--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Mon Feb 21 18:29:47 2011 +0100
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Mon Feb 21 23:47:19 2011 +0100
@@ -253,53 +253,53 @@
\<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
case (2 a b c' n' p' n0 n1)
- from prems have th1: "isnpolyh (C (a,b)) (Suc n')" by simp
- from prems(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" by simp_all
+ from 2 have th1: "isnpolyh (C (a,b)) (Suc n')" by simp
+ from 2(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" by simp_all
with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
- with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
+ with 2(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
- thus ?case using prems th3 by simp
+ thus ?case using 2 th3 by simp
next
case (3 c' n' p' a b n1 n0)
- from prems have th1: "isnpolyh (C (a,b)) (Suc n')" by simp
- from prems(2) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" by simp_all
+ from 3 have th1: "isnpolyh (C (a,b)) (Suc n')" by simp
+ from 3(2) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1" by simp_all
with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
- with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
+ with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
- thus ?case using prems th3 by simp
+ thus ?case using 3 th3 by simp
next
case (4 c n p c' n' p' n0 n1)
hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
- from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
- from prems have ngen0: "n \<ge> n0" by simp
- from prems have n'gen1: "n' \<ge> n1" by simp
+ from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
+ from 4 have ngen0: "n \<ge> n0" by simp
+ from 4 have n'gen1: "n' \<ge> n1" by simp
have "n < n' \<or> n' < n \<or> n = n'" by auto
moreover {assume eq: "n = n'"
- with prems(2)[OF nc nc']
+ with 4(2)[OF nc nc']
have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
- from eq prems(1)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
+ from eq 4(1)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
- from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
+ from minle npp' ncc'n01 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
moreover {assume lt: "n < n'"
have "min n0 n1 \<le> n0" by simp
- with prems have th1:"min n0 n1 \<le> n" by auto
- from prems have th21: "isnpolyh c (Suc n)" by simp
- from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
+ with 4 have th1:"min n0 n1 \<le> n" by auto
+ from 4 have th21: "isnpolyh c (Suc n)" by simp
+ from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
from lt have th23: "min (Suc n) n' = Suc n" by arith
- from prems(4)[OF th21 th22]
+ from 4(4)[OF th21 th22]
have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
- with prems th1 have ?case by simp }
+ with 4 lt th1 have ?case by simp }
moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
have "min n0 n1 \<le> n1" by simp
- with prems have th1:"min n0 n1 \<le> n'" by auto
- from prems have th21: "isnpolyh c' (Suc n')" by simp_all
- from prems have th22: "isnpolyh (CN c n p) n" by simp
+ with 4 have th1:"min n0 n1 \<le> n'" by auto
+ from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
+ from 4 have th22: "isnpolyh (CN c n p) n" by simp
from gt have th23: "min n (Suc n') = Suc n'" by arith
- from prems(3)[OF th22 th21]
+ from 4(3)[OF th22 th21]
have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
- with prems th1 have ?case by simp}
+ with 4 gt th1 have ?case by simp}
ultimately show ?case by blast
qed auto
@@ -370,14 +370,15 @@
\<Longrightarrow> degreen p m = degreen q m"
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
case (4 c n p c' n' p' m n0 n1 x)
- {assume nn': "n' < n" hence ?case using prems by simp}
+ {assume nn': "n' < n" hence ?case using 4 by simp}
moreover
{assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
- moreover {assume "n < n'" with prems have ?case by simp }
- moreover {assume eq: "n = n'" hence ?case using prems
+ moreover {assume "n < n'" with 4 have ?case by simp }
+ moreover {assume eq: "n = n'" hence ?case using 4
apply (cases "p +\<^sub>p p' = 0\<^sub>p")
apply (auto simp add: Let_def)
- by blast
+ apply blast
+ done
}
ultimately have ?case by blast}
ultimately show ?case by blast
@@ -664,13 +665,13 @@
qed
lemma polypow_normh:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
proof (induct k arbitrary: n rule: polypow.induct)
case (2 k n)
let ?q = "polypow (Suc k div 2) p"
let ?d = "polymul (?q,?q)"
- from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
+ from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
from dn on show ?case by (simp add: Let_def)
@@ -695,7 +696,7 @@
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
-by (simp add: shift1_def polymul)
+ by (simp add: shift1_def)
lemma shift1_isnpoly:
assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
@@ -713,7 +714,7 @@
using f np by (induct k arbitrary: p, auto)
lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
- by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
+ by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
@@ -731,10 +732,10 @@
using np
proof (induct p arbitrary: n rule: behead.induct)
case (1 c p n) hence pn: "isnpolyh p n" by simp
- from prems(2)[OF pn]
+ from 1(1)[OF pn]
have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
- by (simp_all add: th[symmetric] field_simps power_Suc)
+ by (simp_all add: th[symmetric] field_simps)
qed (auto simp add: Let_def)
lemma behead_isnpolyh:
@@ -747,7 +748,7 @@
case (goal1 c n p n')
hence "n = Suc (n - 1)" by simp
hence "isnpolyh p (Suc (n - 1))" using `isnpolyh p n` by simp
- with prems(2) show ?case by simp
+ with goal1(2) show ?case by simp
qed
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
@@ -838,7 +839,7 @@
qed
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
- by (induct p, auto)
+ by (induct p) auto
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
unfolding wf_bs_def by simp
@@ -1033,7 +1034,7 @@
isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
lemma polymul_1[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
@@ -1101,17 +1102,17 @@
next
case (2 a b c' n' p')
let ?c = "(a,b)"
- from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
+ from 2 have "degree (C ?c) = degree (CN c' n' p')" by simp
hence nz:"n' > 0" by (cases n', auto)
hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
- with prems show ?case by simp
+ with 2 show ?case by simp
next
case (3 c n p a' b')
let ?c' = "(a',b')"
- from prems have "degree (C ?c') = degree (CN c n p)" by simp
+ from 3 have "degree (C ?c') = degree (CN c n p)" by simp
hence nz:"n > 0" by (cases n, auto)
hence "head (CN c n p) = CN c n p" by (cases n, auto)
- with prems show ?case by simp
+ with 3 show ?case by simp
next
case (4 c n p c' n' p')
hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
@@ -1126,36 +1127,40 @@
moreover
{assume nn': "n = n'"
have "n = 0 \<or> n >0" by arith
- moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
+ moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
moreover {assume nz: "n > 0"
- with nn' H(3) have cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
- hence ?case using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def split_def fst_conv snd_conv] using nn' prems by (simp add: Let_def)}
+ with nn' H(3) have cc': "c = c'" and pp': "p = p'" by (cases n, auto)+
+ hence ?case
+ using polysub_same_0 [OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
+ polysub_same_0 [OF c'nh, simplified polysub_def split_def fst_conv snd_conv]
+ using 4 nn' by (simp add: Let_def) }
ultimately have ?case by blast}
moreover
{assume nn': "n < n'" hence n'p: "n' > 0" by simp
hence headcnp':"head (CN c' n' p') = CN c' n' p'" by (cases n', simp_all)
- have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using prems by (cases n', simp_all)
- hence "n > 0" by (cases n, simp_all)
- hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
- from H(3) headcnp headcnp' nn' have ?case by auto}
+ have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
+ using 4 nn' by (cases n', simp_all)
+ hence "n > 0" by (cases n) simp_all
+ hence headcnp: "head (CN c n p) = CN c n p" by (cases n) auto
+ from H(3) headcnp headcnp' nn' have ?case by auto }
moreover
{assume nn': "n > n'" hence np: "n > 0" by simp
- hence headcnp:"head (CN c n p) = CN c n p" by (cases n, simp_all)
- from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
- from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
+ hence headcnp: "head (CN c n p) = CN c n p" by (cases n) simp_all
+ from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
+ from np have degcnp: "degree (CN c n p) = 0" by (cases n) simp_all
with degcnpeq have "n' > 0" by (cases n', simp_all)
- hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
- from H(3) headcnp headcnp' nn' have ?case by auto}
+ hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
+ from H(3) headcnp headcnp' nn' have ?case by auto }
ultimately show ?case by blast
qed auto
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
-by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
+ by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
proof(induct k arbitrary: n0 p)
case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
- with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
+ with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
and "head (shift1 p) = head p" by (simp_all add: shift1_head)
thus ?case by (simp add: funpow_swap1)
qed auto
@@ -1194,19 +1199,20 @@
apply (metis head_nz)
apply (metis head_nz)
apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
-by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
+apply (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
+done
lemma polymul_head_polyeq:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
case (2 a b c' n' p' n0 n1)
hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)" by (simp_all add: head_isnpolyh)
- thus ?case using prems by (cases n', auto)
+ thus ?case using 2 by (cases n') auto
next
case (3 c n p a' b' n0 n1)
hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')" by (simp_all add: head_isnpolyh)
- thus ?case using prems by (cases n, auto)
+ thus ?case using 3 by (cases n) auto
next
case (4 c n p c' n' p' n0 n1)
hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
@@ -1215,15 +1221,14 @@
have "n < n' \<or> n' < n \<or> n = n'" by arith
moreover
{assume nn': "n < n'" hence ?case
- thm prems
- using norm
- prems(6)[rule_format, OF nn' norm(1,6)]
- prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
+ using norm
+ 4(5)[rule_format, OF nn' norm(1,6)]
+ 4(6)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all) }
moreover {assume nn': "n'< n"
hence stupid: "n' < n \<and> \<not> n < n'" by simp
- hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)]
- prems(5)[rule_format, OF stupid norm(5,4)]
- by (simp,cases n',simp,cases n,auto)}
+ hence ?case using norm 4(3) [rule_format, OF stupid norm(5,3)]
+ 4(4)[rule_format, OF stupid norm(5,4)]
+ by (simp,cases n',simp,cases n,auto) }
moreover {assume nn': "n' = n"
hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp
from nn' polymul_normh[OF norm(5,4)]
@@ -1247,8 +1252,8 @@
have dth:"degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
hence dth':"degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
- have ?case using norm prems(2)[rule_format, OF stupid norm(5,3)]
- prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
+ have ?case using norm 4(1)[rule_format, OF stupid norm(5,3)]
+ 4(2)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
ultimately have ?case by (cases n) auto}
ultimately show ?case by blast
qed simp_all
@@ -1603,12 +1608,13 @@
definition "swapnorm n m t = polynate (swap n m t)"
-lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
+lemma swapnorm:
+ assumes nbs: "n < length bs" and mbs: "m < length bs"
shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
- using swap[OF prems] swapnorm_def by simp
+ using swap[OF assms] swapnorm_def by simp
lemma swapnorm_isnpoly[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
shows "isnpoly (swapnorm n m p)"
unfolding swapnorm_def by simp
@@ -1625,9 +1631,9 @@
"isweaknpoly p = False"
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
- by (induct p arbitrary: n0, auto)
+ by (induct p arbitrary: n0) auto
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
- by (induct p, auto)
+ by (induct p) auto
end
\ No newline at end of file