author | wenzelm |
Fri, 11 May 2007 17:54:36 +0200 | |
changeset 22936 | 284b56463da8 |
parent 22916 | 8caf6da610e2 |
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permissions | -rw-r--r-- |
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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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||
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notation |
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div (infixl "\<^loc>div" 70) |
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||
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notation |
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mod (infixl "\<^loc>mod" 70) |
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||
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end |
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notation |
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div (infixl "div" 70) |
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||
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notation |
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mod (infixl "mod" 70) |
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||
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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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||
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definition |
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(*The definition of dvd is polymorphic!*) |
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dvd :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where |
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dvd_def: "m dvd n \<longleftrightarrow> (\<exists>k. n = m*k)" |
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definition |
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quorem :: "(nat*nat) * (nat*nat) => bool" where |
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(*This definition helps prove the harder properties of div and mod. |
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It is copied from IntDiv.thy; should it be overloaded?*) |
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"quorem = (%((a,b), (q,r)). |
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a = b*q + r & |
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(if 0<b then 0\<le>r & r<b else b<r & r \<le>0))" |
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subsection{*Initial Lemmas*} |
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lemmas wf_less_trans = |
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def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl], |
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standard] |
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lemma mod_eq: "(%m. m mod n) = |
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wfrec (pred_nat^+) (%f j. if j<n | n=0 then j else f (j-n))" |
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by (simp add: mod_def) |
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lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+) |
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(%f j. if j<n | n=0 then 0 else Suc (f (j-n)))" |
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by (simp add: div_def) |
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(** Aribtrary definitions for division by zero. Useful to simplify |
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certain equations **) |
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lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)" |
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by (rule div_eq [THEN wf_less_trans], simp) |
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lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)" |
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by (rule mod_eq [THEN wf_less_trans], simp) |
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subsection{*Remainder*} |
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lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)" |
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by (rule mod_eq [THEN wf_less_trans]) simp |
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lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n" |
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apply (cases "n=0") |
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apply simp |
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apply (rule mod_eq [THEN wf_less_trans]) |
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apply (simp add: cut_apply less_eq) |
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done |
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(*Avoids the ugly ~m<n above*) |
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lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n" |
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by (simp add: mod_geq linorder_not_less) |
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lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)" |
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by (simp add: mod_geq) |
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lemma mod_1 [simp]: "m mod Suc 0 = 0" |
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by (induct m) (simp_all add: mod_geq) |
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lemma mod_self [simp]: "n mod n = (0::nat)" |
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by (cases "n = 0") (simp_all add: mod_geq) |
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lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)" |
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apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n") |
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apply (simp add: add_commute) |
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apply (subst mod_geq [symmetric], simp_all) |
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done |
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lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)" |
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by (simp add: add_commute mod_add_self2) |
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lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)" |
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by (induct k) (simp_all add: add_left_commute [of _ n]) |
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lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)" |
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by (simp add: mult_commute mod_mult_self1) |
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lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)" |
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apply (cases "n = 0", simp) |
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apply (cases "k = 0", simp) |
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apply (induct m rule: nat_less_induct) |
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apply (subst mod_if, simp) |
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apply (simp add: mod_geq diff_mult_distrib) |
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done |
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lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" |
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by (simp add: mult_commute [of k] mod_mult_distrib) |
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lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)" |
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apply (cases "n = 0", simp) |
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apply (induct m, simp) |
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apply (rename_tac k) |
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apply (cut_tac m = "k * n" and n = n in mod_add_self2) |
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apply (simp add: add_commute) |
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done |
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lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)" |
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by (simp add: mult_commute mod_mult_self_is_0) |
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subsection{*Quotient*} |
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lemma div_less [simp]: "m<n ==> m div n = (0::nat)" |
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by (rule div_eq [THEN wf_less_trans], simp) |
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lemma div_geq: "[| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)" |
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apply (rule div_eq [THEN wf_less_trans]) |
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apply (simp add: cut_apply less_eq) |
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done |
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(*Avoids the ugly ~m<n above*) |
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lemma le_div_geq: "[| 0<n; n\<le>m |] ==> m div n = Suc((m-n) div n)" |
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by (simp add: div_geq linorder_not_less) |
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lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))" |
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by (simp add: div_geq) |
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(*Main Result about quotient and remainder.*) |
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lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)" |
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apply (cases "n = 0", simp) |
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apply (induct m rule: nat_less_induct) |
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apply (subst mod_if) |
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apply (simp_all add: add_assoc div_geq add_diff_inverse) |
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done |
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lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)" |
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apply (cut_tac m = m and n = n in mod_div_equality) |
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apply (simp add: mult_commute) |
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done |
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subsection{*Simproc for Cancelling Div and Mod*} |
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lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k" |
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by (simp add: mod_div_equality) |
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lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k" |
22718 | 180 |
by (simp add: mod_div_equality2) |
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|
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ML |
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183 |
{* |
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structure CancelDivModData = |
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struct |
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|
22718 | 187 |
val div_name = @{const_name Divides.div}; |
188 |
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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|
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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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|
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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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|
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structure CancelDivMod = CancelDivModFun(CancelDivModData); |
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|
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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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212 |
|
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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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|
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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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|
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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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|
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(*mod_mult_distrib2 above is the counterpart for remainder*) |
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|
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|
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240 |
subsection{*Proving facts about Quotient and Remainder*} |
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|
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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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|
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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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260 |
==> 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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268 |
|
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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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|
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272 |
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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274 |
|
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(** A dividend of zero **) |
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276 |
|
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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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279 |
|
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280 |
lemma mod_0 [simp]: "0 mod m = (0::nat)" |
22718 | 281 |
by (cases "m = 0") simp_all |
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282 |
|
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283 |
(** proving (a*b) div c = a * (b div c) + a * (b mod c) **) |
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284 |
|
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285 |
lemma quorem_mult1_eq: |
22718 | 286 |
"[| quorem((b,c),(q,r)); 0 < c |] |
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287 |
==> 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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289 |
|
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290 |
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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294 |
|
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295 |
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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299 |
|
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300 |
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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306 |
|
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307 |
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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311 |
|
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312 |
(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **) |
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313 |
|
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314 |
lemma quorem_add1_eq: |
22718 | 315 |
"[| quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); 0 < c |] |
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316 |
==> 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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318 |
|
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319 |
(*NOT suitable for rewriting: the RHS has an instance of the LHS*) |
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320 |
lemma div_add1_eq: |
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321 |
"(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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325 |
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326 |
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 |
|
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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diff
changeset
|
330 |
|
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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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diff
changeset
|
331 |
|
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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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diff
changeset
|
332 |
subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*} |
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
333 |
|
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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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diff
changeset
|
334 |
(** first, a lemma to bound the remainder **) |
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parents:
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diff
changeset
|
335 |
|
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parents:
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diff
changeset
|
336 |
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 |
|
10559
d3fd54fc659b
many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents:
10214
diff
changeset
|
342 |
|
22718 | 343 |
lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r)); 0 < b; 0 < c |] |
14267
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parents:
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diff
changeset
|
344 |
==> 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) |
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
|
346 |
|
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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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diff
changeset
|
347 |
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 |
|
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
|
352 |
|
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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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diff
changeset
|
353 |
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 |
|
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
|
358 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
359 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
360 |
subsection{*Cancellation of Common Factors in Division*} |
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
361 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
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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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
365 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
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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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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changeset
|
370 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
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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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
375 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
376 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
377 |
(*Distribution of Factors over Remainders: |
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
378 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
379 |
Could prove these as in Integ/IntDiv.ML, but we already have |
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
380 |
mod_mult_distrib and mod_mult_distrib2 above! |
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
381 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
382 |
Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)" |
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
383 |
qed "mod_mult_mult1"; |
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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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diff
changeset
|
384 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
385 |
Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)"; |
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
386 |
qed "mod_mult_mult2"; |
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
387 |
***) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
388 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
389 |
subsection{*Further Facts about Quotient and Remainder*} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
390 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
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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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
393 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
394 |
lemma div_self [simp]: "0<n ==> n div n = (1::nat)" |
22718 | 395 |
by (simp add: div_geq) |
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
|
396 |
|
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
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+m-n) div n) ") |
399 |
apply (simp add: add_commute) |
|
400 |
apply (subst div_geq [symmetric], simp_all) |
|
401 |
done |
|
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
|
402 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
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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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
405 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
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
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
410 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
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
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
413 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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diff
changeset
|
414 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
415 |
(* Monotonicity of div in first argument *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
416 |
lemma div_le_mono [rule_format (no_asm)]: |
22718 | 417 |
"\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)" |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
418 |
apply (case_tac "k=0", simp) |
15251 | 419 |
apply (induct "n" rule: nat_less_induct, clarify) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
420 |
apply (case_tac "n<k") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
421 |
(* 1 case n<k *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
422 |
apply simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
423 |
(* 2 case n >= k *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
424 |
apply (case_tac "m<k") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
425 |
(* 2.1 case m<k *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
426 |
apply simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
427 |
(* 2.2 case m>=k *) |
15439 | 428 |
apply (simp add: div_geq diff_le_mono) |
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
429 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
430 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
431 |
(* Antimonotonicity of div in second argument *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
432 |
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
433 |
apply (subgoal_tac "0<n") |
22718 | 434 |
prefer 2 apply simp |
15251 | 435 |
apply (induct_tac k rule: nat_less_induct) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
436 |
apply (rename_tac "k") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
437 |
apply (case_tac "k<n", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
438 |
apply (subgoal_tac "~ (k<m) ") |
22718 | 439 |
prefer 2 apply simp |
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
440 |
apply (simp add: div_geq) |
15251 | 441 |
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n") |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
442 |
prefer 2 |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
443 |
apply (blast intro: div_le_mono diff_le_mono2) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
444 |
apply (rule le_trans, simp) |
15439 | 445 |
apply (simp) |
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
446 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
447 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
448 |
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
449 |
apply (case_tac "n=0", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
450 |
apply (subgoal_tac "m div n \<le> m div 1", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
451 |
apply (rule div_le_mono2) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
452 |
apply (simp_all (no_asm_simp)) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
453 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
454 |
|
22718 | 455 |
(* Similar for "less than" *) |
17085 | 456 |
lemma div_less_dividend [rule_format]: |
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
457 |
"!!n::nat. 1<n ==> 0 < m --> m div n < m" |
15251 | 458 |
apply (induct_tac m rule: nat_less_induct) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
459 |
apply (rename_tac "m") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
460 |
apply (case_tac "m<n", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
461 |
apply (subgoal_tac "0<n") |
22718 | 462 |
prefer 2 apply simp |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
463 |
apply (simp add: div_geq) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
464 |
apply (case_tac "n<m") |
15251 | 465 |
apply (subgoal_tac "(m-n) div n < (m-n) ") |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
466 |
apply (rule impI less_trans_Suc)+ |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
467 |
apply assumption |
15439 | 468 |
apply (simp_all) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
469 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
470 |
|
17085 | 471 |
declare div_less_dividend [simp] |
472 |
||
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
473 |
text{*A fact for the mutilated chess board*} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
474 |
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
475 |
apply (case_tac "n=0", simp) |
15251 | 476 |
apply (induct "m" rule: nat_less_induct) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
477 |
apply (case_tac "Suc (na) <n") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
478 |
(* case Suc(na) < n *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
479 |
apply (frule lessI [THEN less_trans], simp add: less_not_refl3) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
480 |
(* case n \<le> Suc(na) *) |
16796 | 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
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
483 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
484 |
|
14437 | 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
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
491 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
492 |
subsection{*The Divides Relation*} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
493 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
494 |
lemma dvdI [intro?]: "n = m * k ==> m dvd n" |
22718 | 495 |
unfolding dvd_def by blast |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
496 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
497 |
lemma dvdE [elim?]: "!!P. [|m dvd n; !!k. n = m*k ==> P|] ==> P" |
22718 | 498 |
unfolding dvd_def by blast |
13152 | 499 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
500 |
lemma dvd_0_right [iff]: "m dvd (0::nat)" |
22718 | 501 |
unfolding dvd_def by (blast intro: mult_0_right [symmetric]) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
502 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
503 |
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)" |
22718 | 504 |
by (force simp add: dvd_def) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
505 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
506 |
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)" |
22718 | 507 |
by (blast intro: dvd_0_left) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
508 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
509 |
lemma dvd_1_left [iff]: "Suc 0 dvd k" |
22718 | 510 |
unfolding dvd_def by simp |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
511 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
512 |
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)" |
22718 | 513 |
by (simp add: dvd_def) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
514 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
515 |
lemma dvd_refl [simp]: "m dvd (m::nat)" |
22718 | 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 (m-n :: 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 m-n; 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 m-n; 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 (p-k)" (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) = p-n" 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 "p-n < 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=p-k" 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 |
|
22845 | 901 |
definition [code func del]: |
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 |