src/HOL/Divides.thy
author haftmann
Tue Mar 20 08:27:15 2007 +0100 (2007-03-20)
changeset 22473 753123c89d72
parent 22261 9e185f78e7d4
child 22718 936f7580937d
permissions -rw-r--r--
explizit "type" superclass
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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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consts
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  quorem :: "(nat*nat) * (nat*nat) => bool"
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defs
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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_def: "quorem \<equiv> (%((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 (case_tac "n=0", 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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apply (induct "m")
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apply (simp_all (no_asm_simp) add: mod_geq)
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done
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lemma mod_self [simp]: "n mod n = (0::nat)"
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apply (case_tac "n=0")
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apply (simp_all add: mod_geq)
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done
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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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apply (induct "k")
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apply (simp_all add: add_left_commute [of _ n])
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done
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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 (case_tac "n=0", simp)
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apply (case_tac "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 (case_tac "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 (case_tac "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 (no_asm_simp) 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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apply(simp add: mod_div_equality)
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done
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lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
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apply(simp add: mod_div_equality2)
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done
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ML
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{*
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val div_mod_equality = thm "div_mod_equality";
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val div_mod_equality2 = thm "div_mod_equality2";
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structure CancelDivModData =
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struct
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val div_name = "Divides.div";
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val mod_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 [div_mod_equality,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 = add_0 :: add_0_right :: 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 (case_tac "na<n", simp) 
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txt{*case @{term "n \<le> na"}*}
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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 (case_tac "m=0", simp_all)
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lemma mod_0 [simp]: "0 mod m = (0::nat)"
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by (case_tac "m=0", simp_all)
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(** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
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lemma quorem_mult1_eq:
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     "[| quorem((b,c),(q,r));  0 < c |]  
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      ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
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apply (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
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done
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lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
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apply (case_tac "c = 0", simp)
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apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
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done
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lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
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apply (case_tac "c = 0", simp)
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apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
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done
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lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
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apply (rule trans)
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apply (rule_tac s = "b*a mod c" in trans)
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apply (rule_tac [2] mod_mult1_eq)
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apply (simp_all (no_asm) add: mult_commute)
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done
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lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
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apply (rule mod_mult1_eq' [THEN trans])
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apply (rule mod_mult1_eq)
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done
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(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
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lemma quorem_add1_eq:
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     "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]  
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      ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
paulson@14267
   330
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
paulson@14267
   331
paulson@14267
   332
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   333
lemma div_add1_eq:
paulson@14267
   334
     "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
paulson@14267
   335
apply (case_tac "c = 0", simp)
paulson@14267
   336
apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
paulson@14267
   337
done
paulson@14267
   338
paulson@14267
   339
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
paulson@14267
   340
apply (case_tac "c = 0", simp)
paulson@14267
   341
apply (blast intro: quorem_div_mod quorem_div_mod
paulson@14267
   342
                    quorem_add1_eq [THEN quorem_mod])
paulson@14267
   343
done
paulson@14267
   344
paulson@14267
   345
paulson@14267
   346
subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
paulson@14267
   347
paulson@14267
   348
(** first, a lemma to bound the remainder **)
paulson@14267
   349
paulson@14267
   350
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
paulson@14267
   351
apply (cut_tac m = q and n = c in mod_less_divisor)
paulson@14267
   352
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
paulson@14267
   353
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
paulson@14267
   354
apply (simp add: add_mult_distrib2)
paulson@14267
   355
done
paulson@10559
   356
paulson@14267
   357
lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]  
paulson@14267
   358
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
paulson@14267
   359
apply (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
   360
done
paulson@14267
   361
paulson@14267
   362
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
paulson@14267
   363
apply (case_tac "b=0", simp)
paulson@14267
   364
apply (case_tac "c=0", simp)
paulson@14267
   365
apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
paulson@14267
   366
done
paulson@14267
   367
paulson@14267
   368
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
paulson@14267
   369
apply (case_tac "b=0", simp)
paulson@14267
   370
apply (case_tac "c=0", simp)
paulson@14267
   371
apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
paulson@14267
   372
done
paulson@14267
   373
paulson@14267
   374
paulson@14267
   375
subsection{*Cancellation of Common Factors in Division*}
paulson@14267
   376
paulson@14267
   377
lemma div_mult_mult_lemma:
paulson@14267
   378
     "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
paulson@14267
   379
by (auto simp add: div_mult2_eq)
paulson@14267
   380
paulson@14267
   381
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
paulson@14267
   382
apply (case_tac "b = 0")
paulson@14267
   383
apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
paulson@14267
   384
done
paulson@14267
   385
paulson@14267
   386
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
paulson@14267
   387
apply (drule div_mult_mult1)
paulson@14267
   388
apply (auto simp add: mult_commute)
paulson@14267
   389
done
paulson@14267
   390
paulson@14267
   391
paulson@14267
   392
(*Distribution of Factors over Remainders:
paulson@14267
   393
paulson@14267
   394
Could prove these as in Integ/IntDiv.ML, but we already have
paulson@14267
   395
mod_mult_distrib and mod_mult_distrib2 above!
paulson@14267
   396
paulson@14267
   397
Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"
paulson@14267
   398
qed "mod_mult_mult1";
paulson@14267
   399
paulson@14267
   400
Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";
paulson@14267
   401
qed "mod_mult_mult2";
paulson@14267
   402
 ***)
paulson@14267
   403
paulson@14267
   404
subsection{*Further Facts about Quotient and Remainder*}
paulson@14267
   405
paulson@14267
   406
lemma div_1 [simp]: "m div Suc 0 = m"
paulson@15251
   407
apply (induct "m")
paulson@14267
   408
apply (simp_all (no_asm_simp) add: div_geq)
paulson@14267
   409
done
paulson@14267
   410
paulson@14267
   411
lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
paulson@14267
   412
by (simp add: div_geq)
paulson@14267
   413
paulson@14267
   414
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
paulson@15251
   415
apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
paulson@14267
   416
apply (simp add: add_commute)
paulson@14267
   417
apply (subst div_geq [symmetric], simp_all)
paulson@14267
   418
done
paulson@14267
   419
paulson@14267
   420
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
paulson@14267
   421
by (simp add: add_commute div_add_self2)
paulson@14267
   422
paulson@14267
   423
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
paulson@14267
   424
apply (subst div_add1_eq)
paulson@14267
   425
apply (subst div_mult1_eq, simp)
paulson@14267
   426
done
paulson@14267
   427
paulson@14267
   428
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
paulson@14267
   429
by (simp add: mult_commute div_mult_self1)
paulson@14267
   430
paulson@14267
   431
paulson@14267
   432
(* Monotonicity of div in first argument *)
paulson@14267
   433
lemma div_le_mono [rule_format (no_asm)]:
paulson@14267
   434
     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
   435
apply (case_tac "k=0", simp)
paulson@15251
   436
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
   437
apply (case_tac "n<k")
paulson@14267
   438
(* 1  case n<k *)
paulson@14267
   439
apply simp
paulson@14267
   440
(* 2  case n >= k *)
paulson@14267
   441
apply (case_tac "m<k")
paulson@14267
   442
(* 2.1  case m<k *)
paulson@14267
   443
apply simp
paulson@14267
   444
(* 2.2  case m>=k *)
nipkow@15439
   445
apply (simp add: div_geq diff_le_mono)
paulson@14267
   446
done
paulson@14267
   447
paulson@14267
   448
(* Antimonotonicity of div in second argument *)
paulson@14267
   449
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   450
apply (subgoal_tac "0<n")
paulson@14267
   451
 prefer 2 apply simp 
paulson@15251
   452
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   453
apply (rename_tac "k")
paulson@14267
   454
apply (case_tac "k<n", simp)
paulson@14267
   455
apply (subgoal_tac "~ (k<m) ")
paulson@14267
   456
 prefer 2 apply simp 
paulson@14267
   457
apply (simp add: div_geq)
paulson@15251
   458
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   459
 prefer 2
paulson@14267
   460
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   461
apply (rule le_trans, simp)
nipkow@15439
   462
apply (simp)
paulson@14267
   463
done
paulson@14267
   464
paulson@14267
   465
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   466
apply (case_tac "n=0", simp)
paulson@14267
   467
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   468
apply (rule div_le_mono2)
paulson@14267
   469
apply (simp_all (no_asm_simp))
paulson@14267
   470
done
paulson@14267
   471
paulson@14267
   472
(* Similar for "less than" *) 
paulson@17085
   473
lemma div_less_dividend [rule_format]:
paulson@14267
   474
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
paulson@15251
   475
apply (induct_tac m rule: nat_less_induct)
paulson@14267
   476
apply (rename_tac "m")
paulson@14267
   477
apply (case_tac "m<n", simp)
paulson@14267
   478
apply (subgoal_tac "0<n")
paulson@14267
   479
 prefer 2 apply simp 
paulson@14267
   480
apply (simp add: div_geq)
paulson@14267
   481
apply (case_tac "n<m")
paulson@15251
   482
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   483
  apply (rule impI less_trans_Suc)+
paulson@14267
   484
apply assumption
nipkow@15439
   485
  apply (simp_all)
paulson@14267
   486
done
paulson@14267
   487
paulson@17085
   488
declare div_less_dividend [simp]
paulson@17085
   489
paulson@14267
   490
text{*A fact for the mutilated chess board*}
paulson@14267
   491
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   492
apply (case_tac "n=0", simp)
paulson@15251
   493
apply (induct "m" rule: nat_less_induct)
paulson@14267
   494
apply (case_tac "Suc (na) <n")
paulson@14267
   495
(* case Suc(na) < n *)
paulson@14267
   496
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   497
(* case n \<le> Suc(na) *)
paulson@16796
   498
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   499
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   500
done
paulson@14267
   501
paulson@14437
   502
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
paulson@14437
   503
by (case_tac "n=0", auto)
paulson@14437
   504
paulson@14437
   505
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
paulson@14437
   506
by (case_tac "n=0", auto)
paulson@14437
   507
paulson@14267
   508
paulson@14267
   509
subsection{*The Divides Relation*}
paulson@14267
   510
paulson@14267
   511
lemma dvdI [intro?]: "n = m * k ==> m dvd n"
paulson@14267
   512
by (unfold dvd_def, blast)
paulson@14267
   513
paulson@14267
   514
lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
paulson@14267
   515
by (unfold dvd_def, blast)
nipkow@13152
   516
paulson@14267
   517
lemma dvd_0_right [iff]: "m dvd (0::nat)"
paulson@14267
   518
apply (unfold dvd_def)
paulson@14267
   519
apply (blast intro: mult_0_right [symmetric])
paulson@14267
   520
done
paulson@14267
   521
paulson@14267
   522
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
paulson@14267
   523
by (force simp add: dvd_def)
paulson@14267
   524
paulson@14267
   525
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
paulson@14267
   526
by (blast intro: dvd_0_left)
paulson@14267
   527
paulson@14267
   528
lemma dvd_1_left [iff]: "Suc 0 dvd k"
paulson@14267
   529
by (unfold dvd_def, simp)
paulson@14267
   530
paulson@14267
   531
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
paulson@14267
   532
by (simp add: dvd_def)
paulson@14267
   533
paulson@14267
   534
lemma dvd_refl [simp]: "m dvd (m::nat)"
paulson@14267
   535
apply (unfold dvd_def)
paulson@14267
   536
apply (blast intro: mult_1_right [symmetric])
paulson@14267
   537
done
paulson@14267
   538
paulson@14267
   539
lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
paulson@14267
   540
apply (unfold dvd_def)
paulson@14267
   541
apply (blast intro: mult_assoc)
paulson@14267
   542
done
paulson@14267
   543
paulson@14267
   544
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
paulson@14267
   545
apply (unfold dvd_def)
paulson@14267
   546
apply (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
paulson@14267
   547
done
paulson@14267
   548
paulson@14267
   549
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
paulson@14267
   550
apply (unfold dvd_def)
paulson@14267
   551
apply (blast intro: add_mult_distrib2 [symmetric])
paulson@14267
   552
done
paulson@14267
   553
paulson@14267
   554
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
paulson@14267
   555
apply (unfold dvd_def)
paulson@14267
   556
apply (blast intro: diff_mult_distrib2 [symmetric])
paulson@14267
   557
done
paulson@14267
   558
paulson@14267
   559
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
paulson@16796
   560
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
paulson@14267
   561
apply (blast intro: dvd_add)
paulson@14267
   562
done
paulson@14267
   563
paulson@14267
   564
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
paulson@14267
   565
by (drule_tac m = m in dvd_diff, auto)
paulson@14267
   566
paulson@14267
   567
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
paulson@14267
   568
apply (unfold dvd_def)
paulson@14267
   569
apply (blast intro: mult_left_commute)
paulson@14267
   570
done
paulson@14267
   571
paulson@14267
   572
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
paulson@14267
   573
apply (subst mult_commute)
paulson@14267
   574
apply (erule dvd_mult)
paulson@14267
   575
done
paulson@14267
   576
paulson@17084
   577
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
paulson@17084
   578
by (rule dvd_refl [THEN dvd_mult])
paulson@17084
   579
paulson@17084
   580
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
paulson@17084
   581
by (rule dvd_refl [THEN dvd_mult2])
paulson@14267
   582
paulson@14267
   583
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
paulson@14267
   584
apply (rule iffI)
paulson@14267
   585
apply (erule_tac [2] dvd_add)
paulson@14267
   586
apply (rule_tac [2] dvd_refl)
paulson@14267
   587
apply (subgoal_tac "n = (n+k) -k")
paulson@14267
   588
 prefer 2 apply simp 
paulson@14267
   589
apply (erule ssubst)
paulson@14267
   590
apply (erule dvd_diff)
paulson@14267
   591
apply (rule dvd_refl)
paulson@14267
   592
done
paulson@14267
   593
paulson@14267
   594
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
paulson@14267
   595
apply (unfold dvd_def)
paulson@14267
   596
apply (case_tac "n=0", auto)
paulson@14267
   597
apply (blast intro: mod_mult_distrib2 [symmetric])
paulson@14267
   598
done
paulson@14267
   599
paulson@14267
   600
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
paulson@14267
   601
apply (subgoal_tac "k dvd (m div n) *n + m mod n")
paulson@14267
   602
 apply (simp add: mod_div_equality)
paulson@14267
   603
apply (simp only: dvd_add dvd_mult)
paulson@14267
   604
done
paulson@14267
   605
paulson@14267
   606
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
paulson@14267
   607
by (blast intro: dvd_mod_imp_dvd dvd_mod)
paulson@14267
   608
paulson@14267
   609
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
paulson@14267
   610
apply (unfold dvd_def)
paulson@14267
   611
apply (erule exE)
paulson@14267
   612
apply (simp add: mult_ac)
paulson@14267
   613
done
paulson@14267
   614
paulson@14267
   615
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
paulson@14267
   616
apply auto
paulson@14267
   617
apply (subgoal_tac "m*n dvd m*1")
paulson@14267
   618
apply (drule dvd_mult_cancel, auto)
paulson@14267
   619
done
paulson@14267
   620
paulson@14267
   621
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
paulson@14267
   622
apply (subst mult_commute)
paulson@14267
   623
apply (erule dvd_mult_cancel1)
paulson@14267
   624
done
paulson@14267
   625
paulson@14267
   626
lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
paulson@14267
   627
apply (unfold dvd_def, clarify)
paulson@14267
   628
apply (rule_tac x = "k*ka" in exI)
paulson@14267
   629
apply (simp add: mult_ac)
paulson@14267
   630
done
paulson@14267
   631
paulson@14267
   632
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
paulson@14267
   633
by (simp add: dvd_def mult_assoc, blast)
paulson@14267
   634
paulson@14267
   635
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
paulson@14267
   636
apply (unfold dvd_def, clarify)
paulson@14267
   637
apply (rule_tac x = "i*k" in exI)
paulson@14267
   638
apply (simp add: mult_ac)
paulson@14267
   639
done
paulson@14267
   640
paulson@14267
   641
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
paulson@14267
   642
apply (unfold dvd_def, clarify)
paulson@14267
   643
apply (simp_all (no_asm_use) add: zero_less_mult_iff)
paulson@14267
   644
apply (erule conjE)
paulson@14267
   645
apply (rule le_trans)
paulson@14267
   646
apply (rule_tac [2] le_refl [THEN mult_le_mono])
paulson@14267
   647
apply (erule_tac [2] Suc_leI, simp)
paulson@14267
   648
done
paulson@14267
   649
paulson@14267
   650
lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
paulson@14267
   651
apply (unfold dvd_def)
paulson@14267
   652
apply (case_tac "k=0", simp, safe)
paulson@14267
   653
apply (simp add: mult_commute)
paulson@14267
   654
apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
paulson@14267
   655
apply (subst mult_commute, simp)
paulson@14267
   656
done
paulson@14267
   657
paulson@14267
   658
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
paulson@14267
   659
apply (subgoal_tac "m mod n = 0")
paulson@14267
   660
 apply (simp add: mult_div_cancel)
paulson@14267
   661
apply (simp only: dvd_eq_mod_eq_0)
paulson@14267
   662
done
paulson@14267
   663
haftmann@21408
   664
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
haftmann@21408
   665
apply (unfold dvd_def)
haftmann@21408
   666
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@21408
   667
apply (simp add: power_add)
haftmann@21408
   668
done
haftmann@21408
   669
haftmann@21408
   670
lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
haftmann@21408
   671
by (induct "n", auto)
haftmann@21408
   672
haftmann@21408
   673
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
haftmann@21408
   674
apply (induct "j")
haftmann@21408
   675
apply (simp_all add: le_Suc_eq)
haftmann@21408
   676
apply (blast dest!: dvd_mult_right)
haftmann@21408
   677
done
haftmann@21408
   678
haftmann@21408
   679
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
haftmann@21408
   680
apply (rule power_le_imp_le_exp, assumption)
haftmann@21408
   681
apply (erule dvd_imp_le, simp)
haftmann@21408
   682
done
haftmann@21408
   683
paulson@14267
   684
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
paulson@14267
   685
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
   686
paulson@17084
   687
lemmas mod_eq_0D = mod_eq_0_iff [THEN iffD1]
paulson@17084
   688
declare mod_eq_0D [dest!]
paulson@14267
   689
paulson@14267
   690
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
   691
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
paulson@14267
   692
apply (cut_tac m = m in mod_div_equality)
paulson@14267
   693
apply (simp only: add_ac)
paulson@14267
   694
apply (blast intro: sym)
paulson@14267
   695
done
paulson@14267
   696
paulson@14131
   697
nipkow@13152
   698
lemma split_div:
nipkow@13189
   699
 "P(n div k :: nat) =
nipkow@13189
   700
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
   701
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   702
proof
nipkow@13189
   703
  assume P: ?P
nipkow@13189
   704
  show ?Q
nipkow@13189
   705
  proof (cases)
nipkow@13189
   706
    assume "k = 0"
nipkow@13189
   707
    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   708
  next
nipkow@13189
   709
    assume not0: "k \<noteq> 0"
nipkow@13189
   710
    thus ?Q
nipkow@13189
   711
    proof (simp, intro allI impI)
nipkow@13189
   712
      fix i j
nipkow@13189
   713
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
   714
      show "P i"
nipkow@13189
   715
      proof (cases)
nipkow@13189
   716
	assume "i = 0"
nipkow@13189
   717
	with n j P show "P i" by simp
nipkow@13189
   718
      next
nipkow@13189
   719
	assume "i \<noteq> 0"
nipkow@13189
   720
	with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
   721
      qed
nipkow@13189
   722
    qed
nipkow@13189
   723
  qed
nipkow@13189
   724
next
nipkow@13189
   725
  assume Q: ?Q
nipkow@13189
   726
  show ?P
nipkow@13189
   727
  proof (cases)
nipkow@13189
   728
    assume "k = 0"
nipkow@13189
   729
    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   730
  next
nipkow@13189
   731
    assume not0: "k \<noteq> 0"
nipkow@13189
   732
    with Q have R: ?R by simp
nipkow@13189
   733
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   734
    show ?P by simp
nipkow@13189
   735
  qed
nipkow@13189
   736
qed
nipkow@13189
   737
berghofe@13882
   738
lemma split_div_lemma:
paulson@14267
   739
  "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
berghofe@13882
   740
  apply (rule iffI)
berghofe@13882
   741
  apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
nipkow@16733
   742
prefer 3; apply assumption
webertj@20432
   743
  apply (simp_all add: quorem_def) apply arith
berghofe@13882
   744
  apply (rule conjI)
berghofe@13882
   745
  apply (rule_tac P="%x. n * (m div n) \<le> x" in
berghofe@13882
   746
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
   747
  apply (simp only: add: mult_ac)
berghofe@13882
   748
  apply (rule_tac P="%x. x < n + n * (m div n)" in
berghofe@13882
   749
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
   750
  apply (simp only: add: mult_ac add_ac)
paulson@14208
   751
  apply (rule add_less_mono1, simp)
berghofe@13882
   752
  done
berghofe@13882
   753
berghofe@13882
   754
theorem split_div':
berghofe@13882
   755
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   756
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
   757
  apply (case_tac "0 < n")
berghofe@13882
   758
  apply (simp only: add: split_div_lemma)
berghofe@13882
   759
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
berghofe@13882
   760
  done
berghofe@13882
   761
nipkow@13189
   762
lemma split_mod:
nipkow@13189
   763
 "P(n mod k :: nat) =
nipkow@13189
   764
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   765
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   766
proof
nipkow@13189
   767
  assume P: ?P
nipkow@13189
   768
  show ?Q
nipkow@13189
   769
  proof (cases)
nipkow@13189
   770
    assume "k = 0"
nipkow@13189
   771
    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   772
  next
nipkow@13189
   773
    assume not0: "k \<noteq> 0"
nipkow@13189
   774
    thus ?Q
nipkow@13189
   775
    proof (simp, intro allI impI)
nipkow@13189
   776
      fix i j
nipkow@13189
   777
      assume "n = k*i + j" "j < k"
nipkow@13189
   778
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
   779
    qed
nipkow@13189
   780
  qed
nipkow@13189
   781
next
nipkow@13189
   782
  assume Q: ?Q
nipkow@13189
   783
  show ?P
nipkow@13189
   784
  proof (cases)
nipkow@13189
   785
    assume "k = 0"
nipkow@13189
   786
    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   787
  next
nipkow@13189
   788
    assume not0: "k \<noteq> 0"
nipkow@13189
   789
    with Q have R: ?R by simp
nipkow@13189
   790
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   791
    show ?P by simp
nipkow@13189
   792
  qed
nipkow@13189
   793
qed
nipkow@13189
   794
berghofe@13882
   795
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
berghofe@13882
   796
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
berghofe@13882
   797
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
   798
  apply arith
berghofe@13882
   799
  done
berghofe@13882
   800
paulson@14640
   801
subsection {*An ``induction'' law for modulus arithmetic.*}
paulson@14640
   802
paulson@14640
   803
lemma mod_induct_0:
paulson@14640
   804
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
   805
  and base: "P i" and i: "i<p"
paulson@14640
   806
  shows "P 0"
paulson@14640
   807
proof (rule ccontr)
paulson@14640
   808
  assume contra: "\<not>(P 0)"
paulson@14640
   809
  from i have p: "0<p" by simp
paulson@14640
   810
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
   811
  proof
paulson@14640
   812
    fix k
paulson@14640
   813
    show "?A k"
paulson@14640
   814
    proof (induct k)
paulson@14640
   815
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
   816
    next
paulson@14640
   817
      fix n
paulson@14640
   818
      assume ih: "?A n"
paulson@14640
   819
      show "?A (Suc n)"
paulson@14640
   820
      proof (clarsimp)
paulson@14640
   821
	assume y: "P (p - Suc n)"
paulson@14640
   822
	have n: "Suc n < p"
paulson@14640
   823
	proof (rule ccontr)
paulson@14640
   824
	  assume "\<not>(Suc n < p)"
paulson@14640
   825
	  hence "p - Suc n = 0"
paulson@14640
   826
	    by simp
paulson@14640
   827
	  with y contra show "False"
paulson@14640
   828
	    by simp
paulson@14640
   829
	qed
paulson@14640
   830
	hence n2: "Suc (p - Suc n) = p-n" by arith
paulson@14640
   831
	from p have "p - Suc n < p" by arith
paulson@14640
   832
	with y step have z: "P ((Suc (p - Suc n)) mod p)"
paulson@14640
   833
	  by blast
paulson@14640
   834
	show "False"
paulson@14640
   835
	proof (cases "n=0")
paulson@14640
   836
	  case True
paulson@14640
   837
	  with z n2 contra show ?thesis by simp
paulson@14640
   838
	next
paulson@14640
   839
	  case False
paulson@14640
   840
	  with p have "p-n < p" by arith
paulson@14640
   841
	  with z n2 False ih show ?thesis by simp
paulson@14640
   842
	qed
paulson@14640
   843
      qed
paulson@14640
   844
    qed
paulson@14640
   845
  qed
paulson@14640
   846
  moreover
paulson@14640
   847
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
   848
    by (blast dest: less_imp_add_positive)
paulson@14640
   849
  hence "0<k \<and> i=p-k" by auto
paulson@14640
   850
  moreover
paulson@14640
   851
  note base
paulson@14640
   852
  ultimately
paulson@14640
   853
  show "False" by blast
paulson@14640
   854
qed
paulson@14640
   855
paulson@14640
   856
lemma mod_induct:
paulson@14640
   857
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
   858
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
   859
  shows "P j"
paulson@14640
   860
proof -
paulson@14640
   861
  have "\<forall>j<p. P j"
paulson@14640
   862
  proof
paulson@14640
   863
    fix j
paulson@14640
   864
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
   865
    proof (induct j)
paulson@14640
   866
      from step base i show "?A 0"
paulson@14640
   867
	by (auto elim: mod_induct_0)
paulson@14640
   868
    next
paulson@14640
   869
      fix k
paulson@14640
   870
      assume ih: "?A k"
paulson@14640
   871
      show "?A (Suc k)"
paulson@14640
   872
      proof
paulson@14640
   873
	assume suc: "Suc k < p"
paulson@14640
   874
	hence k: "k<p" by simp
paulson@14640
   875
	with ih have "P k" ..
paulson@14640
   876
	with step k have "P (Suc k mod p)"
paulson@14640
   877
	  by blast
paulson@14640
   878
	moreover
paulson@14640
   879
	from suc have "Suc k mod p = Suc k"
paulson@14640
   880
	  by simp
paulson@14640
   881
	ultimately
paulson@14640
   882
	show "P (Suc k)" by simp
paulson@14640
   883
      qed
paulson@14640
   884
    qed
paulson@14640
   885
  qed
paulson@14640
   886
  with j show ?thesis by blast
paulson@14640
   887
qed
paulson@14640
   888
paulson@14640
   889
chaieb@18202
   890
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
chaieb@18202
   891
  apply (rule trans [symmetric])
chaieb@18202
   892
  apply (rule mod_add1_eq, simp)
chaieb@18202
   893
  apply (rule mod_add1_eq [symmetric])
chaieb@18202
   894
  done
chaieb@18202
   895
chaieb@18202
   896
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
chaieb@18202
   897
apply (rule trans [symmetric])
chaieb@18202
   898
apply (rule mod_add1_eq, simp)
chaieb@18202
   899
apply (rule mod_add1_eq [symmetric])
chaieb@18202
   900
done
chaieb@18202
   901
haftmann@20589
   902
haftmann@20589
   903
subsection {* Code generation for div and mod *}
haftmann@20589
   904
haftmann@20589
   905
definition
haftmann@20589
   906
  "divmod (m\<Colon>nat) n = (m div n, m mod n)"
haftmann@20589
   907
haftmann@20589
   908
lemma divmod_zero [code]:
haftmann@20589
   909
  "divmod m 0 = (0, m)"
haftmann@20589
   910
  unfolding divmod_def by simp
haftmann@20589
   911
haftmann@20589
   912
lemma divmod_succ [code]:
haftmann@20589
   913
  "divmod m (Suc k) = (if m < Suc k then (0, m) else
haftmann@20589
   914
    let
haftmann@20589
   915
      (p, q) = divmod (m - Suc k) (Suc k)
haftmann@20589
   916
    in (Suc p, q)
haftmann@20589
   917
  )"
haftmann@20589
   918
  unfolding divmod_def Let_def split_def
haftmann@20589
   919
  by (auto intro: div_geq mod_geq)
haftmann@20589
   920
haftmann@20589
   921
lemma div_divmod [code]:
haftmann@20589
   922
  "m div n = fst (divmod m n)"
haftmann@20589
   923
  unfolding divmod_def by simp
haftmann@20589
   924
haftmann@20589
   925
lemma mod_divmod [code]:
haftmann@20589
   926
  "m mod n = snd (divmod m n)"
haftmann@20589
   927
  unfolding divmod_def by simp
haftmann@20589
   928
haftmann@21191
   929
code_modulename SML
haftmann@21191
   930
  Divides Integer
haftmann@20640
   931
haftmann@21911
   932
code_modulename OCaml
haftmann@21911
   933
  Divides Integer
haftmann@21911
   934
haftmann@20589
   935
hide (open) const divmod
haftmann@20589
   936
haftmann@20589
   937
haftmann@20589
   938
subsection {* Legacy bindings *}
haftmann@20589
   939
paulson@14267
   940
ML
paulson@14267
   941
{*
paulson@14267
   942
val div_def = thm "div_def"
paulson@14267
   943
val mod_def = thm "mod_def"
paulson@14267
   944
val dvd_def = thm "dvd_def"
paulson@14267
   945
val quorem_def = thm "quorem_def"
paulson@14267
   946
paulson@14267
   947
val wf_less_trans = thm "wf_less_trans";
paulson@14267
   948
val mod_eq = thm "mod_eq";
paulson@14267
   949
val div_eq = thm "div_eq";
paulson@14267
   950
val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
paulson@14267
   951
val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
paulson@14267
   952
val mod_less = thm "mod_less";
paulson@14267
   953
val mod_geq = thm "mod_geq";
paulson@14267
   954
val le_mod_geq = thm "le_mod_geq";
paulson@14267
   955
val mod_if = thm "mod_if";
paulson@14267
   956
val mod_1 = thm "mod_1";
paulson@14267
   957
val mod_self = thm "mod_self";
paulson@14267
   958
val mod_add_self2 = thm "mod_add_self2";
paulson@14267
   959
val mod_add_self1 = thm "mod_add_self1";
paulson@14267
   960
val mod_mult_self1 = thm "mod_mult_self1";
paulson@14267
   961
val mod_mult_self2 = thm "mod_mult_self2";
paulson@14267
   962
val mod_mult_distrib = thm "mod_mult_distrib";
paulson@14267
   963
val mod_mult_distrib2 = thm "mod_mult_distrib2";
paulson@14267
   964
val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
paulson@14267
   965
val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
paulson@14267
   966
val div_less = thm "div_less";
paulson@14267
   967
val div_geq = thm "div_geq";
paulson@14267
   968
val le_div_geq = thm "le_div_geq";
paulson@14267
   969
val div_if = thm "div_if";
paulson@14267
   970
val mod_div_equality = thm "mod_div_equality";
paulson@14267
   971
val mod_div_equality2 = thm "mod_div_equality2";
paulson@14267
   972
val div_mod_equality = thm "div_mod_equality";
paulson@14267
   973
val div_mod_equality2 = thm "div_mod_equality2";
paulson@14267
   974
val mult_div_cancel = thm "mult_div_cancel";
paulson@14267
   975
val mod_less_divisor = thm "mod_less_divisor";
paulson@14267
   976
val div_mult_self_is_m = thm "div_mult_self_is_m";
paulson@14267
   977
val div_mult_self1_is_m = thm "div_mult_self1_is_m";
paulson@14267
   978
val unique_quotient_lemma = thm "unique_quotient_lemma";
paulson@14267
   979
val unique_quotient = thm "unique_quotient";
paulson@14267
   980
val unique_remainder = thm "unique_remainder";
paulson@14267
   981
val div_0 = thm "div_0";
paulson@14267
   982
val mod_0 = thm "mod_0";
paulson@14267
   983
val div_mult1_eq = thm "div_mult1_eq";
paulson@14267
   984
val mod_mult1_eq = thm "mod_mult1_eq";
paulson@14267
   985
val mod_mult1_eq' = thm "mod_mult1_eq'";
paulson@14267
   986
val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
paulson@14267
   987
val div_add1_eq = thm "div_add1_eq";
paulson@14267
   988
val mod_add1_eq = thm "mod_add1_eq";
chaieb@18202
   989
val mod_add_left_eq = thm "mod_add_left_eq";
chaieb@18202
   990
 val mod_add_right_eq = thm "mod_add_right_eq";
paulson@14267
   991
val mod_lemma = thm "mod_lemma";
paulson@14267
   992
val div_mult2_eq = thm "div_mult2_eq";
paulson@14267
   993
val mod_mult2_eq = thm "mod_mult2_eq";
paulson@14267
   994
val div_mult_mult_lemma = thm "div_mult_mult_lemma";
paulson@14267
   995
val div_mult_mult1 = thm "div_mult_mult1";
paulson@14267
   996
val div_mult_mult2 = thm "div_mult_mult2";
paulson@14267
   997
val div_1 = thm "div_1";
paulson@14267
   998
val div_self = thm "div_self";
paulson@14267
   999
val div_add_self2 = thm "div_add_self2";
paulson@14267
  1000
val div_add_self1 = thm "div_add_self1";
paulson@14267
  1001
val div_mult_self1 = thm "div_mult_self1";
paulson@14267
  1002
val div_mult_self2 = thm "div_mult_self2";
paulson@14267
  1003
val div_le_mono = thm "div_le_mono";
paulson@14267
  1004
val div_le_mono2 = thm "div_le_mono2";
paulson@14267
  1005
val div_le_dividend = thm "div_le_dividend";
paulson@14267
  1006
val div_less_dividend = thm "div_less_dividend";
paulson@14267
  1007
val mod_Suc = thm "mod_Suc";
paulson@14267
  1008
val dvdI = thm "dvdI";
paulson@14267
  1009
val dvdE = thm "dvdE";
paulson@14267
  1010
val dvd_0_right = thm "dvd_0_right";
paulson@14267
  1011
val dvd_0_left = thm "dvd_0_left";
paulson@14267
  1012
val dvd_0_left_iff = thm "dvd_0_left_iff";
paulson@14267
  1013
val dvd_1_left = thm "dvd_1_left";
paulson@14267
  1014
val dvd_1_iff_1 = thm "dvd_1_iff_1";
paulson@14267
  1015
val dvd_refl = thm "dvd_refl";
paulson@14267
  1016
val dvd_trans = thm "dvd_trans";
paulson@14267
  1017
val dvd_anti_sym = thm "dvd_anti_sym";
paulson@14267
  1018
val dvd_add = thm "dvd_add";
paulson@14267
  1019
val dvd_diff = thm "dvd_diff";
paulson@14267
  1020
val dvd_diffD = thm "dvd_diffD";
paulson@14267
  1021
val dvd_diffD1 = thm "dvd_diffD1";
paulson@14267
  1022
val dvd_mult = thm "dvd_mult";
paulson@14267
  1023
val dvd_mult2 = thm "dvd_mult2";
paulson@14267
  1024
val dvd_reduce = thm "dvd_reduce";
paulson@14267
  1025
val dvd_mod = thm "dvd_mod";
paulson@14267
  1026
val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
paulson@14267
  1027
val dvd_mod_iff = thm "dvd_mod_iff";
paulson@14267
  1028
val dvd_mult_cancel = thm "dvd_mult_cancel";
paulson@14267
  1029
val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
paulson@14267
  1030
val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
paulson@14267
  1031
val mult_dvd_mono = thm "mult_dvd_mono";
paulson@14267
  1032
val dvd_mult_left = thm "dvd_mult_left";
paulson@14267
  1033
val dvd_mult_right = thm "dvd_mult_right";
paulson@14267
  1034
val dvd_imp_le = thm "dvd_imp_le";
paulson@14267
  1035
val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
paulson@14267
  1036
val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
paulson@14267
  1037
val mod_eq_0_iff = thm "mod_eq_0_iff";
paulson@14267
  1038
val mod_eqD = thm "mod_eqD";
paulson@14267
  1039
*}
paulson@14267
  1040
paulson@14267
  1041
nipkow@13189
  1042
(*
nipkow@13189
  1043
lemma split_div:
nipkow@13152
  1044
assumes m: "m \<noteq> 0"
nipkow@13152
  1045
shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
nipkow@13152
  1046
       (is "?P = ?Q")
nipkow@13152
  1047
proof
nipkow@13152
  1048
  assume P: ?P
nipkow@13152
  1049
  show ?Q
nipkow@13152
  1050
  proof (intro allI impI)
nipkow@13152
  1051
    fix i j
nipkow@13152
  1052
    assume n: "n = m*i + j" and j: "j < m"
nipkow@13152
  1053
    show "P i"
nipkow@13152
  1054
    proof (cases)
nipkow@13152
  1055
      assume "i = 0"
nipkow@13152
  1056
      with n j P show "P i" by simp
nipkow@13152
  1057
    next
nipkow@13152
  1058
      assume "i \<noteq> 0"
nipkow@13152
  1059
      with n j P show "P i" by (simp add:add_ac div_mult_self1)
nipkow@13152
  1060
    qed
nipkow@13152
  1061
  qed
nipkow@13152
  1062
next
nipkow@13152
  1063
  assume Q: ?Q
nipkow@13152
  1064
  from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
nipkow@13517
  1065
  show ?P by simp
nipkow@13152
  1066
qed
nipkow@13152
  1067
nipkow@13152
  1068
lemma split_mod:
nipkow@13152
  1069
assumes m: "m \<noteq> 0"
nipkow@13152
  1070
shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
nipkow@13152
  1071
       (is "?P = ?Q")
nipkow@13152
  1072
proof
nipkow@13152
  1073
  assume P: ?P
nipkow@13152
  1074
  show ?Q
nipkow@13152
  1075
  proof (intro allI impI)
nipkow@13152
  1076
    fix i j
nipkow@13152
  1077
    assume "n = m*i + j" "j < m"
nipkow@13152
  1078
    thus "P j" using m P by(simp add:add_ac mult_ac)
nipkow@13152
  1079
  qed
nipkow@13152
  1080
next
nipkow@13152
  1081
  assume Q: ?Q
nipkow@13152
  1082
  from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
nipkow@13517
  1083
  show ?P by simp
nipkow@13152
  1084
qed
nipkow@13189
  1085
*)
paulson@3366
  1086
end