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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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"
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  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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notation
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  div (infixl "\<^loc>div" 70)
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notation
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  mod (infixl "\<^loc>mod" 70)
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end
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notation
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  div (infixl "div" 70)
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notation
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  mod (infixl "mod" 70)
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instance nat :: "Divides.div"
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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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  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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definition
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  (*The definition of dvd is polymorphic!*)
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  dvd  :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
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  dvd_def: "m dvd n \<longleftrightarrow> (\<exists>k. n = m*k)"
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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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diff changeset
   286
     "[| quorem((b,c),(q,r));  0 < c |]
14267
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diff changeset
   287
      ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   288
  by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
14267
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diff changeset
   289
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diff changeset
   290
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   291
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   292
  apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   293
  done
14267
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diff changeset
   294
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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diff changeset
   295
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   296
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   297
  apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   298
  done
14267
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diff changeset
   299
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diff changeset
   300
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   301
  apply (rule trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   302
   apply (rule_tac s = "b*a mod c" in trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   303
    apply (rule_tac [2] mod_mult1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   304
   apply (simp_all add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   305
  done
14267
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parents: 14208
diff changeset
   306
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   307
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
   308
  apply (rule mod_mult1_eq' [THEN trans])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   309
  apply (rule mod_mult1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   310
  done
14267
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parents: 14208
diff changeset
   311
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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diff changeset
   312
(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
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parents: 14208
diff changeset
   313
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   314
lemma quorem_add1_eq:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   315
     "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]
14267
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parents: 14208
diff changeset
   316
      ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   317
  by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
14267
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paulson
parents: 14208
diff changeset
   318
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   319
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   320
lemma div_add1_eq:
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paulson
parents: 14208
diff changeset
   321
     "(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
   322
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   323
  apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   324
  done
14267
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parents: 14208
diff changeset
   325
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   326
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
   327
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   328
  apply (blast intro: quorem_div_mod quorem_div_mod quorem_add1_eq [THEN quorem_mod])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   329
  done
14267
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parents: 14208
diff changeset
   330
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   331
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   332
subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   333
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   334
(** first, a lemma to bound the remainder **)
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parents: 14208
diff changeset
   335
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   336
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   337
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   338
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   339
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   340
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   341
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   342
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   343
lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
14267
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parents: 14208
diff changeset
   344
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   345
  by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
14267
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parents: 14208
diff changeset
   346
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   347
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   348
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   349
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   350
  apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   351
  done
14267
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parents: 14208
diff changeset
   352
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   353
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
   354
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   355
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   356
  apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   357
  done
14267
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parents: 14208
diff changeset
   358
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paulson
parents: 14208
diff changeset
   359
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   360
subsection{*Cancellation of Common Factors in Division*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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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_mult_lemma:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   363
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   364
  by (auto simp add: div_mult2_eq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   365
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   366
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
   367
  apply (cases "b = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   368
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   369
  done
14267
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paulson
parents: 14208
diff changeset
   370
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   371
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
   372
  apply (drule div_mult_mult1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   373
  apply (auto simp add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   374
  done
14267
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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
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   377
(*Distribution of Factors over Remainders:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   378
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   379
Could prove these as in Integ/IntDiv.ML, but we already have
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   380
mod_mult_distrib and mod_mult_distrib2 above!
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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
Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   383
qed "mod_mult_mult1";
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
Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   386
qed "mod_mult_mult2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   387
 ***)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   388
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   389
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
   390
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   391
lemma div_1 [simp]: "m div Suc 0 = m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   392
  by (induct m) (simp_all add: div_geq)
14267
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
lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   395
  by (simp add: div_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   396
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   397
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   398
  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   399
   apply (simp add: add_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   400
  apply (subst div_geq [symmetric], simp_all)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   401
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   402
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   403
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   404
  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
   405
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   406
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
   407
  apply (subst div_add1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   408
  apply (subst div_mult1_eq, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   409
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   410
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   411
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
   412
  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
   413
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   414
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   415
(* Monotonicity of div in first argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   416
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   417
    "\<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
   418
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   419
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
   420
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   421
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   422
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   423
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   424
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   425
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   426
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   427
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   428
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
   429
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   430
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   431
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   432
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
   433
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   434
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   435
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
   436
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   437
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   438
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   439
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   440
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   441
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
   442
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   443
 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
   444
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   445
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   446
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   447
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   448
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
   449
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   450
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
   451
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   452
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   453
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   454
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   455
(* Similar for "less than" *)
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   456
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
   457
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   458
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
   459
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   460
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   461
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   462
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   463
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   464
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   465
 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
   466
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   467
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   468
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   469
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   470
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   471
declare div_less_dividend [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   472
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   473
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
   474
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
   475
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   476
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
   477
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   478
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   479
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
   480
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   481
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   482
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
   483
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   484
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   485
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   486
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   487
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   488
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   489
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   490
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   491
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   492
subsection{*The Divides Relation*}
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 dvdI [intro?]: "n = m * k ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   495
  unfolding dvd_def by blast
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 dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   498
  unfolding dvd_def by blast
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   499
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   500
lemma dvd_0_right [iff]: "m dvd (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   501
  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
   502
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   503
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   504
  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
   505
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   506
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   507
  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
   508
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   509
lemma dvd_1_left [iff]: "Suc 0 dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   510
  unfolding dvd_def by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   511
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   512
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   513
  by (simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   514
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   515
lemma dvd_refl [simp]: "m dvd (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   516
  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
   517
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   518
lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   519
  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
   520
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   521
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   522
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   523
  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
   524
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   525
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   526
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   527
  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
   528
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   529
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   530
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   531
  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
   532
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   533
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
   534
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   535
  apply (blast intro: dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   536
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   537
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   538
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
   539
  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
   540
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   541
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   542
  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
   543
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   544
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   545
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   546
  apply (erule dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   547
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   548
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   549
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   550
  by (rule dvd_refl [THEN dvd_mult])
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   551
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   552
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   553
  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
   554
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   555
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   556
  apply (rule iffI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   557
   apply (erule_tac [2] dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   558
   apply (rule_tac [2] dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   559
  apply (subgoal_tac "n = (n+k) -k")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   560
   prefer 2 apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   561
  apply (erule ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   562
  apply (erule dvd_diff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   563
  apply (rule dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   564
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   565
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   566
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
   567
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   568
  apply (case_tac "n = 0", auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   569
  apply (blast intro: mod_mult_distrib2 [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   570
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   571
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   572
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
   573
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   574
   apply (simp add: mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   575
  apply (simp only: dvd_add dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   576
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   577
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   578
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
   579
  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
   580
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   581
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
   582
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   583
  apply (erule exE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   584
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   585
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   586
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   587
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   588
  apply auto
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   589
   apply (subgoal_tac "m*n dvd m*1")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   590
   apply (drule dvd_mult_cancel, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   591
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   592
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   593
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   594
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   595
  apply (erule dvd_mult_cancel1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   596
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   597
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   598
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
   599
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   600
  apply (rule_tac x = "k*ka" in exI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   601
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   602
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   603
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   604
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   605
  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
   606
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   607
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   608
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   609
  apply (rule_tac x = "i*k" in exI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   610
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   611
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   612
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   613
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   614
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   615
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   616
  apply (erule conjE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   617
  apply (rule le_trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   618
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   619
   apply (erule_tac [2] Suc_leI, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   620
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   621
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   622
lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   623
  apply (unfold dvd_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   624
  apply (case_tac "k=0", simp, safe)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   625
   apply (simp add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   626
  apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   627
  apply (subst mult_commute, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   628
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   629
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   630
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   631
  apply (subgoal_tac "m mod n = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   632
   apply (simp add: mult_div_cancel)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   633
  apply (simp only: dvd_eq_mod_eq_0)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   634
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   635
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   636
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   637
  apply (unfold dvd_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   638
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   639
  apply (simp add: power_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   640
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   641
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   642
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
   643
  by (induct n) auto
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   644
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   645
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
   646
  apply (induct j)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   647
   apply (simp_all add: le_Suc_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   648
  apply (blast dest!: dvd_mult_right)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   649
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   650
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   651
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
   652
  apply (rule power_le_imp_le_exp, assumption)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   653
  apply (erule dvd_imp_le, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   654
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   655
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   656
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   657
  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
   658
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   659
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
   660
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   661
(*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
   662
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   663
  apply (cut_tac m = m in mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   664
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   665
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   666
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   667
14131
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13882
diff changeset
   668
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   669
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   670
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   671
 ((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
   672
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   673
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   674
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   675
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   676
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   677
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   678
    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
   679
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   680
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   681
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   682
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   683
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   684
      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
   685
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   686
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   687
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   688
        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
   689
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   690
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   691
        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
   692
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   693
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   694
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   695
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   696
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   697
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   698
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   699
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   700
    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
   701
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   702
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   703
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   704
    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
   705
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   706
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   707
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   708
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   709
lemma split_div_lemma:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   710
  "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
   711
  apply (rule iffI)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   712
  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
   713
prefer 3; apply assumption
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20380
diff changeset
   714
  apply (simp_all add: quorem_def) apply arith
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   715
  apply (rule conjI)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   716
  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
   717
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   718
  apply (simp only: add: mult_ac)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   719
  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
   720
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   721
  apply (simp only: add: mult_ac add_ac)
14208
144f45277d5a misc tidying
paulson
parents: 14131
diff changeset
   722
  apply (rule add_less_mono1, simp)
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   723
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   724
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   725
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   726
  "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
   727
   (\<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
   728
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   729
  apply (simp only: add: split_div_lemma)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   730
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   731
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   732
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   733
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   734
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   735
 ((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
   736
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   737
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   738
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   739
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   740
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   741
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   742
    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
   743
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   744
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   745
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   746
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   747
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   748
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   749
      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
   750
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   751
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   752
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   753
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   754
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   755
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   756
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   757
    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
   758
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   759
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   760
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   761
    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
   762
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   763
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   764
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   765
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   766
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
   767
  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
   768
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   769
  apply arith
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   770
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   771
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   772
lemma div_mod_equality':
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   773
  fixes m n :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   774
  shows "m div n * n = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   775
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   776
  have "m mod n \<le> m mod n" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   777
  from div_mod_equality have 
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   778
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   779
  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
   780
    "m div n * n + (m mod n - m mod n) = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   781
    by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   782
  then show ?thesis by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   783
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   784
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   785
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   786
subsection {*An ``induction'' law for modulus arithmetic.*}
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   787
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   788
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   789
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   790
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   791
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   792
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   793
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   794
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   795
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   796
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   797
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   798
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   799
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   800
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   801
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   802
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   803
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   804
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   805
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   806
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   807
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   808
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   809
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   810
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   811
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   812
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   813
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   814
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   815
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   816
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   817
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   818
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   819
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   820
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   821
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   822
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   823
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   824
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   825
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   826
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   827
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   828
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   829
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   830
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   831
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   832
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   833
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   834
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   835
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   836
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   837
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   838
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   839
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   840
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   841
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   842
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   843
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   844
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   845
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   846
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   847
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   848
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   849
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   850
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   851
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   852
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   853
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   854
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   855
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   856
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   857
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   858
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   859
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   860
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   861
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   862
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   863
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   864
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   865
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   866
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   867
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   868
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   869
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   870
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   871
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   872
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   873
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   874
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   875
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
   876
  apply (rule trans [symmetric])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   877
   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
   878
  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
   879
  done
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   880
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   881
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
   882
  apply (rule trans [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   883
   apply (rule mod_add1_eq, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   884
  apply (rule mod_add1_eq [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   885
  done
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   886
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   887
lemma mod_div_decomp:
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   888
  fixes n k :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   889
  obtains m q where "m = n div k" and "q = n mod k"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   890
    and "n = m * k + q"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   891
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   892
  from mod_div_equality have "n = n div k * k + n mod k" by auto
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   893
  moreover have "n div k = n div k" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   894
  moreover have "n mod k = n mod k" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   895
  note that ultimately show thesis by blast
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   896
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   897
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   898
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   899
subsection {* Code generation for div, mod and dvd on nat *}
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   900
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   901
definition [code nofunc]:
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   902
  "divmod (m\<Colon>nat) n = (m div n, m mod n)"
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   903
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   904
lemma divmod_zero [code]: "divmod m 0 = (0, m)"
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
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   907
lemma divmod_succ [code]:
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   908
  "divmod m (Suc k) = (if m < Suc k then (0, m) else
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   909
    let
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   910
      (p, q) = divmod (m - Suc k) (Suc k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   911
    in (Suc p, q))"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   912
  unfolding divmod_def Let_def split_def
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   913
  by (auto intro: div_geq mod_geq)
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   914
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   915
lemma div_divmod [code]: "m div n = fst (divmod m n)"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   916
  unfolding divmod_def by simp
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   917
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   918
lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   919
  unfolding divmod_def by simp
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   920
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   921
definition
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   922
  dvd_nat :: "nat \<Rightarrow> nat \<Rightarrow> bool"
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   923
where
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   924
  "dvd_nat m n \<longleftrightarrow> n mod m = (0 \<Colon> nat)"
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   925
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   926
lemma [code inline]:
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   927
  "op dvd = dvd_nat"
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   928
  by (auto simp add: dvd_nat_def dvd_eq_mod_eq_0 expand_fun_eq)
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   929
21191
c00161fbf990 code generator module naming improved
haftmann
parents: 20640
diff changeset
   930
code_modulename SML
c00161fbf990 code generator module naming improved
haftmann
parents: 20640
diff changeset
   931
  Divides Integer
20640
05e6042394b9 name shifts
haftmann
parents: 20589
diff changeset
   932
21911
e29bcab0c81c added OCaml code generation (without dictionaries)
haftmann
parents: 21408
diff changeset
   933
code_modulename OCaml
e29bcab0c81c added OCaml code generation (without dictionaries)
haftmann
parents: 21408
diff changeset
   934
  Divides Integer
e29bcab0c81c added OCaml code generation (without dictionaries)
haftmann
parents: 21408
diff changeset
   935
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22718
diff changeset
   936
hide (open) const divmod dvd_nat
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   937
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   938
subsection {* Legacy bindings *}
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
   939
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   940
ML
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   941
{*
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   942
val div_def = thm "div_def"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   943
val mod_def = thm "mod_def"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   944
val dvd_def = thm "dvd_def"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   945
val quorem_def = thm "quorem_def"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   946
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   947
val wf_less_trans = thm "wf_less_trans";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   948
val mod_eq = thm "mod_eq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   949
val div_eq = thm "div_eq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   950
val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   951
val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   952
val mod_less = thm "mod_less";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   953
val mod_geq = thm "mod_geq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   954
val le_mod_geq = thm "le_mod_geq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   955
val mod_if = thm "mod_if";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   956
val mod_1 = thm "mod_1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   957
val mod_self = thm "mod_self";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   958
val mod_add_self2 = thm "mod_add_self2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   959
val mod_add_self1 = thm "mod_add_self1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   960
val mod_mult_self1 = thm "mod_mult_self1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   961
val mod_mult_self2 = thm "mod_mult_self2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   962
val mod_mult_distrib = thm "mod_mult_distrib";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   963
val mod_mult_distrib2 = thm "mod_mult_distrib2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   964
val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   965
val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   966
val div_less = thm "div_less";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   967
val div_geq = thm "div_geq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   968
val le_div_geq = thm "le_div_geq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   969
val div_if = thm "div_if";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   970
val mod_div_equality = thm "mod_div_equality";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   971
val mod_div_equality2 = thm "mod_div_equality2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   972
val div_mod_equality = thm "div_mod_equality";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   973
val div_mod_equality2 = thm "div_mod_equality2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   974
val mult_div_cancel = thm "mult_div_cancel";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   975
val mod_less_divisor = thm "mod_less_divisor";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   976
val div_mult_self_is_m = thm "div_mult_self_is_m";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   977
val div_mult_self1_is_m = thm "div_mult_self1_is_m";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   978
val unique_quotient_lemma = thm "unique_quotient_lemma";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   979
val unique_quotient = thm "unique_quotient";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   980
val unique_remainder = thm "unique_remainder";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   981
val div_0 = thm "div_0";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   982
val mod_0 = thm "mod_0";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   983
val div_mult1_eq = thm "div_mult1_eq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   984
val mod_mult1_eq = thm "mod_mult1_eq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   985
val mod_mult1_eq' = thm "mod_mult1_eq'";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   986
val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   987
val div_add1_eq = thm "div_add1_eq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   988
val mod_add1_eq = thm "mod_add1_eq";
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   989
val mod_add_left_eq = thm "mod_add_left_eq";
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   990
 val mod_add_right_eq = thm "mod_add_right_eq";
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   991
val mod_lemma = thm "mod_lemma";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   992
val div_mult2_eq = thm "div_mult2_eq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   993
val mod_mult2_eq = thm "mod_mult2_eq";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   994
val div_mult_mult_lemma = thm "div_mult_mult_lemma";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   995
val div_mult_mult1 = thm "div_mult_mult1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   996
val div_mult_mult2 = thm "div_mult_mult2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   997
val div_1 = thm "div_1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   998
val div_self = thm "div_self";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   999
val div_add_self2 = thm "div_add_self2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1000
val div_add_self1 = thm "div_add_self1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1001
val div_mult_self1 = thm "div_mult_self1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1002
val div_mult_self2 = thm "div_mult_self2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1003
val div_le_mono = thm "div_le_mono";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1004
val div_le_mono2 = thm "div_le_mono2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1005
val div_le_dividend = thm "div_le_dividend";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1006
val div_less_dividend = thm "div_less_dividend";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1007
val mod_Suc = thm "mod_Suc";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1008
val dvdI = thm "dvdI";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1009
val dvdE = thm "dvdE";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1010
val dvd_0_right = thm "dvd_0_right";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1011
val dvd_0_left = thm "dvd_0_left";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1012
val dvd_0_left_iff = thm "dvd_0_left_iff";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1013
val dvd_1_left = thm "dvd_1_left";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1014
val dvd_1_iff_1 = thm "dvd_1_iff_1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1015
val dvd_refl = thm "dvd_refl";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1016
val dvd_trans = thm "dvd_trans";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1017
val dvd_anti_sym = thm "dvd_anti_sym";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1018
val dvd_add = thm "dvd_add";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1019
val dvd_diff = thm "dvd_diff";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1020
val dvd_diffD = thm "dvd_diffD";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1021
val dvd_diffD1 = thm "dvd_diffD1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1022
val dvd_mult = thm "dvd_mult";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1023
val dvd_mult2 = thm "dvd_mult2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1024
val dvd_reduce = thm "dvd_reduce";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1025
val dvd_mod = thm "dvd_mod";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1026
val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1027
val dvd_mod_iff = thm "dvd_mod_iff";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1028
val dvd_mult_cancel = thm "dvd_mult_cancel";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1029
val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1030
val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1031
val mult_dvd_mono = thm "mult_dvd_mono";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1032
val dvd_mult_left = thm "dvd_mult_left";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1033
val dvd_mult_right = thm "dvd_mult_right";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1034
val dvd_imp_le = thm "dvd_imp_le";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1035
val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1036
val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1037
val mod_eq_0_iff = thm "mod_eq_0_iff";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1038
val mod_eqD = thm "mod_eqD";
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1039
*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1040
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1041
(*
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1042
lemma split_div:
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1043
assumes m: "m \<noteq> 0"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1044
shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1045
       (is "?P = ?Q")
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1046
proof
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1047
  assume P: ?P
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1048
  show ?Q
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1049
  proof (intro allI impI)
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1050
    fix i j
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1051
    assume n: "n = m*i + j" and j: "j < m"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1052
    show "P i"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1053
    proof (cases)
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1054
      assume "i = 0"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1055
      with n j P show "P i" by simp
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1056
    next
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1057
      assume "i \<noteq> 0"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1058
      with n j P show "P i" by (simp add:add_ac div_mult_self1)
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1059
    qed
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1060
  qed
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1061
next
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1062
  assume Q: ?Q
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1063
  from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
  1064
  show ?P by simp
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1065
qed
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1066
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1067
lemma split_mod:
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1068
assumes m: "m \<noteq> 0"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1069
shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1070
       (is "?P = ?Q")
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1071
proof
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1072
  assume P: ?P
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1073
  show ?Q
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1074
  proof (intro allI impI)
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1075
    fix i j
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1076
    assume "n = m*i + j" "j < m"
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1077
    thus "P j" using m P by(simp add:add_ac mult_ac)
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1078
  qed
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1079
next
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1080
  assume Q: ?Q
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1081
  from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
  1082
  show ?P by simp
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
  1083
qed
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1084
*)
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
  1085
end