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author | krauss |

Mon, 21 Feb 2011 23:14:36 +0100 | |

changeset 41815 | 9a0cacbcd825 |

parent 41814 | 3848eb635eab |

child 41816 | 7a55699805dc |

eliminated global prems

src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy | file | annotate | diff | comparison | revisions |

--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Mon Feb 21 23:14:36 2011 +0100 +++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Mon Feb 21 23:14:36 2011 +0100 @@ -258,26 +258,26 @@ \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)" proof(induct p q arbitrary: n0 n1 rule: polyadd.induct) case (2 ab c' n' p' n0 n1) - from prems have th1: "isnpolyh (C ab) (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 ab) (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 ab +\<^sub>p c') (Suc n')" by simp + with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^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' ab n1 n0) - from prems have th1: "isnpolyh (C ab) (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 ab) (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 ab) (Suc n')" by simp + with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (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 "4.hyps"(3)[OF nc nc'] @@ -286,25 +286,25 @@ using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto from eq "4.hyps"(4)[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 4 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 lt 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 "4.hyps"(1)[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 gt 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 "4.hyps"(2)[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 @@ -375,11 +375,11 @@ \<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 @@ -645,7 +645,7 @@ 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) @@ -706,7 +706,7 @@ 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) @@ -722,7 +722,7 @@ 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" @@ -1074,16 +1074,16 @@ case (1 c c') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) next case (2 c c' n' p') - 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 c') - from prems have "degree (C c') = degree (CN c n p)" by simp + hence "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" @@ -1098,22 +1098,22 @@ 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)} + 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] using nn' 4 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) + 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 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) @@ -1127,7 +1127,7 @@ 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 @@ -1174,11 +1174,11 @@ proof (induct p q arbitrary: n0 n1 rule: polymul.induct) case (2 c c' n' p' n0 n1) hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c" 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 c' n0 n1) hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'" 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')" @@ -1574,7 +1574,7 @@ 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})"