author wenzelm Thu Mar 03 21:43:06 2011 +0100 (2011-03-03) changeset 41891 d37babdf5cae parent 41890 550a8ecffe0c child 41892 2386fb64feaf
tuned proofs -- eliminated prems;
```     1.1 --- a/src/HOL/Decision_Procs/MIR.thy	Thu Mar 03 18:43:15 2011 +0100
1.2 +++ b/src/HOL/Decision_Procs/MIR.thy	Thu Mar 03 21:43:06 2011 +0100
1.3 @@ -1480,7 +1480,7 @@
1.4    let ?at = "snd (zsplit0 t)"
1.5    have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using 5
1.6      by (simp add: Let_def split_def)
1.7 -  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
1.8 +  from abj 5 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
1.9    from th2[simplified] th[simplified] show ?case by simp
1.10  next
1.11    case (6 s t n a)
1.12 @@ -1490,12 +1490,12 @@
1.13    let ?at = "snd (zsplit0 t)"
1.14    have abjs: "zsplit0 s = (?ns,?as)" by simp
1.15    moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp
1.16 -  ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using prems
1.17 +  ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using 6
1.18      by (simp add: Let_def split_def)
1.19    from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
1.20 -  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
1.21 +  from 6 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
1.22    with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
1.23 -  from abjs prems  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
1.24 +  from abjs 6  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
1.25    from th3[simplified] th2[simplified] th[simplified] show ?case
1.26      by (simp add: left_distrib)
1.27  next
1.28 @@ -1506,31 +1506,31 @@
1.29    let ?at = "snd (zsplit0 t)"
1.30    have abjs: "zsplit0 s = (?ns,?as)" by simp
1.31    moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp
1.32 -  ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using prems
1.33 +  ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using 7
1.34      by (simp add: Let_def split_def)
1.35    from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
1.36 -  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
1.37 +  from 7 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
1.38    with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
1.39 -  from abjs prems  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
1.40 +  from abjs 7 have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
1.41    from th3[simplified] th2[simplified] th[simplified] show ?case
1.42      by (simp add: left_diff_distrib)
1.43  next
1.44    case (8 i t n a)
1.45    let ?nt = "fst (zsplit0 t)"
1.46    let ?at = "snd (zsplit0 t)"
1.47 -  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using prems
1.48 +  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using 8
1.49      by (simp add: Let_def split_def)
1.50 -  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
1.51 -  hence " ?I x (Mul i t) = (real i) * ?I x (CN 0 ?nt ?at)" by simp
1.52 +  from abj 8 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
1.53 +  hence "?I x (Mul i t) = (real i) * ?I x (CN 0 ?nt ?at)" by simp
1.54    also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: right_distrib)
1.55    finally show ?case using th th2 by simp
1.56  next
1.57    case (9 t n a)
1.58    let ?nt = "fst (zsplit0 t)"
1.59    let ?at = "snd (zsplit0 t)"
1.60 -  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a= Floor ?at \<and> n=?nt" using prems
1.61 +  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a= Floor ?at \<and> n=?nt" using 9
1.62      by (simp add: Let_def split_def)
1.63 -  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
1.64 +  from abj 9 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
1.65    hence na: "?N a" using th by simp
1.66    have th': "(real ?nt)*(real x) = real (?nt * x)" by simp
1.67    have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp
1.68 @@ -1864,8 +1864,8 @@
1.69    let ?N = "\<lambda> t. Inum (real i#bs) t"
1.70    have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
1.71    moreover
1.72 -  {assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def)
1.73 -    hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)}
1.74 +  { assume j: "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def)
1.75 +    hence ?case using 11 j by (simp del: zlfm.simps add: rdvd_left_0_eq)}
1.76    moreover
1.77    {assume "?c=0" and "j\<noteq>0" hence ?case
1.78        using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"]
1.79 @@ -1910,8 +1910,8 @@
1.80    let ?N = "\<lambda> t. Inum (real i#bs) t"
1.81    have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
1.82    moreover
1.83 -  {assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def)
1.84 -    hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)}
1.85 +  {assume j: "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def)
1.86 +    hence ?case using 12 j by (simp del: zlfm.simps add: rdvd_left_0_eq)}
1.87    moreover
1.88    {assume "?c=0" and "j\<noteq>0" hence ?case
1.89        using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"]
1.90 @@ -2012,20 +2012,21 @@
1.91  proof (induct p rule: iszlfm.induct)
1.92    case (1 p q)
1.93    let ?d = "\<delta> (And p q)"
1.94 -  from prems lcm_pos_int have dp: "?d >0" by simp
1.95 -  have d1: "\<delta> p dvd \<delta> (And p q)" using prems by simp
1.96 -   hence th: "d\<delta> p ?d"
1.97 -     using delta_mono prems by(simp only: iszlfm.simps) blast
1.98 -  have "\<delta> q dvd \<delta> (And p q)" using prems  by simp
1.99 -  hence th': "d\<delta> q ?d" using delta_mono prems by(simp only: iszlfm.simps) blast
1.100 +  from 1 lcm_pos_int have dp: "?d >0" by simp
1.101 +  have d1: "\<delta> p dvd \<delta> (And p q)" using 1 by simp
1.102 +  hence th: "d\<delta> p ?d"
1.103 +    using delta_mono 1 by (simp only: iszlfm.simps) blast
1.104 +  have "\<delta> q dvd \<delta> (And p q)" using 1 by simp
1.105 +  hence th': "d\<delta> q ?d" using delta_mono 1 by (simp only: iszlfm.simps) blast
1.106    from th th' dp show ?case by simp
1.107  next
1.108    case (2 p q)
1.109    let ?d = "\<delta> (And p q)"
1.110 -  from prems lcm_pos_int have dp: "?d >0" by simp
1.111 -  have "\<delta> p dvd \<delta> (And p q)" using prems by simp hence th: "d\<delta> p ?d" using delta_mono prems
1.112 -    by(simp only: iszlfm.simps) blast
1.113 -  have "\<delta> q dvd \<delta> (And p q)" using prems by simp hence th': "d\<delta> q ?d" using delta_mono prems by(simp only: iszlfm.simps) blast
1.114 +  from 2 lcm_pos_int have dp: "?d >0" by simp
1.115 +  have "\<delta> p dvd \<delta> (And p q)" using 2 by simp
1.116 +  hence th: "d\<delta> p ?d" using delta_mono 2 by (simp only: iszlfm.simps) blast
1.117 +  have "\<delta> q dvd \<delta> (And p q)" using 2 by simp
1.118 +  hence th': "d\<delta> q ?d" using delta_mono 2 by (simp only: iszlfm.simps) blast
1.119    from th th' dp show ?case by simp
1.120  qed simp_all
1.121
1.122 @@ -2037,25 +2038,27 @@
1.123  using linp
1.124  proof (induct p rule: minusinf.induct)
1.125    case (1 f g)
1.126 -  from prems have "?P f" by simp
1.127 +  then have "?P f" by simp
1.128    then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast
1.129 -  from prems have "?P g" by simp
1.130 +  with 1 have "?P g" by simp
1.131    then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast
1.132    let ?z = "min z1 z2"
1.133    from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (And f g)) = ?I x (And f g)" by simp
1.134    thus ?case by blast
1.135  next
1.136 -  case (2 f g)   from prems have "?P f" by simp
1.137 +  case (2 f g)
1.138 +  then have "?P f" by simp
1.139    then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast
1.140 -  from prems have "?P g" by simp
1.141 +  with 2 have "?P g" by simp
1.142    then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast
1.143    let ?z = "min z1 z2"
1.144    from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (Or f g)) = ?I x (Or f g)" by simp
1.145    thus ?case by blast
1.146  next
1.147    case (3 c e)
1.148 -  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.149 -  from prems have nbe: "numbound0 e" by simp
1.150 +  then have "c > 0" by simp
1.151 +  hence rcpos: "real c > 0" by simp
1.152 +  from 3 have nbe: "numbound0 e" by simp
1.153    fix y
1.154    have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))"
1.155    proof (simp add: less_floor_eq , rule allI, rule impI)
1.156 @@ -2071,8 +2074,8 @@
1.157    thus ?case by blast
1.158  next
1.159    case (4 c e)
1.160 -  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.161 -  from prems have nbe: "numbound0 e" by simp
1.162 +  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.163 +  from 4 have nbe: "numbound0 e" by simp
1.164    fix y
1.165    have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))"
1.166    proof (simp add: less_floor_eq , rule allI, rule impI)
1.167 @@ -2088,8 +2091,8 @@
1.168    thus ?case by blast
1.169  next
1.170    case (5 c e)
1.171 -  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.172 -  from prems have nbe: "numbound0 e" by simp
1.173 +  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.174 +  from 5 have nbe: "numbound0 e" by simp
1.175    fix y
1.176    have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))"
1.177    proof (simp add: less_floor_eq , rule allI, rule impI)
1.178 @@ -2104,8 +2107,8 @@
1.179    thus ?case by blast
1.180  next
1.181    case (6 c e)
1.182 -  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.183 -  from prems have nbe: "numbound0 e" by simp
1.184 +  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.185 +  from 6 have nbe: "numbound0 e" by simp
1.186    fix y
1.187    have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))"
1.188    proof (simp add: less_floor_eq , rule allI, rule impI)
1.189 @@ -2120,8 +2123,8 @@
1.190    thus ?case by blast
1.191  next
1.192    case (7 c e)
1.193 -  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.194 -  from prems have nbe: "numbound0 e" by simp
1.195 +  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.196 +  from 7 have nbe: "numbound0 e" by simp
1.197    fix y
1.198    have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))"
1.199    proof (simp add: less_floor_eq , rule allI, rule impI)
1.200 @@ -2136,8 +2139,8 @@
1.201    thus ?case by blast
1.202  next
1.203    case (8 c e)
1.204 -  from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.205 -  from prems have nbe: "numbound0 e" by simp
1.206 +  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
1.207 +  from 8 have nbe: "numbound0 e" by simp
1.208    fix y
1.209    have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))"
1.210    proof (simp add: less_floor_eq , rule allI, rule impI)
1.211 @@ -2336,15 +2339,15 @@
1.212    have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
1.213         (real j rdvd - (real c * real x - Inum (real x # bs) e))"
1.214      by (simp only: rdvd_minus[symmetric])
1.215 -  from prems th show  ?case
1.216 +  from 9 th show ?case
1.217      by (simp add: algebra_simps
1.218        numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
1.219  next
1.220 -    case (10 j c e)
1.221 +  case (10 j c e)
1.222    have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
1.223         (real j rdvd - (real c * real x - Inum (real x # bs) e))"
1.224      by (simp only: rdvd_minus[symmetric])
1.225 -  from prems th show  ?case
1.226 +  from 10 th show  ?case
1.227      by (simp add: algebra_simps
1.228        numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
1.229  qed (auto simp add: numbound0_I[where bs="bs" and b="real x" and b'="- real x"])
1.230 @@ -2396,16 +2399,16 @@
1.231  using linp
1.232  proof(induct p rule: iszlfm.induct)
1.233    case (1 p q)
1.234 -  from prems have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
1.235 -  from prems have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
1.236 -  from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
1.237 +  then  have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
1.238 +  from 1 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
1.239 +  from 1 d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
1.240      d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"]
1.241      dl1 dl2 show ?case by (auto simp add: lcm_pos_int)
1.242  next
1.243    case (2 p q)
1.244 -  from prems have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
1.245 -  from prems have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
1.246 -  from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
1.247 +  then have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
1.248 +  from 2 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
1.249 +  from 2 d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
1.250      d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"]
1.251      dl1 dl2 show ?case by (auto simp add: lcm_pos_int)
1.252  qed (auto simp add: lcm_pos_int)
1.253 @@ -2577,19 +2580,20 @@
1.254    shows "?P (x - d)"
1.255  using lp u d dp nob p
1.256  proof(induct p rule: iszlfm.induct)
1.257 -  case (5 c e) hence c1: "c=1" and  bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
1.258 -    with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems
1.259 -    show ?case by (simp del: real_of_int_minus)
1.260 +  case (5 c e) hence c1: "c=1" and  bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp_all
1.261 +  with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] 5
1.262 +  show ?case by (simp del: real_of_int_minus)
1.263  next
1.264 -  case (6 c e)  hence c1: "c=1" and  bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
1.265 -    with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems
1.266 -    show ?case by (simp del: real_of_int_minus)
1.267 +  case (6 c e)  hence c1: "c=1" and  bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp_all
1.268 +  with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] 6
1.269 +  show ?case by (simp del: real_of_int_minus)
1.270  next
1.271 -  case (7 c e) hence p: "Ifm (real x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" and ie1:"isint e (a#bs)" using dvd1_eq1[where x="c"] by simp+
1.272 -    let ?e = "Inum (real x # bs) e"
1.273 -    from ie1 have ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"]
1.274 +  case (7 c e) hence p: "Ifm (real x #bs) (Gt (CN 0 c e))" and c1: "c=1"
1.275 +    and bn:"numbound0 e" and ie1:"isint e (a#bs)" using dvd1_eq1[where x="c"] by simp_all
1.276 +  let ?e = "Inum (real x # bs) e"
1.277 +  from ie1 have ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"]
1.278        numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]
1.279 -      by (simp add: isint_iff)
1.280 +    by (simp add: isint_iff)
1.281      {assume "real (x-d) +?e > 0" hence ?case using c1
1.282        numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"]
1.283          by (simp del: real_of_int_minus)}
1.284 @@ -2597,7 +2601,7 @@
1.285      {assume H: "\<not> real (x-d) + ?e > 0"
1.286        let ?v="Neg e"
1.287        have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp
1.288 -      from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]]
1.289 +      from 7(5)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]]
1.290        have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x =  - ?e + real j)" by auto
1.291        from H p have "real x + ?e > 0 \<and> real x + ?e \<le> real d" by (simp add: c1)
1.292        hence "real (x + floor ?e) > real (0::int) \<and> real (x + floor ?e) \<le> real d"
1.293 @@ -2623,7 +2627,7 @@
1.294      {assume H: "\<not> real (x-d) + ?e \<ge> 0"
1.295        let ?v="Sub (C -1) e"
1.296        have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp
1.297 -      from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]]
1.298 +      from 8(5)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]]
1.299        have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x =  - ?e - 1 + real j)" by auto
1.300        from H p have "real x + ?e \<ge> 0 \<and> real x + ?e < real d" by (simp add: c1)
1.301        hence "real (x + floor ?e) \<ge> real (0::int) \<and> real (x + floor ?e) < real d"
1.302 @@ -2643,7 +2647,7 @@
1.303      let ?e = "Inum (real x # bs) e"
1.304      let ?v="(Sub (C -1) e)"
1.305      have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp
1.306 -    from p have "real x= - ?e" by (simp add: c1) with prems(11) show ?case using dp
1.307 +    from p have "real x= - ?e" by (simp add: c1) with 3(5) show ?case using dp
1.308        by simp (erule ballE[where x="1"],
1.309          simp_all add:algebra_simps numbound0_I[OF bn,where b="real x"and b'="a"and bs="bs"])
1.310  next
1.311 @@ -2659,47 +2663,49 @@
1.312        hence "real x = - Inum ((real (x -d)) # bs) e + real d" by simp
1.313        hence "real x = - Inum (a # bs) e + real d"
1.314          by (simp add: numbound0_I[OF bn,where b="real x - real d"and b'="a"and bs="bs"])
1.315 -       with prems(11) have ?case using dp by simp}
1.316 +       with 4(5) have ?case using dp by simp}
1.317    ultimately show ?case by blast
1.318  next
1.319    case (9 j c e) hence p: "Ifm (real x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1"
1.320      and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
1.321 -    let ?e = "Inum (real x # bs) e"
1.322 -    from prems have "isint e (a #bs)"  by simp
1.323 -    hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real x)#bs"] numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"]
1.324 -      by (simp add: isint_iff)
1.325 -    from prems have id: "j dvd d" by simp
1.326 -    from c1 ie[symmetric] have "?p x = (real j rdvd real (x+ floor ?e))" by simp
1.327 -    also have "\<dots> = (j dvd x + floor ?e)"
1.328 -      using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
1.329 -    also have "\<dots> = (j dvd x - d + floor ?e)"
1.330 -      using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
1.331 -    also have "\<dots> = (real j rdvd real (x - d + floor ?e))"
1.332 -      using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
1.333 +  let ?e = "Inum (real x # bs) e"
1.334 +  from 9 have "isint e (a #bs)"  by simp
1.335 +  hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real x)#bs"] numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"]
1.336 +    by (simp add: isint_iff)
1.337 +  from 9 have id: "j dvd d" by simp
1.338 +  from c1 ie[symmetric] have "?p x = (real j rdvd real (x+ floor ?e))" by simp
1.339 +  also have "\<dots> = (j dvd x + floor ?e)"
1.340 +    using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
1.341 +  also have "\<dots> = (j dvd x - d + floor ?e)"
1.342 +    using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
1.343 +  also have "\<dots> = (real j rdvd real (x - d + floor ?e))"
1.344 +    using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
1.345        ie by simp
1.346 -    also have "\<dots> = (real j rdvd real x - real d + ?e)"
1.347 -      using ie by simp
1.348 -    finally show ?case
1.349 -      using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
1.350 +  also have "\<dots> = (real j rdvd real x - real d + ?e)"
1.351 +    using ie by simp
1.352 +  finally show ?case
1.353 +    using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
1.354  next
1.355    case (10 j c e) hence p: "Ifm (real x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
1.356 -    let ?e = "Inum (real x # bs) e"
1.357 -    from prems have "isint e (a#bs)"  by simp
1.358 -    hence ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"]
1.359 -      by (simp add: isint_iff)
1.360 -    from prems have id: "j dvd d" by simp
1.361 -    from c1 ie[symmetric] have "?p x = (\<not> real j rdvd real (x+ floor ?e))" by simp
1.362 -    also have "\<dots> = (\<not> j dvd x + floor ?e)"
1.363 -      using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
1.364 -    also have "\<dots> = (\<not> j dvd x - d + floor ?e)"
1.365 -      using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
1.366 -    also have "\<dots> = (\<not> real j rdvd real (x - d + floor ?e))"
1.367 -      using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
1.368 +  let ?e = "Inum (real x # bs) e"
1.369 +  from 10 have "isint e (a#bs)"  by simp
1.370 +  hence ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"]
1.371 +    by (simp add: isint_iff)
1.372 +  from 10 have id: "j dvd d" by simp
1.373 +  from c1 ie[symmetric] have "?p x = (\<not> real j rdvd real (x+ floor ?e))" by simp
1.374 +  also have "\<dots> = (\<not> j dvd x + floor ?e)"
1.375 +    using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
1.376 +  also have "\<dots> = (\<not> j dvd x - d + floor ?e)"
1.377 +    using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
1.378 +  also have "\<dots> = (\<not> real j rdvd real (x - d + floor ?e))"
1.379 +    using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
1.380        ie by simp
1.381 -    also have "\<dots> = (\<not> real j rdvd real x - real d + ?e)"
1.382 -      using ie by simp
1.383 -    finally show ?case using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
1.384 -qed (auto simp add: numbound0_I[where bs="bs" and b="real (x - d)" and b'="real x"] simp del: real_of_int_diff)
1.385 +  also have "\<dots> = (\<not> real j rdvd real x - real d + ?e)"
1.386 +    using ie by simp
1.387 +  finally show ?case
1.388 +    using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
1.389 +qed (auto simp add: numbound0_I[where bs="bs" and b="real (x - d)" and b'="real x"]
1.390 +  simp del: real_of_int_diff)
1.391
1.392  lemma \<beta>':
1.393    assumes lp: "iszlfm p (a #bs)"
1.394 @@ -2834,179 +2840,213 @@
1.395  using linp kpos tnb
1.396  proof(induct p rule: \<sigma>\<rho>.induct)
1.397    case (3 c e)
1.398 -  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
1.399 -    {assume kdc: "k dvd c"
1.400 -      from kpos have knz: "k\<noteq>0" by simp
1.401 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.402 -      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.403 -        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.404 +  from 3 have cp: "c > 0" and nb: "numbound0 e" by auto
1.405 +  { assume kdc: "k dvd c"
1.406 +    from kpos have knz: "k\<noteq>0" by simp
1.407 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.408 +    from kdc have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.409 +      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.410        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
1.411 -    moreover
1.412 -    {assume "\<not> k dvd c"
1.413 -      from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
1.414 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.415 -      from prems have "?I (real x) (?s (Eq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k = 0)"
1.416 -        using real_of_int_div[OF knz kdt]
1.417 -          numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.418 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
1.419 -      also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.420 +  moreover
1.421 +  { assume *: "\<not> k dvd c"
1.422 +    from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
1.423 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t"
1.424 +      using isint_def by simp
1.425 +    from assms * have "?I (real x) (?s (Eq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k = 0)"
1.426 +      using real_of_int_div[OF knz kdt]
1.427 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.428 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.429 +      by (simp add: ti algebra_simps)
1.430 +      also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))"
1.431 +        using nonzero_eq_divide_eq[OF knz',
1.432 +            where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
1.433 +          real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.434            numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.435          by (simp add: ti)
1.436        finally have ?case . }
1.437      ultimately show ?case by blast
1.438  next
1.439    case (4 c e)
1.440 -  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
1.441 -    {assume kdc: "k dvd c"
1.442 -      from kpos have knz: "k\<noteq>0" by simp
1.443 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.444 -      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.445 -        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.446 +  then have cp: "c > 0" and nb: "numbound0 e" by auto
1.447 +  { assume kdc: "k dvd c"
1.448 +    from kpos have knz: "k\<noteq>0" by simp
1.449 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.450 +    from kdc have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.451 +      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.452        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
1.453 -    moreover
1.454 -    {assume "\<not> k dvd c"
1.455 -      from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
1.456 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.457 -      from prems have "?I (real x) (?s (NEq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<noteq> 0)"
1.458 -        using real_of_int_div[OF knz kdt]
1.459 -          numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.460 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
1.461 -      also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.462 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.463 -        by (simp add: ti)
1.464 -      finally have ?case . }
1.465 -    ultimately show ?case by blast
1.466 +  moreover
1.467 +  { assume *: "\<not> k dvd c"
1.468 +    from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
1.469 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.470 +    from assms * have "?I (real x) (?s (NEq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<noteq> 0)"
1.471 +      using real_of_int_div[OF knz kdt]
1.472 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.473 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.474 +      by (simp add: ti algebra_simps)
1.475 +    also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))"
1.476 +      using nonzero_eq_divide_eq[OF knz',
1.477 +          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
1.478 +        real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.479 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.480 +      by (simp add: ti)
1.481 +    finally have ?case . }
1.482 +  ultimately show ?case by blast
1.483  next
1.484    case (5 c e)
1.485 -  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
1.486 -    {assume kdc: "k dvd c"
1.487 -      from kpos have knz: "k\<noteq>0" by simp
1.488 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.489 -      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.490 -        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.491 +  then have cp: "c > 0" and nb: "numbound0 e" by auto
1.492 +  { assume kdc: "k dvd c"
1.493 +    from kpos have knz: "k\<noteq>0" by simp
1.494 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.495 +    from kdc have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.496 +      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.497        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
1.498 -    moreover
1.499 -    {assume "\<not> k dvd c"
1.500 -      from kpos have knz: "k\<noteq>0" by simp
1.501 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.502 -      from prems have "?I (real x) (?s (Lt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k < 0)"
1.503 -        using real_of_int_div[OF knz kdt]
1.504 -          numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.505 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
1.506 -      also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))" using pos_less_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.507 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.508 -        by (simp add: ti)
1.509 -      finally have ?case . }
1.510 -    ultimately show ?case by blast
1.511 +  moreover
1.512 +  { assume *: "\<not> k dvd c"
1.513 +    from kpos have knz: "k\<noteq>0" by simp
1.514 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.515 +    from assms * have "?I (real x) (?s (Lt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k < 0)"
1.516 +      using real_of_int_div[OF knz kdt]
1.517 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.518 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.519 +      by (simp add: ti algebra_simps)
1.520 +    also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))"
1.521 +      using pos_less_divide_eq[OF kpos,
1.522 +          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
1.523 +        real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.524 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.525 +      by (simp add: ti)
1.526 +    finally have ?case . }
1.527 +  ultimately show ?case by blast
1.528  next
1.529    case (6 c e)
1.530 -  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
1.531 -    {assume kdc: "k dvd c"
1.532 -      from kpos have knz: "k\<noteq>0" by simp
1.533 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.534 -      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.535 -        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.536 +  then have cp: "c > 0" and nb: "numbound0 e" by auto
1.537 +  { assume kdc: "k dvd c"
1.538 +    from kpos have knz: "k\<noteq>0" by simp
1.539 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.540 +    from kdc have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.541 +      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.542        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
1.543 -    moreover
1.544 -    {assume "\<not> k dvd c"
1.545 -      from kpos have knz: "k\<noteq>0" by simp
1.546 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.547 -      from prems have "?I (real x) (?s (Le (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<le> 0)"
1.548 -        using real_of_int_div[OF knz kdt]
1.549 -          numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.550 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
1.551 -      also have "\<dots> = (?I ?tk (Le (CN 0 c e)))" using pos_le_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.552 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.553 -        by (simp add: ti)
1.554 -      finally have ?case . }
1.555 -    ultimately show ?case by blast
1.556 +  moreover
1.557 +  { assume *: "\<not> k dvd c"
1.558 +    from kpos have knz: "k\<noteq>0" by simp
1.559 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.560 +    from assms * have "?I (real x) (?s (Le (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<le> 0)"
1.561 +      using real_of_int_div[OF knz kdt]
1.562 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.563 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.564 +      by (simp add: ti algebra_simps)
1.565 +    also have "\<dots> = (?I ?tk (Le (CN 0 c e)))"
1.566 +      using pos_le_divide_eq[OF kpos,
1.567 +          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
1.568 +        real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.569 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.570 +      by (simp add: ti)
1.571 +    finally have ?case . }
1.572 +  ultimately show ?case by blast
1.573  next
1.574    case (7 c e)
1.575 -  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
1.576 -    {assume kdc: "k dvd c"
1.577 -      from kpos have knz: "k\<noteq>0" by simp
1.578 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.579 -      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.580 -        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.581 +  then have cp: "c > 0" and nb: "numbound0 e" by auto
1.582 +  { assume kdc: "k dvd c"
1.583 +    from kpos have knz: "k\<noteq>0" by simp
1.584 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.585 +    from kdc have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.586 +      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.587        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
1.588 -    moreover
1.589 -    {assume "\<not> k dvd c"
1.590 -      from kpos have knz: "k\<noteq>0" by simp
1.591 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.592 -      from prems have "?I (real x) (?s (Gt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k > 0)"
1.593 -        using real_of_int_div[OF knz kdt]
1.594 -          numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.595 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
1.596 -      also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))" using pos_divide_less_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.597 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.598 -        by (simp add: ti)
1.599 -      finally have ?case . }
1.600 -    ultimately show ?case by blast
1.601 +  moreover
1.602 +  { assume *: "\<not> k dvd c"
1.603 +    from kpos have knz: "k\<noteq>0" by simp
1.604 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.605 +    from assms * have "?I (real x) (?s (Gt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k > 0)"
1.606 +      using real_of_int_div[OF knz kdt]
1.607 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.608 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.609 +      by (simp add: ti algebra_simps)
1.610 +    also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))"
1.611 +      using pos_divide_less_eq[OF kpos,
1.612 +          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
1.613 +        real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.614 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.615 +      by (simp add: ti)
1.616 +    finally have ?case . }
1.617 +  ultimately show ?case by blast
1.618  next
1.619    case (8 c e)
1.620 -  from prems have cp: "c > 0" and nb: "numbound0 e" by auto
1.621 -    {assume kdc: "k dvd c"
1.622 -      from kpos have knz: "k\<noteq>0" by simp
1.623 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.624 -      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.625 -        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.626 +  then have cp: "c > 0" and nb: "numbound0 e" by auto
1.627 +  { assume kdc: "k dvd c"
1.628 +    from kpos have knz: "k\<noteq>0" by simp
1.629 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.630 +    from kdc have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.631 +      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.632        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
1.633 -    moreover
1.634 -    {assume "\<not> k dvd c"
1.635 -      from kpos have knz: "k\<noteq>0" by simp
1.636 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.637 -      from prems have "?I (real x) (?s (Ge (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<ge> 0)"
1.638 -        using real_of_int_div[OF knz kdt]
1.639 -          numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.640 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
1.641 -      also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))" using pos_divide_le_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.642 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.643 -        by (simp add: ti)
1.644 -      finally have ?case . }
1.645 -    ultimately show ?case by blast
1.646 +  moreover
1.647 +  { assume *: "\<not> k dvd c"
1.648 +    from kpos have knz: "k\<noteq>0" by simp
1.649 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.650 +    from assms * have "?I (real x) (?s (Ge (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<ge> 0)"
1.651 +      using real_of_int_div[OF knz kdt]
1.652 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.653 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.654 +      by (simp add: ti algebra_simps)
1.655 +    also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))"
1.656 +      using pos_divide_le_eq[OF kpos,
1.657 +          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
1.658 +        real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.659 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.660 +      by (simp add: ti)
1.661 +    finally have ?case . }
1.662 +  ultimately show ?case by blast
1.663  next
1.664 -  case (9 i c e)   from prems have cp: "c > 0" and nb: "numbound0 e" by auto
1.665 -    {assume kdc: "k dvd c"
1.666 -      from kpos have knz: "k\<noteq>0" by simp
1.667 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.668 -      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.669 -        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.670 +  case (9 i c e)
1.671 +  then have cp: "c > 0" and nb: "numbound0 e" by auto
1.672 +  { assume kdc: "k dvd c"
1.673 +    from kpos have knz: "k\<noteq>0" by simp
1.674 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.675 +    from kdc have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.676 +      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.677        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
1.678 -    moreover
1.679 -    {assume "\<not> k dvd c"
1.680 -      from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
1.681 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.682 -      from prems have "?I (real x) (?s (Dvd i (CN 0 c e))) = (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k)"
1.683 -        using real_of_int_div[OF knz kdt]
1.684 -          numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.685 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
1.686 -      also have "\<dots> = (?I ?tk (Dvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.687 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.688 -        by (simp add: ti)
1.689 -      finally have ?case . }
1.690 -    ultimately show ?case by blast
1.691 +  moreover
1.692 +  { assume *: "\<not> k dvd c"
1.693 +    from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
1.694 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.695 +    from assms * have "?I (real x) (?s (Dvd i (CN 0 c e))) = (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k)"
1.696 +      using real_of_int_div[OF knz kdt]
1.697 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.698 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.699 +      by (simp add: ti algebra_simps)
1.700 +    also have "\<dots> = (?I ?tk (Dvd i (CN 0 c e)))"
1.701 +      using rdvd_mult[OF knz, where n="i"]
1.702 +        real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.703 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.704 +      by (simp add: ti)
1.705 +    finally have ?case . }
1.706 +  ultimately show ?case by blast
1.707  next
1.708 -  case (10 i c e)    from prems have cp: "c > 0" and nb: "numbound0 e" by auto
1.709 -    {assume kdc: "k dvd c"
1.710 -      from kpos have knz: "k\<noteq>0" by simp
1.711 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.712 -      from prems have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.713 -        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.714 +  case (10 i c e)
1.715 +  then have cp: "c > 0" and nb: "numbound0 e" by auto
1.716 +  { assume kdc: "k dvd c"
1.717 +    from kpos have knz: "k\<noteq>0" by simp
1.718 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.719 +    from kdc have  ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
1.720 +      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.721        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
1.722 -    moreover
1.723 -    {assume "\<not> k dvd c"
1.724 -      from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
1.725 -      from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.726 -      from prems have "?I (real x) (?s (NDvd i (CN 0 c e))) = (\<not> (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k))"
1.727 -        using real_of_int_div[OF knz kdt]
1.728 -          numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.729 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
1.730 -      also have "\<dots> = (?I ?tk (NDvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.731 -          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.732 -        by (simp add: ti)
1.733 -      finally have ?case . }
1.734 -    ultimately show ?case by blast
1.735 -qed (simp_all add: bound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"] numbound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"])
1.736 +  moreover
1.737 +  { assume *: "\<not> k dvd c"
1.738 +    from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
1.739 +    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
1.740 +    from assms * have "?I (real x) (?s (NDvd i (CN 0 c e))) = (\<not> (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k))"
1.741 +      using real_of_int_div[OF knz kdt]
1.742 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.743 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.744 +      by (simp add: ti algebra_simps)
1.745 +    also have "\<dots> = (?I ?tk (NDvd i (CN 0 c e)))"
1.746 +      using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt]
1.747 +        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
1.748 +        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
1.749 +      by (simp add: ti)
1.750 +    finally have ?case . }
1.751 +  ultimately show ?case by blast
1.752 +qed (simp_all add: bound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"]
1.753 +  numbound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"])
1.754
1.755
1.756  lemma \<sigma>\<rho>_nb: assumes lp:"iszlfm p (a#bs)" and nb: "numbound0 t"
1.757 @@ -3054,16 +3094,16 @@
1.758    ultimately show ?case by blast
1.759  next
1.760    case (5 c e) hence cp: "c > 0" by simp
1.761 -  from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric]
1.762 +  from 5 mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric]
1.763      real_of_int_mult]
1.764 -  show ?case using prems dp
1.765 +  show ?case using 5 dp
1.766      by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"]
1.767        algebra_simps)
1.768  next
1.769 -  case (6 c e)  hence cp: "c > 0" by simp
1.770 -  from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric]
1.771 +  case (6 c e) hence cp: "c > 0" by simp
1.772 +  from 6 mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric]
1.773      real_of_int_mult]
1.774 -  show ?case using prems dp
1.775 +  show ?case using 6 dp
1.776      by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"]
1.777        algebra_simps)
1.778  next
1.779 @@ -3118,45 +3158,48 @@
1.780    ultimately show ?case by blast
1.781  next
1.782    case (9 j c e)  hence p: "real j rdvd real (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"  by simp+
1.783 -    let ?e = "Inum (real i # bs) e"
1.784 -    from prems have "isint e (real i #bs)"  by simp
1.785 -    hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
1.786 -      by (simp add: isint_iff)
1.787 -    from prems have id: "j dvd d" by simp
1.788 -    from ie[symmetric] have "?p i = (real j rdvd real (c*i+ floor ?e))" by simp
1.789 -    also have "\<dots> = (j dvd c*i + floor ?e)"
1.790 -      using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
1.791 -    also have "\<dots> = (j dvd c*i - c*d + floor ?e)"
1.792 -      using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
1.793 -    also have "\<dots> = (real j rdvd real (c*i - c*d + floor ?e))"
1.794 -      using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
1.795 +  let ?e = "Inum (real i # bs) e"
1.796 +  from 9 have "isint e (real i #bs)"  by simp
1.797 +  hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
1.798 +    by (simp add: isint_iff)
1.799 +  from 9 have id: "j dvd d" by simp
1.800 +  from ie[symmetric] have "?p i = (real j rdvd real (c*i+ floor ?e))" by simp
1.801 +  also have "\<dots> = (j dvd c*i + floor ?e)"
1.802 +    using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
1.803 +  also have "\<dots> = (j dvd c*i - c*d + floor ?e)"
1.804 +    using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
1.805 +  also have "\<dots> = (real j rdvd real (c*i - c*d + floor ?e))"
1.806 +    using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
1.807        ie by simp
1.808 -    also have "\<dots> = (real j rdvd real (c*(i - d)) + ?e)"
1.809 -      using ie by (simp add:algebra_simps)
1.810 -    finally show ?case
1.811 -      using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p
1.812 -      by (simp add: algebra_simps)
1.813 +  also have "\<dots> = (real j rdvd real (c*(i - d)) + ?e)"
1.814 +    using ie by (simp add:algebra_simps)
1.815 +  finally show ?case
1.816 +    using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p
1.817 +    by (simp add: algebra_simps)
1.818  next
1.819 -  case (10 j c e)   hence p: "\<not> (real j rdvd real (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"  by simp+
1.820 -    let ?e = "Inum (real i # bs) e"
1.821 -    from prems have "isint e (real i #bs)"  by simp
1.822 -    hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
1.823 -      by (simp add: isint_iff)
1.824 -    from prems have id: "j dvd d" by simp
1.825 -    from ie[symmetric] have "?p i = (\<not> (real j rdvd real (c*i+ floor ?e)))" by simp
1.826 -    also have "\<dots> = Not (j dvd c*i + floor ?e)"
1.827 -      using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
1.828 -    also have "\<dots> = Not (j dvd c*i - c*d + floor ?e)"
1.829 -      using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
1.830 -    also have "\<dots> = Not (real j rdvd real (c*i - c*d + floor ?e))"
1.831 -      using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
1.832 +  case (10 j c e)
1.833 +  hence p: "\<not> (real j rdvd real (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"
1.834 +    by simp+
1.835 +  let ?e = "Inum (real i # bs) e"
1.836 +  from 10 have "isint e (real i #bs)"  by simp
1.837 +  hence ie: "real (floor ?e) = ?e"
1.838 +    using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
1.839 +    by (simp add: isint_iff)
1.840 +  from 10 have id: "j dvd d" by simp
1.841 +  from ie[symmetric] have "?p i = (\<not> (real j rdvd real (c*i+ floor ?e)))" by simp
1.842 +  also have "\<dots> = Not (j dvd c*i + floor ?e)"
1.843 +    using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
1.844 +  also have "\<dots> = Not (j dvd c*i - c*d + floor ?e)"
1.845 +    using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
1.846 +  also have "\<dots> = Not (real j rdvd real (c*i - c*d + floor ?e))"
1.847 +    using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
1.848        ie by simp
1.849 -    also have "\<dots> = Not (real j rdvd real (c*(i - d)) + ?e)"
1.850 -      using ie by (simp add:algebra_simps)
1.851 -    finally show ?case
1.852 -      using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p
1.853 -      by (simp add: algebra_simps)
1.854 -qed(auto simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"])
1.855 +  also have "\<dots> = Not (real j rdvd real (c*(i - d)) + ?e)"
1.856 +    using ie by (simp add:algebra_simps)
1.857 +  finally show ?case
1.858 +    using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p
1.859 +    by (simp add: algebra_simps)
1.860 +qed (auto simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"])
1.861
1.862  lemma \<sigma>_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t"
1.863    shows "bound0 (\<sigma> p k t)"
1.864 @@ -3361,10 +3404,10 @@
1.865      (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
1.866      (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))"    by blast
1.867    show ?case
1.868 -    proof(simp only: FS, clarsimp simp del: Ifm.simps Inum.simps, -)
1.869 -      fix p n s
1.870 -      let ?ths = "(?I p \<longrightarrow> (?N (Floor a) = ?N (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p"
1.871 -      assume "(\<exists>ba. (p, 0, ba) \<in> set (rsplit0 a) \<and> n = 0 \<and> s = Floor ba) \<or>
1.872 +  proof(simp only: FS, clarsimp simp del: Ifm.simps Inum.simps, -)
1.873 +    fix p n s
1.874 +    let ?ths = "(?I p \<longrightarrow> (?N (Floor a) = ?N (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p"
1.875 +    assume "(\<exists>ba. (p, 0, ba) \<in> set (rsplit0 a) \<and> n = 0 \<and> s = Floor ba) \<or>
1.876         (\<exists>ab ac ba.
1.877             (ab, ac, ba) \<in> set (rsplit0 a) \<and>
1.878             0 < ac \<and>
1.879 @@ -3375,70 +3418,70 @@
1.880             ac < 0 \<and>
1.881             (\<exists>j. p = fp ab ac ba j \<and>
1.882                  n = 0 \<and> s = Add (Floor ba) (C j) \<and> ac \<le> j \<and> j \<le> 0))"
1.883 -      moreover
1.884 -      {fix s'
1.885 -        assume "(p, 0, s') \<in> ?SS a" and "n = 0" and "s = Floor s'"
1.886 -        hence ?ths using prems by auto}
1.887 -      moreover
1.888 -      { fix p' n' s' j
1.889 -        assume pns: "(p', n', s') \<in> ?SS a"
1.890 -          and np: "0 < n'"
1.891 -          and p_def: "p = ?p (p',n',s') j"
1.892 -          and n0: "n = 0"
1.893 -          and s_def: "s = (Add (Floor s') (C j))"
1.894 -          and jp: "0 \<le> j" and jn: "j \<le> n'"
1.895 -        from prems pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow>
1.896 +    moreover
1.897 +    { fix s'
1.898 +      assume "(p, 0, s') \<in> ?SS a" and "n = 0" and "s = Floor s'"
1.899 +      hence ?ths using 5(1) by auto }
1.900 +    moreover
1.901 +    { fix p' n' s' j
1.902 +      assume pns: "(p', n', s') \<in> ?SS a"
1.903 +        and np: "0 < n'"
1.904 +        and p_def: "p = ?p (p',n',s') j"
1.905 +        and n0: "n = 0"
1.906 +        and s_def: "s = (Add (Floor s') (C j))"
1.907 +        and jp: "0 \<le> j" and jn: "j \<le> n'"
1.908 +      from 5 pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow>
1.909            Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and>
1.910            numbound0 s' \<and> isrlfm p'" by blast
1.911 -        hence nb: "numbound0 s'" by simp
1.912 -        from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numsub_nb)
1.913 -        let ?nxs = "CN 0 n' s'"
1.914 -        let ?l = "floor (?N s') + j"
1.915 -        from H
1.916 -        have "?I (?p (p',n',s') j) \<longrightarrow>
1.917 +      hence nb: "numbound0 s'" by simp
1.918 +      from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numsub_nb)
1.919 +      let ?nxs = "CN 0 n' s'"
1.920 +      let ?l = "floor (?N s') + j"
1.921 +      from H
1.922 +      have "?I (?p (p',n',s') j) \<longrightarrow>
1.923            (((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))"
1.924 -          by (simp add: fp_def np algebra_simps numsub numadd numfloor)
1.925 -        also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
1.926 -          using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
1.927 -        moreover
1.928 -        have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp
1.929 -        ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
1.930 -          by blast
1.931 -        with s_def n0 p_def nb nf have ?ths by auto}
1.932 +        by (simp add: fp_def np algebra_simps numsub numadd numfloor)
1.933 +      also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
1.934 +        using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
1.935        moreover
1.936 -      {fix p' n' s' j
1.937 -        assume pns: "(p', n', s') \<in> ?SS a"
1.938 -          and np: "n' < 0"
1.939 -          and p_def: "p = ?p (p',n',s') j"
1.940 -          and n0: "n = 0"
1.941 -          and s_def: "s = (Add (Floor s') (C j))"
1.942 -          and jp: "n' \<le> j" and jn: "j \<le> 0"
1.943 -        from prems pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow>
1.944 +      have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp
1.945 +      ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
1.946 +        by blast
1.947 +      with s_def n0 p_def nb nf have ?ths by auto}
1.948 +    moreover
1.949 +    { fix p' n' s' j
1.950 +      assume pns: "(p', n', s') \<in> ?SS a"
1.951 +        and np: "n' < 0"
1.952 +        and p_def: "p = ?p (p',n',s') j"
1.953 +        and n0: "n = 0"
1.954 +        and s_def: "s = (Add (Floor s') (C j))"
1.955 +        and jp: "n' \<le> j" and jn: "j \<le> 0"
1.956 +      from 5 pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow>
1.957            Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and>
1.958            numbound0 s' \<and> isrlfm p'" by blast
1.959 -        hence nb: "numbound0 s'" by simp
1.960 -        from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numneg_nb)
1.961 -        let ?nxs = "CN 0 n' s'"
1.962 -        let ?l = "floor (?N s') + j"
1.963 -        from H
1.964 -        have "?I (?p (p',n',s') j) \<longrightarrow>
1.965 +      hence nb: "numbound0 s'" by simp
1.966 +      from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numneg_nb)
1.967 +      let ?nxs = "CN 0 n' s'"
1.968 +      let ?l = "floor (?N s') + j"
1.969 +      from H
1.970 +      have "?I (?p (p',n',s') j) \<longrightarrow>
1.971            (((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))"
1.972 -          by (simp add: np fp_def algebra_simps numneg numfloor numadd numsub)
1.973 -        also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
1.974 -          using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
1.975 -        moreover
1.976 -        have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"  by simp
1.977 -        ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
1.978 -          by blast
1.979 -        with s_def n0 p_def nb nf have ?ths by auto}
1.980 -      ultimately show ?ths by auto
1.981 -    qed
1.982 +        by (simp add: np fp_def algebra_simps numneg numfloor numadd numsub)
1.983 +      also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
1.984 +        using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
1.985 +      moreover
1.986 +      have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"  by simp
1.987 +      ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
1.988 +        by blast
1.989 +      with s_def n0 p_def nb nf have ?ths by auto}
1.990 +    ultimately show ?ths by auto
1.991 +  qed
1.992  next
1.993    case (3 a b) then show ?case
1.994 -  apply auto
1.995 -  apply (erule_tac x = "(aa, aaa, ba)" in ballE) apply simp_all
1.996 -  apply (erule_tac x = "(ab, ac, baa)" in ballE) apply simp_all
1.997 -  done
1.998 +    apply auto
1.999 +    apply (erule_tac x = "(aa, aaa, ba)" in ballE) apply simp_all
1.1000 +    apply (erule_tac x = "(ab, ac, baa)" in ballE) apply simp_all
1.1001 +    done
1.1002  qed (auto simp add: Let_def split_def algebra_simps conj_rl)
1.1003
1.1004  lemma real_in_int_intervals:
1.1005 @@ -3452,9 +3495,9 @@
1.1006    shows "\<exists> (p,n,s) \<in> set (rsplit0 t). Ifm (x#bs) p" (is "\<exists> (p,n,s) \<in> ?SS t. ?I p")
1.1007  proof(induct t rule: rsplit0.induct)
1.1008    case (2 a b)
1.1009 -  from prems have "\<exists> (pa,na,sa) \<in> ?SS a. ?I pa" by auto
1.1010 +  then have "\<exists> (pa,na,sa) \<in> ?SS a. ?I pa" by auto
1.1011    then obtain "pa" "na" "sa" where pa: "(pa,na,sa)\<in> ?SS a \<and> ?I pa" by blast
1.1012 -  from prems have "\<exists> (pb,nb,sb) \<in> ?SS b. ?I pb" by blast
1.1013 +  with 2 have "\<exists> (pb,nb,sb) \<in> ?SS b. ?I pb" by blast
1.1014    then obtain "pb" "nb" "sb" where pb: "(pb,nb,sb)\<in> ?SS b \<and> ?I pb" by blast
1.1015    from pa pb have th: "((pa,na,sa),(pb,nb,sb)) \<in> set[(x,y). x\<leftarrow>rsplit0 a, y\<leftarrow>rsplit0 b]"
1.1016      by (auto)
1.1017 @@ -3515,8 +3558,9 @@
1.1018    have FS: "?SS (Floor a) =
1.1019      ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
1.1020      (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
1.1021 -    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))"    by blast
1.1022 -  from prems have "\<exists> (p,n,s) \<in> ?SS a. ?I p" by auto
1.1023 +    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))"
1.1024 +    by blast
1.1025 +  from 5 have "\<exists> (p,n,s) \<in> ?SS a. ?I p" by auto
1.1026    then obtain "p" "n" "s" where pns: "(p,n,s) \<in> ?SS a \<and> ?I p" by blast
1.1027    let ?N = "\<lambda> t. Inum (x#bs) t"
1.1028    from rsplit0_cs[rule_format] pns have ans:"(?N a = ?N (CN 0 n s)) \<and> numbound0 s \<and> isrlfm p"
1.1029 @@ -3933,19 +3977,18 @@
1.1030      have "isatom (simpfm (Lt a))" by (cases "simpnum a", auto simp add: Let_def)
1.1031      with bn bound0at_l have ?case by blast}
1.1032    moreover
1.1033 -  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1034 -    {
1.1035 -      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1036 +  { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1037 +    { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1038        with numgcd_pos[where t="CN 0 c (simpnum e)"]
1.1039        have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
1.1040 -      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1041 +      from `c > 0` have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1042          by (simp add: numgcd_def)
1.1043 -      from prems have th': "c\<noteq>0" by auto
1.1044 -      from prems have cp: "c \<ge> 0" by simp
1.1045 +      from `c > 0` have th': "c\<noteq>0" by auto
1.1046 +      from `c > 0` have cp: "c \<ge> 0" by simp
1.1047        from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
1.1048 -        have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1049 +      have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1050      }
1.1051 -    with prems have ?case
1.1052 +    with Lt a have ?case
1.1053        by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
1.1054    ultimately show ?case by blast
1.1055  next
1.1056 @@ -3953,24 +3996,23 @@
1.1057    hence "bound0 (Le a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
1.1058      by (cases a,simp_all, case_tac "nat", simp_all)
1.1059    moreover
1.1060 -  {assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))"
1.1061 +  { assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))"
1.1062        using simpfm_bound0 by blast
1.1063      have "isatom (simpfm (Le a))" by (cases "simpnum a", auto simp add: Let_def)
1.1064      with bn bound0at_l have ?case by blast}
1.1065    moreover
1.1066 -  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1067 -    {
1.1068 -      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1069 +  { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1070 +    { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1071        with numgcd_pos[where t="CN 0 c (simpnum e)"]
1.1072        have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
1.1073 -      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1074 +      from `c > 0` have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1075          by (simp add: numgcd_def)
1.1076 -      from prems have th': "c\<noteq>0" by auto
1.1077 -      from prems have cp: "c \<ge> 0" by simp
1.1078 +      from `c > 0` have th': "c\<noteq>0" by auto
1.1079 +      from `c > 0` have cp: "c \<ge> 0" by simp
1.1080        from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
1.1081 -        have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1082 +      have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1083      }
1.1084 -    with prems have ?case
1.1085 +    with Le a have ?case
1.1086        by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
1.1087    ultimately show ?case by blast
1.1088  next
1.1089 @@ -3983,19 +4025,18 @@
1.1090      have "isatom (simpfm (Gt a))" by (cases "simpnum a", auto simp add: Let_def)
1.1091      with bn bound0at_l have ?case by blast}
1.1092    moreover
1.1093 -  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1094 -    {
1.1095 -      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1096 +  { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1097 +    { assume cn1: "numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1098        with numgcd_pos[where t="CN 0 c (simpnum e)"]
1.1099        have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
1.1100 -      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1101 +      from `c > 0` have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1102          by (simp add: numgcd_def)
1.1103 -      from prems have th': "c\<noteq>0" by auto
1.1104 -      from prems have cp: "c \<ge> 0" by simp
1.1105 +      from `c > 0` have th': "c\<noteq>0" by auto
1.1106 +      from `c > 0` have cp: "c \<ge> 0" by simp
1.1107        from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
1.1108 -        have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1109 +      have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1110      }
1.1111 -    with prems have ?case
1.1112 +    with Gt a have ?case
1.1113        by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
1.1114    ultimately show ?case by blast
1.1115  next
1.1116 @@ -4003,24 +4044,23 @@
1.1117    hence "bound0 (Ge a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
1.1118      by (cases a,simp_all, case_tac "nat", simp_all)
1.1119    moreover
1.1120 -  {assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))"
1.1121 +  { assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))"
1.1122        using simpfm_bound0 by blast
1.1123      have "isatom (simpfm (Ge a))" by (cases "simpnum a", auto simp add: Let_def)
1.1124      with bn bound0at_l have ?case by blast}
1.1125    moreover
1.1126 -  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1127 -    {
1.1128 -      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1129 +  { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1130 +    { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1131        with numgcd_pos[where t="CN 0 c (simpnum e)"]
1.1132        have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
1.1133 -      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1134 +      from `c > 0` have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1135          by (simp add: numgcd_def)
1.1136 -      from prems have th': "c\<noteq>0" by auto
1.1137 -      from prems have cp: "c \<ge> 0" by simp
1.1138 +      from `c > 0` have th': "c\<noteq>0" by auto
1.1139 +      from `c > 0` have cp: "c \<ge> 0" by simp
1.1140        from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
1.1141 -        have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1142 +      have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1143      }
1.1144 -    with prems have ?case
1.1145 +    with Ge a have ?case
1.1146        by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
1.1147    ultimately show ?case by blast
1.1148  next
1.1149 @@ -4028,24 +4068,23 @@
1.1150    hence "bound0 (Eq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
1.1151      by (cases a,simp_all, case_tac "nat", simp_all)
1.1152    moreover
1.1153 -  {assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))"
1.1154 +  { assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))"
1.1155        using simpfm_bound0 by blast
1.1156      have "isatom (simpfm (Eq a))" by (cases "simpnum a", auto simp add: Let_def)
1.1157      with bn bound0at_l have ?case by blast}
1.1158    moreover
1.1159 -  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1160 -    {
1.1161 -      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1162 +  { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1163 +    { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1164        with numgcd_pos[where t="CN 0 c (simpnum e)"]
1.1165        have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
1.1166 -      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1167 +      from `c > 0` have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1168          by (simp add: numgcd_def)
1.1169 -      from prems have th': "c\<noteq>0" by auto
1.1170 -      from prems have cp: "c \<ge> 0" by simp
1.1171 +      from `c > 0` have th': "c\<noteq>0" by auto
1.1172 +      from `c > 0` have cp: "c \<ge> 0" by simp
1.1173        from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
1.1174 -        have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1175 +      have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1176      }
1.1177 -    with prems have ?case
1.1178 +    with Eq a have ?case
1.1179        by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
1.1180    ultimately show ?case by blast
1.1181  next
1.1182 @@ -4058,19 +4097,18 @@
1.1183      have "isatom (simpfm (NEq a))" by (cases "simpnum a", auto simp add: Let_def)
1.1184      with bn bound0at_l have ?case by blast}
1.1185    moreover
1.1186 -  {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1187 -    {
1.1188 -      assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1189 +  { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
1.1190 +    { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
1.1191        with numgcd_pos[where t="CN 0 c (simpnum e)"]
1.1192        have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
1.1193 -      from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1194 +      from `c > 0` have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
1.1195          by (simp add: numgcd_def)
1.1196 -      from prems have th': "c\<noteq>0" by auto
1.1197 -      from prems have cp: "c \<ge> 0" by simp
1.1198 +      from `c > 0` have th': "c\<noteq>0" by auto
1.1199 +      from `c > 0` have cp: "c \<ge> 0" by simp
1.1200        from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
1.1201 -        have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1202 +      have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
1.1203      }
1.1204 -    with prems have ?case
1.1205 +    with NEq a have ?case
1.1206        by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
1.1207    ultimately show ?case by blast
1.1208  next
1.1209 @@ -4111,8 +4149,8 @@
1.1210    case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
1.1211  next
1.1212    case (3 c e)
1.1213 -  from prems have nb: "numbound0 e" by simp
1.1214 -  from prems have cp: "real c > 0" by simp
1.1215 +  from 3 have nb: "numbound0 e" by simp
1.1216 +  from 3 have cp: "real c > 0" by simp
1.1217    fix a
1.1218    let ?e="Inum (a#bs) e"
1.1219    let ?z = "(- ?e) / real c"
1.1220 @@ -4128,8 +4166,8 @@
1.1221    thus ?case by blast
1.1222  next
1.1223    case (4 c e)
1.1224 -  from prems have nb: "numbound0 e" by simp
1.1225 -  from prems have cp: "real c > 0" by simp
1.1226 +  from 4 have nb: "numbound0 e" by simp
1.1227 +  from 4 have cp: "real c > 0" by simp
1.1228    fix a
1.1229    let ?e="Inum (a#bs) e"
1.1230    let ?z = "(- ?e) / real c"
1.1231 @@ -4145,8 +4183,8 @@
1.1232    thus ?case by blast
1.1233  next
1.1234    case (5 c e)
1.1235 -    from prems have nb: "numbound0 e" by simp
1.1236 -  from prems have cp: "real c > 0" by simp
1.1237 +  from 5 have nb: "numbound0 e" by simp
1.1238 +  from 5 have cp: "real c > 0" by simp
1.1239    fix a
1.1240    let ?e="Inum (a#bs) e"
1.1241    let ?z = "(- ?e) / real c"
1.1242 @@ -4161,8 +4199,8 @@
1.1243    thus ?case by blast
1.1244  next
1.1245    case (6 c e)
1.1246 -    from prems have nb: "numbound0 e" by simp
1.1247 -  from prems have cp: "real c > 0" by simp
1.1248 +  from 6 have nb: "numbound0 e" by simp
1.1249 +  from 6 have cp: "real c > 0" by simp
1.1250    fix a
1.1251    let ?e="Inum (a#bs) e"
1.1252    let ?z = "(- ?e) / real c"
1.1253 @@ -4177,8 +4215,8 @@
1.1254    thus ?case by blast
1.1255  next
1.1256    case (7 c e)
1.1257 -    from prems have nb: "numbound0 e" by simp
1.1258 -  from prems have cp: "real c > 0" by simp
1.1259 +  from 7 have nb: "numbound0 e" by simp
1.1260 +  from 7 have cp: "real c > 0" by simp
1.1261    fix a
1.1262    let ?e="Inum (a#bs) e"
1.1263    let ?z = "(- ?e) / real c"
1.1264 @@ -4193,8 +4231,8 @@
1.1265    thus ?case by blast
1.1266  next
1.1267    case (8 c e)
1.1268 -    from prems have nb: "numbound0 e" by simp
1.1269 -  from prems have cp: "real c > 0" by simp
1.1270 +  from 8 have nb: "numbound0 e" by simp
1.1271 +  from 8 have cp: "real c > 0" by simp
1.1272    fix a
1.1273    let ?e="Inum (a#bs) e"
1.1274    let ?z = "(- ?e) / real c"
1.1275 @@ -4219,8 +4257,8 @@
1.1276    case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
1.1277  next
1.1278    case (3 c e)
1.1279 -  from prems have nb: "numbound0 e" by simp
1.1280 -  from prems have cp: "real c > 0" by simp
1.1281 +  from 3 have nb: "numbound0 e" by simp
1.1282 +  from 3 have cp: "real c > 0" by simp
1.1283    fix a
1.1284    let ?e="Inum (a#bs) e"
1.1285    let ?z = "(- ?e) / real c"
1.1286 @@ -4236,8 +4274,8 @@
1.1287    thus ?case by blast
1.1288  next
1.1289    case (4 c e)
1.1290 -  from prems have nb: "numbound0 e" by simp
1.1291 -  from prems have cp: "real c > 0" by simp
1.1292 +  from 4 have nb: "numbound0 e" by simp
1.1293 +  from 4 have cp: "real c > 0" by simp
1.1294    fix a
1.1295    let ?e="Inum (a#bs) e"
1.1296    let ?z = "(- ?e) / real c"
1.1297 @@ -4253,8 +4291,8 @@
1.1298    thus ?case by blast
1.1299  next
1.1300    case (5 c e)
1.1301 -  from prems have nb: "numbound0 e" by simp
1.1302 -  from prems have cp: "real c > 0" by simp
1.1303 +  from 5 have nb: "numbound0 e" by simp
1.1304 +  from 5 have cp: "real c > 0" by simp
1.1305    fix a
1.1306    let ?e="Inum (a#bs) e"
1.1307    let ?z = "(- ?e) / real c"
1.1308 @@ -4269,8 +4307,8 @@
1.1309    thus ?case by blast
1.1310  next
1.1311    case (6 c e)
1.1312 -  from prems have nb: "numbound0 e" by simp
1.1313 -  from prems have cp: "real c > 0" by simp
1.1314 +  from 6 have nb: "numbound0 e" by simp
1.1315 +  from 6 have cp: "real c > 0" by simp
1.1316    fix a
1.1317    let ?e="Inum (a#bs) e"
1.1318    let ?z = "(- ?e) / real c"
1.1319 @@ -4285,8 +4323,8 @@
1.1320    thus ?case by blast
1.1321  next
1.1322    case (7 c e)
1.1323 -  from prems have nb: "numbound0 e" by simp
1.1324 -  from prems have cp: "real c > 0" by simp
1.1325 +  from 7 have nb: "numbound0 e" by simp
1.1326 +  from 7 have cp: "real c > 0" by simp
1.1327    fix a
1.1328    let ?e="Inum (a#bs) e"
1.1329    let ?z = "(- ?e) / real c"
1.1330 @@ -4301,8 +4339,8 @@
1.1331    thus ?case by blast
1.1332  next
1.1333    case (8 c e)
1.1334 -  from prems have nb: "numbound0 e" by simp
1.1335 -  from prems have cp: "real c > 0" by simp
1.1336 +  from 8 have nb: "numbound0 e" by simp
1.1337 +  from 8 have cp: "real c > 0" by simp
1.1338    fix a
1.1339    let ?e="Inum (a#bs) e"
1.1340    let ?z = "(- ?e) / real c"
1.1341 @@ -4387,7 +4425,8 @@
1.1342    shows "(Ifm (x#bs) (\<upsilon> p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (\<upsilon> p (t,n))" (is "(?I x (\<upsilon> p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
1.1343    using lp
1.1344  proof(induct p rule: \<upsilon>.induct)
1.1345 -  case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
1.1346 +  case (5 c e)
1.1347 +  from 5 have cp: "c >0" and nb: "numbound0 e" by simp_all
1.1348    have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
1.1349      using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
1.1350    also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
1.1351 @@ -4397,7 +4436,8 @@
1.1352      using np by simp
1.1353    finally show ?case using nbt nb by (simp add: algebra_simps)
1.1354  next
1.1355 -  case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
1.1356 +  case (6 c e)
1.1357 +  from 6 have cp: "c >0" and nb: "numbound0 e" by simp_all
1.1358    have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
1.1359      using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
1.1360    also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
1.1361 @@ -4407,7 +4447,8 @@
1.1362      using np by simp
1.1363    finally show ?case using nbt nb by (simp add: algebra_simps)
1.1364  next
1.1365 -  case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
1.1366 +  case (7 c e)
1.1367 +  from 7 have cp: "c >0" and nb: "numbound0 e" by simp_all
1.1368    have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
1.1369      using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
1.1370    also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
1.1371 @@ -4417,7 +4458,8 @@
1.1372      using np by simp
1.1373    finally show ?case using nbt nb by (simp add: algebra_simps)
1.1374  next
1.1375 -  case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
1.1376 +  case (8 c e)
1.1377 +  from 8 have cp: "c >0" and nb: "numbound0 e" by simp_all
1.1378    have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
1.1379      using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
1.1380    also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
1.1381 @@ -4427,7 +4469,8 @@
1.1382      using np by simp
1.1383    finally show ?case using nbt nb by (simp add: algebra_simps)
1.1384  next
1.1385 -  case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
1.1386 +  case (3 c e)
1.1387 +  from 3 have cp: "c >0" and nb: "numbound0 e" by simp_all
1.1388    from np have np: "real n \<noteq> 0" by simp
1.1389    have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
1.1390      using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
1.1391 @@ -4438,7 +4481,8 @@
1.1392      using np by simp
1.1393    finally show ?case using nbt nb by (simp add: algebra_simps)
1.1394  next
1.1395 -  case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
1.1396 +  case (4 c e)
1.1397 +  from 4 have cp: "c >0" and nb: "numbound0 e" by simp_all
1.1398    from np have np: "real n \<noteq> 0" by simp
1.1399    have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
1.1400      using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
1.1401 @@ -4497,100 +4541,100 @@
1.1402    shows "Ifm (y#bs) p"
1.1403  using lp px noS
1.1404  proof (induct p rule: isrlfm.induct)
1.1405 -  case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
1.1406 -    from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
1.1407 -    hence pxc: "x < (- ?N x e) / real c"
1.1408 -      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
1.1409 -    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1410 -    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1411 -    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
1.1412 -    moreover {assume y: "y < (-?N x e)/ real c"
1.1413 -      hence "y * real c < - ?N x e"
1.1414 -        by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
1.1415 -      hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
1.1416 -      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
1.1417 -    moreover {assume y: "y > (- ?N x e) / real c"
1.1418 -      with yu have eu: "u > (- ?N x e) / real c" by auto
1.1419 -      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
1.1420 -      with lx pxc have "False" by auto
1.1421 -      hence ?case by simp }
1.1422 -    ultimately show ?case by blast
1.1423 +  case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
1.1424 +  from 5 have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
1.1425 +  hence pxc: "x < (- ?N x e) / real c"
1.1426 +    by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
1.1427 +  from 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1428 +  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1429 +  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
1.1430 +  moreover {assume y: "y < (-?N x e)/ real c"
1.1431 +    hence "y * real c < - ?N x e"
1.1432 +      by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
1.1433 +    hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
1.1434 +    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
1.1435 +  moreover {assume y: "y > (- ?N x e) / real c"
1.1436 +    with yu have eu: "u > (- ?N x e) / real c" by auto
1.1437 +    with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
1.1438 +    with lx pxc have "False" by auto
1.1439 +    hence ?case by simp }
1.1440 +  ultimately show ?case by blast
1.1441  next
1.1442 -  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
1.1443 -    from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
1.1444 -    hence pxc: "x \<le> (- ?N x e) / real c"
1.1445 -      by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
1.1446 -    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1447 -    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1448 -    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
1.1449 -    moreover {assume y: "y < (-?N x e)/ real c"
1.1450 -      hence "y * real c < - ?N x e"
1.1451 -        by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
1.1452 -      hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
1.1453 -      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
1.1454 -    moreover {assume y: "y > (- ?N x e) / real c"
1.1455 -      with yu have eu: "u > (- ?N x e) / real c" by auto
1.1456 -      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
1.1457 -      with lx pxc have "False" by auto
1.1458 -      hence ?case by simp }
1.1459 -    ultimately show ?case by blast
1.1460 +  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
1.1461 +  from 6 have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
1.1462 +  hence pxc: "x \<le> (- ?N x e) / real c"
1.1463 +    by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
1.1464 +  from 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1465 +  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1466 +  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
1.1467 +  moreover {assume y: "y < (-?N x e)/ real c"
1.1468 +    hence "y * real c < - ?N x e"
1.1469 +      by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
1.1470 +    hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
1.1471 +    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
1.1472 +  moreover {assume y: "y > (- ?N x e) / real c"
1.1473 +    with yu have eu: "u > (- ?N x e) / real c" by auto
1.1474 +    with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
1.1475 +    with lx pxc have "False" by auto
1.1476 +    hence ?case by simp }
1.1477 +  ultimately show ?case by blast
1.1478  next
1.1479 -  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
1.1480 -    from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
1.1481 -    hence pxc: "x > (- ?N x e) / real c"
1.1482 -      by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
1.1483 -    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1484 -    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1485 -    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
1.1486 -    moreover {assume y: "y > (-?N x e)/ real c"
1.1487 -      hence "y * real c > - ?N x e"
1.1488 -        by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
1.1489 -      hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
1.1490 -      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
1.1491 -    moreover {assume y: "y < (- ?N x e) / real c"
1.1492 -      with ly have eu: "l < (- ?N x e) / real c" by auto
1.1493 -      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
1.1494 -      with xu pxc have "False" by auto
1.1495 -      hence ?case by simp }
1.1496 -    ultimately show ?case by blast
1.1497 +  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
1.1498 +  from 7 have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
1.1499 +  hence pxc: "x > (- ?N x e) / real c"
1.1500 +    by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
1.1501 +  from 7 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1502 +  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1503 +  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
1.1504 +  moreover {assume y: "y > (-?N x e)/ real c"
1.1505 +    hence "y * real c > - ?N x e"
1.1506 +      by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
1.1507 +    hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
1.1508 +    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
1.1509 +  moreover {assume y: "y < (- ?N x e) / real c"
1.1510 +    with ly have eu: "l < (- ?N x e) / real c" by auto
1.1511 +    with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
1.1512 +    with xu pxc have "False" by auto
1.1513 +    hence ?case by simp }
1.1514 +  ultimately show ?case by blast
1.1515  next
1.1516 -  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
1.1517 -    from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
1.1518 -    hence pxc: "x \<ge> (- ?N x e) / real c"
1.1519 -      by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
1.1520 -    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1521 -    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1522 -    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
1.1523 -    moreover {assume y: "y > (-?N x e)/ real c"
1.1524 -      hence "y * real c > - ?N x e"
1.1525 -        by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
1.1526 -      hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
1.1527 -      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
1.1528 -    moreover {assume y: "y < (- ?N x e) / real c"
1.1529 -      with ly have eu: "l < (- ?N x e) / real c" by auto
1.1530 -      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
1.1531 -      with xu pxc have "False" by auto
1.1532 -      hence ?case by simp }
1.1533 -    ultimately show ?case by blast
1.1534 +  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
1.1535 +  from 8 have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
1.1536 +  hence pxc: "x \<ge> (- ?N x e) / real c"
1.1537 +    by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
1.1538 +  from 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1539 +  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1540 +  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
1.1541 +  moreover {assume y: "y > (-?N x e)/ real c"
1.1542 +    hence "y * real c > - ?N x e"
1.1543 +      by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
1.1544 +    hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
1.1545 +    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
1.1546 +  moreover {assume y: "y < (- ?N x e) / real c"
1.1547 +    with ly have eu: "l < (- ?N x e) / real c" by auto
1.1548 +    with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
1.1549 +    with xu pxc have "False" by auto
1.1550 +    hence ?case by simp }
1.1551 +  ultimately show ?case by blast
1.1552  next
1.1553 -  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
1.1554 -    from cp have cnz: "real c \<noteq> 0" by simp
1.1555 -    from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
1.1556 -    hence pxc: "x = (- ?N x e) / real c"
1.1557 -      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
1.1558 -    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1559 -    with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
1.1560 -    with pxc show ?case by simp
1.1561 +  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
1.1562 +  from cp have cnz: "real c \<noteq> 0" by simp
1.1563 +  from 3 have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
1.1564 +  hence pxc: "x = (- ?N x e) / real c"
1.1565 +    by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
1.1566 +  from 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1567 +  with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
1.1568 +  with pxc show ?case by simp
1.1569  next
1.1570 -  case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
1.1571 -    from cp have cnz: "real c \<noteq> 0" by simp
1.1572 -    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1573 -    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1574 -    hence "y* real c \<noteq> -?N x e"
1.1575 -      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
1.1576 -    hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
1.1577 -    thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
1.1578 -      by (simp add: algebra_simps)
1.1579 +  case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
1.1580 +  from cp have cnz: "real c \<noteq> 0" by simp
1.1581 +  from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
1.1582 +  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
1.1583 +  hence "y* real c \<noteq> -?N x e"
1.1584 +    by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
1.1585 +  hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
1.1586 +  thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
1.1587 +    by (simp add: algebra_simps)
1.1588  qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"])
1.1589
1.1590  lemma rinf_\<Upsilon>:
1.1591 @@ -4598,7 +4642,8 @@
1.1592    and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
1.1593    and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
1.1594    and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
1.1595 -  shows "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p"
1.1596 +  shows "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p).
1.1597 +    ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p"
1.1598  proof-
1.1599    let ?N = "\<lambda> x t. Inum (x#bs) t"
1.1600    let ?U = "set (\<Upsilon> p)"
1.1601 @@ -5602,23 +5647,18 @@
1.1602  setup "Mir_Tac.setup"
1.1603
1.1604  lemma "ALL (x::real). (\<lfloor>x\<rfloor> = \<lceil>x\<rceil> = (x = real \<lfloor>x\<rfloor>))"
1.1605 -apply mir
1.1606 -done
1.1607 +  by mir
1.1608
1.1609  lemma "ALL (x::real). real (2::int)*x - (real (1::int)) < real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil> \<and> real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil>  \<le> real (2::int)*x + (real (1::int))"
1.1610 -apply mir
1.1611 -done
1.1612 +  by mir
1.1613
1.1614  lemma "ALL (x::real). 2*\<lfloor>x\<rfloor> \<le> \<lfloor>2*x\<rfloor> \<and> \<lfloor>2*x\<rfloor> \<le> 2*\<lfloor>x+1\<rfloor>"
1.1615 -apply mir
1.1616 -done
1.1617 +  by mir
1.1618
1.1619  lemma "ALL (x::real). \<exists>y \<le> x. (\<lfloor>x\<rfloor> = \<lceil>y\<rceil>)"
1.1620 -apply mir
1.1621 -done
1.1622 +  by mir
1.1623
1.1624  lemma "ALL (x::real) (y::real). \<lfloor>x\<rfloor> = \<lfloor>y\<rfloor> \<longrightarrow> 0 \<le> abs (y - x) \<and> abs (y - x) \<le> 1"
1.1625 -apply mir
1.1626 -done
1.1627 +  by mir
1.1628
1.1629  end
```
```     2.1 --- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Thu Mar 03 18:43:15 2011 +0100
2.2 +++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Thu Mar 03 21:43:06 2011 +0100
2.3 @@ -644,7 +644,7 @@
2.4
2.5  lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
2.6    shows "\<forall>n \<ge> m. d n < e m"
2.7 -  using prems apply auto
2.8 +  using assms apply auto
2.9    apply (erule_tac x="n" in allE)
2.10    apply (erule_tac x="n" in allE)
2.11    apply auto
```
```     3.1 --- a/src/HOL/Multivariate_Analysis/L2_Norm.thy	Thu Mar 03 18:43:15 2011 +0100
3.2 +++ b/src/HOL/Multivariate_Analysis/L2_Norm.thy	Thu Mar 03 21:43:06 2011 +0100
3.3 @@ -44,7 +44,7 @@
3.4    assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
3.5    shows "setL2 f K \<le> setL2 g K"
3.6    unfolding setL2_def
3.7 -  by (simp add: setsum_nonneg setsum_mono power_mono prems)
3.8 +  by (simp add: setsum_nonneg setsum_mono power_mono assms)
3.9
3.10  lemma setL2_strict_mono:
3.11    assumes "finite K" and "K \<noteq> {}"
```