   750
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   751
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   752
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   753
  apply arith
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   754
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   755
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   756
lemma div_mod_equality':
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   757
  fixes m n :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   758
  shows "m div n * n = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   759
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   760
  have "m mod n \<le> m mod n" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   761
  from div_mod_equality have 
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   762
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   763
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   764
    "m div n * n + (m mod n - m mod n) = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   765
    by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   766
  then show ?thesis by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   767
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   768
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   769
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   770
subsection {*An ``induction'' law for modulus arithmetic.*}
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   771
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   772
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   773
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   774
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   775
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   776
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   777
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   778
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   779
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   780
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   781
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   782
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   783
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   784
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   785
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   786
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   787
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   788
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   789
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   790
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   791
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   792
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   793
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   794
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   795
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   796
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   797
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   798
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   799
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   800
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   801
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   802
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   803
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   804
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   805
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   806
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   807
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   808
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   809
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   810
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   811
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   812
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   813
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   814
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   815
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   816
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   817
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   818
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   819
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   820
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   821
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   822
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   823
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   824
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   825
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   826
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   827
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   828
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   829
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   830
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   831
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   832
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   833
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   834
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   835
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   836
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   837
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   838
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   839
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   840
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   841
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   842
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   843
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   844
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   845
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   846
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   847
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   848
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   849
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   850
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   851
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   852
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   853
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   854
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   855
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   856
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   857
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   858
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   859
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   860
  apply (rule trans [symmetric])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   861
   apply (rule mod_add1_eq, simp)
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   862
  apply (rule mod_add1_eq [symmetric])
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   863
  done
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   864
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   865
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   866
  apply (rule trans [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   867
   apply (rule mod_add1_eq, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   868
  apply (rule mod_add1_eq [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   869
  done
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   870
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   871
lemma mod_div_decomp:
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   872
  fixes n k :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   873
  obtains m q where "m = n div k" and "q = n mod k"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   874
    and "n = m * k + q"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   875
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   876
  from mod_div_equality have "n = n div k * k + n mod k" by auto
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   877
  moreover have "n div k = n div k" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   878
  moreover have "n mod k = n mod k" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   879
  note that ultimately show thesis by blast
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   880
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   881
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   882
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   883
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   884
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   885
subsection {* Code generation for div, mod and dvd on nat *}
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   886
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22800
diff changeset
   887
definition [code func del]:
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   888
  "divmod (m\<Colon>nat) n = (m div n, m mod n)"
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   889
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   890
lemma divmod_zero [code]: "divmod m 0 = (0, m)"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   891
  unfolding divmod_def by simp
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   892
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   893
lemma divmod_succ [code]:
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   894
  "divmod m (Suc k) = (if m < Suc k then (0, m) else
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   895
    let
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   896
      (p, q) = divmod (m - Suc k) (Suc k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   897
    in (Suc p, q))"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   898
  unfolding divmod_def Let_def split_def
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   899
  by (auto intro: div_geq mod_geq)
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   900
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   901
lemma div_divmod [code]: "m div n = fst (divmod m n)"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   902
  unfolding divmod_def by simp
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   903
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   904
lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   905
  unfolding divmod_def by simp
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   906
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   907
instance nat :: dvd_mod
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   908
  by default (simp add: times_class.dvd [symmetric] dvd_eq_mod_eq_0)
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   909
21191
c00161fbf990 code generator module naming improved
haftmann
parents: 20640
diff changeset
   910
code_modulename SML
23017
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
   911
  Divides Nat
20640
05e6042394b9 name shifts
haftmann
parents: 20589
diff changeset
   912
21911
e29bcab0c81c added OCaml code generation (without dictionaries)
haftmann
parents: 21408
diff changeset
   913
code_modulename OCaml
23017
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
   914
  Divides Nat
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
   915
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
   916
code_modulename Haskell
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
   917
  Divides Nat
21911
e29bcab0c81c added OCaml code generation (without dictionaries)
haftmann
parents: 21408
diff changeset
   918
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   919
hide (open) const divmod
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   920
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
   921
end