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permissions  rwrr 
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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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15131  9 
theory Divides 
21408  10 
imports Datatype Power 
15131  11 
begin 
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8902  13 
(*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 + 
21408  16 
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) 

25 

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end 

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21408  28 
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 (jn)) 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 (jn))) 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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22718  45 
definition 
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quorem :: "(nat*nat) * (nat*nat) => bool" where 

21408  47 
(*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?*) 

22718  49 
"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 (jn))" 
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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 (jn)))" 
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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 = (mn) 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 = (mn) 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 (mn) mod n)" 
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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+mn) 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((mn) 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((mn) div n)" 
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lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((mn) div n))" 
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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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22718  195 
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 = 
22718  200 
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)" 
22718  216 
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)" 
22718  219 
apply (induct m rule: nat_less_induct) 
220 
apply (rename_tac m) 

221 
apply (case_tac "m<n", simp) 

222 
txt{*case @{term "n \<le> m"}*} 

223 
apply (simp add: mod_geq) 

224 
done 

15439  225 

226 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 

22718  227 
apply (drule mod_less_divisor [where m = m]) 
228 
apply simp 

229 
done 

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lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
22718  232 
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)" 
22718  235 
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: 
22718  243 
"[ b*q' + r' \<le> b*q + r; x < b; r < b ] 
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==> q' \<le> (q::nat)" 
22718  245 
apply (rule leI) 
246 
apply (subst less_iff_Suc_add) 

247 
apply (auto simp add: add_mult_distrib2) 

248 
done 

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lemma unique_quotient: 
22718  251 
"[ quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); 0 < b ] 
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==> q = q'" 
22718  253 
apply (simp add: split_ifs quorem_def) 
254 
apply (blast intro: order_antisym 

255 
dest: order_eq_refl [THEN unique_quotient_lemma] sym) 

256 
done 

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lemma unique_remainder: 
22718  259 
"[ quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); 0 < b ] 
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==> r = r'" 
22718  261 
apply (subgoal_tac "q = q'") 
262 
prefer 2 apply (blast intro: unique_quotient) 

263 
apply (simp add: quorem_def) 

264 
done 

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lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))" 
22718  267 
unfolding quorem_def by simp 
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lemma quorem_div: "[ quorem((a,b),(q,r)); 0 < b ] ==> a div b = q" 
22718  270 
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" 
22718  273 
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)" 
22718  278 
by (cases "m = 0") simp_all 
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lemma mod_0 [simp]: "0 mod m = (0::nat)" 
22718  281 
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: 
22718  286 
"[ 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))" 
22718  288 
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2) 
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lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)" 
22718  291 
apply (cases "c = 0", simp) 
292 
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div]) 

293 
done 

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lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)" 
22718  296 
apply (cases "c = 0", simp) 
297 
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod]) 

298 
done 

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lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c" 
22718  301 
apply (rule trans) 
302 
apply (rule_tac s = "b*a mod c" in trans) 

303 
apply (rule_tac [2] mod_mult1_eq) 

304 
apply (simp_all add: mult_commute) 

305 
done 

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lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c" 
22718  308 
apply (rule mod_mult1_eq' [THEN trans]) 
309 
apply (rule mod_mult1_eq) 

310 
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: 
22718  315 
"[ 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))" 
22718  317 
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2) 
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(*NOT suitable for rewriting: the RHS has an instance of the LHS*) 
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lemma div_add1_eq: 
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"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" 
22718  322 
apply (cases "c = 0", simp) 
323 
apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod) 

324 
done 

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lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c" 
22718  327 
apply (cases "c = 0", simp) 
328 
apply (blast intro: quorem_div_mod quorem_div_mod quorem_add1_eq [THEN quorem_mod]) 

329 
done 

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subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*} 
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(** first, a lemma to bound the remainder **) 
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lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  337 
apply (cut_tac m = q and n = c in mod_less_divisor) 
338 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

339 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

340 
apply (simp add: add_mult_distrib2) 

341 
done 

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22718  343 
lemma quorem_mult2_eq: "[ quorem ((a,b), (q,r)); 0 < b; 0 < c ] 
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==> quorem ((a, b*c), (q div c, b*(q mod c) + r))" 
22718  345 
by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma) 
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lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  348 
apply (cases "b = 0", simp) 
349 
apply (cases "c = 0", simp) 

350 
apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div]) 

351 
done 

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lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  354 
apply (cases "b = 0", simp) 
355 
apply (cases "c = 0", simp) 

356 
apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod]) 

357 
done 

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358 

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359 

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360 
subsection{*Cancellation of Common Factors in Division*} 
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362 
lemma div_mult_mult_lemma: 
22718  363 
"[ (0::nat) < b; 0 < c ] ==> (c*a) div (c*b) = a div b" 
364 
by (auto simp add: div_mult2_eq) 

14267
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365 

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366 
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b" 
22718  367 
apply (cases "b = 0") 
368 
apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma) 

369 
done 

14267
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370 

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371 
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b" 
22718  372 
apply (drule div_mult_mult1) 
373 
apply (auto simp add: mult_commute) 

374 
done 

14267
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375 

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376 

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377 
(*Distribution of Factors over Remainders: 
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378 

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379 
Could prove these as in Integ/IntDiv.ML, but we already have 
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380 
mod_mult_distrib and mod_mult_distrib2 above! 
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381 

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382 
Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)" 
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383 
qed "mod_mult_mult1"; 
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384 

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385 
Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)"; 
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386 
qed "mod_mult_mult2"; 
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387 
***) 
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388 

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389 
subsection{*Further Facts about Quotient and Remainder*} 
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390 

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391 
lemma div_1 [simp]: "m div Suc 0 = m" 
22718  392 
by (induct m) (simp_all add: div_geq) 
14267
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393 

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394 
lemma div_self [simp]: "0<n ==> n div n = (1::nat)" 
22718  395 
by (simp add: div_geq) 
14267
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396 

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397 
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)" 
22718  398 
apply (subgoal_tac "(n + m) div n = Suc ((n+mn) div n) ") 
399 
apply (simp add: add_commute) 

400 
apply (subst div_geq [symmetric], simp_all) 

401 
done 

14267
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changeset

402 

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403 
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)" 
22718  404 
by (simp add: add_commute div_add_self2) 
14267
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parents:
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changeset

405 

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406 
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n" 
22718  407 
apply (subst div_add1_eq) 
408 
apply (subst div_mult1_eq, simp) 

409 
done 

14267
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parents:
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changeset

410 

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411 
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)" 
22718  412 
by (simp add: mult_commute div_mult_self1) 
14267
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parents:
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changeset

413 

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parents:
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diff
changeset

414 

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415 
(* Monotonicity of div in first argument *) 
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416 
lemma div_le_mono [rule_format (no_asm)]: 
22718  417 
"\<forall>m::nat. m \<le> n > (m div k) \<le> (n div k)" 
14267
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418 
apply (case_tac "k=0", simp) 
15251  419 
apply (induct "n" rule: nat_less_induct, clarify) 
14267
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420 
apply (case_tac "n<k") 
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421 
(* 1 case n<k *) 
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422 
apply simp 
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423 
(* 2 case n >= k *) 
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424 
apply (case_tac "m<k") 
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425 
(* 2.1 case m<k *) 
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426 
apply simp 
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427 
(* 2.2 case m>=k *) 
15439  428 
apply (simp add: div_geq diff_le_mono) 
14267
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429 
done 
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parents:
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430 

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431 
(* Antimonotonicity of div in second argument *) 
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432 
lemma div_le_mono2: "!!m::nat. [ 0<m; m\<le>n ] ==> (k div n) \<le> (k div m)" 
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433 
apply (subgoal_tac "0<n") 
22718  434 
prefer 2 apply simp 
15251  435 
apply (induct_tac k rule: nat_less_induct) 
14267
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436 
apply (rename_tac "k") 
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437 
apply (case_tac "k<n", simp) 
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438 
apply (subgoal_tac "~ (k<m) ") 
22718  439 
prefer 2 apply simp 
14267
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440 
apply (simp add: div_geq) 
15251  441 
apply (subgoal_tac "(kn) div n \<le> (km) div n") 
14267
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442 
prefer 2 
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443 
apply (blast intro: div_le_mono diff_le_mono2) 
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444 
apply (rule le_trans, simp) 
15439  445 
apply (simp) 
14267
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446 
done 
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parents:
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diff
changeset

447 

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parents:
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448 
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)" 
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449 
apply (case_tac "n=0", simp) 
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changeset

450 
apply (subgoal_tac "m div n \<le> m div 1", simp) 
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451 
apply (rule div_le_mono2) 
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452 
apply (simp_all (no_asm_simp)) 
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453 
done 
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454 

22718  455 
(* Similar for "less than" *) 
17085  456 
lemma div_less_dividend [rule_format]: 
14267
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paulson
parents:
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457 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  458 
apply (induct_tac m rule: nat_less_induct) 
14267
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parents:
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changeset

459 
apply (rename_tac "m") 
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diff
changeset

460 
apply (case_tac "m<n", simp) 
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461 
apply (subgoal_tac "0<n") 
22718  462 
prefer 2 apply simp 
14267
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paulson
parents:
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diff
changeset

463 
apply (simp add: div_geq) 
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changeset

464 
apply (case_tac "n<m") 
15251  465 
apply (subgoal_tac "(mn) div n < (mn) ") 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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changeset

466 
apply (rule impI less_trans_Suc)+ 
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diff
changeset

467 
apply assumption 
15439  468 
apply (simp_all) 
14267
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parents:
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changeset

469 
done 
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parents:
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470 

17085  471 
declare div_less_dividend [simp] 
472 

14267
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parents:
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changeset

473 
text{*A fact for the mutilated chess board*} 
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changeset

474 
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))" 
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changeset

475 
apply (case_tac "n=0", simp) 
15251  476 
apply (induct "m" rule: nat_less_induct) 
14267
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parents:
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changeset

477 
apply (case_tac "Suc (na) <n") 
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478 
(* case Suc(na) < n *) 
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changeset

479 
apply (frule lessI [THEN less_trans], simp add: less_not_refl3) 
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changeset

480 
(* case n \<le> Suc(na) *) 
16796  481 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  482 
apply (auto simp add: Suc_diff_le le_mod_geq) 
14267
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diff
changeset

483 
done 
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parents:
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484 

14437  485 
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)" 
22718  486 
by (cases "n = 0") auto 
14437  487 

488 
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)" 

22718  489 
by (cases "n = 0") auto 
14437  490 

14267
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parents:
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changeset

491 

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parents:
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changeset

492 
subsection{*The Divides Relation*} 
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changeset

493 

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changeset

494 
lemma dvdI [intro?]: "n = m * k ==> m dvd n" 
22718  495 
unfolding dvd_def by blast 
14267
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paulson
parents:
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diff
changeset

496 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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changeset

497 
lemma dvdE [elim?]: "!!P. [m dvd n; !!k. n = m*k ==> P] ==> P" 
22718  498 
unfolding dvd_def by blast 
13152  499 

14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset

500 
lemma dvd_0_right [iff]: "m dvd (0::nat)" 
22718  501 
unfolding dvd_def by (blast intro: mult_0_right [symmetric]) 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset

502 

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parents:
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diff
changeset

503 
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)" 
22718  504 
by (force simp add: dvd_def) 
14267
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paulson
parents:
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diff
changeset

505 

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parents:
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diff
changeset

506 
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)" 
22718  507 
by (blast intro: dvd_0_left) 
14267
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paulson
parents:
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diff
changeset

508 

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parents:
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diff
changeset

509 
lemma dvd_1_left [iff]: "Suc 0 dvd k" 
22718  510 
unfolding dvd_def by simp 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset

511 

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parents:
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diff
changeset

512 
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)" 
22718  513 
by (simp add: dvd_def) 
14267
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parents:
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diff
changeset

514 

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parents:
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diff
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515 
lemma dvd_refl [simp]: "m dvd (m::nat)" 
22718  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  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  522 
unfolding dvd_def 
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  526 
unfolding dvd_def 
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 (mn :: nat)" 
22718  530 
unfolding dvd_def 
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 mn; k dvd n; n\<le>m ] ==> k dvd (m::nat)" 
22718  534 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 
535 
apply (blast intro: dvd_add) 

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 mn; k dvd m; n\<le>m ] ==> k dvd (n::nat)" 
22718  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  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  545 
apply (subst mult_commute) 
546 
apply (erule dvd_mult) 

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  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  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  556 
apply (rule iffI) 
557 
apply (erule_tac [2] dvd_add) 

558 
apply (rule_tac [2] dvd_refl) 

559 
apply (subgoal_tac "n = (n+k) k") 

560 
prefer 2 apply simp 

561 
apply (erule ssubst) 

562 
apply (erule dvd_diff) 

563 
apply (rule dvd_refl) 

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  567 
unfolding dvd_def 
568 
apply (case_tac "n = 0", auto) 

569 
apply (blast intro: mod_mult_distrib2 [symmetric]) 

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  573 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
574 
apply (simp add: mod_div_equality) 

575 
apply (simp only: dvd_add dvd_mult) 

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  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  582 
unfolding dvd_def 
583 
apply (erule exE) 

584 
apply (simp add: mult_ac) 

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  588 
apply auto 
589 
apply (subgoal_tac "m*n dvd m*1") 

590 
apply (drule dvd_mult_cancel, auto) 

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  594 
apply (subst mult_commute) 
595 
apply (erule dvd_mult_cancel1) 

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  599 
apply (unfold dvd_def, clarify) 
600 
apply (rule_tac x = "k*ka" in exI) 

601 
apply (simp add: mult_ac) 

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  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  608 
apply (unfold dvd_def, clarify) 
609 
apply (rule_tac x = "i*k" in exI) 

610 
apply (simp add: mult_ac) 

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  614 
apply (unfold dvd_def, clarify) 
615 
apply (simp_all (no_asm_use) add: zero_less_mult_iff) 

616 
apply (erule conjE) 

617 
apply (rule le_trans) 

618 
apply (rule_tac [2] le_refl [THEN mult_le_mono]) 

619 
apply (erule_tac [2] Suc_leI, simp) 

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  623 
apply (unfold dvd_def) 
624 
apply (case_tac "k=0", simp, safe) 

625 
apply (simp add: mult_commute) 

626 
apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst]) 

627 
apply (subst mult_commute, simp) 

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  631 
apply (subgoal_tac "m mod n = 0") 
632 
apply (simp add: mult_div_cancel) 

633 
apply (simp only: dvd_eq_mod_eq_0) 

634 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

635 

21408  636 
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n" 
22718  637 
apply (unfold dvd_def) 
638 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 

639 
apply (simp add: power_add) 

640 
done 

21408  641 

642 
lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat)  n=0)" 

22718  643 
by (induct n) auto 
21408  644 

645 
lemma power_le_dvd [rule_format]: "k^j dvd n > i\<le>j > k^i dvd (n::nat)" 

22718  646 
apply (induct j) 
647 
apply (simp_all add: le_Suc_eq) 

648 
apply (blast dest!: dvd_mult_right) 

649 
done 

21408  650 

651 
lemma power_dvd_imp_le: "[i^m dvd i^n; (1::nat) < i] ==> m \<le> n" 

22718  652 
apply (rule power_le_imp_le_exp, assumption) 
653 
apply (erule dvd_imp_le, simp) 

654 
done 

21408  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  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  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  663 
apply (cut_tac m = m in mod_div_equality) 
664 
apply (simp only: add_ac) 

665 
apply (blast intro: sym) 

666 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

667 

14131  668 

13152  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  687 
assume "i = 0" 
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  690 
assume "i \<noteq> 0" 
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  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  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  711 
apply (rule iffI) 
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  715 
apply (rule conjI) 
716 
apply (rule_tac P="%x. n * (m div n) \<le> x" in 

717 
subst [OF mod_div_equality [of _ n]]) 

718 
apply (simp only: add: mult_ac) 

719 
apply (rule_tac P="%x. x < n + n * (m div n)" in 

720 
subst [OF mod_div_equality [of _ n]]) 

721 
apply (simp only: add: mult_ac add_ac) 

14208  722 
apply (rule add_less_mono1, simp) 
13882  723 
done 
724 

725 
theorem split_div': 

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  728 
apply (case_tac "0 < n") 
729 
apply (simp only: add: split_div_lemma) 

730 
apply (simp_all add: DIVISION_BY_ZERO_DIV) 

731 
done 

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  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  766 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
767 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

768 
subst [OF mod_div_equality [of _ n]]) 

769 
apply arith 

770 
done 

771 

22800  772 
lemma div_mod_equality': 
773 
fixes m n :: nat 

774 
shows "m div n * n = m  m mod n" 

775 
proof  

776 
have "m mod n \<le> m mod n" .. 

777 
from div_mod_equality have 

778 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

779 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

780 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

781 
by simp 

782 
then show ?thesis by simp 

783 
qed 

784 

785 

14640  786 
subsection {*An ``induction'' law for modulus arithmetic.*} 
787 

788 
lemma mod_induct_0: 

789 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

790 
and base: "P i" and i: "i<p" 

791 
shows "P 0" 

792 
proof (rule ccontr) 

793 
assume contra: "\<not>(P 0)" 

794 
from i have p: "0<p" by simp 

795 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

796 
proof 

797 
fix k 

798 
show "?A k" 

799 
proof (induct k) 

800 
show "?A 0" by simp  "by contradiction" 

801 
next 

802 
fix n 

803 
assume ih: "?A n" 

804 
show "?A (Suc n)" 

805 
proof (clarsimp) 

22718  806 
assume y: "P (p  Suc n)" 
807 
have n: "Suc n < p" 

808 
proof (rule ccontr) 

809 
assume "\<not>(Suc n < p)" 

810 
hence "p  Suc n = 0" 

811 
by simp 

812 
with y contra show "False" 

813 
by simp 

814 
qed 

815 
hence n2: "Suc (p  Suc n) = pn" by arith 

816 
from p have "p  Suc n < p" by arith 

817 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

818 
by blast 

819 
show "False" 

820 
proof (cases "n=0") 

821 
case True 

822 
with z n2 contra show ?thesis by simp 

823 
next 

824 
case False 

825 
with p have "pn < p" by arith 

826 
with z n2 False ih show ?thesis by simp 

827 
qed 

14640  828 
qed 
829 
qed 

830 
qed 

831 
moreover 

832 
from i obtain k where "0<k \<and> i+k=p" 

833 
by (blast dest: less_imp_add_positive) 

834 
hence "0<k \<and> i=pk" by auto 

835 
moreover 

836 
note base 

837 
ultimately 

838 
show "False" by blast 

839 
qed 

840 

841 
lemma mod_induct: 

842 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

843 
and base: "P i" and i: "i<p" and j: "j<p" 

844 
shows "P j" 

845 
proof  

846 
have "\<forall>j<p. P j" 

847 
proof 

848 
fix j 

849 
show "j<p \<longrightarrow> P j" (is "?A j") 

850 
proof (induct j) 

851 
from step base i show "?A 0" 

22718  852 
by (auto elim: mod_induct_0) 
14640  853 
next 
854 
fix k 

855 
assume ih: "?A k" 

856 
show "?A (Suc k)" 

857 
proof 

22718  858 
assume suc: "Suc k < p" 
859 
hence k: "k<p" by simp 

860 
with ih have "P k" .. 

861 
with step k have "P (Suc k mod p)" 

862 
by blast 

863 
moreover 

864 
from suc have "Suc k mod p = Suc k" 

865 
by simp 

866 
ultimately 

867 
show "P (Suc k)" by simp 

14640  868 
qed 
869 
qed 

870 
qed 

871 
with j show ?thesis by blast 

872 
qed 

873 

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  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  882 
apply (rule trans [symmetric]) 
883 
apply (rule mod_add1_eq, simp) 

884 
apply (rule mod_add1_eq [symmetric]) 

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  887 
lemma mod_div_decomp: 
888 
fixes n k :: nat 

889 
obtains m q where "m = n div k" and "q = n mod k" 

890 
and "n = m * k + q" 

891 
proof  

892 
from mod_div_equality have "n = n div k * k + n mod k" by auto 

893 
moreover have "n div k = n div k" .. 

894 
moreover have "n mod k = n mod k" .. 

895 
note that ultimately show thesis by blast 

896 
qed 

897 

20589  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  900 

22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22718
diff
changeset

901 
definition [code nofunc]: 
20589  902 
"divmod (m\<Colon>nat) n = (m div n, m mod n)" 
903 

22718  904 
lemma divmod_zero [code]: "divmod m 0 = (0, m)" 
20589  905 
unfolding divmod_def by simp 
906 

907 
lemma divmod_succ [code]: 

908 
"divmod m (Suc k) = (if m < Suc k then (0, m) else 

909 
let 

910 
(p, q) = divmod (m  Suc k) (Suc k) 

22718  911 
in (Suc p, q))" 
20589  912 
unfolding divmod_def Let_def split_def 
913 
by (auto intro: div_geq mod_geq) 

914 

22718  915 
lemma div_divmod [code]: "m div n = fst (divmod m n)" 
20589  916 
unfolding divmod_def by simp 
917 

22718  918 
lemma mod_divmod [code]: "m mod n = snd (divmod m n)" 
20589  919 
unfolding divmod_def by simp 
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  930 
code_modulename SML 
931 
Divides Integer 

20640  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  937 

938 
subsection {* Legacy bindings *} 

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  1043 
assumes m: "m \<noteq> 0" 
1044 
shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)" 

1045 
(is "?P = ?Q") 

1046 
proof 

1047 
assume P: ?P 

1048 
show ?Q 

1049 
proof (intro allI impI) 

1050 
fix i j 

1051 
assume n: "n = m*i + j" and j: "j < m" 

1052 
show "P i" 

1053 
proof (cases) 

1054 
assume "i = 0" 

1055 
with n j P show "P i" by simp 

1056 
next 

1057 
assume "i \<noteq> 0" 

1058 
with n j P show "P i" by (simp add:add_ac div_mult_self1) 

1059 
qed 

1060 
qed 

1061 
next 

1062 
assume Q: ?Q 

1063 
from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"] 

13517  1064 
show ?P by simp 
13152  1065 
qed 
1066 

1067 
lemma split_mod: 

1068 
assumes m: "m \<noteq> 0" 

1069 
shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)" 

1070 
(is "?P = ?Q") 

1071 
proof 

1072 
assume P: ?P 

1073 
show ?Q 

1074 
proof (intro allI impI) 

1075 
fix i j 

1076 
assume "n = m*i + j" "j < m" 

1077 
thus "P j" using m P by(simp add:add_ac mult_ac) 

1078 
qed 

1079 
next 

1080 
assume Q: ?Q 

1081 
from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"] 

13517  1082 
show ?P by simp 
13152  1083 
qed 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1084 
*) 
3366  1085 
end 