--- a/src/HOL/Decision_Procs/Ferrack.thy Tue Jun 16 22:07:39 2009 -0700
+++ b/src/HOL/Decision_Procs/Ferrack.thy Wed Jun 17 16:55:01 2009 -0700
@@ -478,8 +478,8 @@
by (induct t rule: maxcoeff.induct, auto)
recdef numgcdh "measure size"
- "numgcdh (C i) = (\<lambda>g. zgcd i g)"
- "numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))"
+ "numgcdh (C i) = (\<lambda>g. gcd i g)"
+ "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
"numgcdh t = (\<lambda>g. 1)"
defs numgcd_def [code]: "numgcd t \<equiv> numgcdh t (maxcoeff t)"
@@ -512,11 +512,11 @@
assumes g0: "numgcd t = 0"
shows "Inum bs t = 0"
using g0[simplified numgcd_def]
- by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)
+ by (induct t rule: numgcdh.induct, auto simp add: natabs0 max_def maxcoeff_pos)
lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
using gp
- by (induct t rule: numgcdh.induct, auto simp add: zgcd_def)
+ by (induct t rule: numgcdh.induct, auto)
lemma numgcd_pos: "numgcd t \<ge>0"
by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
@@ -551,15 +551,15 @@
from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
qed simp_all
-lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
- apply (cases "abs i = 0", simp_all add: zgcd_def)
+lemma zgcd_gt1: "gcd i j > (1::int) \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
+ apply (cases "abs i = 0", simp_all add: gcd_int_def)
apply (cases "abs j = 0", simp_all)
apply (cases "abs i = 1", simp_all)
apply (cases "abs j = 1", simp_all)
apply auto
done
lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow> m =0"
- by (induct t rule: numgcdh.induct, auto simp add:zgcd0)
+ by (induct t rule: numgcdh.induct, auto)
lemma dvdnumcoeff_aux:
assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
@@ -568,22 +568,22 @@
proof(induct t rule: numgcdh.induct)
case (2 n c t)
let ?g = "numgcdh t m"
- from prems have th:"zgcd c ?g > 1" by simp
+ from prems have th:"gcd c ?g > 1" by simp
from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
moreover {assume "abs c > 1" and gp: "?g > 1" with prems
have th: "dvdnumcoeff t ?g" by simp
- have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
- from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
+ have th': "gcd c ?g dvd ?g" by simp
+ from dvdnumcoeff_trans[OF th' th] have ?case by simp }
moreover {assume "abs c = 0 \<and> ?g > 1"
with prems have th: "dvdnumcoeff t ?g" by simp
- have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
- from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
+ have th': "gcd c ?g dvd ?g" by simp
+ from dvdnumcoeff_trans[OF th' th] have ?case by simp
hence ?case by simp }
moreover {assume "abs c > 1" and g0:"?g = 0"
from numgcdh0[OF g0] have "m=0". with prems have ?case by simp }
ultimately show ?case by blast
-qed(auto simp add: zgcd_zdvd1)
+qed auto
lemma dvdnumcoeff_aux2:
assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
@@ -727,7 +727,7 @@
constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
"simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
(let t' = simpnum t ; g = numgcd t' in
- if g > 1 then (let g' = zgcd n g in
+ if g > 1 then (let g' = gcd n g in
if g' = 1 then (t',n)
else (reducecoeffh t' g', n div g'))
else (t',n))))"
@@ -738,23 +738,23 @@
proof-
let ?t' = "simpnum t"
let ?g = "numgcd ?t'"
- let ?g' = "zgcd n ?g"
+ let ?g' = "gcd n ?g"
{assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
moreover
{ assume nnz: "n \<noteq> 0"
{assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
- from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
- hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
+ from g1 nnz have gp0: "?g' \<noteq> 0" by simp
+ hence g'p: "?g' > 0" using int_gcd_ge_0[where x="n" and y="numgcd ?t'"] by arith
hence "?g'= 1 \<or> ?g' > 1" by arith
moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
moreover {assume g'1:"?g'>1"
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
let ?tt = "reducecoeffh ?t' ?g'"
let ?t = "Inum bs ?tt"
- have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
- have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
+ have gpdg: "?g' dvd ?g" by simp
+ have gpdd: "?g' dvd n" by simp
have gpdgp: "?g' dvd ?g'" by simp
from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
have th2:"real ?g' * ?t = Inum bs ?t'" by simp
@@ -774,21 +774,21 @@
proof-
let ?t' = "simpnum t"
let ?g = "numgcd ?t'"
- let ?g' = "zgcd n ?g"
+ let ?g' = "gcd n ?g"
{assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
moreover
{ assume nnz: "n \<noteq> 0"
{assume "\<not> ?g > 1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
- from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
- hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
+ from g1 nnz have gp0: "?g' \<noteq> 0" by simp
+ hence g'p: "?g' > 0" using int_gcd_ge_0[where x="n" and y="numgcd ?t'"] by arith
hence "?g'= 1 \<or> ?g' > 1" by arith
moreover {assume "?g'=1" hence ?thesis using prems
by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
moreover {assume g'1:"?g'>1"
- have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
- have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
+ have gpdg: "?g' dvd ?g" by simp
+ have gpdd: "?g' dvd n" by simp
have gpdgp: "?g' dvd ?g'" by simp
from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
--- a/src/HOL/Decision_Procs/MIR.thy Tue Jun 16 22:07:39 2009 -0700
+++ b/src/HOL/Decision_Procs/MIR.thy Wed Jun 17 16:55:01 2009 -0700
@@ -642,9 +642,9 @@
done
recdef numgcdh "measure size"
- "numgcdh (C i) = (\<lambda>g. zgcd i g)"
- "numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))"
- "numgcdh (CF c s t) = (\<lambda>g. zgcd c (numgcdh t g))"
+ "numgcdh (C i) = (\<lambda>g. gcd i g)"
+ "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
+ "numgcdh (CF c s t) = (\<lambda>g. gcd c (numgcdh t g))"
"numgcdh t = (\<lambda>g. 1)"
definition
@@ -687,13 +687,13 @@
shows "Inum bs t = 0"
proof-
have "\<And>x. numgcdh t x= 0 \<Longrightarrow> Inum bs t = 0"
- by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)
+ by (induct t rule: numgcdh.induct, auto)
thus ?thesis using g0[simplified numgcd_def] by blast
qed
lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
using gp
- by (induct t rule: numgcdh.induct, auto simp add: zgcd_def)
+ by (induct t rule: numgcdh.induct, auto)
lemma numgcd_pos: "numgcd t \<ge>0"
by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
@@ -738,8 +738,8 @@
from ismaxcoeff_mono[OF H1 thh1] show ?case by (simp add: le_maxI1)
qed simp_all
-lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
- apply (unfold zgcd_def)
+lemma zgcd_gt1: "gcd i j > (1::int) \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
+ apply (unfold gcd_int_def)
apply (cases "i = 0", simp_all)
apply (cases "j = 0", simp_all)
apply (cases "abs i = 1", simp_all)
@@ -747,7 +747,7 @@
apply auto
done
lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow> m =0"
- by (induct t rule: numgcdh.induct, auto simp add:zgcd0)
+ by (induct t rule: numgcdh.induct, auto)
lemma dvdnumcoeff_aux:
assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
@@ -756,17 +756,17 @@
proof(induct t rule: numgcdh.induct)
case (2 n c t)
let ?g = "numgcdh t m"
- from prems have th:"zgcd c ?g > 1" by simp
+ from prems have th:"gcd c ?g > 1" by simp
from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
moreover {assume "abs c > 1" and gp: "?g > 1" with prems
have th: "dvdnumcoeff t ?g" by simp
- have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
- from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
+ have th': "gcd c ?g dvd ?g" by simp
+ from dvdnumcoeff_trans[OF th' th] have ?case by simp }
moreover {assume "abs c = 0 \<and> ?g > 1"
with prems have th: "dvdnumcoeff t ?g" by simp
- have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
- from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
+ have th': "gcd c ?g dvd ?g" by simp
+ from dvdnumcoeff_trans[OF th' th] have ?case by simp
hence ?case by simp }
moreover {assume "abs c > 1" and g0:"?g = 0"
from numgcdh0[OF g0] have "m=0". with prems have ?case by simp }
@@ -774,22 +774,22 @@
next
case (3 c s t)
let ?g = "numgcdh t m"
- from prems have th:"zgcd c ?g > 1" by simp
+ from prems have th:"gcd c ?g > 1" by simp
from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
moreover {assume "abs c > 1" and gp: "?g > 1" with prems
have th: "dvdnumcoeff t ?g" by simp
- have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
- from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
+ have th': "gcd c ?g dvd ?g" by simp
+ from dvdnumcoeff_trans[OF th' th] have ?case by simp }
moreover {assume "abs c = 0 \<and> ?g > 1"
with prems have th: "dvdnumcoeff t ?g" by simp
- have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
- from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
+ have th': "gcd c ?g dvd ?g" by simp
+ from dvdnumcoeff_trans[OF th' th] have ?case by simp
hence ?case by simp }
moreover {assume "abs c > 1" and g0:"?g = 0"
from numgcdh0[OF g0] have "m=0". with prems have ?case by simp }
ultimately show ?case by blast
-qed(auto simp add: zgcd_zdvd1)
+qed auto
lemma dvdnumcoeff_aux2:
assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
@@ -1041,7 +1041,7 @@
constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
"simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
(let t' = simpnum t ; g = numgcd t' in
- if g > 1 then (let g' = zgcd n g in
+ if g > 1 then (let g' = gcd n g in
if g' = 1 then (t',n)
else (reducecoeffh t' g', n div g'))
else (t',n))))"
@@ -1052,23 +1052,23 @@
proof-
let ?t' = "simpnum t"
let ?g = "numgcd ?t'"
- let ?g' = "zgcd n ?g"
+ let ?g' = "gcd n ?g"
{assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
moreover
{ assume nnz: "n \<noteq> 0"
{assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
- from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
- hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
+ from g1 nnz have gp0: "?g' \<noteq> 0" by simp
+ hence g'p: "?g' > 0" using int_gcd_ge_0[where x="n" and y="numgcd ?t'"] by arith
hence "?g'= 1 \<or> ?g' > 1" by arith
moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
moreover {assume g'1:"?g'>1"
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
let ?tt = "reducecoeffh ?t' ?g'"
let ?t = "Inum bs ?tt"
- have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
- have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
+ have gpdg: "?g' dvd ?g" by simp
+ have gpdd: "?g' dvd n" by simp
have gpdgp: "?g' dvd ?g'" by simp
from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
have th2:"real ?g' * ?t = Inum bs ?t'" by simp
@@ -1088,21 +1088,21 @@
proof-
let ?t' = "simpnum t"
let ?g = "numgcd ?t'"
- let ?g' = "zgcd n ?g"
+ let ?g' = "gcd n ?g"
{assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
moreover
{ assume nnz: "n \<noteq> 0"
{assume "\<not> ?g > 1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def)}
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
- from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
- hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
+ from g1 nnz have gp0: "?g' \<noteq> 0" by simp
+ hence g'p: "?g' > 0" using int_gcd_ge_0[where x="n" and y="numgcd ?t'"] by arith
hence "?g'= 1 \<or> ?g' > 1" by arith
moreover {assume "?g'=1" hence ?thesis using prems
by (auto simp add: Let_def simp_num_pair_def)}
moreover {assume g'1:"?g'>1"
- have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
- have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
+ have gpdg: "?g' dvd ?g" by simp
+ have gpdd: "?g' dvd n" by simp
have gpdgp: "?g' dvd ?g'" by simp
from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
@@ -1219,7 +1219,7 @@
constdefs simpdvd:: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)"
"simpdvd d t \<equiv>
(let g = numgcd t in
- if g > 1 then (let g' = zgcd d g in
+ if g > 1 then (let g' = gcd d g in
if g' = 1 then (d, t)
else (d div g',reducecoeffh t g'))
else (d, t))"
@@ -1228,20 +1228,20 @@
shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)"
proof-
let ?g = "numgcd t"
- let ?g' = "zgcd d ?g"
+ let ?g' = "gcd d ?g"
{assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simpdvd_def)}
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
- from zgcd0 g1 dnz have gp0: "?g' \<noteq> 0" by simp
- hence g'p: "?g' > 0" using zgcd_pos[where i="d" and j="numgcd t"] by arith
+ from g1 dnz have gp0: "?g' \<noteq> 0" by simp
+ hence g'p: "?g' > 0" using int_gcd_ge_0[where x="d" and y="numgcd t"] by arith
hence "?g'= 1 \<or> ?g' > 1" by arith
moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simpdvd_def)}
moreover {assume g'1:"?g'>1"
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" ..
let ?tt = "reducecoeffh t ?g'"
let ?t = "Inum bs ?tt"
- have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
- have gpdd: "?g' dvd d" by (simp add: zgcd_zdvd1)
+ have gpdg: "?g' dvd ?g" by simp
+ have gpdd: "?g' dvd d" by simp
have gpdgp: "?g' dvd ?g'" by simp
from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
have th2:"real ?g' * ?t = Inum bs t" by simp
@@ -2093,8 +2093,8 @@
"plusinf p = p"
recdef \<delta> "measure size"
- "\<delta> (And p q) = zlcm (\<delta> p) (\<delta> q)"
- "\<delta> (Or p q) = zlcm (\<delta> p) (\<delta> q)"
+ "\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)"
+ "\<delta> (Or p q) = lcm (\<delta> p) (\<delta> q)"
"\<delta> (Dvd i (CN 0 c e)) = i"
"\<delta> (NDvd i (CN 0 c e)) = i"
"\<delta> p = 1"
@@ -2126,20 +2126,20 @@
proof (induct p rule: iszlfm.induct)
case (1 p q)
let ?d = "\<delta> (And p q)"
- from prems zlcm_pos have dp: "?d >0" by simp
+ from prems int_lcm_pos have dp: "?d >0" by simp
have d1: "\<delta> p dvd \<delta> (And p q)" using prems by simp
hence th: "d\<delta> p ?d"
- using delta_mono prems by (auto simp del: dvd_zlcm_self1)
+ using delta_mono prems by (auto simp del: int_lcm_dvd1)
have "\<delta> q dvd \<delta> (And p q)" using prems by simp
- hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: dvd_zlcm_self2)
+ hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: int_lcm_dvd2)
from th th' dp show ?case by simp
next
case (2 p q)
let ?d = "\<delta> (And p q)"
- from prems zlcm_pos have dp: "?d >0" by simp
+ from prems int_lcm_pos have dp: "?d >0" by simp
have "\<delta> p dvd \<delta> (And p q)" using prems by simp hence th: "d\<delta> p ?d" using delta_mono prems
- by (auto simp del: dvd_zlcm_self1)
- have "\<delta> q dvd \<delta> (And p q)" using prems by simp hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: dvd_zlcm_self2)
+ by (auto simp del: int_lcm_dvd1)
+ have "\<delta> q dvd \<delta> (And p q)" using prems by simp hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: int_lcm_dvd2)
from th th' dp show ?case by simp
qed simp_all
@@ -2388,8 +2388,8 @@
"d\<beta> p = (\<lambda> k. True)"
recdef \<zeta> "measure size"
- "\<zeta> (And p q) = zlcm (\<zeta> p) (\<zeta> q)"
- "\<zeta> (Or p q) = zlcm (\<zeta> p) (\<zeta> q)"
+ "\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)"
+ "\<zeta> (Or p q) = lcm (\<zeta> p) (\<zeta> q)"
"\<zeta> (Eq (CN 0 c e)) = c"
"\<zeta> (NEq (CN 0 c e)) = c"
"\<zeta> (Lt (CN 0 c e)) = c"
@@ -2510,19 +2510,19 @@
using linp
proof(induct p rule: iszlfm.induct)
case (1 p q)
- from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
- from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
- from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
- d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
- dl1 dl2 show ?case by (auto simp add: zlcm_pos)
+ from prems have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
+ from prems have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
+ from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
+ d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"]
+ dl1 dl2 show ?case by (auto simp add: int_lcm_pos)
next
case (2 p q)
- from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
- from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
- from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
- d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
- dl1 dl2 show ?case by (auto simp add: zlcm_pos)
-qed (auto simp add: zlcm_pos)
+ from prems have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
+ from prems have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
+ from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
+ d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"]
+ dl1 dl2 show ?case by (auto simp add: int_lcm_pos)
+qed (auto simp add: int_lcm_pos)
lemma a\<beta>: assumes linp: "iszlfm p (a #bs)" and d: "d\<beta> p l" and lp: "l > 0"
shows "iszlfm (a\<beta> p l) (a #bs) \<and> d\<beta> (a\<beta> p l) 1 \<and> (Ifm (real (l * x) #bs) (a\<beta> p l) = Ifm ((real x)#bs) p)"
@@ -4173,9 +4173,9 @@
lemma bound0at_l : "\<lbrakk>isatom p ; bound0 p\<rbrakk> \<Longrightarrow> isrlfm p"
by (induct p rule: isrlfm.induct, auto)
-lemma zgcd_le1: assumes ip: "0 < i" shows "zgcd i j \<le> i"
+lemma zgcd_le1: assumes ip: "(0::int) < i" shows "gcd i j \<le> i"
proof-
- from zgcd_zdvd1 have th: "zgcd i j dvd i" by blast
+ from int_gcd_dvd1 have th: "gcd i j dvd i" by blast
from zdvd_imp_le[OF th ip] show ?thesis .
qed
@@ -5119,9 +5119,9 @@
let ?M = "?I x (minusinf ?rq)"
let ?P = "?I x (plusinf ?rq)"
have MF: "?M = False"
- apply (simp add: Let_def reducecoeff_def numgcd_def zgcd_def rsplit_def ge_def lt_def conj_def disj_def)
+ apply (simp add: Let_def reducecoeff_def numgcd_def rsplit_def ge_def lt_def conj_def disj_def)
by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all)
- have PF: "?P = False" apply (simp add: Let_def reducecoeff_def numgcd_def zgcd_def rsplit_def ge_def lt_def conj_def disj_def)
+ have PF: "?P = False" apply (simp add: Let_def reducecoeff_def numgcd_def rsplit_def ge_def lt_def conj_def disj_def)
by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all)
have "(\<exists> x. ?I x ?q ) =
((?I x (minusinf ?rq)) \<or> (?I x (plusinf ?rq )) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> ?rq). \<exists> (s,m) \<in> set (\<Upsilon> ?rq ). ?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))))"
@@ -5286,7 +5286,7 @@
let ?F = "\<lambda> p. \<exists> a \<in> set (\<Upsilon> p). \<exists> b \<in> set (\<Upsilon> p). ?I x (\<upsilon> p (?g(a,b)))"
let ?ep = "evaldjf (\<upsilon> ?q) ?Y"
from rlfm_l[OF qf] have lq: "isrlfm ?q"
- by (simp add: rsplit_def lt_def ge_def conj_def disj_def Let_def reducecoeff_def numgcd_def zgcd_def)
+ by (simp add: rsplit_def lt_def ge_def conj_def disj_def Let_def reducecoeff_def numgcd_def)
from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
from \<Upsilon>_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
from U_l UpU
--- a/src/HOL/GCD.thy Tue Jun 16 22:07:39 2009 -0700
+++ b/src/HOL/GCD.thy Wed Jun 17 16:55:01 2009 -0700
@@ -1,202 +1,563 @@
-(* Title: HOL/GCD.thy
- Author: Christophe Tabacznyj and Lawrence C Paulson
- Copyright 1996 University of Cambridge
+(* Title: GCD.thy
+ Authors: Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
+ Thomas M. Rasmussen, Jeremy Avigad
+
+
+This file deals with the functions gcd and lcm, and properties of
+primes. Definitions and lemmas are proved uniformly for the natural
+numbers and integers.
+
+This file combines and revises a number of prior developments.
+
+The original theories "GCD" and "Primes" were by Christophe Tabacznyj
+and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
+gcd, lcm, and prime for the natural numbers.
+
+The original theory "IntPrimes" was by Thomas M. Rasmussen, and
+extended gcd, lcm, primes to the integers. Amine Chaieb provided
+another extension of the notions to the integers, and added a number
+of results to "Primes" and "GCD". IntPrimes also defined and developed
+the congruence relations on the integers. The notion was extended to
+the natural numbers by Chiaeb.
+
*)
-header {* The Greatest Common Divisor *}
+
+header {* GCD *}
theory GCD
-imports Main
+imports NatTransfer
+begin
+
+declare One_nat_def [simp del]
+
+subsection {* gcd *}
+
+class gcd = one +
+
+fixes
+ gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and
+ lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+
begin
-text {*
- See \cite{davenport92}. \bigskip
-*}
+abbreviation
+ coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+where
+ "coprime x y == (gcd x y = 1)"
+
+end
+
+class prime = one +
+
+fixes
+ prime :: "'a \<Rightarrow> bool"
+
+
+(* definitions for the natural numbers *)
+
+instantiation nat :: gcd
+
+begin
-subsection {* Specification of GCD on nats *}
+fun
+ gcd_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+ "gcd_nat x y =
+ (if y = 0 then x else gcd y (x mod y))"
+
+definition
+ lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+ "lcm_nat x y = x * y div (gcd x y)"
+
+instance proof qed
+
+end
+
+
+instantiation nat :: prime
+
+begin
definition
- is_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where -- {* @{term gcd} as a relation *}
- [code del]: "is_gcd m n p \<longleftrightarrow> p dvd m \<and> p dvd n \<and>
- (\<forall>d. d dvd m \<longrightarrow> d dvd n \<longrightarrow> d dvd p)"
+ prime_nat :: "nat \<Rightarrow> bool"
+where
+ "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
+
+instance proof qed
-text {* Uniqueness *}
+end
+
-lemma is_gcd_unique: "is_gcd a b m \<Longrightarrow> is_gcd a b n \<Longrightarrow> m = n"
- by (simp add: is_gcd_def) (blast intro: dvd_anti_sym)
+(* definitions for the integers *)
+
+instantiation int :: gcd
-text {* Connection to divides relation *}
+begin
-lemma is_gcd_dvd: "is_gcd a b m \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd m"
- by (auto simp add: is_gcd_def)
+definition
+ gcd_int :: "int \<Rightarrow> int \<Rightarrow> int"
+where
+ "gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
-text {* Commutativity *}
+definition
+ lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
+where
+ "lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
-lemma is_gcd_commute: "is_gcd m n k = is_gcd n m k"
- by (auto simp add: is_gcd_def)
+instance proof qed
+
+end
-subsection {* GCD on nat by Euclid's algorithm *}
+instantiation int :: prime
+
+begin
+
+definition
+ prime_int :: "int \<Rightarrow> bool"
+where
+ "prime_int p = prime (nat p)"
+
+instance proof qed
+
+end
+
+
+subsection {* Set up Transfer *}
+
+
+lemma transfer_nat_int_gcd:
+ "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
+ "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
+ "(x::int) >= 0 \<Longrightarrow> prime (nat x) = prime x"
+ unfolding gcd_int_def lcm_int_def prime_int_def
+ by auto
-fun
- gcd :: "nat => nat => nat"
-where
- "gcd m n = (if n = 0 then m else gcd n (m mod n))"
-lemma gcd_induct [case_names "0" rec]:
+lemma transfer_nat_int_gcd_closures:
+ "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
+ "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
+ by (auto simp add: gcd_int_def lcm_int_def)
+
+declare TransferMorphism_nat_int[transfer add return:
+ transfer_nat_int_gcd transfer_nat_int_gcd_closures]
+
+lemma transfer_int_nat_gcd:
+ "gcd (int x) (int y) = int (gcd x y)"
+ "lcm (int x) (int y) = int (lcm x y)"
+ "prime (int x) = prime x"
+ by (unfold gcd_int_def lcm_int_def prime_int_def, auto)
+
+lemma transfer_int_nat_gcd_closures:
+ "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
+ "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
+ by (auto simp add: gcd_int_def lcm_int_def)
+
+declare TransferMorphism_int_nat[transfer add return:
+ transfer_int_nat_gcd transfer_int_nat_gcd_closures]
+
+
+
+subsection {* GCD *}
+
+(* was gcd_induct *)
+lemma nat_gcd_induct:
fixes m n :: nat
assumes "\<And>m. P m 0"
and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
shows "P m n"
-proof (induct m n rule: gcd.induct)
- case (1 m n) with assms show ?case by (cases "n = 0") simp_all
-qed
+ apply (rule gcd_nat.induct)
+ apply (case_tac "y = 0")
+ using assms apply simp_all
+done
+
+(* specific to int *)
+
+lemma int_gcd_neg1 [simp]: "gcd (-x::int) y = gcd x y"
+ by (simp add: gcd_int_def)
+
+lemma int_gcd_neg2 [simp]: "gcd (x::int) (-y) = gcd x y"
+ by (simp add: gcd_int_def)
+
+lemma int_gcd_abs: "gcd (x::int) y = gcd (abs x) (abs y)"
+ by (simp add: gcd_int_def)
+
+lemma int_gcd_cases:
+ fixes x :: int and y
+ assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
+ and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
+ and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
+ and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
+ shows "P (gcd x y)"
+by (insert prems, auto simp add: int_gcd_neg1 int_gcd_neg2, arith)
-lemma gcd_0 [simp, algebra]: "gcd m 0 = m"
- by simp
+lemma int_gcd_ge_0 [simp]: "gcd (x::int) y >= 0"
+ by (simp add: gcd_int_def)
+
+lemma int_lcm_neg1: "lcm (-x::int) y = lcm x y"
+ by (simp add: lcm_int_def)
+
+lemma int_lcm_neg2: "lcm (x::int) (-y) = lcm x y"
+ by (simp add: lcm_int_def)
+
+lemma int_lcm_abs: "lcm (x::int) y = lcm (abs x) (abs y)"
+ by (simp add: lcm_int_def)
-lemma gcd_0_left [simp,algebra]: "gcd 0 m = m"
+lemma int_lcm_cases:
+ fixes x :: int and y
+ assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
+ and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
+ and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
+ and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
+ shows "P (lcm x y)"
+by (insert prems, auto simp add: int_lcm_neg1 int_lcm_neg2, arith)
+
+lemma int_lcm_ge_0 [simp]: "lcm (x::int) y >= 0"
+ by (simp add: lcm_int_def)
+
+(* was gcd_0, etc. *)
+lemma nat_gcd_0 [simp]: "gcd (x::nat) 0 = x"
by simp
-lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd m n = gcd n (m mod n)"
+(* was igcd_0, etc. *)
+lemma int_gcd_0 [simp]: "gcd (x::int) 0 = abs x"
+ by (unfold gcd_int_def, auto)
+
+lemma nat_gcd_0_left [simp]: "gcd 0 (x::nat) = x"
by simp
-lemma gcd_1 [simp, algebra]: "gcd m (Suc 0) = Suc 0"
+lemma int_gcd_0_left [simp]: "gcd 0 (x::int) = abs x"
+ by (unfold gcd_int_def, auto)
+
+lemma nat_gcd_red: "gcd (x::nat) y = gcd y (x mod y)"
+ by (case_tac "y = 0", auto)
+
+(* weaker, but useful for the simplifier *)
+
+lemma nat_gcd_non_0: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
+ by simp
+
+lemma nat_gcd_1 [simp]: "gcd (m::nat) 1 = 1"
by simp
-lemma nat_gcd_1_right [simp, algebra]: "gcd m 1 = 1"
- unfolding One_nat_def by (rule gcd_1)
+lemma nat_gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
+ by (simp add: One_nat_def)
+
+lemma int_gcd_1 [simp]: "gcd (m::int) 1 = 1"
+ by (simp add: gcd_int_def)
-declare gcd.simps [simp del]
+lemma nat_gcd_self [simp]: "gcd (x::nat) x = x"
+ by simp
+
+lemma int_gcd_def [simp]: "gcd (x::int) x = abs x"
+ by (subst int_gcd_abs, auto simp add: gcd_int_def)
+
+declare gcd_nat.simps [simp del]
text {*
\medskip @{term "gcd m n"} divides @{text m} and @{text n}. The
conjunctions don't seem provable separately.
*}
-lemma gcd_dvd1 [iff, algebra]: "gcd m n dvd m"
- and gcd_dvd2 [iff, algebra]: "gcd m n dvd n"
- apply (induct m n rule: gcd_induct)
- apply (simp_all add: gcd_non_0)
+lemma nat_gcd_dvd1 [iff]: "(gcd (m::nat)) n dvd m"
+ and nat_gcd_dvd2 [iff]: "(gcd m n) dvd n"
+ apply (induct m n rule: nat_gcd_induct)
+ apply (simp_all add: nat_gcd_non_0)
apply (blast dest: dvd_mod_imp_dvd)
- done
+done
+
+thm nat_gcd_dvd1 [transferred]
+
+lemma int_gcd_dvd1 [iff]: "gcd (x::int) y dvd x"
+ apply (subst int_gcd_abs)
+ apply (rule dvd_trans)
+ apply (rule nat_gcd_dvd1 [transferred])
+ apply auto
+done
-text {*
- \medskip Maximality: for all @{term m}, @{term n}, @{term k}
- naturals, if @{term k} divides @{term m} and @{term k} divides
- @{term n} then @{term k} divides @{term "gcd m n"}.
-*}
+lemma int_gcd_dvd2 [iff]: "gcd (x::int) y dvd y"
+ apply (subst int_gcd_abs)
+ apply (rule dvd_trans)
+ apply (rule nat_gcd_dvd2 [transferred])
+ apply auto
+done
+
+lemma nat_gcd_le1 [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
+ by (rule dvd_imp_le, auto)
+
+lemma nat_gcd_le2 [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
+ by (rule dvd_imp_le, auto)
+
+lemma int_gcd_le1 [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
+ by (rule zdvd_imp_le, auto)
-lemma gcd_greatest: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
- by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod)
+lemma int_gcd_le2 [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
+ by (rule zdvd_imp_le, auto)
+
+lemma nat_gcd_greatest: "(k::nat) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
+ by (induct m n rule: nat_gcd_induct) (simp_all add: nat_gcd_non_0 dvd_mod)
+
+lemma int_gcd_greatest:
+ assumes "(k::int) dvd m" and "k dvd n"
+ shows "k dvd gcd m n"
+
+ apply (subst int_gcd_abs)
+ apply (subst abs_dvd_iff [symmetric])
+ apply (rule nat_gcd_greatest [transferred])
+ using prems apply auto
+done
-text {*
- \medskip Function gcd yields the Greatest Common Divisor.
-*}
+lemma nat_gcd_greatest_iff [iff]: "(k dvd gcd (m::nat) n) =
+ (k dvd m & k dvd n)"
+ by (blast intro!: nat_gcd_greatest intro: dvd_trans)
+
+lemma int_gcd_greatest_iff: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
+ by (blast intro!: int_gcd_greatest intro: dvd_trans)
-lemma is_gcd: "is_gcd m n (gcd m n) "
- by (simp add: is_gcd_def gcd_greatest)
+lemma nat_gcd_zero [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
+ by (simp only: dvd_0_left_iff [symmetric] nat_gcd_greatest_iff)
+lemma int_gcd_zero [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
+ by (auto simp add: gcd_int_def)
-subsection {* Derived laws for GCD *}
+lemma nat_gcd_pos [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
+ by (insert nat_gcd_zero [of m n], arith)
-lemma gcd_greatest_iff [iff, algebra]: "k dvd gcd m n \<longleftrightarrow> k dvd m \<and> k dvd n"
- by (blast intro!: gcd_greatest intro: dvd_trans)
+lemma int_gcd_pos [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
+ by (insert int_gcd_zero [of m n], insert int_gcd_ge_0 [of m n], arith)
+
+lemma nat_gcd_commute: "gcd (m::nat) n = gcd n m"
+ by (rule dvd_anti_sym, auto)
-lemma gcd_zero[algebra]: "gcd m n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
- by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff)
+lemma int_gcd_commute: "gcd (m::int) n = gcd n m"
+ by (auto simp add: gcd_int_def nat_gcd_commute)
+
+lemma nat_gcd_assoc: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
+ apply (rule dvd_anti_sym)
+ apply (blast intro: dvd_trans)+
+done
-lemma gcd_commute: "gcd m n = gcd n m"
- apply (rule is_gcd_unique)
- apply (rule is_gcd)
- apply (subst is_gcd_commute)
- apply (simp add: is_gcd)
- done
+lemma int_gcd_assoc: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
+ by (auto simp add: gcd_int_def nat_gcd_assoc)
+
+lemma nat_gcd_left_commute: "gcd (k::nat) (gcd m n) = gcd m (gcd k n)"
+ apply (rule nat_gcd_commute [THEN trans])
+ apply (rule nat_gcd_assoc [THEN trans])
+ apply (rule nat_gcd_commute [THEN arg_cong])
+done
+
+lemma int_gcd_left_commute: "gcd (k::int) (gcd m n) = gcd m (gcd k n)"
+ apply (rule int_gcd_commute [THEN trans])
+ apply (rule int_gcd_assoc [THEN trans])
+ apply (rule int_gcd_commute [THEN arg_cong])
+done
+
+lemmas nat_gcd_ac = nat_gcd_assoc nat_gcd_commute nat_gcd_left_commute
+ -- {* gcd is an AC-operator *}
-lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)"
- apply (rule is_gcd_unique)
- apply (rule is_gcd)
- apply (simp add: is_gcd_def)
- apply (blast intro: dvd_trans)
- done
+lemmas int_gcd_ac = int_gcd_assoc int_gcd_commute int_gcd_left_commute
+
+lemma nat_gcd_1_left [simp]: "gcd (1::nat) m = 1"
+ by (subst nat_gcd_commute, simp)
+
+lemma nat_gcd_Suc_0_left [simp]: "gcd (Suc 0) m = Suc 0"
+ by (subst nat_gcd_commute, simp add: One_nat_def)
+
+lemma int_gcd_1_left [simp]: "gcd (1::int) m = 1"
+ by (subst int_gcd_commute, simp)
-lemma gcd_1_left [simp, algebra]: "gcd (Suc 0) m = Suc 0"
- by (simp add: gcd_commute)
+lemma nat_gcd_unique: "(d::nat) dvd a \<and> d dvd b \<and>
+ (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
+ apply auto
+ apply (rule dvd_anti_sym)
+ apply (erule (1) nat_gcd_greatest)
+ apply auto
+done
-lemma nat_gcd_1_left [simp, algebra]: "gcd 1 m = 1"
- unfolding One_nat_def by (rule gcd_1_left)
+lemma int_gcd_unique: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
+ (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
+ apply (case_tac "d = 0")
+ apply force
+ apply (rule iffI)
+ apply (rule zdvd_anti_sym)
+ apply arith
+ apply (subst int_gcd_pos)
+ apply clarsimp
+ apply (drule_tac x = "d + 1" in spec)
+ apply (frule zdvd_imp_le)
+ apply (auto intro: int_gcd_greatest)
+done
text {*
\medskip Multiplication laws
*}
-lemma gcd_mult_distrib2: "k * gcd m n = gcd (k * m) (k * n)"
+lemma nat_gcd_mult_distrib: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
-- {* \cite[page 27]{davenport92} *}
- apply (induct m n rule: gcd_induct)
- apply simp
+ apply (induct m n rule: nat_gcd_induct)
+ apply simp
apply (case_tac "k = 0")
- apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
- done
+ apply (simp_all add: mod_geq nat_gcd_non_0 mod_mult_distrib2)
+done
-lemma gcd_mult [simp, algebra]: "gcd k (k * n) = k"
- apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
- done
+lemma int_gcd_mult_distrib: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
+ apply (subst (1 2) int_gcd_abs)
+ apply (simp add: abs_mult)
+ apply (rule nat_gcd_mult_distrib [transferred])
+ apply auto
+done
-lemma gcd_self [simp, algebra]: "gcd k k = k"
- apply (rule gcd_mult [of k 1, simplified])
- done
+lemma nat_gcd_mult [simp]: "gcd (k::nat) (k * n) = k"
+ by (rule nat_gcd_mult_distrib [of k 1 n, simplified, symmetric])
-lemma relprime_dvd_mult: "gcd k n = 1 ==> k dvd m * n ==> k dvd m"
- apply (insert gcd_mult_distrib2 [of m k n])
+lemma int_gcd_mult [simp]: "gcd (k::int) (k * n) = abs k"
+ by (rule int_gcd_mult_distrib [of k 1 n, simplified, symmetric])
+
+lemma nat_coprime_dvd_mult: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
+ apply (insert nat_gcd_mult_distrib [of m k n])
apply simp
apply (erule_tac t = m in ssubst)
apply simp
done
-lemma relprime_dvd_mult_iff: "gcd k n = 1 ==> (k dvd m * n) = (k dvd m)"
- by (auto intro: relprime_dvd_mult dvd_mult2)
+lemma int_coprime_dvd_mult:
+ assumes "coprime (k::int) n" and "k dvd m * n"
+ shows "k dvd m"
-lemma gcd_mult_cancel: "gcd k n = 1 ==> gcd (k * m) n = gcd m n"
+ using prems
+ apply (subst abs_dvd_iff [symmetric])
+ apply (subst dvd_abs_iff [symmetric])
+ apply (subst (asm) int_gcd_abs)
+ apply (rule nat_coprime_dvd_mult [transferred])
+ apply auto
+ apply (subst abs_mult [symmetric], auto)
+done
+
+lemma nat_coprime_dvd_mult_iff: "coprime (k::nat) n \<Longrightarrow>
+ (k dvd m * n) = (k dvd m)"
+ by (auto intro: nat_coprime_dvd_mult)
+
+lemma int_coprime_dvd_mult_iff: "coprime (k::int) n \<Longrightarrow>
+ (k dvd m * n) = (k dvd m)"
+ by (auto intro: int_coprime_dvd_mult)
+
+lemma nat_gcd_mult_cancel: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
apply (rule dvd_anti_sym)
- apply (rule gcd_greatest)
- apply (rule_tac n = k in relprime_dvd_mult)
- apply (simp add: gcd_assoc)
- apply (simp add: gcd_commute)
- apply (simp_all add: mult_commute)
- done
+ apply (rule nat_gcd_greatest)
+ apply (rule_tac n = k in nat_coprime_dvd_mult)
+ apply (simp add: nat_gcd_assoc)
+ apply (simp add: nat_gcd_commute)
+ apply (simp_all add: mult_commute)
+done
+lemma int_gcd_mult_cancel:
+ assumes "coprime (k::int) n"
+ shows "gcd (k * m) n = gcd m n"
+
+ using prems
+ apply (subst (1 2) int_gcd_abs)
+ apply (subst abs_mult)
+ apply (rule nat_gcd_mult_cancel [transferred])
+ apply (auto simp add: int_gcd_abs [symmetric])
+done
text {* \medskip Addition laws *}
-lemma gcd_add1 [simp, algebra]: "gcd (m + n) n = gcd m n"
- by (cases "n = 0") (auto simp add: gcd_non_0)
+lemma nat_gcd_add1 [simp]: "gcd ((m::nat) + n) n = gcd m n"
+ apply (case_tac "n = 0")
+ apply (simp_all add: nat_gcd_non_0)
+done
+
+lemma nat_gcd_add2 [simp]: "gcd (m::nat) (m + n) = gcd m n"
+ apply (subst (1 2) nat_gcd_commute)
+ apply (subst add_commute)
+ apply simp
+done
+
+(* to do: add the other variations? *)
+
+lemma nat_gcd_diff1: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
+ by (subst nat_gcd_add1 [symmetric], auto)
+
+lemma nat_gcd_diff2: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
+ apply (subst nat_gcd_commute)
+ apply (subst nat_gcd_diff1 [symmetric])
+ apply auto
+ apply (subst nat_gcd_commute)
+ apply (subst nat_gcd_diff1)
+ apply assumption
+ apply (rule nat_gcd_commute)
+done
+
+lemma int_gcd_non_0: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
+ apply (frule_tac b = y and a = x in pos_mod_sign)
+ apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
+ apply (auto simp add: nat_gcd_non_0 nat_mod_distrib [symmetric]
+ zmod_zminus1_eq_if)
+ apply (frule_tac a = x in pos_mod_bound)
+ apply (subst (1 2) nat_gcd_commute)
+ apply (simp del: pos_mod_bound add: nat_diff_distrib nat_gcd_diff2
+ nat_le_eq_zle)
+done
-lemma gcd_add2 [simp, algebra]: "gcd m (m + n) = gcd m n"
-proof -
- have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute)
- also have "... = gcd (n + m) m" by (simp add: add_commute)
- also have "... = gcd n m" by simp
- also have "... = gcd m n" by (rule gcd_commute)
- finally show ?thesis .
-qed
+lemma int_gcd_red: "gcd (x::int) y = gcd y (x mod y)"
+ apply (case_tac "y = 0")
+ apply force
+ apply (case_tac "y > 0")
+ apply (subst int_gcd_non_0, auto)
+ apply (insert int_gcd_non_0 [of "-y" "-x"])
+ apply (auto simp add: int_gcd_neg1 int_gcd_neg2)
+done
+
+lemma int_gcd_add1 [simp]: "gcd ((m::int) + n) n = gcd m n"
+ apply (case_tac "n = 0", force)
+ apply (subst (1 2) int_gcd_red)
+ apply auto
+done
+
+lemma int_gcd_add2 [simp]: "gcd m ((m::int) + n) = gcd m n"
+ apply (subst int_gcd_commute)
+ apply (subst add_commute)
+ apply (subst int_gcd_add1)
+ apply (subst int_gcd_commute)
+ apply (rule refl)
+done
-lemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n"
- apply (subst add_commute)
- apply (rule gcd_add2)
- done
+lemma nat_gcd_add_mult: "gcd (m::nat) (k * m + n) = gcd m n"
+ by (induct k, simp_all add: ring_simps)
-lemma gcd_add_mult[algebra]: "gcd m (k * m + n) = gcd m n"
- by (induct k) (simp_all add: add_assoc)
+lemma int_gcd_add_mult: "gcd (m::int) (k * m + n) = gcd m n"
+ apply (subgoal_tac "ALL s. ALL m. ALL n.
+ gcd m (int (s::nat) * m + n) = gcd m n")
+ apply (case_tac "k >= 0")
+ apply (drule_tac x = "nat k" in spec, force)
+ apply (subst (1 2) int_gcd_neg2 [symmetric])
+ apply (drule_tac x = "nat (- k)" in spec)
+ apply (drule_tac x = "m" in spec)
+ apply (drule_tac x = "-n" in spec)
+ apply auto
+ apply (rule nat_induct)
+ apply auto
+ apply (auto simp add: left_distrib)
+ apply (subst add_assoc)
+ apply simp
+done
-lemma gcd_dvd_prod: "gcd m n dvd m * n"
+(* to do: differences, and all variations of addition rules
+ as simplification rules for nat and int *)
+
+lemma nat_gcd_dvd_prod [iff]: "gcd (m::nat) n dvd k * n"
using mult_dvd_mono [of 1] by auto
-text {*
- \medskip Division by gcd yields rrelatively primes.
-*}
+(* to do: add the three variations of these, and for ints? *)
+
-lemma div_gcd_relprime:
- assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
- shows "gcd (a div gcd a b) (b div gcd a b) = 1"
+subsection {* Coprimality *}
+
+lemma nat_div_gcd_coprime [intro]:
+ assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
+ shows "coprime (a div gcd a b) (b div gcd a b)"
proof -
let ?g = "gcd a b"
let ?a' = "a div ?g"
@@ -207,42 +568,551 @@
from dvdg dvdg' obtain ka kb ka' kb' where
kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
unfolding dvd_def by blast
- then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all
+ then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
+ by simp_all
then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
- have "?g \<noteq> 0" using nz by (simp add: gcd_zero)
- then have gp: "?g > 0" by simp
- from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
+ have "?g \<noteq> 0" using nz by (simp add: nat_gcd_zero)
+ then have gp: "?g > 0" by arith
+ from nat_gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
qed
+lemma int_div_gcd_coprime [intro]:
+ assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
+ shows "coprime (a div gcd a b) (b div gcd a b)"
-lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
-proof(auto)
- assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d"
- from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b]
- have th: "gcd a b dvd d" by blast
- from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]] show "d = gcd a b" by blast
+ apply (subst (1 2 3) int_gcd_abs)
+ apply (subst (1 2) abs_div)
+ apply auto
+ prefer 3
+ apply (rule nat_div_gcd_coprime [transferred])
+ using nz apply (auto simp add: int_gcd_abs [symmetric])
+done
+
+lemma nat_coprime: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
+ using nat_gcd_unique[of 1 a b, simplified] by auto
+
+lemma nat_coprime_Suc_0:
+ "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
+ using nat_coprime by (simp add: One_nat_def)
+
+lemma int_coprime: "coprime (a::int) b \<longleftrightarrow>
+ (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
+ using int_gcd_unique [of 1 a b]
+ apply clarsimp
+ apply (erule subst)
+ apply (rule iffI)
+ apply force
+ apply (drule_tac x = "abs e" in exI)
+ apply (case_tac "e >= 0")
+ apply force
+ apply force
+done
+
+lemma nat_gcd_coprime:
+ assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
+ b: "b = b' * gcd a b"
+ shows "coprime a' b'"
+
+ apply (subgoal_tac "a' = a div gcd a b")
+ apply (erule ssubst)
+ apply (subgoal_tac "b' = b div gcd a b")
+ apply (erule ssubst)
+ apply (rule nat_div_gcd_coprime)
+ using prems
+ apply force
+ apply (subst (1) b)
+ using z apply force
+ apply (subst (1) a)
+ using z apply force
+done
+
+lemma int_gcd_coprime:
+ assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
+ b: "b = b' * gcd a b"
+ shows "coprime a' b'"
+
+ apply (subgoal_tac "a' = a div gcd a b")
+ apply (erule ssubst)
+ apply (subgoal_tac "b' = b div gcd a b")
+ apply (erule ssubst)
+ apply (rule int_div_gcd_coprime)
+ using prems
+ apply force
+ apply (subst (1) b)
+ using z apply force
+ apply (subst (1) a)
+ using z apply force
+done
+
+lemma nat_coprime_mult: assumes da: "coprime (d::nat) a" and db: "coprime d b"
+ shows "coprime d (a * b)"
+ apply (subst nat_gcd_commute)
+ using da apply (subst nat_gcd_mult_cancel)
+ apply (subst nat_gcd_commute, assumption)
+ apply (subst nat_gcd_commute, rule db)
+done
+
+lemma int_coprime_mult: assumes da: "coprime (d::int) a" and db: "coprime d b"
+ shows "coprime d (a * b)"
+ apply (subst int_gcd_commute)
+ using da apply (subst int_gcd_mult_cancel)
+ apply (subst int_gcd_commute, assumption)
+ apply (subst int_gcd_commute, rule db)
+done
+
+lemma nat_coprime_lmult:
+ assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
+proof -
+ have "gcd d a dvd gcd d (a * b)"
+ by (rule nat_gcd_greatest, auto)
+ with dab show ?thesis
+ by auto
+qed
+
+lemma int_coprime_lmult:
+ assumes dab: "coprime (d::int) (a * b)" shows "coprime d a"
+proof -
+ have "gcd d a dvd gcd d (a * b)"
+ by (rule int_gcd_greatest, auto)
+ with dab show ?thesis
+ by auto
+qed
+
+lemma nat_coprime_rmult:
+ assumes dab: "coprime (d::nat) (a * b)" shows "coprime d b"
+proof -
+ have "gcd d b dvd gcd d (a * b)"
+ by (rule nat_gcd_greatest, auto intro: dvd_mult)
+ with dab show ?thesis
+ by auto
+qed
+
+lemma int_coprime_rmult:
+ assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
+proof -
+ have "gcd d b dvd gcd d (a * b)"
+ by (rule int_gcd_greatest, auto intro: dvd_mult)
+ with dab show ?thesis
+ by auto
+qed
+
+lemma nat_coprime_mul_eq: "coprime (d::nat) (a * b) \<longleftrightarrow>
+ coprime d a \<and> coprime d b"
+ using nat_coprime_rmult[of d a b] nat_coprime_lmult[of d a b]
+ nat_coprime_mult[of d a b]
+ by blast
+
+lemma int_coprime_mul_eq: "coprime (d::int) (a * b) \<longleftrightarrow>
+ coprime d a \<and> coprime d b"
+ using int_coprime_rmult[of d a b] int_coprime_lmult[of d a b]
+ int_coprime_mult[of d a b]
+ by blast
+
+lemma nat_gcd_coprime_exists:
+ assumes nz: "gcd (a::nat) b \<noteq> 0"
+ shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
+ apply (rule_tac x = "a div gcd a b" in exI)
+ apply (rule_tac x = "b div gcd a b" in exI)
+ using nz apply (auto simp add: nat_div_gcd_coprime dvd_div_mult)
+done
+
+lemma int_gcd_coprime_exists:
+ assumes nz: "gcd (a::int) b \<noteq> 0"
+ shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
+ apply (rule_tac x = "a div gcd a b" in exI)
+ apply (rule_tac x = "b div gcd a b" in exI)
+ using nz apply (auto simp add: int_div_gcd_coprime dvd_div_mult_self)
+done
+
+lemma nat_coprime_exp: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
+ by (induct n, simp_all add: nat_coprime_mult)
+
+lemma int_coprime_exp: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
+ by (induct n, simp_all add: int_coprime_mult)
+
+lemma nat_coprime_exp2 [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
+ apply (rule nat_coprime_exp)
+ apply (subst nat_gcd_commute)
+ apply (rule nat_coprime_exp)
+ apply (subst nat_gcd_commute, assumption)
+done
+
+lemma int_coprime_exp2 [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
+ apply (rule int_coprime_exp)
+ apply (subst int_gcd_commute)
+ apply (rule int_coprime_exp)
+ apply (subst int_gcd_commute, assumption)
+done
+
+lemma nat_gcd_exp: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
+proof (cases)
+ assume "a = 0 & b = 0"
+ thus ?thesis by simp
+ next assume "~(a = 0 & b = 0)"
+ hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
+ by auto
+ hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
+ ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
+ apply (subst (1 2) mult_commute)
+ apply (subst nat_gcd_mult_distrib [symmetric])
+ apply simp
+ done
+ also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
+ apply (subst div_power)
+ apply auto
+ apply (rule dvd_div_mult_self)
+ apply (rule dvd_power_same)
+ apply auto
+ done
+ also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
+ apply (subst div_power)
+ apply auto
+ apply (rule dvd_div_mult_self)
+ apply (rule dvd_power_same)
+ apply auto
+ done
+ finally show ?thesis .
+qed
+
+lemma int_gcd_exp: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
+ apply (subst (1 2) int_gcd_abs)
+ apply (subst (1 2) power_abs)
+ apply (rule nat_gcd_exp [where n = n, transferred])
+ apply auto
+done
+
+lemma nat_coprime_divprod: "(d::nat) dvd a * b \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
+ using nat_coprime_dvd_mult_iff[of d a b]
+ by (auto simp add: mult_commute)
+
+lemma int_coprime_divprod: "(d::int) dvd a * b \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
+ using int_coprime_dvd_mult_iff[of d a b]
+ by (auto simp add: mult_commute)
+
+lemma nat_division_decomp: assumes dc: "(a::nat) dvd b * c"
+ shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
+proof-
+ let ?g = "gcd a b"
+ {assume "?g = 0" with dc have ?thesis by auto}
+ moreover
+ {assume z: "?g \<noteq> 0"
+ from nat_gcd_coprime_exists[OF z]
+ obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
+ by blast
+ have thb: "?g dvd b" by auto
+ from ab'(1) have "a' dvd a" unfolding dvd_def by blast
+ with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
+ from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
+ hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
+ with z have th_1: "a' dvd b' * c" by auto
+ from nat_coprime_dvd_mult[OF ab'(3)] th_1
+ have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
+ from ab' have "a = ?g*a'" by algebra
+ with thb thc have ?thesis by blast }
+ ultimately show ?thesis by blast
+qed
+
+lemma int_division_decomp: assumes dc: "(a::int) dvd b * c"
+ shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
+proof-
+ let ?g = "gcd a b"
+ {assume "?g = 0" with dc have ?thesis by auto}
+ moreover
+ {assume z: "?g \<noteq> 0"
+ from int_gcd_coprime_exists[OF z]
+ obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
+ by blast
+ have thb: "?g dvd b" by auto
+ from ab'(1) have "a' dvd a" unfolding dvd_def by blast
+ with dc have th0: "a' dvd b*c"
+ using dvd_trans[of a' a "b*c"] by simp
+ from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
+ hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
+ with z have th_1: "a' dvd b' * c" by auto
+ from int_coprime_dvd_mult[OF ab'(3)] th_1
+ have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
+ from ab' have "a = ?g*a'" by algebra
+ with thb thc have ?thesis by blast }
+ ultimately show ?thesis by blast
qed
-lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v"
- shows "gcd x y = gcd u v"
+lemma nat_pow_divides_pow:
+ assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
+ shows "a dvd b"
+proof-
+ let ?g = "gcd a b"
+ from n obtain m where m: "n = Suc m" by (cases n, simp_all)
+ {assume "?g = 0" with ab n have ?thesis by auto }
+ moreover
+ {assume z: "?g \<noteq> 0"
+ hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
+ from nat_gcd_coprime_exists[OF z]
+ obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
+ by blast
+ from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
+ by (simp add: ab'(1,2)[symmetric])
+ hence "?g^n*a'^n dvd ?g^n *b'^n"
+ by (simp only: power_mult_distrib mult_commute)
+ with zn z n have th0:"a'^n dvd b'^n" by auto
+ have "a' dvd a'^n" by (simp add: m)
+ with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
+ hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
+ from nat_coprime_dvd_mult[OF nat_coprime_exp [OF ab'(3), of m]] th1
+ have "a' dvd b'" by (subst (asm) mult_commute, blast)
+ hence "a'*?g dvd b'*?g" by simp
+ with ab'(1,2) have ?thesis by simp }
+ ultimately show ?thesis by blast
+qed
+
+lemma int_pow_divides_pow:
+ assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
+ shows "a dvd b"
proof-
- from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd u v" by simp
- with gcd_unique[of "gcd u v" x y] show ?thesis by auto
+ let ?g = "gcd a b"
+ from n obtain m where m: "n = Suc m" by (cases n, simp_all)
+ {assume "?g = 0" with ab n have ?thesis by auto }
+ moreover
+ {assume z: "?g \<noteq> 0"
+ hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
+ from int_gcd_coprime_exists[OF z]
+ obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
+ by blast
+ from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
+ by (simp add: ab'(1,2)[symmetric])
+ hence "?g^n*a'^n dvd ?g^n *b'^n"
+ by (simp only: power_mult_distrib mult_commute)
+ with zn z n have th0:"a'^n dvd b'^n" by auto
+ have "a' dvd a'^n" by (simp add: m)
+ with th0 have "a' dvd b'^n"
+ using dvd_trans[of a' "a'^n" "b'^n"] by simp
+ hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
+ from int_coprime_dvd_mult[OF int_coprime_exp [OF ab'(3), of m]] th1
+ have "a' dvd b'" by (subst (asm) mult_commute, blast)
+ hence "a'*?g dvd b'*?g" by simp
+ with ab'(1,2) have ?thesis by simp }
+ ultimately show ?thesis by blast
+qed
+
+lemma nat_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
+ by (auto intro: nat_pow_divides_pow dvd_power_same)
+
+lemma int_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
+ by (auto intro: int_pow_divides_pow dvd_power_same)
+
+lemma nat_divides_mult:
+ assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
+ shows "m * n dvd r"
+proof-
+ from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
+ unfolding dvd_def by blast
+ from mr n' have "m dvd n'*n" by (simp add: mult_commute)
+ hence "m dvd n'" using nat_coprime_dvd_mult_iff[OF mn] by simp
+ then obtain k where k: "n' = m*k" unfolding dvd_def by blast
+ from n' k show ?thesis unfolding dvd_def by auto
+qed
+
+lemma int_divides_mult:
+ assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
+ shows "m * n dvd r"
+proof-
+ from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
+ unfolding dvd_def by blast
+ from mr n' have "m dvd n'*n" by (simp add: mult_commute)
+ hence "m dvd n'" using int_coprime_dvd_mult_iff[OF mn] by simp
+ then obtain k where k: "n' = m*k" unfolding dvd_def by blast
+ from n' k show ?thesis unfolding dvd_def by auto
qed
-lemma ind_euclid:
- assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
- and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
+lemma nat_coprime_plus_one [simp]: "coprime ((n::nat) + 1) n"
+ apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
+ apply force
+ apply (rule nat_dvd_diff)
+ apply auto
+done
+
+lemma nat_coprime_Suc [simp]: "coprime (Suc n) n"
+ using nat_coprime_plus_one by (simp add: One_nat_def)
+
+lemma int_coprime_plus_one [simp]: "coprime ((n::int) + 1) n"
+ apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
+ apply force
+ apply (rule dvd_diff)
+ apply auto
+done
+
+lemma nat_coprime_minus_one: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
+ using nat_coprime_plus_one [of "n - 1"]
+ nat_gcd_commute [of "n - 1" n] by auto
+
+lemma int_coprime_minus_one: "coprime ((n::int) - 1) n"
+ using int_coprime_plus_one [of "n - 1"]
+ int_gcd_commute [of "n - 1" n] by auto
+
+lemma nat_setprod_coprime [rule_format]:
+ "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
+ apply (case_tac "finite A")
+ apply (induct set: finite)
+ apply (auto simp add: nat_gcd_mult_cancel)
+done
+
+lemma int_setprod_coprime [rule_format]:
+ "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
+ apply (case_tac "finite A")
+ apply (induct set: finite)
+ apply (auto simp add: int_gcd_mult_cancel)
+done
+
+lemma nat_prime_odd: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
+ unfolding prime_nat_def
+ apply (subst even_mult_two_ex)
+ apply clarify
+ apply (drule_tac x = 2 in spec)
+ apply auto
+done
+
+lemma int_prime_odd: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
+ unfolding prime_int_def
+ apply (frule nat_prime_odd)
+ apply (auto simp add: even_nat_def)
+done
+
+lemma nat_coprime_common_divisor: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
+ x dvd b \<Longrightarrow> x = 1"
+ apply (subgoal_tac "x dvd gcd a b")
+ apply simp
+ apply (erule (1) nat_gcd_greatest)
+done
+
+lemma int_coprime_common_divisor: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
+ x dvd b \<Longrightarrow> abs x = 1"
+ apply (subgoal_tac "x dvd gcd a b")
+ apply simp
+ apply (erule (1) int_gcd_greatest)
+done
+
+lemma nat_coprime_divisors: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
+ coprime d e"
+ apply (auto simp add: dvd_def)
+ apply (frule int_coprime_lmult)
+ apply (subst int_gcd_commute)
+ apply (subst (asm) (2) int_gcd_commute)
+ apply (erule int_coprime_lmult)
+done
+
+lemma nat_invertible_coprime: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
+apply (metis nat_coprime_lmult nat_gcd_1 nat_gcd_commute nat_gcd_red)
+done
+
+lemma int_invertible_coprime: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
+apply (metis int_coprime_lmult int_gcd_1 int_gcd_commute int_gcd_red)
+done
+
+
+subsection {* Bezout's theorem *}
+
+(* Function bezw returns a pair of witnesses to Bezout's theorem --
+ see the theorems that follow the definition. *)
+fun
+ bezw :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
+where
+ "bezw x y =
+ (if y = 0 then (1, 0) else
+ (snd (bezw y (x mod y)),
+ fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
+
+lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
+
+lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
+ fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
+ by simp
+
+declare bezw.simps [simp del]
+
+lemma bezw_aux [rule_format]:
+ "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
+proof (induct x y rule: nat_gcd_induct)
+ fix m :: nat
+ show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
+ by auto
+ next fix m :: nat and n
+ assume ngt0: "n > 0" and
+ ih: "fst (bezw n (m mod n)) * int n +
+ snd (bezw n (m mod n)) * int (m mod n) =
+ int (gcd n (m mod n))"
+ thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
+ apply (simp add: bezw_non_0 nat_gcd_non_0)
+ apply (erule subst)
+ apply (simp add: ring_simps)
+ apply (subst mod_div_equality [of m n, symmetric])
+ (* applying simp here undoes the last substitution!
+ what is procedure cancel_div_mod? *)
+ apply (simp only: ring_simps zadd_int [symmetric]
+ zmult_int [symmetric])
+ done
+qed
+
+lemma int_bezout:
+ fixes x y
+ shows "EX u v. u * (x::int) + v * y = gcd x y"
+proof -
+ have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
+ EX u v. u * x + v * y = gcd x y"
+ apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
+ apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
+ apply (unfold gcd_int_def)
+ apply simp
+ apply (subst bezw_aux [symmetric])
+ apply auto
+ done
+ have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
+ (x \<le> 0 \<and> y \<le> 0)"
+ by auto
+ moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
+ by (erule (1) bezout_aux)
+ moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
+ apply (insert bezout_aux [of x "-y"])
+ apply auto
+ apply (rule_tac x = u in exI)
+ apply (rule_tac x = "-v" in exI)
+ apply (subst int_gcd_neg2 [symmetric])
+ apply auto
+ done
+ moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
+ apply (insert bezout_aux [of "-x" y])
+ apply auto
+ apply (rule_tac x = "-u" in exI)
+ apply (rule_tac x = v in exI)
+ apply (subst int_gcd_neg1 [symmetric])
+ apply auto
+ done
+ moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
+ apply (insert bezout_aux [of "-x" "-y"])
+ apply auto
+ apply (rule_tac x = "-u" in exI)
+ apply (rule_tac x = "-v" in exI)
+ apply (subst int_gcd_neg1 [symmetric])
+ apply (subst int_gcd_neg2 [symmetric])
+ apply auto
+ done
+ ultimately show ?thesis by blast
+qed
+
+text {* versions of Bezout for nat, by Amine Chaieb *}
+
+lemma ind_euclid:
+ assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
+ and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
shows "P a b"
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
fix n a b
assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
have "a = b \<or> a < b \<or> b < a" by arith
moreover {assume eq: "a= b"
- from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
+ from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
+ by simp}
moreover
{assume lt: "a < b"
hence "a + b - a < n \<or> a = 0" using H(2) by arith
@@ -269,65 +1139,67 @@
ultimately show "P a b" by blast
qed
-lemma bezout_lemma:
- assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
- shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
-using ex
-apply clarsimp
-apply (rule_tac x="d" in exI, simp add: dvd_add)
-apply (case_tac "a * x = b * y + d" , simp_all)
-apply (rule_tac x="x + y" in exI)
-apply (rule_tac x="y" in exI)
-apply algebra
-apply (rule_tac x="x" in exI)
-apply (rule_tac x="x + y" in exI)
-apply algebra
+lemma nat_bezout_lemma:
+ assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
+ (a * x = b * y + d \<or> b * x = a * y + d)"
+ shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
+ (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
+ using ex
+ apply clarsimp
+ apply (rule_tac x="d" in exI, simp add: dvd_add)
+ apply (case_tac "a * x = b * y + d" , simp_all)
+ apply (rule_tac x="x + y" in exI)
+ apply (rule_tac x="y" in exI)
+ apply algebra
+ apply (rule_tac x="x" in exI)
+ apply (rule_tac x="x + y" in exI)
+ apply algebra
done
-lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
-apply(induct a b rule: ind_euclid)
-apply blast
-apply clarify
-apply (rule_tac x="a" in exI, simp add: dvd_add)
-apply clarsimp
-apply (rule_tac x="d" in exI)
-apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
-apply (rule_tac x="x+y" in exI)
-apply (rule_tac x="y" in exI)
-apply algebra
-apply (rule_tac x="x" in exI)
-apply (rule_tac x="x+y" in exI)
-apply algebra
+lemma nat_bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
+ (a * x = b * y + d \<or> b * x = a * y + d)"
+ apply(induct a b rule: ind_euclid)
+ apply blast
+ apply clarify
+ apply (rule_tac x="a" in exI, simp add: dvd_add)
+ apply clarsimp
+ apply (rule_tac x="d" in exI)
+ apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
+ apply (rule_tac x="x+y" in exI)
+ apply (rule_tac x="y" in exI)
+ apply algebra
+ apply (rule_tac x="x" in exI)
+ apply (rule_tac x="x+y" in exI)
+ apply algebra
done
-lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
-using bezout_add[of a b]
-apply clarsimp
-apply (rule_tac x="d" in exI, simp)
-apply (rule_tac x="x" in exI)
-apply (rule_tac x="y" in exI)
-apply auto
+lemma nat_bezout1: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
+ (a * x - b * y = d \<or> b * x - a * y = d)"
+ using nat_bezout_add[of a b]
+ apply clarsimp
+ apply (rule_tac x="d" in exI, simp)
+ apply (rule_tac x="x" in exI)
+ apply (rule_tac x="y" in exI)
+ apply auto
done
-
-text {* We can get a stronger version with a nonzeroness assumption. *}
-lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)
-
-lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
+lemma nat_bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
proof-
- from nz have ap: "a > 0" by simp
- from bezout_add[of a b]
- have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
+ from nz have ap: "a > 0" by simp
+ from nat_bezout_add[of a b]
+ have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
+ (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
moreover
- {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
- from H have ?thesis by blast }
+ {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
+ from H have ?thesis by blast }
moreover
{fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
{assume b0: "b = 0" with H have ?thesis by simp}
- moreover
+ moreover
{assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
- from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast
+ from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
+ by auto
moreover
{assume db: "d=b"
from prems have ?thesis apply simp
@@ -335,18 +1207,22 @@
apply (rule exI[where x = b])
by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
moreover
- {assume db: "d < b"
+ {assume db: "d < b"
{assume "x=0" hence ?thesis using prems by simp }
moreover
{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
-
from db have "d \<le> b - 1" by simp
hence "d*b \<le> b*(b - 1)" by simp
with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
- from H (3) have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
+ from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
+ by simp
+ hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
+ by (simp only: mult_assoc right_distrib)
+ hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
+ by algebra
hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
- hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
+ hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
by (simp only: diff_add_assoc[OF dble, of d, symmetric])
hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
by (simp only: diff_mult_distrib2 add_commute mult_ac)
@@ -361,156 +1237,156 @@
ultimately show ?thesis by blast
qed
-
-lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd a b \<or> b * x - a * y = gcd a b"
-proof-
- let ?g = "gcd a b"
- from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast
- from d(1,2) have "d dvd ?g" by simp
- then obtain k where k: "?g = d*k" unfolding dvd_def by blast
- from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast
- hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k"
- by (algebra add: diff_mult_distrib)
- hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g"
- by (simp add: k mult_assoc)
- thus ?thesis by blast
-qed
-
-lemma bezout_gcd_strong: assumes a: "a \<noteq> 0"
+lemma nat_bezout: assumes a: "(a::nat) \<noteq> 0"
shows "\<exists>x y. a * x = b * y + gcd a b"
proof-
let ?g = "gcd a b"
- from bezout_add_strong[OF a, of b]
+ from nat_bezout_add_strong[OF a, of b]
obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
from d(1,2) have "d dvd ?g" by simp
then obtain k where k: "?g = d*k" unfolding dvd_def by blast
- from d(3) have "a * x * k = (b * y + d) *k " by algebra
+ from d(3) have "a * x * k = (b * y + d) *k " by auto
hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
thus ?thesis by blast
qed
-lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"
-by(simp add: gcd_mult_distrib2 mult_commute)
+
+subsection {* LCM *}
+
+lemma int_lcm_altdef: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
+ by (simp add: lcm_int_def lcm_nat_def zdiv_int
+ zmult_int [symmetric] gcd_int_def)
+
+lemma nat_prod_gcd_lcm: "(m::nat) * n = gcd m n * lcm m n"
+ unfolding lcm_nat_def
+ by (simp add: dvd_mult_div_cancel [OF nat_gcd_dvd_prod])
+
+lemma int_prod_gcd_lcm: "abs(m::int) * abs n = gcd m n * lcm m n"
+ unfolding lcm_int_def gcd_int_def
+ apply (subst int_mult [symmetric])
+ apply (subst nat_prod_gcd_lcm [symmetric])
+ apply (subst nat_abs_mult_distrib [symmetric])
+ apply (simp, simp add: abs_mult)
+done
+
+lemma nat_lcm_0 [simp]: "lcm (m::nat) 0 = 0"
+ unfolding lcm_nat_def by simp
+
+lemma int_lcm_0 [simp]: "lcm (m::int) 0 = 0"
+ unfolding lcm_int_def by simp
+
+lemma nat_lcm_1 [simp]: "lcm (m::nat) 1 = m"
+ unfolding lcm_nat_def by simp
+
+lemma nat_lcm_Suc_0 [simp]: "lcm (m::nat) (Suc 0) = m"
+ unfolding lcm_nat_def by (simp add: One_nat_def)
+
+lemma int_lcm_1 [simp]: "lcm (m::int) 1 = abs m"
+ unfolding lcm_int_def by simp
+
+lemma nat_lcm_0_left [simp]: "lcm (0::nat) n = 0"
+ unfolding lcm_nat_def by simp
-lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd a b dvd d"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
- let ?g = "gcd a b"
- {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
- from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g"
- by blast
- hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto
- hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k"
- by (simp only: diff_mult_distrib)
- hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d"
- by (simp add: k[symmetric] mult_assoc)
- hence ?lhs by blast}
+lemma int_lcm_0_left [simp]: "lcm (0::int) n = 0"
+ unfolding lcm_int_def by simp
+
+lemma nat_lcm_1_left [simp]: "lcm (1::nat) m = m"
+ unfolding lcm_nat_def by simp
+
+lemma nat_lcm_Suc_0_left [simp]: "lcm (Suc 0) m = m"
+ unfolding lcm_nat_def by (simp add: One_nat_def)
+
+lemma int_lcm_1_left [simp]: "lcm (1::int) m = abs m"
+ unfolding lcm_int_def by simp
+
+lemma nat_lcm_commute: "lcm (m::nat) n = lcm n m"
+ unfolding lcm_nat_def by (simp add: nat_gcd_commute ring_simps)
+
+lemma int_lcm_commute: "lcm (m::int) n = lcm n m"
+ unfolding lcm_int_def by (subst nat_lcm_commute, rule refl)
+
+(* to do: show lcm is associative, and then declare ac simps *)
+
+lemma nat_lcm_pos:
+ assumes mpos: "(m::nat) > 0"
+ and npos: "n>0"
+ shows "lcm m n > 0"
+proof(rule ccontr, simp add: lcm_nat_def nat_gcd_zero)
+ assume h:"m*n div gcd m n = 0"
+ from mpos npos have "gcd m n \<noteq> 0" using nat_gcd_zero by simp
+ hence gcdp: "gcd m n > 0" by simp
+ with h
+ have "m*n < gcd m n"
+ by (cases "m * n < gcd m n")
+ (auto simp add: div_if[OF gcdp, where m="m*n"])
moreover
- {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
- have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y"
- using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
- from nat_dvd_diff[OF dv(1,2)] nat_dvd_diff[OF dv(3,4)] H
- have ?rhs by auto}
- ultimately show ?thesis by blast
-qed
-
-lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d"
-proof-
- let ?g = "gcd a b"
- have dv: "?g dvd a*x" "?g dvd b * y"
- using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
- from dvd_add[OF dv] H
- show ?thesis by auto
+ have "gcd m n dvd m" by simp
+ with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
+ with npos have t1:"gcd m n*n \<le> m*n" by simp
+ have "gcd m n \<le> gcd m n*n" using npos by simp
+ with t1 have "gcd m n \<le> m*n" by arith
+ ultimately show "False" by simp
qed
-lemma gcd_mult': "gcd b (a * b) = b"
-by (simp add: gcd_mult mult_commute[of a b])
-
-lemma gcd_add: "gcd(a + b) b = gcd a b"
- "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"
-apply (simp_all add: gcd_add1)
-by (simp add: gcd_commute gcd_add1)
-
-lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b"
-proof-
- {fix a b assume H: "b \<le> (a::nat)"
- hence th: "a - b + b = a" by arith
- from gcd_add(1)[of "a - b" b] th have "gcd(a - b) b = gcd a b" by simp}
- note th = this
-{
- assume ab: "b \<le> a"
- from th[OF ab] show "gcd (a - b) b = gcd a b" by blast
-next
- assume ab: "a \<le> b"
- from th[OF ab] show "gcd a (b - a) = gcd a b"
- by (simp add: gcd_commute)}
-qed
-
+lemma int_lcm_pos:
+ assumes mneq0: "(m::int) ~= 0"
+ and npos: "n ~= 0"
+ shows "lcm m n > 0"
-subsection {* LCM defined by GCD *}
-
-
-definition
- lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
- lcm_def: "lcm m n = m * n div gcd m n"
-
-lemma prod_gcd_lcm:
- "m * n = gcd m n * lcm m n"
- unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
+ apply (subst int_lcm_abs)
+ apply (rule nat_lcm_pos [transferred])
+ using prems apply auto
+done
-lemma lcm_0 [simp]: "lcm m 0 = 0"
- unfolding lcm_def by simp
-
-lemma lcm_1 [simp]: "lcm m 1 = m"
- unfolding lcm_def by simp
-
-lemma lcm_0_left [simp]: "lcm 0 n = 0"
- unfolding lcm_def by simp
-
-lemma lcm_1_left [simp]: "lcm 1 m = m"
- unfolding lcm_def by simp
-
-lemma dvd_pos:
+lemma nat_dvd_pos:
fixes n m :: nat
assumes "n > 0" and "m dvd n"
shows "m > 0"
using assms by (cases m) auto
-lemma lcm_least:
- assumes "m dvd k" and "n dvd k"
+lemma nat_lcm_least:
+ assumes "(m::nat) dvd k" and "n dvd k"
shows "lcm m n dvd k"
proof (cases k)
case 0 then show ?thesis by auto
next
case (Suc _) then have pos_k: "k > 0" by auto
- from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
- with gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
+ from assms nat_dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
+ with nat_gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
from assms obtain p where k_m: "k = m * p" using dvd_def by blast
from assms obtain q where k_n: "k = n * q" using dvd_def by blast
from pos_k k_m have pos_p: "p > 0" by auto
from pos_k k_n have pos_q: "q > 0" by auto
have "k * k * gcd q p = k * gcd (k * q) (k * p)"
- by (simp add: mult_ac gcd_mult_distrib2)
+ by (simp add: mult_ac nat_gcd_mult_distrib)
also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
by (simp add: k_m [symmetric] k_n [symmetric])
also have "\<dots> = k * p * q * gcd m n"
- by (simp add: mult_ac gcd_mult_distrib2)
+ by (simp add: mult_ac nat_gcd_mult_distrib)
finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
by (simp only: k_m [symmetric] k_n [symmetric])
then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
by (simp add: mult_ac)
with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
by simp
- with prod_gcd_lcm [of m n]
+ with nat_prod_gcd_lcm [of m n]
have "lcm m n * gcd q p * gcd m n = k * gcd m n"
by (simp add: mult_ac)
- with pos_gcd have "lcm m n * gcd q p = k" by simp
+ with pos_gcd have "lcm m n * gcd q p = k" by auto
then show ?thesis using dvd_def by auto
qed
-lemma lcm_dvd1 [iff]:
- "m dvd lcm m n"
+lemma int_lcm_least:
+ assumes "(m::int) dvd k" and "n dvd k"
+ shows "lcm m n dvd k"
+
+ apply (subst int_lcm_abs)
+ apply (rule dvd_trans)
+ apply (rule nat_lcm_least [transferred, of _ "abs k" _])
+ using prems apply auto
+done
+
+lemma nat_lcm_dvd1 [iff]: "(m::nat) dvd lcm m n"
proof (cases m)
case 0 then show ?thesis by simp
next
@@ -524,264 +1400,323 @@
then have npos: "n > 0" by simp
have "gcd m n dvd n" by simp
then obtain k where "n = gcd m n * k" using dvd_def by auto
- then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n" by (simp add: mult_ac)
- also have "\<dots> = m * k" using mpos npos gcd_zero by simp
- finally show ?thesis by (simp add: lcm_def)
- qed
-qed
-
-lemma lcm_dvd2 [iff]:
- "n dvd lcm m n"
-proof (cases n)
- case 0 then show ?thesis by simp
-next
- case (Suc _)
- then have npos: "n > 0" by simp
- show ?thesis
- proof (cases m)
- case 0 then show ?thesis by simp
- next
- case (Suc _)
- then have mpos: "m > 0" by simp
- have "gcd m n dvd m" by simp
- then obtain k where "m = gcd m n * k" using dvd_def by auto
- then have "m * n div gcd m n = (gcd m n * k) * n div gcd m n" by (simp add: mult_ac)
- also have "\<dots> = n * k" using mpos npos gcd_zero by simp
- finally show ?thesis by (simp add: lcm_def)
+ then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
+ by (simp add: mult_ac)
+ also have "\<dots> = m * k" using mpos npos nat_gcd_zero by simp
+ finally show ?thesis by (simp add: lcm_nat_def)
qed
qed
-lemma gcd_add1_eq: "gcd (m + k) k = gcd (m + k) m"
- by (simp add: gcd_commute)
+lemma int_lcm_dvd1 [iff]: "(m::int) dvd lcm m n"
+ apply (subst int_lcm_abs)
+ apply (rule dvd_trans)
+ prefer 2
+ apply (rule nat_lcm_dvd1 [transferred])
+ apply auto
+done
+
+lemma nat_lcm_dvd2 [iff]: "(n::nat) dvd lcm m n"
+ by (subst nat_lcm_commute, rule nat_lcm_dvd1)
+
+lemma int_lcm_dvd2 [iff]: "(n::int) dvd lcm m n"
+ by (subst int_lcm_commute, rule int_lcm_dvd1)
+
+lemma nat_lcm_unique: "(a::nat) dvd d \<and> b dvd d \<and>
+ (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
+ by (auto intro: dvd_anti_sym nat_lcm_least)
-lemma gcd_diff2: "m \<le> n ==> gcd n (n - m) = gcd n m"
- apply (subgoal_tac "n = m + (n - m)")
- apply (erule ssubst, rule gcd_add1_eq, simp)
- done
+lemma int_lcm_unique: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
+ (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
+ by (auto intro: dvd_anti_sym [transferred] int_lcm_least)
+
+lemma nat_lcm_dvd_eq [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
+ apply (rule sym)
+ apply (subst nat_lcm_unique [symmetric])
+ apply auto
+done
+
+lemma int_lcm_dvd_eq [simp]: "0 <= y \<Longrightarrow> (x::int) dvd y \<Longrightarrow> lcm x y = y"
+ apply (rule sym)
+ apply (subst int_lcm_unique [symmetric])
+ apply auto
+done
+
+lemma nat_lcm_dvd_eq' [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
+ by (subst nat_lcm_commute, erule nat_lcm_dvd_eq)
+
+lemma int_lcm_dvd_eq' [simp]: "y >= 0 \<Longrightarrow> (x::int) dvd y \<Longrightarrow> lcm y x = y"
+ by (subst int_lcm_commute, erule (1) int_lcm_dvd_eq)
+
-subsection {* GCD and LCM on integers *}
-
-definition
- zgcd :: "int \<Rightarrow> int \<Rightarrow> int" where
- "zgcd i j = int (gcd (nat (abs i)) (nat (abs j)))"
+subsection {* Primes *}
-lemma zgcd_zdvd1 [iff,simp, algebra]: "zgcd i j dvd i"
-by (simp add: zgcd_def int_dvd_iff)
+(* Is there a better way to handle these, rather than making them
+ elim rules? *)
-lemma zgcd_zdvd2 [iff,simp, algebra]: "zgcd i j dvd j"
-by (simp add: zgcd_def int_dvd_iff)
+lemma nat_prime_ge_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
+ by (unfold prime_nat_def, auto)
-lemma zgcd_pos: "zgcd i j \<ge> 0"
-by (simp add: zgcd_def)
+lemma nat_prime_gt_0 [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
+ by (unfold prime_nat_def, auto)
-lemma zgcd0 [simp,algebra]: "(zgcd i j = 0) = (i = 0 \<and> j = 0)"
-by (simp add: zgcd_def gcd_zero)
+lemma nat_prime_ge_1 [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
+ by (unfold prime_nat_def, auto)
-lemma zgcd_commute: "zgcd i j = zgcd j i"
-unfolding zgcd_def by (simp add: gcd_commute)
+lemma nat_prime_gt_1 [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
+ by (unfold prime_nat_def, auto)
-lemma zgcd_zminus [simp, algebra]: "zgcd (- i) j = zgcd i j"
-unfolding zgcd_def by simp
+lemma nat_prime_ge_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
+ by (unfold prime_nat_def, auto)
-lemma zgcd_zminus2 [simp, algebra]: "zgcd i (- j) = zgcd i j"
-unfolding zgcd_def by simp
+lemma nat_prime_gt_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
+ by (unfold prime_nat_def, auto)
+
+lemma nat_prime_ge_2 [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
+ by (unfold prime_nat_def, auto)
+
+lemma int_prime_ge_0 [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
+ by (unfold prime_int_def prime_nat_def, auto)
- (* should be solved by algebra*)
-lemma zrelprime_dvd_mult: "zgcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
- unfolding zgcd_def
-proof -
- assume "int (gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>)) = 1" "i dvd k * j"
- then have g: "gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>) = 1" by simp
- from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast
- have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
- unfolding dvd_def
- by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric])
- from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
- unfolding dvd_def by blast
- from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp
- then have "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
- then show ?thesis
- apply (subst abs_dvd_iff [symmetric])
- apply (subst dvd_abs_iff [symmetric])
- apply (unfold dvd_def)
- apply (rule_tac x = "int h'" in exI, simp)
- done
-qed
+lemma int_prime_gt_0 [elim]: "prime (p::int) \<Longrightarrow> p > 0"
+ by (unfold prime_int_def prime_nat_def, auto)
+
+lemma int_prime_ge_1 [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
+ by (unfold prime_int_def prime_nat_def, auto)
-lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
+lemma int_prime_gt_1 [elim]: "prime (p::int) \<Longrightarrow> p > 1"
+ by (unfold prime_int_def prime_nat_def, auto)
+
+lemma int_prime_ge_2 [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
+ by (unfold prime_int_def prime_nat_def, auto)
-lemma zgcd_greatest:
- assumes "k dvd m" and "k dvd n"
- shows "k dvd zgcd m n"
-proof -
- let ?k' = "nat \<bar>k\<bar>"
- let ?m' = "nat \<bar>m\<bar>"
- let ?n' = "nat \<bar>n\<bar>"
- from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
- unfolding zdvd_int by (simp_all only: int_nat_abs abs_dvd_iff dvd_abs_iff)
- from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd zgcd m n"
- unfolding zgcd_def by (simp only: zdvd_int)
- then have "\<bar>k\<bar> dvd zgcd m n" by (simp only: int_nat_abs)
- then show "k dvd zgcd m n" by simp
-qed
+thm prime_nat_def;
+thm prime_nat_def [transferred];
+
+lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
+ m = 1 \<or> m = p))"
+ using prime_nat_def [transferred]
+ apply (case_tac "p >= 0")
+ by (blast, auto simp add: int_prime_ge_0)
+
+(* To do: determine primality of any numeral *)
+
+lemma nat_zero_not_prime [simp]: "~prime (0::nat)"
+ by (simp add: prime_nat_def)
+
+lemma int_zero_not_prime [simp]: "~prime (0::int)"
+ by (simp add: prime_int_def)
+
+lemma nat_one_not_prime [simp]: "~prime (1::nat)"
+ by (simp add: prime_nat_def)
-lemma div_zgcd_relprime:
- assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
- shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1"
-proof -
- from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by arith
- let ?g = "zgcd a b"
- let ?a' = "a div ?g"
- let ?b' = "b div ?g"
- let ?g' = "zgcd ?a' ?b'"
- have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
- have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
- from dvdg dvdg' obtain ka kb ka' kb' where
- kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
- unfolding dvd_def by blast
- then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
- then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
- by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)]
- zdvd_mult_div_cancel [OF dvdg(2)] dvd_def)
- have "?g \<noteq> 0" using nz by simp
- then have gp: "?g \<noteq> 0" using zgcd_pos[where i="a" and j="b"] by arith
- from zgcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
- with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp
- with zgcd_pos show "?g' = 1" by simp
-qed
+lemma nat_Suc_0_not_prime [simp]: "~prime (Suc 0)"
+ by (simp add: prime_nat_def One_nat_def)
+
+lemma int_one_not_prime [simp]: "~prime (1::int)"
+ by (simp add: prime_int_def)
+
+lemma nat_two_is_prime [simp]: "prime (2::nat)"
+ apply (auto simp add: prime_nat_def)
+ apply (case_tac m)
+ apply (auto dest!: dvd_imp_le)
+ done
-lemma zgcd_0 [simp, algebra]: "zgcd m 0 = abs m"
- by (simp add: zgcd_def abs_if)
-
-lemma zgcd_0_left [simp, algebra]: "zgcd 0 m = abs m"
- by (simp add: zgcd_def abs_if)
+lemma int_two_is_prime [simp]: "prime (2::int)"
+ by (rule nat_two_is_prime [transferred direction: nat "op <= (0::int)"])
-lemma zgcd_non_0: "0 < n ==> zgcd m n = zgcd n (m mod n)"
- apply (frule_tac b = n and a = m in pos_mod_sign)
- apply (simp del: pos_mod_sign add: zgcd_def abs_if nat_mod_distrib)
- apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
- apply (frule_tac a = m in pos_mod_bound)
- apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2 nat_le_eq_zle)
+lemma nat_prime_imp_coprime: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
+ apply (unfold prime_nat_def)
+ apply (metis nat_gcd_dvd1 nat_gcd_dvd2)
+ done
+
+lemma int_prime_imp_coprime: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
+ apply (unfold prime_int_altdef)
+ apply (metis int_gcd_dvd1 int_gcd_dvd2 int_gcd_ge_0)
done
-lemma zgcd_eq: "zgcd m n = zgcd n (m mod n)"
- apply (case_tac "n = 0", simp add: DIVISION_BY_ZERO)
- apply (auto simp add: linorder_neq_iff zgcd_non_0)
- apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0, auto)
- done
+lemma nat_prime_dvd_mult: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
+ by (blast intro: nat_coprime_dvd_mult nat_prime_imp_coprime)
+
+lemma int_prime_dvd_mult: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
+ by (blast intro: int_coprime_dvd_mult int_prime_imp_coprime)
+
+lemma nat_prime_dvd_mult_eq [simp]: "prime (p::nat) \<Longrightarrow>
+ p dvd m * n = (p dvd m \<or> p dvd n)"
+ by (rule iffI, rule nat_prime_dvd_mult, auto)
-lemma zgcd_1 [simp, algebra]: "zgcd m 1 = 1"
- by (simp add: zgcd_def abs_if)
+lemma int_prime_dvd_mult_eq [simp]: "prime (p::int) \<Longrightarrow>
+ p dvd m * n = (p dvd m \<or> p dvd n)"
+ by (rule iffI, rule int_prime_dvd_mult, auto)
-lemma zgcd_0_1_iff [simp, algebra]: "zgcd 0 m = 1 \<longleftrightarrow> \<bar>m\<bar> = 1"
- by (simp add: zgcd_def abs_if)
-
-lemma zgcd_greatest_iff[algebra]: "k dvd zgcd m n = (k dvd m \<and> k dvd n)"
- by (simp add: zgcd_def abs_if int_dvd_iff dvd_int_iff nat_dvd_iff)
+lemma nat_not_prime_eq_prod: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
+ EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
+ unfolding prime_nat_def dvd_def apply auto
+ apply (subgoal_tac "k > 1")
+ apply force
+ apply (subgoal_tac "k ~= 0")
+ apply force
+ apply (rule notI)
+ apply force
+done
-lemma zgcd_1_left [simp, algebra]: "zgcd 1 m = 1"
- by (simp add: zgcd_def gcd_1_left)
-
-lemma zgcd_assoc: "zgcd (zgcd k m) n = zgcd k (zgcd m n)"
- by (simp add: zgcd_def gcd_assoc)
+(* there's a lot of messing around with signs of products here --
+ could this be made more automatic? *)
+lemma int_not_prime_eq_prod: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
+ EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
+ unfolding prime_int_altdef dvd_def
+ apply auto
+ apply (rule_tac x = m in exI)
+ apply (rule_tac x = k in exI)
+ apply (auto simp add: mult_compare_simps)
+ apply (subgoal_tac "k > 0")
+ apply arith
+ apply (case_tac "k <= 0")
+ apply (subgoal_tac "m * k <= 0")
+ apply force
+ apply (subst zero_compare_simps(8))
+ apply auto
+ apply (subgoal_tac "m * k <= 0")
+ apply force
+ apply (subst zero_compare_simps(8))
+ apply auto
+done
-lemma zgcd_left_commute: "zgcd k (zgcd m n) = zgcd m (zgcd k n)"
- apply (rule zgcd_commute [THEN trans])
- apply (rule zgcd_assoc [THEN trans])
- apply (rule zgcd_commute [THEN arg_cong])
- done
+lemma nat_prime_dvd_power [rule_format]: "prime (p::nat) -->
+ n > 0 --> (p dvd x^n --> p dvd x)"
+ by (induct n rule: nat_induct, auto)
-lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute
- -- {* addition is an AC-operator *}
+lemma int_prime_dvd_power [rule_format]: "prime (p::int) -->
+ n > 0 --> (p dvd x^n --> p dvd x)"
+ apply (induct n rule: nat_induct, auto)
+ apply (frule int_prime_ge_0)
+ apply auto
+done
+
+lemma nat_prime_imp_power_coprime: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
+ coprime a (p^m)"
+ apply (rule nat_coprime_exp)
+ apply (subst nat_gcd_commute)
+ apply (erule (1) nat_prime_imp_coprime)
+done
-lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd m n = zgcd (k * m) (k * n)"
- by (simp del: minus_mult_right [symmetric]
- add: minus_mult_right nat_mult_distrib zgcd_def abs_if
- mult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
+lemma int_prime_imp_power_coprime: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
+ coprime a (p^m)"
+ apply (rule int_coprime_exp)
+ apply (subst int_gcd_commute)
+ apply (erule (1) int_prime_imp_coprime)
+done
-lemma zgcd_zmult_distrib2_abs: "zgcd (k * m) (k * n) = abs k * zgcd m n"
- by (simp add: abs_if zgcd_zmult_distrib2)
+lemma nat_primes_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
+ apply (rule nat_prime_imp_coprime, assumption)
+ apply (unfold prime_nat_def, auto)
+done
-lemma zgcd_self [simp]: "0 \<le> m ==> zgcd m m = m"
- by (cut_tac k = m and m = 1 and n = 1 in zgcd_zmult_distrib2, simp_all)
+lemma int_primes_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
+ apply (rule int_prime_imp_coprime, assumption)
+ apply (unfold prime_int_altdef, clarify)
+ apply (drule_tac x = q in spec)
+ apply (drule_tac x = p in spec)
+ apply auto
+done
-lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd k (k * n) = k"
- by (cut_tac k = k and m = 1 and n = n in zgcd_zmult_distrib2, simp_all)
+lemma nat_primes_imp_powers_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
+ coprime (p^m) (q^n)"
+ by (rule nat_coprime_exp2, rule nat_primes_coprime)
-lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n) k = k"
- by (cut_tac k = k and m = n and n = 1 in zgcd_zmult_distrib2, simp_all)
+lemma int_primes_imp_powers_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
+ coprime (p^m) (q^n)"
+ by (rule int_coprime_exp2, rule int_primes_coprime)
-
-definition "zlcm i j = int (lcm(nat(abs i)) (nat(abs j)))"
+lemma nat_prime_factor: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
+ apply (induct n rule: nat_less_induct)
+ apply (case_tac "n = 0")
+ using nat_two_is_prime apply blast
+ apply (case_tac "prime n")
+ apply blast
+ apply (subgoal_tac "n > 1")
+ apply (frule (1) nat_not_prime_eq_prod)
+ apply (auto intro: dvd_mult dvd_mult2)
+done
-lemma dvd_zlcm_self1[simp, algebra]: "i dvd zlcm i j"
-by(simp add:zlcm_def dvd_int_iff)
+(* An Isar version:
+
+lemma nat_prime_factor_b:
+ fixes n :: nat
+ assumes "n \<noteq> 1"
+ shows "\<exists>p. prime p \<and> p dvd n"
-lemma dvd_zlcm_self2[simp, algebra]: "j dvd zlcm i j"
-by(simp add:zlcm_def dvd_int_iff)
-
-
-lemma dvd_imp_dvd_zlcm1:
- assumes "k dvd i" shows "k dvd (zlcm i j)"
-proof -
- have "nat(abs k) dvd nat(abs i)" using `k dvd i`
- by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
- thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
+using `n ~= 1`
+proof (induct n rule: nat_less_induct)
+ fix n :: nat
+ assume "n ~= 1" and
+ ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
+ thus "\<exists>p. prime p \<and> p dvd n"
+ proof -
+ {
+ assume "n = 0"
+ moreover note nat_two_is_prime
+ ultimately have ?thesis
+ by (auto simp del: nat_two_is_prime)
+ }
+ moreover
+ {
+ assume "prime n"
+ hence ?thesis by auto
+ }
+ moreover
+ {
+ assume "n ~= 0" and "~ prime n"
+ with `n ~= 1` have "n > 1" by auto
+ with `~ prime n` and nat_not_prime_eq_prod obtain m k where
+ "n = m * k" and "1 < m" and "m < n" by blast
+ with ih obtain p where "prime p" and "p dvd m" by blast
+ with `n = m * k` have ?thesis by auto
+ }
+ ultimately show ?thesis by blast
+ qed
qed
-lemma dvd_imp_dvd_zlcm2:
- assumes "k dvd j" shows "k dvd (zlcm i j)"
-proof -
- have "nat(abs k) dvd nat(abs j)" using `k dvd j`
- by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
- thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
+*)
+
+text {* One property of coprimality is easier to prove via prime factors. *}
+
+lemma nat_prime_divprod_pow:
+ assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
+ shows "p^n dvd a \<or> p^n dvd b"
+proof-
+ {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
+ apply (cases "n=0", simp_all)
+ apply (cases "a=1", simp_all) done}
+ moreover
+ {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
+ then obtain m where m: "n = Suc m" by (cases n, auto)
+ from n have "p dvd p^n" by (intro dvd_power, auto)
+ also note pab
+ finally have pab': "p dvd a * b".
+ from nat_prime_dvd_mult[OF p pab']
+ have "p dvd a \<or> p dvd b" .
+ moreover
+ {assume pa: "p dvd a"
+ have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
+ from nat_coprime_common_divisor [OF ab, OF pa] p have "\<not> p dvd b" by auto
+ with p have "coprime b p"
+ by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
+ hence pnb: "coprime (p^n) b"
+ by (subst nat_gcd_commute, rule nat_coprime_exp)
+ from nat_coprime_divprod[OF pnba pnb] have ?thesis by blast }
+ moreover
+ {assume pb: "p dvd b"
+ have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
+ from nat_coprime_common_divisor [OF ab, of p] pb p have "\<not> p dvd a"
+ by auto
+ with p have "coprime a p"
+ by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
+ hence pna: "coprime (p^n) a"
+ by (subst nat_gcd_commute, rule nat_coprime_exp)
+ from nat_coprime_divprod[OF pab pna] have ?thesis by blast }
+ ultimately have ?thesis by blast}
+ ultimately show ?thesis by blast
qed
-
-lemma zdvd_self_abs1: "(d::int) dvd (abs d)"
-by (case_tac "d <0", simp_all)
-
-lemma zdvd_self_abs2: "(abs (d::int)) dvd d"
-by (case_tac "d<0", simp_all)
-
-(* lcm a b is positive for positive a and b *)
-
-lemma lcm_pos:
- assumes mpos: "m > 0"
- and npos: "n>0"
- shows "lcm m n > 0"
-proof(rule ccontr, simp add: lcm_def gcd_zero)
-assume h:"m*n div gcd m n = 0"
-from mpos npos have "gcd m n \<noteq> 0" using gcd_zero by simp
-hence gcdp: "gcd m n > 0" by simp
-with h
-have "m*n < gcd m n"
- by (cases "m * n < gcd m n") (auto simp add: div_if[OF gcdp, where m="m*n"])
-moreover
-have "gcd m n dvd m" by simp
- with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
- with npos have t1:"gcd m n *n \<le> m*n" by simp
- have "gcd m n \<le> gcd m n*n" using npos by simp
- with t1 have "gcd m n \<le> m*n" by arith
-ultimately show "False" by simp
-qed
-
-lemma zlcm_pos:
- assumes anz: "a \<noteq> 0"
- and bnz: "b \<noteq> 0"
- shows "0 < zlcm a b"
-proof-
- let ?na = "nat (abs a)"
- let ?nb = "nat (abs b)"
- have nap: "?na >0" using anz by simp
- have nbp: "?nb >0" using bnz by simp
- have "0 < lcm ?na ?nb" by (rule lcm_pos[OF nap nbp])
- thus ?thesis by (simp add: zlcm_def)
-qed
-
-lemma zgcd_code [code]:
- "zgcd k l = \<bar>if l = 0 then k else zgcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
- by (simp add: zgcd_def gcd.simps [of "nat \<bar>k\<bar>"] nat_mod_distrib)
-
end
--- a/src/HOL/IsaMakefile Tue Jun 16 22:07:39 2009 -0700
+++ b/src/HOL/IsaMakefile Wed Jun 17 16:55:01 2009 -0700
@@ -284,6 +284,7 @@
Ln.thy \
Log.thy \
MacLaurin.thy \
+ NatTransfer.thy \
NthRoot.thy \
SEQ.thy \
Series.thy \
@@ -300,6 +301,7 @@
Real.thy \
RealVector.thy \
Tools/float_syntax.ML \
+ Tools/transfer_data.ML \
Tools/Qelim/ferrante_rackoff_data.ML \
Tools/Qelim/ferrante_rackoff.ML \
Tools/Qelim/langford_data.ML \
@@ -325,6 +327,7 @@
Library/FrechetDeriv.thy \
Library/Fundamental_Theorem_Algebra.thy \
Library/Inner_Product.thy Library/Lattice_Syntax.thy \
+ Library/Legacy_GCD.thy \
Library/Library.thy Library/List_Prefix.thy Library/State_Monad.thy \
Library/Nat_Int_Bij.thy Library/Multiset.thy Library/Permutation.thy \
Library/Primes.thy Library/Pocklington.thy Library/Quotient.thy \
--- a/src/HOL/Library/Abstract_Rat.thy Tue Jun 16 22:07:39 2009 -0700
+++ b/src/HOL/Library/Abstract_Rat.thy Wed Jun 17 16:55:01 2009 -0700
@@ -21,17 +21,17 @@
definition
isnormNum :: "Num \<Rightarrow> bool"
where
- "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> zgcd a b = 1))"
+ "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1))"
definition
normNum :: "Num \<Rightarrow> Num"
where
"normNum = (\<lambda>(a,b). (if a=0 \<or> b = 0 then (0,0) else
- (let g = zgcd a b
+ (let g = gcd a b
in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
-declare zgcd_zdvd1[presburger]
-declare zgcd_zdvd2[presburger]
+declare int_gcd_dvd1[presburger]
+declare int_gcd_dvd2[presburger]
lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
proof -
have " \<exists> a b. x = (a,b)" by auto
@@ -39,19 +39,19 @@
{assume "a=0 \<or> b = 0" hence ?thesis by (simp add: normNum_def isnormNum_def)}
moreover
{assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
- let ?g = "zgcd a b"
+ let ?g = "gcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
- let ?g' = "zgcd ?a' ?b'"
- from anz bnz have "?g \<noteq> 0" by simp with zgcd_pos[of a b]
+ let ?g' = "gcd ?a' ?b'"
+ from anz bnz have "?g \<noteq> 0" by simp with int_gcd_ge_0[of a b]
have gpos: "?g > 0" by arith
have gdvd: "?g dvd a" "?g dvd b" by arith+
from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)]
anz bnz
- have nz':"?a' \<noteq> 0" "?b' \<noteq> 0"
- by - (rule notI,simp add:zgcd_def)+
+ have nz':"?a' \<noteq> 0" "?b' \<noteq> 0"
+ by - (rule notI, simp)+
from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
- from div_zgcd_relprime[OF stupid] have gp1: "?g' = 1" .
+ from int_div_gcd_coprime[OF stupid] have gp1: "?g' = 1" .
from bnz have "b < 0 \<or> b > 0" by arith
moreover
{assume b: "b > 0"
@@ -67,7 +67,7 @@
have False using b by arith }
hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
from anz bnz nz' b b' gp1 have ?thesis
- by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
+ by (simp add: isnormNum_def normNum_def Let_def split_def)}
ultimately have ?thesis by blast
}
ultimately show ?thesis by blast
@@ -85,7 +85,7 @@
definition
Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60)
where
- "Nmul = (\<lambda>(a,b) (a',b'). let g = zgcd (a*a') (b*b')
+ "Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b')
in (a*a' div g, b*b' div g))"
definition
@@ -121,11 +121,11 @@
then obtain a b a' b' where ab: "x = (a,b)" and ab': "y = (a',b')" by blast
{assume "a = 0"
hence ?thesis using xn ab ab'
- by (simp add: zgcd_def isnormNum_def Let_def Nmul_def split_def)}
+ by (simp add: isnormNum_def Let_def Nmul_def split_def)}
moreover
{assume "a' = 0"
hence ?thesis using yn ab ab'
- by (simp add: zgcd_def isnormNum_def Let_def Nmul_def split_def)}
+ by (simp add: isnormNum_def Let_def Nmul_def split_def)}
moreover
{assume a: "a \<noteq>0" and a': "a'\<noteq>0"
hence bp: "b > 0" "b' > 0" using xn yn ab ab' by (simp_all add: isnormNum_def)
@@ -137,11 +137,11 @@
lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
by (simp add: Ninv_def isnormNum_def split_def)
- (cases "fst x = 0", auto simp add: zgcd_commute)
+ (cases "fst x = 0", auto simp add: int_gcd_commute)
lemma isnormNum_int[simp]:
"isnormNum 0\<^sub>N" "isnormNum (1::int)\<^sub>N" "i \<noteq> 0 \<Longrightarrow> isnormNum i\<^sub>N"
- by (simp_all add: isnormNum_def zgcd_def)
+ by (simp_all add: isnormNum_def)
text {* Relations over Num *}
@@ -202,8 +202,8 @@
from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: isnormNum_def)
from prems have eq:"a * b' = a'*b"
by (simp add: INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
- from prems have gcd1: "zgcd a b = 1" "zgcd b a = 1" "zgcd a' b' = 1" "zgcd b' a' = 1"
- by (simp_all add: isnormNum_def add: zgcd_commute)
+ from prems have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
+ by (simp_all add: isnormNum_def add: int_gcd_commute)
from eq have raw_dvd: "a dvd a'*b" "b dvd b'*a" "a' dvd a*b'" "b' dvd b*a'"
apply -
apply algebra
@@ -211,8 +211,8 @@
apply simp
apply algebra
done
- from zdvd_dvd_eq[OF bz zrelprime_dvd_mult[OF gcd1(2) raw_dvd(2)]
- zrelprime_dvd_mult[OF gcd1(4) raw_dvd(4)]]
+ from zdvd_dvd_eq[OF bz int_coprime_dvd_mult[OF gcd1(2) raw_dvd(2)]
+ int_coprime_dvd_mult[OF gcd1(4) raw_dvd(4)]]
have eq1: "b = b'" using pos by arith
with eq have "a = a'" using pos by simp
with eq1 have ?rhs by simp}
@@ -258,7 +258,7 @@
by (simp add: INum_def normNum_def split_def Let_def)}
moreover
{assume a: "a\<noteq>0" and b: "b\<noteq>0"
- let ?g = "zgcd a b"
+ let ?g = "gcd a b"
from a b have g: "?g \<noteq> 0"by simp
from of_int_div[OF g, where ?'a = 'a]
have ?thesis by (auto simp add: INum_def normNum_def split_def Let_def)}
@@ -294,11 +294,11 @@
from z aa' bb' have ?thesis
by (simp add: th Nadd_def normNum_def INum_def split_def)}
moreover {assume z: "a * b' + b * a' \<noteq> 0"
- let ?g = "zgcd (a * b' + b * a') (b*b')"
+ let ?g = "gcd (a * b' + b * a') (b*b')"
have gz: "?g \<noteq> 0" using z by simp
have ?thesis using aa' bb' z gz
- of_int_div[where ?'a = 'a, OF gz zgcd_zdvd1[where i="a * b' + b * a'" and j="b*b'"]] of_int_div[where ?'a = 'a,
- OF gz zgcd_zdvd2[where i="a * b' + b * a'" and j="b*b'"]]
+ of_int_div[where ?'a = 'a, OF gz int_gcd_dvd1[where x="a * b' + b * a'" and y="b*b'"]] of_int_div[where ?'a = 'a,
+ OF gz int_gcd_dvd2[where x="a * b' + b * a'" and y="b*b'"]]
by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib)}
ultimately have ?thesis using aa' bb'
by (simp add: Nadd_def INum_def normNum_def x y Let_def) }
@@ -317,10 +317,10 @@
done }
moreover
{assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
- let ?g="zgcd (a*a') (b*b')"
+ let ?g="gcd (a*a') (b*b')"
have gz: "?g \<noteq> 0" using z by simp
- from z of_int_div[where ?'a = 'a, OF gz zgcd_zdvd1[where i="a*a'" and j="b*b'"]]
- of_int_div[where ?'a = 'a , OF gz zgcd_zdvd2[where i="a*a'" and j="b*b'"]]
+ from z of_int_div[where ?'a = 'a, OF gz int_gcd_dvd1[where x="a*a'" and y="b*b'"]]
+ of_int_div[where ?'a = 'a , OF gz int_gcd_dvd2[where x="a*a'" and y="b*b'"]]
have ?thesis by (simp add: Nmul_def x y Let_def INum_def)}
ultimately show ?thesis by blast
qed
@@ -478,7 +478,7 @@
qed
lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
- by (simp add: Nmul_def split_def Let_def zgcd_commute mult_commute)
+ by (simp add: Nmul_def split_def Let_def int_gcd_commute mult_commute)
lemma Nmul_assoc:
assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Legacy_GCD.thy Wed Jun 17 16:55:01 2009 -0700
@@ -0,0 +1,787 @@
+(* Title: HOL/GCD.thy
+ Author: Christophe Tabacznyj and Lawrence C Paulson
+ Copyright 1996 University of Cambridge
+*)
+
+header {* The Greatest Common Divisor *}
+
+theory Legacy_GCD
+imports Main
+begin
+
+text {*
+ See \cite{davenport92}. \bigskip
+*}
+
+subsection {* Specification of GCD on nats *}
+
+definition
+ is_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where -- {* @{term gcd} as a relation *}
+ [code del]: "is_gcd m n p \<longleftrightarrow> p dvd m \<and> p dvd n \<and>
+ (\<forall>d. d dvd m \<longrightarrow> d dvd n \<longrightarrow> d dvd p)"
+
+text {* Uniqueness *}
+
+lemma is_gcd_unique: "is_gcd a b m \<Longrightarrow> is_gcd a b n \<Longrightarrow> m = n"
+ by (simp add: is_gcd_def) (blast intro: dvd_anti_sym)
+
+text {* Connection to divides relation *}
+
+lemma is_gcd_dvd: "is_gcd a b m \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd m"
+ by (auto simp add: is_gcd_def)
+
+text {* Commutativity *}
+
+lemma is_gcd_commute: "is_gcd m n k = is_gcd n m k"
+ by (auto simp add: is_gcd_def)
+
+
+subsection {* GCD on nat by Euclid's algorithm *}
+
+fun
+ gcd :: "nat => nat => nat"
+where
+ "gcd m n = (if n = 0 then m else gcd n (m mod n))"
+lemma gcd_induct [case_names "0" rec]:
+ fixes m n :: nat
+ assumes "\<And>m. P m 0"
+ and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
+ shows "P m n"
+proof (induct m n rule: gcd.induct)
+ case (1 m n) with assms show ?case by (cases "n = 0") simp_all
+qed
+
+lemma gcd_0 [simp, algebra]: "gcd m 0 = m"
+ by simp
+
+lemma gcd_0_left [simp,algebra]: "gcd 0 m = m"
+ by simp
+
+lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd m n = gcd n (m mod n)"
+ by simp
+
+lemma gcd_1 [simp, algebra]: "gcd m (Suc 0) = Suc 0"
+ by simp
+
+lemma nat_gcd_1_right [simp, algebra]: "gcd m 1 = 1"
+ unfolding One_nat_def by (rule gcd_1)
+
+declare gcd.simps [simp del]
+
+text {*
+ \medskip @{term "gcd m n"} divides @{text m} and @{text n}. The
+ conjunctions don't seem provable separately.
+*}
+
+lemma gcd_dvd1 [iff, algebra]: "gcd m n dvd m"
+ and gcd_dvd2 [iff, algebra]: "gcd m n dvd n"
+ apply (induct m n rule: gcd_induct)
+ apply (simp_all add: gcd_non_0)
+ apply (blast dest: dvd_mod_imp_dvd)
+ done
+
+text {*
+ \medskip Maximality: for all @{term m}, @{term n}, @{term k}
+ naturals, if @{term k} divides @{term m} and @{term k} divides
+ @{term n} then @{term k} divides @{term "gcd m n"}.
+*}
+
+lemma gcd_greatest: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
+ by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod)
+
+text {*
+ \medskip Function gcd yields the Greatest Common Divisor.
+*}
+
+lemma is_gcd: "is_gcd m n (gcd m n) "
+ by (simp add: is_gcd_def gcd_greatest)
+
+
+subsection {* Derived laws for GCD *}
+
+lemma gcd_greatest_iff [iff, algebra]: "k dvd gcd m n \<longleftrightarrow> k dvd m \<and> k dvd n"
+ by (blast intro!: gcd_greatest intro: dvd_trans)
+
+lemma gcd_zero[algebra]: "gcd m n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
+ by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff)
+
+lemma gcd_commute: "gcd m n = gcd n m"
+ apply (rule is_gcd_unique)
+ apply (rule is_gcd)
+ apply (subst is_gcd_commute)
+ apply (simp add: is_gcd)
+ done
+
+lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)"
+ apply (rule is_gcd_unique)
+ apply (rule is_gcd)
+ apply (simp add: is_gcd_def)
+ apply (blast intro: dvd_trans)
+ done
+
+lemma gcd_1_left [simp, algebra]: "gcd (Suc 0) m = Suc 0"
+ by (simp add: gcd_commute)
+
+lemma nat_gcd_1_left [simp, algebra]: "gcd 1 m = 1"
+ unfolding One_nat_def by (rule gcd_1_left)
+
+text {*
+ \medskip Multiplication laws
+*}
+
+lemma gcd_mult_distrib2: "k * gcd m n = gcd (k * m) (k * n)"
+ -- {* \cite[page 27]{davenport92} *}
+ apply (induct m n rule: gcd_induct)
+ apply simp
+ apply (case_tac "k = 0")
+ apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
+ done
+
+lemma gcd_mult [simp, algebra]: "gcd k (k * n) = k"
+ apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
+ done
+
+lemma gcd_self [simp, algebra]: "gcd k k = k"
+ apply (rule gcd_mult [of k 1, simplified])
+ done
+
+lemma relprime_dvd_mult: "gcd k n = 1 ==> k dvd m * n ==> k dvd m"
+ apply (insert gcd_mult_distrib2 [of m k n])
+ apply simp
+ apply (erule_tac t = m in ssubst)
+ apply simp
+ done
+
+lemma relprime_dvd_mult_iff: "gcd k n = 1 ==> (k dvd m * n) = (k dvd m)"
+ by (auto intro: relprime_dvd_mult dvd_mult2)
+
+lemma gcd_mult_cancel: "gcd k n = 1 ==> gcd (k * m) n = gcd m n"
+ apply (rule dvd_anti_sym)
+ apply (rule gcd_greatest)
+ apply (rule_tac n = k in relprime_dvd_mult)
+ apply (simp add: gcd_assoc)
+ apply (simp add: gcd_commute)
+ apply (simp_all add: mult_commute)
+ done
+
+
+text {* \medskip Addition laws *}
+
+lemma gcd_add1 [simp, algebra]: "gcd (m + n) n = gcd m n"
+ by (cases "n = 0") (auto simp add: gcd_non_0)
+
+lemma gcd_add2 [simp, algebra]: "gcd m (m + n) = gcd m n"
+proof -
+ have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute)
+ also have "... = gcd (n + m) m" by (simp add: add_commute)
+ also have "... = gcd n m" by simp
+ also have "... = gcd m n" by (rule gcd_commute)
+ finally show ?thesis .
+qed
+
+lemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n"
+ apply (subst add_commute)
+ apply (rule gcd_add2)
+ done
+
+lemma gcd_add_mult[algebra]: "gcd m (k * m + n) = gcd m n"
+ by (induct k) (simp_all add: add_assoc)
+
+lemma gcd_dvd_prod: "gcd m n dvd m * n"
+ using mult_dvd_mono [of 1] by auto
+
+text {*
+ \medskip Division by gcd yields rrelatively primes.
+*}
+
+lemma div_gcd_relprime:
+ assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
+ shows "gcd (a div gcd a b) (b div gcd a b) = 1"
+proof -
+ let ?g = "gcd a b"
+ let ?a' = "a div ?g"
+ let ?b' = "b div ?g"
+ let ?g' = "gcd ?a' ?b'"
+ have dvdg: "?g dvd a" "?g dvd b" by simp_all
+ have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
+ from dvdg dvdg' obtain ka kb ka' kb' where
+ kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
+ unfolding dvd_def by blast
+ then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all
+ then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
+ by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
+ dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
+ have "?g \<noteq> 0" using nz by (simp add: gcd_zero)
+ then have gp: "?g > 0" by simp
+ from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
+ with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
+qed
+
+
+lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
+proof(auto)
+ assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d"
+ from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b]
+ have th: "gcd a b dvd d" by blast
+ from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]] show "d = gcd a b" by blast
+qed
+
+lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v"
+ shows "gcd x y = gcd u v"
+proof-
+ from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd u v" by simp
+ with gcd_unique[of "gcd u v" x y] show ?thesis by auto
+qed
+
+lemma ind_euclid:
+ assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
+ and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
+ shows "P a b"
+proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
+ fix n a b
+ assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
+ have "a = b \<or> a < b \<or> b < a" by arith
+ moreover {assume eq: "a= b"
+ from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
+ moreover
+ {assume lt: "a < b"
+ hence "a + b - a < n \<or> a = 0" using H(2) by arith
+ moreover
+ {assume "a =0" with z c have "P a b" by blast }
+ moreover
+ {assume ab: "a + b - a < n"
+ have th0: "a + b - a = a + (b - a)" using lt by arith
+ from add[rule_format, OF H(1)[rule_format, OF ab th0]]
+ have "P a b" by (simp add: th0[symmetric])}
+ ultimately have "P a b" by blast}
+ moreover
+ {assume lt: "a > b"
+ hence "b + a - b < n \<or> b = 0" using H(2) by arith
+ moreover
+ {assume "b =0" with z c have "P a b" by blast }
+ moreover
+ {assume ab: "b + a - b < n"
+ have th0: "b + a - b = b + (a - b)" using lt by arith
+ from add[rule_format, OF H(1)[rule_format, OF ab th0]]
+ have "P b a" by (simp add: th0[symmetric])
+ hence "P a b" using c by blast }
+ ultimately have "P a b" by blast}
+ultimately show "P a b" by blast
+qed
+
+lemma bezout_lemma:
+ assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
+ shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
+using ex
+apply clarsimp
+apply (rule_tac x="d" in exI, simp add: dvd_add)
+apply (case_tac "a * x = b * y + d" , simp_all)
+apply (rule_tac x="x + y" in exI)
+apply (rule_tac x="y" in exI)
+apply algebra
+apply (rule_tac x="x" in exI)
+apply (rule_tac x="x + y" in exI)
+apply algebra
+done
+
+lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
+apply(induct a b rule: ind_euclid)
+apply blast
+apply clarify
+apply (rule_tac x="a" in exI, simp add: dvd_add)
+apply clarsimp
+apply (rule_tac x="d" in exI)
+apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
+apply (rule_tac x="x+y" in exI)
+apply (rule_tac x="y" in exI)
+apply algebra
+apply (rule_tac x="x" in exI)
+apply (rule_tac x="x+y" in exI)
+apply algebra
+done
+
+lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
+using bezout_add[of a b]
+apply clarsimp
+apply (rule_tac x="d" in exI, simp)
+apply (rule_tac x="x" in exI)
+apply (rule_tac x="y" in exI)
+apply auto
+done
+
+
+text {* We can get a stronger version with a nonzeroness assumption. *}
+lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)
+
+lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
+ shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
+proof-
+ from nz have ap: "a > 0" by simp
+ from bezout_add[of a b]
+ have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
+ moreover
+ {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
+ from H have ?thesis by blast }
+ moreover
+ {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
+ {assume b0: "b = 0" with H have ?thesis by simp}
+ moreover
+ {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
+ from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast
+ moreover
+ {assume db: "d=b"
+ from prems have ?thesis apply simp
+ apply (rule exI[where x = b], simp)
+ apply (rule exI[where x = b])
+ by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
+ moreover
+ {assume db: "d < b"
+ {assume "x=0" hence ?thesis using prems by simp }
+ moreover
+ {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
+
+ from db have "d \<le> b - 1" by simp
+ hence "d*b \<le> b*(b - 1)" by simp
+ with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
+ have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
+ from H (3) have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
+ hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
+ hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
+ by (simp only: diff_add_assoc[OF dble, of d, symmetric])
+ hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
+ by (simp only: diff_mult_distrib2 add_commute mult_ac)
+ hence ?thesis using H(1,2)
+ apply -
+ apply (rule exI[where x=d], simp)
+ apply (rule exI[where x="(b - 1) * y"])
+ by (rule exI[where x="x*(b - 1) - d"], simp)}
+ ultimately have ?thesis by blast}
+ ultimately have ?thesis by blast}
+ ultimately have ?thesis by blast}
+ ultimately show ?thesis by blast
+qed
+
+
+lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd a b \<or> b * x - a * y = gcd a b"
+proof-
+ let ?g = "gcd a b"
+ from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast
+ from d(1,2) have "d dvd ?g" by simp
+ then obtain k where k: "?g = d*k" unfolding dvd_def by blast
+ from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast
+ hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k"
+ by (algebra add: diff_mult_distrib)
+ hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g"
+ by (simp add: k mult_assoc)
+ thus ?thesis by blast
+qed
+
+lemma bezout_gcd_strong: assumes a: "a \<noteq> 0"
+ shows "\<exists>x y. a * x = b * y + gcd a b"
+proof-
+ let ?g = "gcd a b"
+ from bezout_add_strong[OF a, of b]
+ obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
+ from d(1,2) have "d dvd ?g" by simp
+ then obtain k where k: "?g = d*k" unfolding dvd_def by blast
+ from d(3) have "a * x * k = (b * y + d) *k " by algebra
+ hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
+ thus ?thesis by blast
+qed
+
+lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"
+by(simp add: gcd_mult_distrib2 mult_commute)
+
+lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd a b dvd d"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+ let ?g = "gcd a b"
+ {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
+ from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g"
+ by blast
+ hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto
+ hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k"
+ by (simp only: diff_mult_distrib)
+ hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d"
+ by (simp add: k[symmetric] mult_assoc)
+ hence ?lhs by blast}
+ moreover
+ {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
+ have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y"
+ using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
+ from nat_dvd_diff[OF dv(1,2)] nat_dvd_diff[OF dv(3,4)] H
+ have ?rhs by auto}
+ ultimately show ?thesis by blast
+qed
+
+lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d"
+proof-
+ let ?g = "gcd a b"
+ have dv: "?g dvd a*x" "?g dvd b * y"
+ using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
+ from dvd_add[OF dv] H
+ show ?thesis by auto
+qed
+
+lemma gcd_mult': "gcd b (a * b) = b"
+by (simp add: gcd_mult mult_commute[of a b])
+
+lemma gcd_add: "gcd(a + b) b = gcd a b"
+ "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"
+apply (simp_all add: gcd_add1)
+by (simp add: gcd_commute gcd_add1)
+
+lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b"
+proof-
+ {fix a b assume H: "b \<le> (a::nat)"
+ hence th: "a - b + b = a" by arith
+ from gcd_add(1)[of "a - b" b] th have "gcd(a - b) b = gcd a b" by simp}
+ note th = this
+{
+ assume ab: "b \<le> a"
+ from th[OF ab] show "gcd (a - b) b = gcd a b" by blast
+next
+ assume ab: "a \<le> b"
+ from th[OF ab] show "gcd a (b - a) = gcd a b"
+ by (simp add: gcd_commute)}
+qed
+
+
+subsection {* LCM defined by GCD *}
+
+
+definition
+ lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+ lcm_def: "lcm m n = m * n div gcd m n"
+
+lemma prod_gcd_lcm:
+ "m * n = gcd m n * lcm m n"
+ unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
+
+lemma lcm_0 [simp]: "lcm m 0 = 0"
+ unfolding lcm_def by simp
+
+lemma lcm_1 [simp]: "lcm m 1 = m"
+ unfolding lcm_def by simp
+
+lemma lcm_0_left [simp]: "lcm 0 n = 0"
+ unfolding lcm_def by simp
+
+lemma lcm_1_left [simp]: "lcm 1 m = m"
+ unfolding lcm_def by simp
+
+lemma dvd_pos:
+ fixes n m :: nat
+ assumes "n > 0" and "m dvd n"
+ shows "m > 0"
+using assms by (cases m) auto
+
+lemma lcm_least:
+ assumes "m dvd k" and "n dvd k"
+ shows "lcm m n dvd k"
+proof (cases k)
+ case 0 then show ?thesis by auto
+next
+ case (Suc _) then have pos_k: "k > 0" by auto
+ from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
+ with gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
+ from assms obtain p where k_m: "k = m * p" using dvd_def by blast
+ from assms obtain q where k_n: "k = n * q" using dvd_def by blast
+ from pos_k k_m have pos_p: "p > 0" by auto
+ from pos_k k_n have pos_q: "q > 0" by auto
+ have "k * k * gcd q p = k * gcd (k * q) (k * p)"
+ by (simp add: mult_ac gcd_mult_distrib2)
+ also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
+ by (simp add: k_m [symmetric] k_n [symmetric])
+ also have "\<dots> = k * p * q * gcd m n"
+ by (simp add: mult_ac gcd_mult_distrib2)
+ finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
+ by (simp only: k_m [symmetric] k_n [symmetric])
+ then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
+ by (simp add: mult_ac)
+ with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
+ by simp
+ with prod_gcd_lcm [of m n]
+ have "lcm m n * gcd q p * gcd m n = k * gcd m n"
+ by (simp add: mult_ac)
+ with pos_gcd have "lcm m n * gcd q p = k" by simp
+ then show ?thesis using dvd_def by auto
+qed
+
+lemma lcm_dvd1 [iff]:
+ "m dvd lcm m n"
+proof (cases m)
+ case 0 then show ?thesis by simp
+next
+ case (Suc _)
+ then have mpos: "m > 0" by simp
+ show ?thesis
+ proof (cases n)
+ case 0 then show ?thesis by simp
+ next
+ case (Suc _)
+ then have npos: "n > 0" by simp
+ have "gcd m n dvd n" by simp
+ then obtain k where "n = gcd m n * k" using dvd_def by auto
+ then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n" by (simp add: mult_ac)
+ also have "\<dots> = m * k" using mpos npos gcd_zero by simp
+ finally show ?thesis by (simp add: lcm_def)
+ qed
+qed
+
+lemma lcm_dvd2 [iff]:
+ "n dvd lcm m n"
+proof (cases n)
+ case 0 then show ?thesis by simp
+next
+ case (Suc _)
+ then have npos: "n > 0" by simp
+ show ?thesis
+ proof (cases m)
+ case 0 then show ?thesis by simp
+ next
+ case (Suc _)
+ then have mpos: "m > 0" by simp
+ have "gcd m n dvd m" by simp
+ then obtain k where "m = gcd m n * k" using dvd_def by auto
+ then have "m * n div gcd m n = (gcd m n * k) * n div gcd m n" by (simp add: mult_ac)
+ also have "\<dots> = n * k" using mpos npos gcd_zero by simp
+ finally show ?thesis by (simp add: lcm_def)
+ qed
+qed
+
+lemma gcd_add1_eq: "gcd (m + k) k = gcd (m + k) m"
+ by (simp add: gcd_commute)
+
+lemma gcd_diff2: "m \<le> n ==> gcd n (n - m) = gcd n m"
+ apply (subgoal_tac "n = m + (n - m)")
+ apply (erule ssubst, rule gcd_add1_eq, simp)
+ done
+
+
+subsection {* GCD and LCM on integers *}
+
+definition
+ zgcd :: "int \<Rightarrow> int \<Rightarrow> int" where
+ "zgcd i j = int (gcd (nat (abs i)) (nat (abs j)))"
+
+lemma zgcd_zdvd1 [iff,simp, algebra]: "zgcd i j dvd i"
+by (simp add: zgcd_def int_dvd_iff)
+
+lemma zgcd_zdvd2 [iff,simp, algebra]: "zgcd i j dvd j"
+by (simp add: zgcd_def int_dvd_iff)
+
+lemma zgcd_pos: "zgcd i j \<ge> 0"
+by (simp add: zgcd_def)
+
+lemma zgcd0 [simp,algebra]: "(zgcd i j = 0) = (i = 0 \<and> j = 0)"
+by (simp add: zgcd_def gcd_zero)
+
+lemma zgcd_commute: "zgcd i j = zgcd j i"
+unfolding zgcd_def by (simp add: gcd_commute)
+
+lemma zgcd_zminus [simp, algebra]: "zgcd (- i) j = zgcd i j"
+unfolding zgcd_def by simp
+
+lemma zgcd_zminus2 [simp, algebra]: "zgcd i (- j) = zgcd i j"
+unfolding zgcd_def by simp
+
+ (* should be solved by algebra*)
+lemma zrelprime_dvd_mult: "zgcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
+ unfolding zgcd_def
+proof -
+ assume "int (gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>)) = 1" "i dvd k * j"
+ then have g: "gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>) = 1" by simp
+ from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast
+ have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
+ unfolding dvd_def
+ by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric])
+ from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
+ unfolding dvd_def by blast
+ from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp
+ then have "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
+ then show ?thesis
+ apply (subst abs_dvd_iff [symmetric])
+ apply (subst dvd_abs_iff [symmetric])
+ apply (unfold dvd_def)
+ apply (rule_tac x = "int h'" in exI, simp)
+ done
+qed
+
+lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
+
+lemma zgcd_greatest:
+ assumes "k dvd m" and "k dvd n"
+ shows "k dvd zgcd m n"
+proof -
+ let ?k' = "nat \<bar>k\<bar>"
+ let ?m' = "nat \<bar>m\<bar>"
+ let ?n' = "nat \<bar>n\<bar>"
+ from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
+ unfolding zdvd_int by (simp_all only: int_nat_abs abs_dvd_iff dvd_abs_iff)
+ from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd zgcd m n"
+ unfolding zgcd_def by (simp only: zdvd_int)
+ then have "\<bar>k\<bar> dvd zgcd m n" by (simp only: int_nat_abs)
+ then show "k dvd zgcd m n" by simp
+qed
+
+lemma div_zgcd_relprime:
+ assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
+ shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1"
+proof -
+ from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by arith
+ let ?g = "zgcd a b"
+ let ?a' = "a div ?g"
+ let ?b' = "b div ?g"
+ let ?g' = "zgcd ?a' ?b'"
+ have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
+ have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
+ from dvdg dvdg' obtain ka kb ka' kb' where
+ kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
+ unfolding dvd_def by blast
+ then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
+ then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
+ by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)]
+ zdvd_mult_div_cancel [OF dvdg(2)] dvd_def)
+ have "?g \<noteq> 0" using nz by simp
+ then have gp: "?g \<noteq> 0" using zgcd_pos[where i="a" and j="b"] by arith
+ from zgcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
+ with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp
+ with zgcd_pos show "?g' = 1" by simp
+qed
+
+lemma zgcd_0 [simp, algebra]: "zgcd m 0 = abs m"
+ by (simp add: zgcd_def abs_if)
+
+lemma zgcd_0_left [simp, algebra]: "zgcd 0 m = abs m"
+ by (simp add: zgcd_def abs_if)
+
+lemma zgcd_non_0: "0 < n ==> zgcd m n = zgcd n (m mod n)"
+ apply (frule_tac b = n and a = m in pos_mod_sign)
+ apply (simp del: pos_mod_sign add: zgcd_def abs_if nat_mod_distrib)
+ apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
+ apply (frule_tac a = m in pos_mod_bound)
+ apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2 nat_le_eq_zle)
+ done
+
+lemma zgcd_eq: "zgcd m n = zgcd n (m mod n)"
+ apply (case_tac "n = 0", simp add: DIVISION_BY_ZERO)
+ apply (auto simp add: linorder_neq_iff zgcd_non_0)
+ apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0, auto)
+ done
+
+lemma zgcd_1 [simp, algebra]: "zgcd m 1 = 1"
+ by (simp add: zgcd_def abs_if)
+
+lemma zgcd_0_1_iff [simp, algebra]: "zgcd 0 m = 1 \<longleftrightarrow> \<bar>m\<bar> = 1"
+ by (simp add: zgcd_def abs_if)
+
+lemma zgcd_greatest_iff[algebra]: "k dvd zgcd m n = (k dvd m \<and> k dvd n)"
+ by (simp add: zgcd_def abs_if int_dvd_iff dvd_int_iff nat_dvd_iff)
+
+lemma zgcd_1_left [simp, algebra]: "zgcd 1 m = 1"
+ by (simp add: zgcd_def gcd_1_left)
+
+lemma zgcd_assoc: "zgcd (zgcd k m) n = zgcd k (zgcd m n)"
+ by (simp add: zgcd_def gcd_assoc)
+
+lemma zgcd_left_commute: "zgcd k (zgcd m n) = zgcd m (zgcd k n)"
+ apply (rule zgcd_commute [THEN trans])
+ apply (rule zgcd_assoc [THEN trans])
+ apply (rule zgcd_commute [THEN arg_cong])
+ done
+
+lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute
+ -- {* addition is an AC-operator *}
+
+lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd m n = zgcd (k * m) (k * n)"
+ by (simp del: minus_mult_right [symmetric]
+ add: minus_mult_right nat_mult_distrib zgcd_def abs_if
+ mult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
+
+lemma zgcd_zmult_distrib2_abs: "zgcd (k * m) (k * n) = abs k * zgcd m n"
+ by (simp add: abs_if zgcd_zmult_distrib2)
+
+lemma zgcd_self [simp]: "0 \<le> m ==> zgcd m m = m"
+ by (cut_tac k = m and m = 1 and n = 1 in zgcd_zmult_distrib2, simp_all)
+
+lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd k (k * n) = k"
+ by (cut_tac k = k and m = 1 and n = n in zgcd_zmult_distrib2, simp_all)
+
+lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n) k = k"
+ by (cut_tac k = k and m = n and n = 1 in zgcd_zmult_distrib2, simp_all)
+
+
+definition "zlcm i j = int (lcm(nat(abs i)) (nat(abs j)))"
+
+lemma dvd_zlcm_self1[simp, algebra]: "i dvd zlcm i j"
+by(simp add:zlcm_def dvd_int_iff)
+
+lemma dvd_zlcm_self2[simp, algebra]: "j dvd zlcm i j"
+by(simp add:zlcm_def dvd_int_iff)
+
+
+lemma dvd_imp_dvd_zlcm1:
+ assumes "k dvd i" shows "k dvd (zlcm i j)"
+proof -
+ have "nat(abs k) dvd nat(abs i)" using `k dvd i`
+ by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
+ thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
+qed
+
+lemma dvd_imp_dvd_zlcm2:
+ assumes "k dvd j" shows "k dvd (zlcm i j)"
+proof -
+ have "nat(abs k) dvd nat(abs j)" using `k dvd j`
+ by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
+ thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
+qed
+
+
+lemma zdvd_self_abs1: "(d::int) dvd (abs d)"
+by (case_tac "d <0", simp_all)
+
+lemma zdvd_self_abs2: "(abs (d::int)) dvd d"
+by (case_tac "d<0", simp_all)
+
+(* lcm a b is positive for positive a and b *)
+
+lemma lcm_pos:
+ assumes mpos: "m > 0"
+ and npos: "n>0"
+ shows "lcm m n > 0"
+proof(rule ccontr, simp add: lcm_def gcd_zero)
+assume h:"m*n div gcd m n = 0"
+from mpos npos have "gcd m n \<noteq> 0" using gcd_zero by simp
+hence gcdp: "gcd m n > 0" by simp
+with h
+have "m*n < gcd m n"
+ by (cases "m * n < gcd m n") (auto simp add: div_if[OF gcdp, where m="m*n"])
+moreover
+have "gcd m n dvd m" by simp
+ with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
+ with npos have t1:"gcd m n *n \<le> m*n" by simp
+ have "gcd m n \<le> gcd m n*n" using npos by simp
+ with t1 have "gcd m n \<le> m*n" by arith
+ultimately show "False" by simp
+qed
+
+lemma zlcm_pos:
+ assumes anz: "a \<noteq> 0"
+ and bnz: "b \<noteq> 0"
+ shows "0 < zlcm a b"
+proof-
+ let ?na = "nat (abs a)"
+ let ?nb = "nat (abs b)"
+ have nap: "?na >0" using anz by simp
+ have nbp: "?nb >0" using bnz by simp
+ have "0 < lcm ?na ?nb" by (rule lcm_pos[OF nap nbp])
+ thus ?thesis by (simp add: zlcm_def)
+qed
+
+lemma zgcd_code [code]:
+ "zgcd k l = \<bar>if l = 0 then k else zgcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
+ by (simp add: zgcd_def gcd.simps [of "nat \<bar>k\<bar>"] nat_mod_distrib)
+
+end
--- a/src/HOL/Library/Primes.thy Tue Jun 16 22:07:39 2009 -0700
+++ b/src/HOL/Library/Primes.thy Wed Jun 17 16:55:01 2009 -0700
@@ -6,9 +6,11 @@
header {* Primality on nat *}
theory Primes
-imports Complex_Main
+imports Complex_Main Legacy_GCD
begin
+hide (open) const GCD.gcd GCD.coprime GCD.prime
+
definition
coprime :: "nat => nat => bool" where
"coprime m n \<longleftrightarrow> gcd m n = 1"
--- a/src/HOL/Rational.thy Tue Jun 16 22:07:39 2009 -0700
+++ b/src/HOL/Rational.thy Wed Jun 17 16:55:01 2009 -0700
@@ -851,11 +851,11 @@
subsection {* Implementation of rational numbers as pairs of integers *}
-lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"
+lemma Fract_norm: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
proof (cases "a = 0 \<or> b = 0")
case True then show ?thesis by (auto simp add: eq_rat)
next
- let ?c = "zgcd a b"
+ let ?c = "gcd a b"
case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
then have "?c \<noteq> 0" by simp
then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)
@@ -870,7 +870,7 @@
definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where
[simp, code del]: "Fract_norm a b = Fract a b"
-lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in
+lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = gcd a b in
if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"
by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)
--- a/src/HOL/RealDef.thy Tue Jun 16 22:07:39 2009 -0700
+++ b/src/HOL/RealDef.thy Wed Jun 17 16:55:01 2009 -0700
@@ -895,14 +895,15 @@
lemma Rats_abs_nat_div_natE:
assumes "x \<in> \<rat>"
- obtains m n where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
+ obtains m n :: nat
+ where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
proof -
from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
by(auto simp add: Rats_eq_int_div_nat)
hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
let ?gcd = "gcd m n"
- from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
+ from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by simp
let ?k = "m div ?gcd"
let ?l = "n div ?gcd"
let ?gcd' = "gcd ?k ?l"
@@ -924,9 +925,9 @@
have "?gcd' = 1"
proof -
have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
- by (rule gcd_mult_distrib2)
+ by (rule nat_gcd_mult_distrib)
with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
- with gcd show ?thesis by simp
+ with gcd show ?thesis by auto
qed
ultimately show ?thesis ..
qed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/transfer_data.ML Wed Jun 17 16:55:01 2009 -0700
@@ -0,0 +1,242 @@
+(* Title: Tools/transfer.ML
+ Author: Amine Chaieb, University of Cambridge, 2009
+ Jeremy Avigad, Carnegie Mellon University
+*)
+
+signature TRANSFER_DATA =
+sig
+ type data
+ type entry
+ val get: Proof.context -> data
+ val del: attribute
+ val add: attribute
+ val setup: theory -> theory
+end;
+
+structure TransferData (* : TRANSFER_DATA*) =
+struct
+type entry = {inj : thm list , emb : thm list , ret : thm list , cong : thm list, guess : bool, hints : string list};
+type data = simpset * (thm * entry) list;
+
+val eq_key = Thm.eq_thm;
+fun eq_data arg = eq_fst eq_key arg;
+
+structure Data = GenericDataFun
+(
+ type T = data;
+ val empty = (HOL_ss, []);
+ val extend = I;
+ fun merge _ ((ss, e), (ss', e')) =
+ (merge_ss (ss, ss'), AList.merge eq_key (K true) (e, e'));
+);
+
+val get = Data.get o Context.Proof;
+
+fun del_data key = apsnd (remove eq_data (key, []));
+
+val del = Thm.declaration_attribute (Data.map o del_data);
+val add_ss = Thm.declaration_attribute
+ (fn th => Data.map (fn (ss,data) => (ss addsimps [th], data)));
+
+val del_ss = Thm.declaration_attribute
+ (fn th => Data.map (fn (ss,data) => (ss delsimps [th], data)));
+
+val transM_pat = (Thm.dest_arg1 o Thm.dest_arg o cprop_of) @{thm TransferMorphism_def};
+
+fun merge_update eq m (k,v) [] = [(k,v)]
+ | merge_update eq m (k,v) ((k',v')::al) =
+ if eq (k,k') then (k',m (v,v')):: al else (k',v') :: merge_update eq m (k,v) al
+
+fun C f x y = f y x
+
+fun simpset_of_entry injonly {inj = inj, emb = emb, ret = ret, cong = cg, guess = g, hints = hints} =
+ HOL_ss addsimps inj addsimps (if injonly then [] else emb@ret) addcongs cg;
+
+fun basic_transfer_rule injonly a0 D0 e leave ctxt0 th =
+ let
+ val ([a,D], ctxt) = apfst (map Drule.dest_term o snd) (Variable.import_thms true (map Drule.mk_term [a0, D0]) ctxt0)
+ val (aT,bT) =
+ let val T = typ_of (ctyp_of_term a)
+ in (Term.range_type T, Term.domain_type T)
+ end
+ val ctxt' = (Variable.declare_term (term_of a) o Variable.declare_term (term_of D) o Variable.declare_thm th) ctxt
+ val ns = filter (fn i => Type.could_unify (snd i, aT) andalso not (fst (fst i) mem_string leave)) (Term.add_vars (prop_of th) [])
+ val (ins, ctxt'') = Variable.variant_fixes (map (fst o fst) ns) ctxt'
+ val cns = map ((cterm_of o ProofContext.theory_of) ctxt'' o Var) ns
+ val cfis = map ((cterm_of o ProofContext.theory_of) ctxt'' o (fn n => Free (n, bT))) ins
+ val cis = map (Thm.capply a) cfis
+ val (hs,ctxt''') = Assumption.add_assumes (map (fn ct => Thm.capply @{cterm "Trueprop"} (Thm.capply D ct)) cfis) ctxt''
+ val th1 = Drule.cterm_instantiate (cns~~ cis) th
+ val th2 = fold (C implies_elim) hs (fold_rev implies_intr (map cprop_of hs) th1)
+ val th3 = Simplifier.asm_full_simplify (Simplifier.context ctxt''' (simpset_of_entry injonly e))
+ (fold_rev implies_intr (map cprop_of hs) th2)
+in hd (Variable.export ctxt''' ctxt0 [th3]) end;
+
+local
+fun transfer_ruleh a D leave ctxt th =
+ let val (ss,al) = get ctxt
+ val a0 = cterm_of (ProofContext.theory_of ctxt) a
+ val D0 = cterm_of (ProofContext.theory_of ctxt) D
+ fun h (th', e) = let val (a',D') = (Thm.dest_binop o Thm.dest_arg o cprop_of) th'
+ in if a0 aconvc a' andalso D0 aconvc D' then SOME e else NONE
+ end
+ in case get_first h al of
+ SOME e => basic_transfer_rule false a0 D0 e leave ctxt th
+ | NONE => error "Transfer: corresponding instance not found in context-data"
+ end
+in fun transfer_rule (a,D) leave (gctxt,th) =
+ (gctxt, transfer_ruleh a D leave (Context.proof_of gctxt) th)
+end;
+
+fun splits P [] = []
+ | splits P (xxs as (x::xs)) =
+ let val pss = filter (P x) xxs
+ val qss = filter_out (P x) xxs
+ in if null pss then [qss] else if null qss then [pss] else pss:: splits P qss
+ end
+
+fun all_transfers leave (gctxt,th) =
+ let
+ val ctxt = Context.proof_of gctxt
+ val tys = map snd (Term.add_vars (prop_of th) [])
+ val _ = if null tys then error "transfer: Unable to guess instance" else ()
+ val tyss = splits (curry Type.could_unify) tys
+ val get_ty = typ_of o ctyp_of_term o fst o Thm.dest_binop o Thm.dest_arg o cprop_of
+ val get_aD = Thm.dest_binop o Thm.dest_arg o cprop_of
+ val insts =
+ map_filter (fn tys =>
+ get_first (fn (k,ss) => if Type.could_unify (hd tys, range_type (get_ty k))
+ then SOME (get_aD k, ss)
+ else NONE) (snd (get ctxt))) tyss
+ val _ = if null insts then error "Transfer guesser: there were no possible instances, use direction: in order to provide a direction" else ()
+ val ths = map (fn ((a,D),e) => basic_transfer_rule false a D e leave ctxt th) insts
+ val cth = Conjunction.intr_balanced ths
+ in (gctxt, cth)
+ end;
+
+fun transfer_rule_by_hint ls leave (gctxt,th) =
+ let
+ val ctxt = Context.proof_of gctxt
+ val get_aD = Thm.dest_binop o Thm.dest_arg o cprop_of
+ val insts =
+ map_filter (fn (k,e) => if exists (fn l => l mem_string (#hints e)) ls
+ then SOME (get_aD k, e) else NONE)
+ (snd (get ctxt))
+ val _ = if null insts then error "Transfer: No labels provided are stored in the context" else ()
+ val ths = map (fn ((a,D),e) => basic_transfer_rule false a D e leave ctxt th) insts
+ val cth = Conjunction.intr_balanced ths
+ in (gctxt, cth)
+ end;
+
+
+fun transferred_attribute ls NONE leave =
+ if null ls then all_transfers leave else transfer_rule_by_hint ls leave
+ | transferred_attribute _ (SOME (a,D)) leave = transfer_rule (a,D) leave
+
+ (* Add data to the context *)
+fun gen_merge_entries {inj = inj0, emb = emb0, ret = ret0, cong = cg0, guess = g0, hints = hints0}
+ ({inj = inj1, emb = emb1, ret = ret1, cong = cg1, guess = g1, hints = hints1},
+ {inj = inj2, emb = emb2, ret = ret2, cong = cg2, guess = g2, hints = hints2})
+ =
+ let fun h xs0 xs ys = subtract Thm.eq_thm xs0 (merge Thm.eq_thm (xs,ys)) in
+ {inj = h inj0 inj1 inj2, emb = h emb0 emb1 emb2,
+ ret = h ret0 ret1 ret2, cong = h cg0 cg1 cg2, guess = g1 andalso g2,
+ hints = subtract (op = : string*string -> bool) hints0
+ (hints1 union_string hints2)}
+ end;
+
+local
+ val h = curry (merge Thm.eq_thm)
+in
+fun merge_entries ({inj = inj1, emb = emb1, ret = ret1, cong = cg1, guess = g1, hints = hints1},
+ {inj = inj2, emb = emb2, ret = ret2, cong = cg2, guess = g2, hints = hints2}) =
+ {inj = h inj1 inj2, emb = h emb1 emb2, ret = h ret1 ret2, cong = h cg1 cg2, guess = g1 andalso g2, hints = hints1 union_string hints2}
+end;
+
+fun add ((inja,injd), (emba,embd), (reta,retd), (cga,cgd), g, (hintsa, hintsd)) =
+ Thm.declaration_attribute (fn key => fn context => context |> Data.map
+ (fn (ss, al) =>
+ let
+ val _ = ((let val _ = Thm.match (transM_pat, (Thm.dest_arg o cprop_of) key)
+ in 0 end)
+ handle MATCH => error "Attribute expected Theorem of the form : TransferMorphism A a B b")
+ val e0 = {inj = inja, emb = emba, ret = reta, cong = cga, guess = g, hints = hintsa}
+ val ed = {inj = injd, emb = embd, ret = retd, cong = cgd, guess = g, hints = hintsd}
+ val entry =
+ if g then
+ let val (a0,D0) = (Thm.dest_binop o Thm.dest_arg o cprop_of) key
+ val ctxt0 = ProofContext.init (Thm.theory_of_thm key)
+ val inj' = if null inja then #inj (case AList.lookup eq_key al key of SOME e => e | NONE => error "Transfer: can not generate return rules on the fly, either add injectivity axiom or force manual mode with mode: manual")
+ else inja
+ val ret' = merge Thm.eq_thm (reta, map (fn th => basic_transfer_rule true a0 D0 {inj = inj', emb = [], ret = [], cong = cga, guess = g, hints = hintsa} [] ctxt0 th RS sym) emba)
+ in {inj = inja, emb = emba, ret = ret', cong = cga, guess = g, hints = hintsa} end
+ else e0
+ in (ss, merge_update eq_key (gen_merge_entries ed) (key, entry) al)
+ end));
+
+
+
+(* concrete syntax *)
+
+local
+
+fun keyword k = Scan.lift (Args.$$$ k) >> K ()
+fun keywordC k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
+
+val congN = "cong"
+val injN = "inj"
+val embedN = "embed"
+val returnN = "return"
+val addN = "add"
+val delN = "del"
+val modeN = "mode"
+val automaticN = "automatic"
+val manualN = "manual"
+val directionN = "direction"
+val labelsN = "labels"
+val leavingN = "leaving"
+
+val any_keyword = keywordC congN || keywordC injN || keywordC embedN || keywordC returnN || keywordC directionN || keywordC modeN || keywordC delN || keywordC labelsN || keywordC leavingN
+
+val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
+val terms = thms >> map Drule.dest_term
+val types = thms >> (Logic.dest_type o HOLogic.dest_Trueprop o prop_of o hd)
+val name = Scan.lift Args.name
+val names = Scan.repeat (Scan.unless any_keyword name)
+fun optional scan = Scan.optional scan []
+fun optional2 scan = Scan.optional scan ([],[])
+
+val mode = keywordC modeN |-- ((Scan.lift (Args.$$$ manualN) >> K false) || (Scan.lift (Args.$$$ automaticN) >> K true))
+val inj = (keywordC injN |-- thms) -- optional (keywordC delN |-- thms)
+val embed = (keywordC embedN |-- thms) -- optional (keywordC delN |-- thms)
+val return = (keywordC returnN |-- thms) -- optional (keywordC delN |-- thms)
+val cong = (keywordC congN |-- thms) -- optional (keywordC delN |-- thms)
+val addscan = Scan.unless any_keyword (keyword addN)
+val labels = (keywordC labelsN |-- names) -- optional (keywordC delN |-- names)
+val entry = Scan.optional mode true -- optional2 inj -- optional2 embed -- optional2 return -- optional2 cong -- optional2 labels
+
+val transf_add = addscan |-- entry
+in
+
+val install_att_syntax =
+ (Scan.lift (Args.$$$ delN >> K del) ||
+ transf_add
+ >> (fn (((((g, inj), embed), ret), cg), hints) => add (inj, embed, ret, cg, g, hints)))
+
+val transferred_att_syntax = (optional names -- Scan.option (keywordC directionN |-- (Args.term -- Args.term)) -- optional (keywordC leavingN |-- names) >> (fn ((hints, aD),leave) => transferred_attribute hints aD leave));
+
+end;
+
+
+(* theory setup *)
+
+
+val setup =
+ Attrib.setup @{binding transfer} install_att_syntax
+ "Installs transfer data" #>
+ Attrib.setup @{binding transfer_simps} (Attrib.add_del add_ss del_ss)
+ "simp rules for transfer" #>
+ Attrib.setup @{binding transferred} transferred_att_syntax
+ "Transfers theorems";
+
+end;