src/HOL/Decision_Procs/Ferrack.thy
author wenzelm
Mon Mar 16 18:24:30 2009 +0100 (2009-03-16)
changeset 30549 d2d7874648bd
parent 30439 57c68b3af2ea
child 30684 c98a64746c69
permissions -rw-r--r--
simplified method setup;
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(*  Title:      HOL/Decision_Procs/Ferrack.thy
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    Author:     Amine Chaieb
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*)
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theory Ferrack
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imports Complex_Main Dense_Linear_Order Efficient_Nat
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uses ("ferrack_tac.ML")
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begin
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section {* Quantifier elimination for @{text "\<real> (0, 1, +, <)"} *}
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  (*********************************************************************************)
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  (*          SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB                      *)
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  (*********************************************************************************)
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consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
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primrec
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  "alluopairs [] = []"
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  "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
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lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
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by (induct xs, auto)
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lemma alluopairs_set:
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  "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
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by (induct xs, auto)
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lemma alluopairs_ex:
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  assumes Pc: "\<forall> x y. P x y = P y x"
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  shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
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proof
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  assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
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  then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
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  from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
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    by auto
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next
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  assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
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  then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
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  from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
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  with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
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qed
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lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
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using Nat.gr0_conv_Suc
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by clarsimp
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lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
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  apply (induct xs, auto) done
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consts remdps:: "'a list \<Rightarrow> 'a list"
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recdef remdps "measure size"
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  "remdps [] = []"
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  "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
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(hints simp add: filter_length[rule_format])
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lemma remdps_set[simp]: "set (remdps xs) = set xs"
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  by (induct xs rule: remdps.induct, auto)
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  (*********************************************************************************)
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  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
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  (*********************************************************************************)
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datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
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  | Mul int num 
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  (* A size for num to make inductive proofs simpler*)
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consts num_size :: "num \<Rightarrow> nat" 
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primrec 
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  "num_size (C c) = 1"
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  "num_size (Bound n) = 1"
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  "num_size (Neg a) = 1 + num_size a"
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  "num_size (Add a b) = 1 + num_size a + num_size b"
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  "num_size (Sub a b) = 3 + num_size a + num_size b"
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  "num_size (Mul c a) = 1 + num_size a"
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  "num_size (CN n c a) = 3 + num_size a "
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  (* Semantics of numeral terms (num) *)
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consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
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primrec
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  "Inum bs (C c) = (real c)"
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  "Inum bs (Bound n) = bs!n"
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  "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
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  "Inum bs (Neg a) = -(Inum bs a)"
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  "Inum bs (Add a b) = Inum bs a + Inum bs b"
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  "Inum bs (Sub a b) = Inum bs a - Inum bs b"
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  "Inum bs (Mul c a) = (real c) * Inum bs a"
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    (* FORMULAE *)
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datatype fm  = 
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  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
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  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
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  (* A size for fm *)
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consts fmsize :: "fm \<Rightarrow> nat"
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recdef fmsize "measure size"
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  "fmsize (NOT p) = 1 + fmsize p"
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  "fmsize (And p q) = 1 + fmsize p + fmsize q"
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  "fmsize (Or p q) = 1 + fmsize p + fmsize q"
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  "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
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  "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
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  "fmsize (E p) = 1 + fmsize p"
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  "fmsize (A p) = 4+ fmsize p"
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  "fmsize p = 1"
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  (* several lemmas about fmsize *)
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lemma fmsize_pos: "fmsize p > 0"
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by (induct p rule: fmsize.induct) simp_all
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  (* Semantics of formulae (fm) *)
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consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
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primrec
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  "Ifm bs T = True"
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  "Ifm bs F = False"
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  "Ifm bs (Lt a) = (Inum bs a < 0)"
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  "Ifm bs (Gt a) = (Inum bs a > 0)"
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  "Ifm bs (Le a) = (Inum bs a \<le> 0)"
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  "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
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  "Ifm bs (Eq a) = (Inum bs a = 0)"
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  "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
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  "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
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  "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
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  "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
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  "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
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  "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
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  "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
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  "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
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lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
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apply simp
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done
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lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
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apply simp
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done
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lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
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apply simp
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done
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lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
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apply simp
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done
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lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
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apply simp
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done
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lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
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apply simp
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done
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lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
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apply simp
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done
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lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
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apply simp
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done
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lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
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apply simp
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done
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lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
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apply simp
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done
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consts not:: "fm \<Rightarrow> fm"
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recdef not "measure size"
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  "not (NOT p) = p"
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  "not T = F"
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  "not F = T"
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  "not p = NOT p"
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lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
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by (cases p) auto
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constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
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  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
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   if p = q then p else And p q)"
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lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
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by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
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constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
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  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
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       else if p=q then p else Or p q)"
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lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
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by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
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constdefs  imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
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  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
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    else Imp p q)"
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lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
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by (cases "p=F \<or> q=T",simp_all add: imp_def) 
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constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
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  "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
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       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
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  Iff p q)"
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lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
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  by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
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lemma conj_simps:
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  "conj F Q = F"
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  "conj P F = F"
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  "conj T Q = Q"
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  "conj P T = P"
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  "conj P P = P"
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  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> conj P Q = And P Q"
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  by (simp_all add: conj_def)
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lemma disj_simps:
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  "disj T Q = T"
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  "disj P T = T"
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  "disj F Q = Q"
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  "disj P F = P"
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  "disj P P = P"
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  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> disj P Q = Or P Q"
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  by (simp_all add: disj_def)
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lemma imp_simps:
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  "imp F Q = T"
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  "imp P T = T"
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  "imp T Q = Q"
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  "imp P F = not P"
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  "imp P P = T"
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  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> imp P Q = Imp P Q"
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  by (simp_all add: imp_def)
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lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
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apply (induct p, auto)
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done
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lemma iff_simps:
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  "iff p p = T"
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  "iff p (NOT p) = F"
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  "iff (NOT p) p = F"
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  "iff p F = not p"
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  "iff F p = not p"
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  "p \<noteq> NOT T \<Longrightarrow> iff T p = p"
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  "p\<noteq> NOT T \<Longrightarrow> iff p T = p"
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  "p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q"
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  using trivNOT
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  by (simp_all add: iff_def, cases p, auto)
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  (* Quantifier freeness *)
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consts qfree:: "fm \<Rightarrow> bool"
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recdef qfree "measure size"
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  "qfree (E p) = False"
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  "qfree (A p) = False"
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  "qfree (NOT p) = qfree p" 
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  "qfree (And p q) = (qfree p \<and> qfree q)" 
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  "qfree (Or  p q) = (qfree p \<and> qfree q)" 
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  "qfree (Imp p q) = (qfree p \<and> qfree q)" 
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  "qfree (Iff p q) = (qfree p \<and> qfree q)"
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  "qfree p = True"
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  (* Boundedness and substitution *)
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consts 
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  numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
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  bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
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primrec
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  "numbound0 (C c) = True"
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  "numbound0 (Bound n) = (n>0)"
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  "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
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  "numbound0 (Neg a) = numbound0 a"
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  "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
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  "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
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  "numbound0 (Mul i a) = numbound0 a"
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lemma numbound0_I:
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  assumes nb: "numbound0 a"
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  shows "Inum (b#bs) a = Inum (b'#bs) a"
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using nb
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by (induct a rule: numbound0.induct,auto simp add: nth_pos2)
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primrec
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  "bound0 T = True"
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  "bound0 F = True"
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  "bound0 (Lt a) = numbound0 a"
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  "bound0 (Le a) = numbound0 a"
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  "bound0 (Gt a) = numbound0 a"
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  "bound0 (Ge a) = numbound0 a"
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  "bound0 (Eq a) = numbound0 a"
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  "bound0 (NEq a) = numbound0 a"
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  "bound0 (NOT p) = bound0 p"
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  "bound0 (And p q) = (bound0 p \<and> bound0 q)"
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  "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
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  "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
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  "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
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  "bound0 (E p) = False"
haftmann@29789
   283
  "bound0 (A p) = False"
haftmann@29789
   284
haftmann@29789
   285
lemma bound0_I:
haftmann@29789
   286
  assumes bp: "bound0 p"
haftmann@29789
   287
  shows "Ifm (b#bs) p = Ifm (b'#bs) p"
haftmann@29789
   288
using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
haftmann@29789
   289
by (induct p rule: bound0.induct) (auto simp add: nth_pos2)
haftmann@29789
   290
haftmann@29789
   291
lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
haftmann@29789
   292
by (cases p, auto)
haftmann@29789
   293
lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
haftmann@29789
   294
by (cases p, auto)
haftmann@29789
   295
haftmann@29789
   296
haftmann@29789
   297
lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
haftmann@29789
   298
using conj_def by auto 
haftmann@29789
   299
lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
haftmann@29789
   300
using conj_def by auto 
haftmann@29789
   301
haftmann@29789
   302
lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
haftmann@29789
   303
using disj_def by auto 
haftmann@29789
   304
lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
haftmann@29789
   305
using disj_def by auto 
haftmann@29789
   306
haftmann@29789
   307
lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
haftmann@29789
   308
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
haftmann@29789
   309
lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
haftmann@29789
   310
using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
haftmann@29789
   311
haftmann@29789
   312
lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
haftmann@29789
   313
  by (unfold iff_def,cases "p=q", auto)
haftmann@29789
   314
lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
haftmann@29789
   315
using iff_def by (unfold iff_def,cases "p=q", auto)
haftmann@29789
   316
haftmann@29789
   317
consts 
haftmann@29789
   318
  decrnum:: "num \<Rightarrow> num" 
haftmann@29789
   319
  decr :: "fm \<Rightarrow> fm"
haftmann@29789
   320
haftmann@29789
   321
recdef decrnum "measure size"
haftmann@29789
   322
  "decrnum (Bound n) = Bound (n - 1)"
haftmann@29789
   323
  "decrnum (Neg a) = Neg (decrnum a)"
haftmann@29789
   324
  "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
haftmann@29789
   325
  "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
haftmann@29789
   326
  "decrnum (Mul c a) = Mul c (decrnum a)"
haftmann@29789
   327
  "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
haftmann@29789
   328
  "decrnum a = a"
haftmann@29789
   329
haftmann@29789
   330
recdef decr "measure size"
haftmann@29789
   331
  "decr (Lt a) = Lt (decrnum a)"
haftmann@29789
   332
  "decr (Le a) = Le (decrnum a)"
haftmann@29789
   333
  "decr (Gt a) = Gt (decrnum a)"
haftmann@29789
   334
  "decr (Ge a) = Ge (decrnum a)"
haftmann@29789
   335
  "decr (Eq a) = Eq (decrnum a)"
haftmann@29789
   336
  "decr (NEq a) = NEq (decrnum a)"
haftmann@29789
   337
  "decr (NOT p) = NOT (decr p)" 
haftmann@29789
   338
  "decr (And p q) = conj (decr p) (decr q)"
haftmann@29789
   339
  "decr (Or p q) = disj (decr p) (decr q)"
haftmann@29789
   340
  "decr (Imp p q) = imp (decr p) (decr q)"
haftmann@29789
   341
  "decr (Iff p q) = iff (decr p) (decr q)"
haftmann@29789
   342
  "decr p = p"
haftmann@29789
   343
haftmann@29789
   344
lemma decrnum: assumes nb: "numbound0 t"
haftmann@29789
   345
  shows "Inum (x#bs) t = Inum bs (decrnum t)"
haftmann@29789
   346
  using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)
haftmann@29789
   347
haftmann@29789
   348
lemma decr: assumes nb: "bound0 p"
haftmann@29789
   349
  shows "Ifm (x#bs) p = Ifm bs (decr p)"
haftmann@29789
   350
  using nb 
haftmann@29789
   351
  by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)
haftmann@29789
   352
haftmann@29789
   353
lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
haftmann@29789
   354
by (induct p, simp_all)
haftmann@29789
   355
haftmann@29789
   356
consts 
haftmann@29789
   357
  isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
haftmann@29789
   358
recdef isatom "measure size"
haftmann@29789
   359
  "isatom T = True"
haftmann@29789
   360
  "isatom F = True"
haftmann@29789
   361
  "isatom (Lt a) = True"
haftmann@29789
   362
  "isatom (Le a) = True"
haftmann@29789
   363
  "isatom (Gt a) = True"
haftmann@29789
   364
  "isatom (Ge a) = True"
haftmann@29789
   365
  "isatom (Eq a) = True"
haftmann@29789
   366
  "isatom (NEq a) = True"
haftmann@29789
   367
  "isatom p = False"
haftmann@29789
   368
haftmann@29789
   369
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
haftmann@29789
   370
by (induct p, simp_all)
haftmann@29789
   371
haftmann@29789
   372
constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
haftmann@29789
   373
  "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
haftmann@29789
   374
  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
haftmann@29789
   375
constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
haftmann@29789
   376
  "evaldjf f ps \<equiv> foldr (djf f) ps F"
haftmann@29789
   377
haftmann@29789
   378
lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
haftmann@29789
   379
by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
haftmann@29789
   380
(cases "f p", simp_all add: Let_def djf_def) 
haftmann@29789
   381
haftmann@29789
   382
haftmann@29789
   383
lemma djf_simps:
haftmann@29789
   384
  "djf f p T = T"
haftmann@29789
   385
  "djf f p F = f p"
haftmann@29789
   386
  "q\<noteq>T \<Longrightarrow> q\<noteq>F \<Longrightarrow> djf f p q = (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)"
haftmann@29789
   387
  by (simp_all add: djf_def)
haftmann@29789
   388
haftmann@29789
   389
lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
haftmann@29789
   390
  by(induct ps, simp_all add: evaldjf_def djf_Or)
haftmann@29789
   391
haftmann@29789
   392
lemma evaldjf_bound0: 
haftmann@29789
   393
  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
haftmann@29789
   394
  shows "bound0 (evaldjf f xs)"
haftmann@29789
   395
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
haftmann@29789
   396
haftmann@29789
   397
lemma evaldjf_qf: 
haftmann@29789
   398
  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
haftmann@29789
   399
  shows "qfree (evaldjf f xs)"
haftmann@29789
   400
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
haftmann@29789
   401
haftmann@29789
   402
consts disjuncts :: "fm \<Rightarrow> fm list"
haftmann@29789
   403
recdef disjuncts "measure size"
haftmann@29789
   404
  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
haftmann@29789
   405
  "disjuncts F = []"
haftmann@29789
   406
  "disjuncts p = [p]"
haftmann@29789
   407
haftmann@29789
   408
lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
haftmann@29789
   409
by(induct p rule: disjuncts.induct, auto)
haftmann@29789
   410
haftmann@29789
   411
lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
haftmann@29789
   412
proof-
haftmann@29789
   413
  assume nb: "bound0 p"
haftmann@29789
   414
  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
haftmann@29789
   415
  thus ?thesis by (simp only: list_all_iff)
haftmann@29789
   416
qed
haftmann@29789
   417
haftmann@29789
   418
lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
haftmann@29789
   419
proof-
haftmann@29789
   420
  assume qf: "qfree p"
haftmann@29789
   421
  hence "list_all qfree (disjuncts p)"
haftmann@29789
   422
    by (induct p rule: disjuncts.induct, auto)
haftmann@29789
   423
  thus ?thesis by (simp only: list_all_iff)
haftmann@29789
   424
qed
haftmann@29789
   425
haftmann@29789
   426
constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
haftmann@29789
   427
  "DJ f p \<equiv> evaldjf f (disjuncts p)"
haftmann@29789
   428
haftmann@29789
   429
lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
haftmann@29789
   430
  and fF: "f F = F"
haftmann@29789
   431
  shows "Ifm bs (DJ f p) = Ifm bs (f p)"
haftmann@29789
   432
proof-
haftmann@29789
   433
  have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
haftmann@29789
   434
    by (simp add: DJ_def evaldjf_ex) 
haftmann@29789
   435
  also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
haftmann@29789
   436
  finally show ?thesis .
haftmann@29789
   437
qed
haftmann@29789
   438
haftmann@29789
   439
lemma DJ_qf: assumes 
haftmann@29789
   440
  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
haftmann@29789
   441
  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
haftmann@29789
   442
proof(clarify)
haftmann@29789
   443
  fix  p assume qf: "qfree p"
haftmann@29789
   444
  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
haftmann@29789
   445
  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
haftmann@29789
   446
  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
haftmann@29789
   447
  
haftmann@29789
   448
  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
haftmann@29789
   449
qed
haftmann@29789
   450
haftmann@29789
   451
lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
haftmann@29789
   452
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
haftmann@29789
   453
proof(clarify)
haftmann@29789
   454
  fix p::fm and bs
haftmann@29789
   455
  assume qf: "qfree p"
haftmann@29789
   456
  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
haftmann@29789
   457
  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
haftmann@29789
   458
  have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
haftmann@29789
   459
    by (simp add: DJ_def evaldjf_ex)
haftmann@29789
   460
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
haftmann@29789
   461
  also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
haftmann@29789
   462
  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
haftmann@29789
   463
qed
haftmann@29789
   464
  (* Simplification *)
haftmann@29789
   465
consts 
haftmann@29789
   466
  numgcd :: "num \<Rightarrow> int"
haftmann@29789
   467
  numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
haftmann@29789
   468
  reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
haftmann@29789
   469
  reducecoeff :: "num \<Rightarrow> num"
haftmann@29789
   470
  dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
haftmann@29789
   471
consts maxcoeff:: "num \<Rightarrow> int"
haftmann@29789
   472
recdef maxcoeff "measure size"
haftmann@29789
   473
  "maxcoeff (C i) = abs i"
haftmann@29789
   474
  "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
haftmann@29789
   475
  "maxcoeff t = 1"
haftmann@29789
   476
haftmann@29789
   477
lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
haftmann@29789
   478
  by (induct t rule: maxcoeff.induct, auto)
haftmann@29789
   479
haftmann@29789
   480
recdef numgcdh "measure size"
haftmann@29789
   481
  "numgcdh (C i) = (\<lambda>g. zgcd i g)"
haftmann@29789
   482
  "numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))"
haftmann@29789
   483
  "numgcdh t = (\<lambda>g. 1)"
haftmann@29789
   484
defs numgcd_def [code]: "numgcd t \<equiv> numgcdh t (maxcoeff t)"
haftmann@29789
   485
haftmann@29789
   486
recdef reducecoeffh "measure size"
haftmann@29789
   487
  "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
haftmann@29789
   488
  "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
haftmann@29789
   489
  "reducecoeffh t = (\<lambda>g. t)"
haftmann@29789
   490
haftmann@29789
   491
defs reducecoeff_def: "reducecoeff t \<equiv> 
haftmann@29789
   492
  (let g = numgcd t in 
haftmann@29789
   493
  if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
haftmann@29789
   494
haftmann@29789
   495
recdef dvdnumcoeff "measure size"
haftmann@29789
   496
  "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
haftmann@29789
   497
  "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
haftmann@29789
   498
  "dvdnumcoeff t = (\<lambda>g. False)"
haftmann@29789
   499
haftmann@29789
   500
lemma dvdnumcoeff_trans: 
haftmann@29789
   501
  assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
haftmann@29789
   502
  shows "dvdnumcoeff t g"
haftmann@29789
   503
  using dgt' gdg 
nipkow@30042
   504
  by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg dvd_trans[OF gdg])
haftmann@29789
   505
nipkow@30042
   506
declare dvd_trans [trans add]
haftmann@29789
   507
haftmann@29789
   508
lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
haftmann@29789
   509
by arith
haftmann@29789
   510
haftmann@29789
   511
lemma numgcd0:
haftmann@29789
   512
  assumes g0: "numgcd t = 0"
haftmann@29789
   513
  shows "Inum bs t = 0"
haftmann@29789
   514
  using g0[simplified numgcd_def] 
haftmann@29789
   515
  by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)
haftmann@29789
   516
haftmann@29789
   517
lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
haftmann@29789
   518
  using gp
haftmann@29789
   519
  by (induct t rule: numgcdh.induct, auto simp add: zgcd_def)
haftmann@29789
   520
haftmann@29789
   521
lemma numgcd_pos: "numgcd t \<ge>0"
haftmann@29789
   522
  by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
haftmann@29789
   523
haftmann@29789
   524
lemma reducecoeffh:
haftmann@29789
   525
  assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
haftmann@29789
   526
  shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
haftmann@29789
   527
  using gt
haftmann@29789
   528
proof(induct t rule: reducecoeffh.induct) 
haftmann@29789
   529
  case (1 i) hence gd: "g dvd i" by simp
haftmann@29789
   530
  from gp have gnz: "g \<noteq> 0" by simp
haftmann@29789
   531
  from prems show ?case by (simp add: real_of_int_div[OF gnz gd])
haftmann@29789
   532
next
haftmann@29789
   533
  case (2 n c t)  hence gd: "g dvd c" by simp
haftmann@29789
   534
  from gp have gnz: "g \<noteq> 0" by simp
haftmann@29789
   535
  from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
haftmann@29789
   536
qed (auto simp add: numgcd_def gp)
haftmann@29789
   537
consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
haftmann@29789
   538
recdef ismaxcoeff "measure size"
haftmann@29789
   539
  "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
haftmann@29789
   540
  "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
haftmann@29789
   541
  "ismaxcoeff t = (\<lambda>x. True)"
haftmann@29789
   542
haftmann@29789
   543
lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
haftmann@29789
   544
by (induct t rule: ismaxcoeff.induct, auto)
haftmann@29789
   545
haftmann@29789
   546
lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
haftmann@29789
   547
proof (induct t rule: maxcoeff.induct)
haftmann@29789
   548
  case (2 n c t)
haftmann@29789
   549
  hence H:"ismaxcoeff t (maxcoeff t)" .
haftmann@29789
   550
  have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2)
haftmann@29789
   551
  from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
haftmann@29789
   552
qed simp_all
haftmann@29789
   553
haftmann@29789
   554
lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
haftmann@29789
   555
  apply (cases "abs i = 0", simp_all add: zgcd_def)
haftmann@29789
   556
  apply (cases "abs j = 0", simp_all)
haftmann@29789
   557
  apply (cases "abs i = 1", simp_all)
haftmann@29789
   558
  apply (cases "abs j = 1", simp_all)
haftmann@29789
   559
  apply auto
haftmann@29789
   560
  done
haftmann@29789
   561
lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
haftmann@29789
   562
  by (induct t rule: numgcdh.induct, auto simp add:zgcd0)
haftmann@29789
   563
haftmann@29789
   564
lemma dvdnumcoeff_aux:
haftmann@29789
   565
  assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
haftmann@29789
   566
  shows "dvdnumcoeff t (numgcdh t m)"
haftmann@29789
   567
using prems
haftmann@29789
   568
proof(induct t rule: numgcdh.induct)
haftmann@29789
   569
  case (2 n c t) 
haftmann@29789
   570
  let ?g = "numgcdh t m"
haftmann@29789
   571
  from prems have th:"zgcd c ?g > 1" by simp
haftmann@29789
   572
  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
haftmann@29789
   573
  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
haftmann@29789
   574
  moreover {assume "abs c > 1" and gp: "?g > 1" with prems
haftmann@29789
   575
    have th: "dvdnumcoeff t ?g" by simp
haftmann@29789
   576
    have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
haftmann@29789
   577
    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
haftmann@29789
   578
  moreover {assume "abs c = 0 \<and> ?g > 1"
haftmann@29789
   579
    with prems have th: "dvdnumcoeff t ?g" by simp
haftmann@29789
   580
    have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
haftmann@29789
   581
    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
haftmann@29789
   582
    hence ?case by simp }
haftmann@29789
   583
  moreover {assume "abs c > 1" and g0:"?g = 0" 
haftmann@29789
   584
    from numgcdh0[OF g0] have "m=0". with prems   have ?case by simp }
haftmann@29789
   585
  ultimately show ?case by blast
haftmann@29789
   586
qed(auto simp add: zgcd_zdvd1)
haftmann@29789
   587
haftmann@29789
   588
lemma dvdnumcoeff_aux2:
haftmann@29789
   589
  assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
haftmann@29789
   590
  using prems 
haftmann@29789
   591
proof (simp add: numgcd_def)
haftmann@29789
   592
  let ?mc = "maxcoeff t"
haftmann@29789
   593
  let ?g = "numgcdh t ?mc"
haftmann@29789
   594
  have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
haftmann@29789
   595
  have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
haftmann@29789
   596
  assume H: "numgcdh t ?mc > 1"
haftmann@29789
   597
  from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
haftmann@29789
   598
qed
haftmann@29789
   599
haftmann@29789
   600
lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
haftmann@29789
   601
proof-
haftmann@29789
   602
  let ?g = "numgcd t"
haftmann@29789
   603
  have "?g \<ge> 0"  by (simp add: numgcd_pos)
haftmann@29789
   604
  hence	"?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
haftmann@29789
   605
  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} 
haftmann@29789
   606
  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} 
haftmann@29789
   607
  moreover { assume g1:"?g > 1"
haftmann@29789
   608
    from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
haftmann@29789
   609
    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis 
haftmann@29789
   610
      by (simp add: reducecoeff_def Let_def)} 
haftmann@29789
   611
  ultimately show ?thesis by blast
haftmann@29789
   612
qed
haftmann@29789
   613
haftmann@29789
   614
lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
haftmann@29789
   615
by (induct t rule: reducecoeffh.induct, auto)
haftmann@29789
   616
haftmann@29789
   617
lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
haftmann@29789
   618
using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
haftmann@29789
   619
haftmann@29789
   620
consts
haftmann@29789
   621
  simpnum:: "num \<Rightarrow> num"
haftmann@29789
   622
  numadd:: "num \<times> num \<Rightarrow> num"
haftmann@29789
   623
  nummul:: "num \<Rightarrow> int \<Rightarrow> num"
haftmann@29789
   624
recdef numadd "measure (\<lambda> (t,s). size t + size s)"
haftmann@29789
   625
  "numadd (CN n1 c1 r1,CN n2 c2 r2) =
haftmann@29789
   626
  (if n1=n2 then 
haftmann@29789
   627
  (let c = c1 + c2
haftmann@29789
   628
  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
haftmann@29789
   629
  else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) 
haftmann@29789
   630
  else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
haftmann@29789
   631
  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
haftmann@29789
   632
  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
haftmann@29789
   633
  "numadd (C b1, C b2) = C (b1+b2)"
haftmann@29789
   634
  "numadd (a,b) = Add a b"
haftmann@29789
   635
haftmann@29789
   636
lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
haftmann@29789
   637
apply (induct t s rule: numadd.induct, simp_all add: Let_def)
haftmann@29789
   638
apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
haftmann@29789
   639
apply (case_tac "n1 = n2", simp_all add: algebra_simps)
haftmann@29789
   640
by (simp only: left_distrib[symmetric],simp)
haftmann@29789
   641
haftmann@29789
   642
lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
haftmann@29789
   643
by (induct t s rule: numadd.induct, auto simp add: Let_def)
haftmann@29789
   644
haftmann@29789
   645
recdef nummul "measure size"
haftmann@29789
   646
  "nummul (C j) = (\<lambda> i. C (i*j))"
haftmann@29789
   647
  "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
haftmann@29789
   648
  "nummul t = (\<lambda> i. Mul i t)"
haftmann@29789
   649
haftmann@29789
   650
lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
haftmann@29789
   651
by (induct t rule: nummul.induct, auto simp add: algebra_simps)
haftmann@29789
   652
haftmann@29789
   653
lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
haftmann@29789
   654
by (induct t rule: nummul.induct, auto )
haftmann@29789
   655
haftmann@29789
   656
constdefs numneg :: "num \<Rightarrow> num"
haftmann@29789
   657
  "numneg t \<equiv> nummul t (- 1)"
haftmann@29789
   658
haftmann@29789
   659
constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
haftmann@29789
   660
  "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
haftmann@29789
   661
haftmann@29789
   662
lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
haftmann@29789
   663
using numneg_def by simp
haftmann@29789
   664
haftmann@29789
   665
lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
haftmann@29789
   666
using numneg_def by simp
haftmann@29789
   667
haftmann@29789
   668
lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
haftmann@29789
   669
using numsub_def by simp
haftmann@29789
   670
haftmann@29789
   671
lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
haftmann@29789
   672
using numsub_def by simp
haftmann@29789
   673
haftmann@29789
   674
recdef simpnum "measure size"
haftmann@29789
   675
  "simpnum (C j) = C j"
haftmann@29789
   676
  "simpnum (Bound n) = CN n 1 (C 0)"
haftmann@29789
   677
  "simpnum (Neg t) = numneg (simpnum t)"
haftmann@29789
   678
  "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
haftmann@29789
   679
  "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
haftmann@29789
   680
  "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
haftmann@29789
   681
  "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
haftmann@29789
   682
haftmann@29789
   683
lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
haftmann@29789
   684
by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)
haftmann@29789
   685
haftmann@29789
   686
lemma simpnum_numbound0[simp]: 
haftmann@29789
   687
  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
haftmann@29789
   688
by (induct t rule: simpnum.induct, auto)
haftmann@29789
   689
haftmann@29789
   690
consts nozerocoeff:: "num \<Rightarrow> bool"
haftmann@29789
   691
recdef nozerocoeff "measure size"
haftmann@29789
   692
  "nozerocoeff (C c) = True"
haftmann@29789
   693
  "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
haftmann@29789
   694
  "nozerocoeff t = True"
haftmann@29789
   695
haftmann@29789
   696
lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
haftmann@29789
   697
by (induct a b rule: numadd.induct,auto simp add: Let_def)
haftmann@29789
   698
haftmann@29789
   699
lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
haftmann@29789
   700
by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
haftmann@29789
   701
haftmann@29789
   702
lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
haftmann@29789
   703
by (simp add: numneg_def nummul_nz)
haftmann@29789
   704
haftmann@29789
   705
lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
haftmann@29789
   706
by (simp add: numsub_def numneg_nz numadd_nz)
haftmann@29789
   707
haftmann@29789
   708
lemma simpnum_nz: "nozerocoeff (simpnum t)"
haftmann@29789
   709
by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz)
haftmann@29789
   710
haftmann@29789
   711
lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
haftmann@29789
   712
proof (induct t rule: maxcoeff.induct)
haftmann@29789
   713
  case (2 n c t)
haftmann@29789
   714
  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
haftmann@29789
   715
  have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
haftmann@29789
   716
  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
haftmann@29789
   717
  with prems show ?case by simp
haftmann@29789
   718
qed auto
haftmann@29789
   719
haftmann@29789
   720
lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
haftmann@29789
   721
proof-
haftmann@29789
   722
  from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
haftmann@29789
   723
  from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
haftmann@29789
   724
  from maxcoeff_nz[OF nz th] show ?thesis .
haftmann@29789
   725
qed
haftmann@29789
   726
haftmann@29789
   727
constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
haftmann@29789
   728
  "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
haftmann@29789
   729
   (let t' = simpnum t ; g = numgcd t' in 
haftmann@29789
   730
      if g > 1 then (let g' = zgcd n g in 
haftmann@29789
   731
        if g' = 1 then (t',n) 
haftmann@29789
   732
        else (reducecoeffh t' g', n div g')) 
haftmann@29789
   733
      else (t',n))))"
haftmann@29789
   734
haftmann@29789
   735
lemma simp_num_pair_ci:
haftmann@29789
   736
  shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
haftmann@29789
   737
  (is "?lhs = ?rhs")
haftmann@29789
   738
proof-
haftmann@29789
   739
  let ?t' = "simpnum t"
haftmann@29789
   740
  let ?g = "numgcd ?t'"
haftmann@29789
   741
  let ?g' = "zgcd n ?g"
haftmann@29789
   742
  {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
haftmann@29789
   743
  moreover
haftmann@29789
   744
  { assume nnz: "n \<noteq> 0"
haftmann@29789
   745
    {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
haftmann@29789
   746
    moreover
haftmann@29789
   747
    {assume g1:"?g>1" hence g0: "?g > 0" by simp
haftmann@29789
   748
      from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
haftmann@29789
   749
      hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith 
haftmann@29789
   750
      hence "?g'= 1 \<or> ?g' > 1" by arith
haftmann@29789
   751
      moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
haftmann@29789
   752
      moreover {assume g'1:"?g'>1"
haftmann@29789
   753
	from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
haftmann@29789
   754
	let ?tt = "reducecoeffh ?t' ?g'"
haftmann@29789
   755
	let ?t = "Inum bs ?tt"
haftmann@29789
   756
	have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
haftmann@29789
   757
	have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) 
haftmann@29789
   758
	have gpdgp: "?g' dvd ?g'" by simp
haftmann@29789
   759
	from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
haftmann@29789
   760
	have th2:"real ?g' * ?t = Inum bs ?t'" by simp
haftmann@29789
   761
	from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
haftmann@29789
   762
	also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
haftmann@29789
   763
	also have "\<dots> = (Inum bs ?t' / real n)"
haftmann@29789
   764
	  using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
haftmann@29789
   765
	finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci)
haftmann@29789
   766
	then have ?thesis using prems by (simp add: simp_num_pair_def)}
haftmann@29789
   767
      ultimately have ?thesis by blast}
haftmann@29789
   768
    ultimately have ?thesis by blast} 
haftmann@29789
   769
  ultimately show ?thesis by blast
haftmann@29789
   770
qed
haftmann@29789
   771
haftmann@29789
   772
lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
haftmann@29789
   773
  shows "numbound0 t' \<and> n' >0"
haftmann@29789
   774
proof-
haftmann@29789
   775
    let ?t' = "simpnum t"
haftmann@29789
   776
  let ?g = "numgcd ?t'"
haftmann@29789
   777
  let ?g' = "zgcd n ?g"
haftmann@29789
   778
  {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
haftmann@29789
   779
  moreover
haftmann@29789
   780
  { assume nnz: "n \<noteq> 0"
haftmann@29789
   781
    {assume "\<not> ?g > 1" hence ?thesis  using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
haftmann@29789
   782
    moreover
haftmann@29789
   783
    {assume g1:"?g>1" hence g0: "?g > 0" by simp
haftmann@29789
   784
      from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
haftmann@29789
   785
      hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
haftmann@29789
   786
      hence "?g'= 1 \<or> ?g' > 1" by arith
haftmann@29789
   787
      moreover {assume "?g'=1" hence ?thesis using prems 
haftmann@29789
   788
	  by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
haftmann@29789
   789
      moreover {assume g'1:"?g'>1"
haftmann@29789
   790
	have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
haftmann@29789
   791
	have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) 
haftmann@29789
   792
	have gpdgp: "?g' dvd ?g'" by simp
haftmann@29789
   793
	from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
haftmann@29789
   794
	from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
haftmann@29789
   795
	have "n div ?g' >0" by simp
haftmann@29789
   796
	hence ?thesis using prems 
haftmann@29789
   797
	  by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)}
haftmann@29789
   798
      ultimately have ?thesis by blast}
haftmann@29789
   799
    ultimately have ?thesis by blast} 
haftmann@29789
   800
  ultimately show ?thesis by blast
haftmann@29789
   801
qed
haftmann@29789
   802
haftmann@29789
   803
consts simpfm :: "fm \<Rightarrow> fm"
haftmann@29789
   804
recdef simpfm "measure fmsize"
haftmann@29789
   805
  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
haftmann@29789
   806
  "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
haftmann@29789
   807
  "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
haftmann@29789
   808
  "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
haftmann@29789
   809
  "simpfm (NOT p) = not (simpfm p)"
haftmann@29789
   810
  "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
haftmann@29789
   811
  | _ \<Rightarrow> Lt a')"
haftmann@29789
   812
  "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le a')"
haftmann@29789
   813
  "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
haftmann@29789
   814
  "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge a')"
haftmann@29789
   815
  "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
haftmann@29789
   816
  "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq a')"
haftmann@29789
   817
  "simpfm p = p"
haftmann@29789
   818
lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p"
haftmann@29789
   819
proof(induct p rule: simpfm.induct)
haftmann@29789
   820
  case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   821
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   822
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   823
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   824
  ultimately show ?case by blast
haftmann@29789
   825
next
haftmann@29789
   826
  case (7 a)  let ?sa = "simpnum a" 
haftmann@29789
   827
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   828
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   829
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   830
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   831
  ultimately show ?case by blast
haftmann@29789
   832
next
haftmann@29789
   833
  case (8 a)  let ?sa = "simpnum a" 
haftmann@29789
   834
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   835
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   836
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   837
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   838
  ultimately show ?case by blast
haftmann@29789
   839
next
haftmann@29789
   840
  case (9 a)  let ?sa = "simpnum a" 
haftmann@29789
   841
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   842
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   843
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   844
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   845
  ultimately show ?case by blast
haftmann@29789
   846
next
haftmann@29789
   847
  case (10 a)  let ?sa = "simpnum a" 
haftmann@29789
   848
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   849
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   850
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   851
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   852
  ultimately show ?case by blast
haftmann@29789
   853
next
haftmann@29789
   854
  case (11 a)  let ?sa = "simpnum a" 
haftmann@29789
   855
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   856
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   857
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   858
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   859
  ultimately show ?case by blast
haftmann@29789
   860
qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
haftmann@29789
   861
haftmann@29789
   862
haftmann@29789
   863
lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
haftmann@29789
   864
proof(induct p rule: simpfm.induct)
haftmann@29789
   865
  case (6 a) hence nb: "numbound0 a" by simp
haftmann@29789
   866
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
haftmann@29789
   867
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
haftmann@29789
   868
next
haftmann@29789
   869
  case (7 a) hence nb: "numbound0 a" by simp
haftmann@29789
   870
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
haftmann@29789
   871
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
haftmann@29789
   872
next
haftmann@29789
   873
  case (8 a) hence nb: "numbound0 a" by simp
haftmann@29789
   874
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
haftmann@29789
   875
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
haftmann@29789
   876
next
haftmann@29789
   877
  case (9 a) hence nb: "numbound0 a" by simp
haftmann@29789
   878
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
haftmann@29789
   879
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
haftmann@29789
   880
next
haftmann@29789
   881
  case (10 a) hence nb: "numbound0 a" by simp
haftmann@29789
   882
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
haftmann@29789
   883
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
haftmann@29789
   884
next
haftmann@29789
   885
  case (11 a) hence nb: "numbound0 a" by simp
haftmann@29789
   886
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
haftmann@29789
   887
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
haftmann@29789
   888
qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
haftmann@29789
   889
haftmann@29789
   890
lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
haftmann@29789
   891
by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
haftmann@29789
   892
 (case_tac "simpnum a",auto)+
haftmann@29789
   893
haftmann@29789
   894
consts prep :: "fm \<Rightarrow> fm"
haftmann@29789
   895
recdef prep "measure fmsize"
haftmann@29789
   896
  "prep (E T) = T"
haftmann@29789
   897
  "prep (E F) = F"
haftmann@29789
   898
  "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
haftmann@29789
   899
  "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
haftmann@29789
   900
  "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
haftmann@29789
   901
  "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
haftmann@29789
   902
  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
haftmann@29789
   903
  "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
haftmann@29789
   904
  "prep (E p) = E (prep p)"
haftmann@29789
   905
  "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
haftmann@29789
   906
  "prep (A p) = prep (NOT (E (NOT p)))"
haftmann@29789
   907
  "prep (NOT (NOT p)) = prep p"
haftmann@29789
   908
  "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
haftmann@29789
   909
  "prep (NOT (A p)) = prep (E (NOT p))"
haftmann@29789
   910
  "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
haftmann@29789
   911
  "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
haftmann@29789
   912
  "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
haftmann@29789
   913
  "prep (NOT p) = not (prep p)"
haftmann@29789
   914
  "prep (Or p q) = disj (prep p) (prep q)"
haftmann@29789
   915
  "prep (And p q) = conj (prep p) (prep q)"
haftmann@29789
   916
  "prep (Imp p q) = prep (Or (NOT p) q)"
haftmann@29789
   917
  "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
haftmann@29789
   918
  "prep p = p"
haftmann@29789
   919
(hints simp add: fmsize_pos)
haftmann@29789
   920
lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
haftmann@29789
   921
by (induct p rule: prep.induct, auto)
haftmann@29789
   922
haftmann@29789
   923
  (* Generic quantifier elimination *)
haftmann@29789
   924
consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
haftmann@29789
   925
recdef qelim "measure fmsize"
haftmann@29789
   926
  "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
haftmann@29789
   927
  "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
haftmann@29789
   928
  "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
haftmann@29789
   929
  "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
haftmann@29789
   930
  "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
haftmann@29789
   931
  "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
haftmann@29789
   932
  "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
haftmann@29789
   933
  "qelim p = (\<lambda> y. simpfm p)"
haftmann@29789
   934
haftmann@29789
   935
lemma qelim_ci:
haftmann@29789
   936
  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
haftmann@29789
   937
  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
haftmann@29789
   938
using qe_inv DJ_qe[OF qe_inv] 
haftmann@29789
   939
by(induct p rule: qelim.induct) 
haftmann@29789
   940
(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
haftmann@29789
   941
  simpfm simpfm_qf simp del: simpfm.simps)
haftmann@29789
   942
haftmann@29789
   943
consts 
haftmann@29789
   944
  plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
haftmann@29789
   945
  minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
haftmann@29789
   946
recdef minusinf "measure size"
haftmann@29789
   947
  "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
haftmann@29789
   948
  "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
haftmann@29789
   949
  "minusinf (Eq  (CN 0 c e)) = F"
haftmann@29789
   950
  "minusinf (NEq (CN 0 c e)) = T"
haftmann@29789
   951
  "minusinf (Lt  (CN 0 c e)) = T"
haftmann@29789
   952
  "minusinf (Le  (CN 0 c e)) = T"
haftmann@29789
   953
  "minusinf (Gt  (CN 0 c e)) = F"
haftmann@29789
   954
  "minusinf (Ge  (CN 0 c e)) = F"
haftmann@29789
   955
  "minusinf p = p"
haftmann@29789
   956
haftmann@29789
   957
recdef plusinf "measure size"
haftmann@29789
   958
  "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
haftmann@29789
   959
  "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
haftmann@29789
   960
  "plusinf (Eq  (CN 0 c e)) = F"
haftmann@29789
   961
  "plusinf (NEq (CN 0 c e)) = T"
haftmann@29789
   962
  "plusinf (Lt  (CN 0 c e)) = F"
haftmann@29789
   963
  "plusinf (Le  (CN 0 c e)) = F"
haftmann@29789
   964
  "plusinf (Gt  (CN 0 c e)) = T"
haftmann@29789
   965
  "plusinf (Ge  (CN 0 c e)) = T"
haftmann@29789
   966
  "plusinf p = p"
haftmann@29789
   967
haftmann@29789
   968
consts
haftmann@29789
   969
  isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
haftmann@29789
   970
recdef isrlfm "measure size"
haftmann@29789
   971
  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" 
haftmann@29789
   972
  "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" 
haftmann@29789
   973
  "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@29789
   974
  "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@29789
   975
  "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@29789
   976
  "isrlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@29789
   977
  "isrlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@29789
   978
  "isrlfm (Ge  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@29789
   979
  "isrlfm p = (isatom p \<and> (bound0 p))"
haftmann@29789
   980
haftmann@29789
   981
  (* splits the bounded from the unbounded part*)
haftmann@29789
   982
consts rsplit0 :: "num \<Rightarrow> int \<times> num" 
haftmann@29789
   983
recdef rsplit0 "measure num_size"
haftmann@29789
   984
  "rsplit0 (Bound 0) = (1,C 0)"
haftmann@29789
   985
  "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b 
haftmann@29789
   986
              in (ca+cb, Add ta tb))"
haftmann@29789
   987
  "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
haftmann@29789
   988
  "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
haftmann@29789
   989
  "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
haftmann@29789
   990
  "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
haftmann@29789
   991
  "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
haftmann@29789
   992
  "rsplit0 t = (0,t)"
haftmann@29789
   993
lemma rsplit0: 
haftmann@29789
   994
  shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
haftmann@29789
   995
proof (induct t rule: rsplit0.induct)
haftmann@29789
   996
  case (2 a b) 
haftmann@29789
   997
  let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
haftmann@29789
   998
  let ?ca = "fst ?sa" let ?cb = "fst ?sb"
haftmann@29789
   999
  let ?ta = "snd ?sa" let ?tb = "snd ?sb"
haftmann@29789
  1000
  from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))" 
haftmann@29789
  1001
    by(cases "rsplit0 a",auto simp add: Let_def split_def)
haftmann@29789
  1002
  have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) = 
haftmann@29789
  1003
    Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
haftmann@29789
  1004
    by (simp add: Let_def split_def algebra_simps)
haftmann@29789
  1005
  also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all)
haftmann@29789
  1006
  finally show ?case using nb by simp 
haftmann@29789
  1007
qed(auto simp add: Let_def split_def algebra_simps , simp add: right_distrib[symmetric])
haftmann@29789
  1008
haftmann@29789
  1009
    (* Linearize a formula*)
haftmann@29789
  1010
definition
haftmann@29789
  1011
  lt :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
  1012
where
haftmann@29789
  1013
  "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) 
haftmann@29789
  1014
    else (Gt (CN 0 (-c) (Neg t))))"
haftmann@29789
  1015
haftmann@29789
  1016
definition
haftmann@29789
  1017
  le :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
  1018
where
haftmann@29789
  1019
  "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) 
haftmann@29789
  1020
    else (Ge (CN 0 (-c) (Neg t))))"
haftmann@29789
  1021
haftmann@29789
  1022
definition
haftmann@29789
  1023
  gt :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
  1024
where
haftmann@29789
  1025
  "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) 
haftmann@29789
  1026
    else (Lt (CN 0 (-c) (Neg t))))"
haftmann@29789
  1027
haftmann@29789
  1028
definition
haftmann@29789
  1029
  ge :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
  1030
where
haftmann@29789
  1031
  "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) 
haftmann@29789
  1032
    else (Le (CN 0 (-c) (Neg t))))"
haftmann@29789
  1033
haftmann@29789
  1034
definition
haftmann@29789
  1035
  eq :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
  1036
where
haftmann@29789
  1037
  "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) 
haftmann@29789
  1038
    else (Eq (CN 0 (-c) (Neg t))))"
haftmann@29789
  1039
haftmann@29789
  1040
definition
haftmann@29789
  1041
  neq :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
  1042
where
haftmann@29789
  1043
  "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) 
haftmann@29789
  1044
    else (NEq (CN 0 (-c) (Neg t))))"
haftmann@29789
  1045
haftmann@29789
  1046
lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
haftmann@29789
  1047
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
  1048
by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
  1049
haftmann@29789
  1050
lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
haftmann@29789
  1051
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
  1052
by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
  1053
haftmann@29789
  1054
lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
haftmann@29789
  1055
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
  1056
by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
  1057
haftmann@29789
  1058
lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
haftmann@29789
  1059
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
  1060
by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
  1061
haftmann@29789
  1062
lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
haftmann@29789
  1063
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
  1064
by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
  1065
haftmann@29789
  1066
lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
haftmann@29789
  1067
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
  1068
by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
  1069
haftmann@29789
  1070
lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
haftmann@29789
  1071
by (auto simp add: conj_def)
haftmann@29789
  1072
lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
haftmann@29789
  1073
by (auto simp add: disj_def)
haftmann@29789
  1074
haftmann@29789
  1075
consts rlfm :: "fm \<Rightarrow> fm"
haftmann@29789
  1076
recdef rlfm "measure fmsize"
haftmann@29789
  1077
  "rlfm (And p q) = conj (rlfm p) (rlfm q)"
haftmann@29789
  1078
  "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
haftmann@29789
  1079
  "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
haftmann@29789
  1080
  "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
haftmann@29789
  1081
  "rlfm (Lt a) = split lt (rsplit0 a)"
haftmann@29789
  1082
  "rlfm (Le a) = split le (rsplit0 a)"
haftmann@29789
  1083
  "rlfm (Gt a) = split gt (rsplit0 a)"
haftmann@29789
  1084
  "rlfm (Ge a) = split ge (rsplit0 a)"
haftmann@29789
  1085
  "rlfm (Eq a) = split eq (rsplit0 a)"
haftmann@29789
  1086
  "rlfm (NEq a) = split neq (rsplit0 a)"
haftmann@29789
  1087
  "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
haftmann@29789
  1088
  "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
haftmann@29789
  1089
  "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
haftmann@29789
  1090
  "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
haftmann@29789
  1091
  "rlfm (NOT (NOT p)) = rlfm p"
haftmann@29789
  1092
  "rlfm (NOT T) = F"
haftmann@29789
  1093
  "rlfm (NOT F) = T"
haftmann@29789
  1094
  "rlfm (NOT (Lt a)) = rlfm (Ge a)"
haftmann@29789
  1095
  "rlfm (NOT (Le a)) = rlfm (Gt a)"
haftmann@29789
  1096
  "rlfm (NOT (Gt a)) = rlfm (Le a)"
haftmann@29789
  1097
  "rlfm (NOT (Ge a)) = rlfm (Lt a)"
haftmann@29789
  1098
  "rlfm (NOT (Eq a)) = rlfm (NEq a)"
haftmann@29789
  1099
  "rlfm (NOT (NEq a)) = rlfm (Eq a)"
haftmann@29789
  1100
  "rlfm p = p" (hints simp add: fmsize_pos)
haftmann@29789
  1101
haftmann@29789
  1102
lemma rlfm_I:
haftmann@29789
  1103
  assumes qfp: "qfree p"
haftmann@29789
  1104
  shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
haftmann@29789
  1105
  using qfp 
haftmann@29789
  1106
by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
haftmann@29789
  1107
haftmann@29789
  1108
    (* Operations needed for Ferrante and Rackoff *)
haftmann@29789
  1109
lemma rminusinf_inf:
haftmann@29789
  1110
  assumes lp: "isrlfm p"
haftmann@29789
  1111
  shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
haftmann@29789
  1112
using lp
haftmann@29789
  1113
proof (induct p rule: minusinf.induct)
haftmann@29789
  1114
  case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto 
haftmann@29789
  1115
next
haftmann@29789
  1116
  case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
haftmann@29789
  1117
next
haftmann@29789
  1118
  case (3 c e) 
haftmann@29789
  1119
  from prems have nb: "numbound0 e" by simp
haftmann@29789
  1120
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1121
  fix a
haftmann@29789
  1122
  let ?e="Inum (a#bs) e"
haftmann@29789
  1123
  let ?z = "(- ?e) / real c"
haftmann@29789
  1124
  {fix x
haftmann@29789
  1125
    assume xz: "x < ?z"
haftmann@29789
  1126
    hence "(real c * x < - ?e)" 
haftmann@29789
  1127
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1128
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1129
    hence "real c * x + ?e \<noteq> 0" by simp
haftmann@29789
  1130
    with xz have "?P ?z x (Eq (CN 0 c e))"
haftmann@29789
  1131
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
haftmann@29789
  1132
  hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
haftmann@29789
  1133
  thus ?case by blast
haftmann@29789
  1134
next
haftmann@29789
  1135
  case (4 c e)   
haftmann@29789
  1136
  from prems have nb: "numbound0 e" by simp
haftmann@29789
  1137
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1138
  fix a
haftmann@29789
  1139
  let ?e="Inum (a#bs) e"
haftmann@29789
  1140
  let ?z = "(- ?e) / real c"
haftmann@29789
  1141
  {fix x
haftmann@29789
  1142
    assume xz: "x < ?z"
haftmann@29789
  1143
    hence "(real c * x < - ?e)" 
haftmann@29789
  1144
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1145
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1146
    hence "real c * x + ?e \<noteq> 0" by simp
haftmann@29789
  1147
    with xz have "?P ?z x (NEq (CN 0 c e))"
haftmann@29789
  1148
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1149
  hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
haftmann@29789
  1150
  thus ?case by blast
haftmann@29789
  1151
next
haftmann@29789
  1152
  case (5 c e) 
haftmann@29789
  1153
    from prems have nb: "numbound0 e" by simp
haftmann@29789
  1154
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1155
  fix a
haftmann@29789
  1156
  let ?e="Inum (a#bs) e"
haftmann@29789
  1157
  let ?z = "(- ?e) / real c"
haftmann@29789
  1158
  {fix x
haftmann@29789
  1159
    assume xz: "x < ?z"
haftmann@29789
  1160
    hence "(real c * x < - ?e)" 
haftmann@29789
  1161
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1162
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1163
    with xz have "?P ?z x (Lt (CN 0 c e))"
haftmann@29789
  1164
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
haftmann@29789
  1165
  hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
haftmann@29789
  1166
  thus ?case by blast
haftmann@29789
  1167
next
haftmann@29789
  1168
  case (6 c e)  
haftmann@29789
  1169
    from prems have nb: "numbound0 e" by simp
haftmann@29789
  1170
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1171
  fix a
haftmann@29789
  1172
  let ?e="Inum (a#bs) e"
haftmann@29789
  1173
  let ?z = "(- ?e) / real c"
haftmann@29789
  1174
  {fix x
haftmann@29789
  1175
    assume xz: "x < ?z"
haftmann@29789
  1176
    hence "(real c * x < - ?e)" 
haftmann@29789
  1177
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1178
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1179
    with xz have "?P ?z x (Le (CN 0 c e))"
haftmann@29789
  1180
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1181
  hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
haftmann@29789
  1182
  thus ?case by blast
haftmann@29789
  1183
next
haftmann@29789
  1184
  case (7 c e)  
haftmann@29789
  1185
    from prems have nb: "numbound0 e" by simp
haftmann@29789
  1186
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1187
  fix a
haftmann@29789
  1188
  let ?e="Inum (a#bs) e"
haftmann@29789
  1189
  let ?z = "(- ?e) / real c"
haftmann@29789
  1190
  {fix x
haftmann@29789
  1191
    assume xz: "x < ?z"
haftmann@29789
  1192
    hence "(real c * x < - ?e)" 
haftmann@29789
  1193
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1194
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1195
    with xz have "?P ?z x (Gt (CN 0 c e))"
haftmann@29789
  1196
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1197
  hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
haftmann@29789
  1198
  thus ?case by blast
haftmann@29789
  1199
next
haftmann@29789
  1200
  case (8 c e)  
haftmann@29789
  1201
    from prems have nb: "numbound0 e" by simp
haftmann@29789
  1202
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1203
  fix a
haftmann@29789
  1204
  let ?e="Inum (a#bs) e"
haftmann@29789
  1205
  let ?z = "(- ?e) / real c"
haftmann@29789
  1206
  {fix x
haftmann@29789
  1207
    assume xz: "x < ?z"
haftmann@29789
  1208
    hence "(real c * x < - ?e)" 
haftmann@29789
  1209
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1210
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1211
    with xz have "?P ?z x (Ge (CN 0 c e))"
haftmann@29789
  1212
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1213
  hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
haftmann@29789
  1214
  thus ?case by blast
haftmann@29789
  1215
qed simp_all
haftmann@29789
  1216
haftmann@29789
  1217
lemma rplusinf_inf:
haftmann@29789
  1218
  assumes lp: "isrlfm p"
haftmann@29789
  1219
  shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
haftmann@29789
  1220
using lp
haftmann@29789
  1221
proof (induct p rule: isrlfm.induct)
haftmann@29789
  1222
  case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
haftmann@29789
  1223
next
haftmann@29789
  1224
  case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
haftmann@29789
  1225
next
haftmann@29789
  1226
  case (3 c e) 
haftmann@29789
  1227
  from prems have nb: "numbound0 e" by simp
haftmann@29789
  1228
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1229
  fix a
haftmann@29789
  1230
  let ?e="Inum (a#bs) e"
haftmann@29789
  1231
  let ?z = "(- ?e) / real c"
haftmann@29789
  1232
  {fix x
haftmann@29789
  1233
    assume xz: "x > ?z"
haftmann@29789
  1234
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1235
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1236
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1237
    hence "real c * x + ?e \<noteq> 0" by simp
haftmann@29789
  1238
    with xz have "?P ?z x (Eq (CN 0 c e))"
haftmann@29789
  1239
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1240
  hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
haftmann@29789
  1241
  thus ?case by blast
haftmann@29789
  1242
next
haftmann@29789
  1243
  case (4 c e) 
haftmann@29789
  1244
  from prems have nb: "numbound0 e" by simp
haftmann@29789
  1245
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1246
  fix a
haftmann@29789
  1247
  let ?e="Inum (a#bs) e"
haftmann@29789
  1248
  let ?z = "(- ?e) / real c"
haftmann@29789
  1249
  {fix x
haftmann@29789
  1250
    assume xz: "x > ?z"
haftmann@29789
  1251
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1252
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1253
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1254
    hence "real c * x + ?e \<noteq> 0" by simp
haftmann@29789
  1255
    with xz have "?P ?z x (NEq (CN 0 c e))"
haftmann@29789
  1256
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1257
  hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
haftmann@29789
  1258
  thus ?case by blast
haftmann@29789
  1259
next
haftmann@29789
  1260
  case (5 c e) 
haftmann@29789
  1261
  from prems have nb: "numbound0 e" by simp
haftmann@29789
  1262
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1263
  fix a
haftmann@29789
  1264
  let ?e="Inum (a#bs) e"
haftmann@29789
  1265
  let ?z = "(- ?e) / real c"
haftmann@29789
  1266
  {fix x
haftmann@29789
  1267
    assume xz: "x > ?z"
haftmann@29789
  1268
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1269
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1270
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1271
    with xz have "?P ?z x (Lt (CN 0 c e))"
haftmann@29789
  1272
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1273
  hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
haftmann@29789
  1274
  thus ?case by blast
haftmann@29789
  1275
next
haftmann@29789
  1276
  case (6 c e) 
haftmann@29789
  1277
  from prems have nb: "numbound0 e" by simp
haftmann@29789
  1278
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1279
  fix a
haftmann@29789
  1280
  let ?e="Inum (a#bs) e"
haftmann@29789
  1281
  let ?z = "(- ?e) / real c"
haftmann@29789
  1282
  {fix x
haftmann@29789
  1283
    assume xz: "x > ?z"
haftmann@29789
  1284
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1285
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1286
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1287
    with xz have "?P ?z x (Le (CN 0 c e))"
haftmann@29789
  1288
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1289
  hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
haftmann@29789
  1290
  thus ?case by blast
haftmann@29789
  1291
next
haftmann@29789
  1292
  case (7 c e) 
haftmann@29789
  1293
  from prems have nb: "numbound0 e" by simp
haftmann@29789
  1294
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1295
  fix a
haftmann@29789
  1296
  let ?e="Inum (a#bs) e"
haftmann@29789
  1297
  let ?z = "(- ?e) / real c"
haftmann@29789
  1298
  {fix x
haftmann@29789
  1299
    assume xz: "x > ?z"
haftmann@29789
  1300
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1301
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1302
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1303
    with xz have "?P ?z x (Gt (CN 0 c e))"
haftmann@29789
  1304
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1305
  hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
haftmann@29789
  1306
  thus ?case by blast
haftmann@29789
  1307
next
haftmann@29789
  1308
  case (8 c e) 
haftmann@29789
  1309
  from prems have nb: "numbound0 e" by simp
haftmann@29789
  1310
  from prems have cp: "real c > 0" by simp
haftmann@29789
  1311
  fix a
haftmann@29789
  1312
  let ?e="Inum (a#bs) e"
haftmann@29789
  1313
  let ?z = "(- ?e) / real c"
haftmann@29789
  1314
  {fix x
haftmann@29789
  1315
    assume xz: "x > ?z"
haftmann@29789
  1316
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1317
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1318
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1319
    with xz have "?P ?z x (Ge (CN 0 c e))"
haftmann@29789
  1320
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
haftmann@29789
  1321
  hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
haftmann@29789
  1322
  thus ?case by blast
haftmann@29789
  1323
qed simp_all
haftmann@29789
  1324
haftmann@29789
  1325
lemma rminusinf_bound0:
haftmann@29789
  1326
  assumes lp: "isrlfm p"
haftmann@29789
  1327
  shows "bound0 (minusinf p)"
haftmann@29789
  1328
  using lp
haftmann@29789
  1329
  by (induct p rule: minusinf.induct) simp_all
haftmann@29789
  1330
haftmann@29789
  1331
lemma rplusinf_bound0:
haftmann@29789
  1332
  assumes lp: "isrlfm p"
haftmann@29789
  1333
  shows "bound0 (plusinf p)"
haftmann@29789
  1334
  using lp
haftmann@29789
  1335
  by (induct p rule: plusinf.induct) simp_all
haftmann@29789
  1336
haftmann@29789
  1337
lemma rminusinf_ex:
haftmann@29789
  1338
  assumes lp: "isrlfm p"
haftmann@29789
  1339
  and ex: "Ifm (a#bs) (minusinf p)"
haftmann@29789
  1340
  shows "\<exists> x. Ifm (x#bs) p"
haftmann@29789
  1341
proof-
haftmann@29789
  1342
  from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
haftmann@29789
  1343
  have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
haftmann@29789
  1344
  from rminusinf_inf[OF lp, where bs="bs"] 
haftmann@29789
  1345
  obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
haftmann@29789
  1346
  from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
haftmann@29789
  1347
  moreover have "z - 1 < z" by simp
haftmann@29789
  1348
  ultimately show ?thesis using z_def by auto
haftmann@29789
  1349
qed
haftmann@29789
  1350
haftmann@29789
  1351
lemma rplusinf_ex:
haftmann@29789
  1352
  assumes lp: "isrlfm p"
haftmann@29789
  1353
  and ex: "Ifm (a#bs) (plusinf p)"
haftmann@29789
  1354
  shows "\<exists> x. Ifm (x#bs) p"
haftmann@29789
  1355
proof-
haftmann@29789
  1356
  from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
haftmann@29789
  1357
  have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
haftmann@29789
  1358
  from rplusinf_inf[OF lp, where bs="bs"] 
haftmann@29789
  1359
  obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
haftmann@29789
  1360
  from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
haftmann@29789
  1361
  moreover have "z + 1 > z" by simp
haftmann@29789
  1362
  ultimately show ?thesis using z_def by auto
haftmann@29789
  1363
qed
haftmann@29789
  1364
haftmann@29789
  1365
consts 
haftmann@29789
  1366
  uset:: "fm \<Rightarrow> (num \<times> int) list"
haftmann@29789
  1367
  usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
haftmann@29789
  1368
recdef uset "measure size"
haftmann@29789
  1369
  "uset (And p q) = (uset p @ uset q)" 
haftmann@29789
  1370
  "uset (Or p q) = (uset p @ uset q)" 
haftmann@29789
  1371
  "uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1372
  "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1373
  "uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1374
  "uset (Le  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1375
  "uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1376
  "uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1377
  "uset p = []"
haftmann@29789
  1378
recdef usubst "measure size"
haftmann@29789
  1379
  "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
haftmann@29789
  1380
  "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
haftmann@29789
  1381
  "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1382
  "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1383
  "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1384
  "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1385
  "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1386
  "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1387
  "usubst p = (\<lambda> (t,n). p)"
haftmann@29789
  1388
haftmann@29789
  1389
lemma usubst_I: assumes lp: "isrlfm p"
haftmann@29789
  1390
  and np: "real n > 0" and nbt: "numbound0 t"
haftmann@29789
  1391
  shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
haftmann@29789
  1392
  using lp
haftmann@29789
  1393
proof(induct p rule: usubst.induct)
haftmann@29789
  1394
  case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
haftmann@29789
  1395
  have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
haftmann@29789
  1396
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1397
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
haftmann@29789
  1398
    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1399
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1400
  also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
haftmann@29789
  1401
    using np by simp 
haftmann@29789
  1402
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1403
next
haftmann@29789
  1404
  case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
haftmann@29789
  1405
  have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
haftmann@29789
  1406
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1407
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
haftmann@29789
  1408
    by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1409
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1410
  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
haftmann@29789
  1411
    using np by simp 
haftmann@29789
  1412
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1413
next
haftmann@29789
  1414
  case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
haftmann@29789
  1415
  have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
haftmann@29789
  1416
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1417
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
haftmann@29789
  1418
    by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1419
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1420
  also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
haftmann@29789
  1421
    using np by simp 
haftmann@29789
  1422
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1423
next
haftmann@29789
  1424
  case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
haftmann@29789
  1425
  have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
haftmann@29789
  1426
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1427
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
haftmann@29789
  1428
    by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1429
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1430
  also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
haftmann@29789
  1431
    using np by simp 
haftmann@29789
  1432
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1433
next
haftmann@29789
  1434
  case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
haftmann@29789
  1435
  from np have np: "real n \<noteq> 0" by simp
haftmann@29789
  1436
  have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
haftmann@29789
  1437
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1438
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
haftmann@29789
  1439
    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1440
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1441
  also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
haftmann@29789
  1442
    using np by simp 
haftmann@29789
  1443
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1444
next
haftmann@29789
  1445
  case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
haftmann@29789
  1446
  from np have np: "real n \<noteq> 0" by simp
haftmann@29789
  1447
  have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
haftmann@29789
  1448
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1449
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
haftmann@29789
  1450
    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1451
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1452
  also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
haftmann@29789
  1453
    using np by simp 
haftmann@29789
  1454
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1455
qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
haftmann@29789
  1456
haftmann@29789
  1457
lemma uset_l:
haftmann@29789
  1458
  assumes lp: "isrlfm p"
haftmann@29789
  1459
  shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
haftmann@29789
  1460
using lp
haftmann@29789
  1461
by(induct p rule: uset.induct,auto)
haftmann@29789
  1462
haftmann@29789
  1463
lemma rminusinf_uset:
haftmann@29789
  1464
  assumes lp: "isrlfm p"
haftmann@29789
  1465
  and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
haftmann@29789
  1466
  and ex: "Ifm (x#bs) p" (is "?I x p")
haftmann@29789
  1467
  shows "\<exists> (s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m")
haftmann@29789
  1468
proof-
haftmann@29789
  1469
  have "\<exists> (s,m) \<in> set (uset p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s")
haftmann@29789
  1470
    using lp nmi ex
haftmann@29789
  1471
    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
haftmann@29789
  1472
  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
haftmann@29789
  1473
  from uset_l[OF lp] smU have mp: "real m > 0" by auto
haftmann@29789
  1474
  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" 
haftmann@29789
  1475
    by (auto simp add: mult_commute)
haftmann@29789
  1476
  thus ?thesis using smU by auto
haftmann@29789
  1477
qed
haftmann@29789
  1478
haftmann@29789
  1479
lemma rplusinf_uset:
haftmann@29789
  1480
  assumes lp: "isrlfm p"
haftmann@29789
  1481
  and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
haftmann@29789
  1482
  and ex: "Ifm (x#bs) p" (is "?I x p")
haftmann@29789
  1483
  shows "\<exists> (s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m")
haftmann@29789
  1484
proof-
haftmann@29789
  1485
  have "\<exists> (s,m) \<in> set (uset p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s")
haftmann@29789
  1486
    using lp nmi ex
haftmann@29789
  1487
    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
haftmann@29789
  1488
  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
haftmann@29789
  1489
  from uset_l[OF lp] smU have mp: "real m > 0" by auto
haftmann@29789
  1490
  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" 
haftmann@29789
  1491
    by (auto simp add: mult_commute)
haftmann@29789
  1492
  thus ?thesis using smU by auto
haftmann@29789
  1493
qed
haftmann@29789
  1494
haftmann@29789
  1495
lemma lin_dense: 
haftmann@29789
  1496
  assumes lp: "isrlfm p"
haftmann@29789
  1497
  and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)" 
haftmann@29789
  1498
  (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
haftmann@29789
  1499
  and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
haftmann@29789
  1500
  and ly: "l < y" and yu: "y < u"
haftmann@29789
  1501
  shows "Ifm (y#bs) p"
haftmann@29789
  1502
using lp px noS
haftmann@29789
  1503
proof (induct p rule: isrlfm.induct)
haftmann@29789
  1504
  case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
haftmann@29789
  1505
    from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
haftmann@29789
  1506
    hence pxc: "x < (- ?N x e) / real c" 
haftmann@29789
  1507
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
haftmann@29789
  1508
    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
haftmann@29789
  1509
    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
haftmann@29789
  1510
    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
haftmann@29789
  1511
    moreover {assume y: "y < (-?N x e)/ real c"
haftmann@29789
  1512
      hence "y * real c < - ?N x e"
haftmann@29789
  1513
	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
haftmann@29789
  1514
      hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
haftmann@29789
  1515
      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
haftmann@29789
  1516
    moreover {assume y: "y > (- ?N x e) / real c" 
haftmann@29789
  1517
      with yu have eu: "u > (- ?N x e) / real c" by auto
haftmann@29789
  1518
      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
haftmann@29789
  1519
      with lx pxc have "False" by auto
haftmann@29789
  1520
      hence ?case by simp }
haftmann@29789
  1521
    ultimately show ?case by blast
haftmann@29789
  1522
next
haftmann@29789
  1523
  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
haftmann@29789
  1524
    from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
haftmann@29789
  1525
    hence pxc: "x \<le> (- ?N x e) / real c" 
haftmann@29789
  1526
      by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
haftmann@29789
  1527
    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
haftmann@29789
  1528
    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
haftmann@29789
  1529
    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
haftmann@29789
  1530
    moreover {assume y: "y < (-?N x e)/ real c"
haftmann@29789
  1531
      hence "y * real c < - ?N x e"
haftmann@29789
  1532
	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
haftmann@29789
  1533
      hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
haftmann@29789
  1534
      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
haftmann@29789
  1535
    moreover {assume y: "y > (- ?N x e) / real c" 
haftmann@29789
  1536
      with yu have eu: "u > (- ?N x e) / real c" by auto
haftmann@29789
  1537
      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
haftmann@29789
  1538
      with lx pxc have "False" by auto
haftmann@29789
  1539
      hence ?case by simp }
haftmann@29789
  1540
    ultimately show ?case by blast
haftmann@29789
  1541
next
haftmann@29789
  1542
  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
haftmann@29789
  1543
    from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
haftmann@29789
  1544
    hence pxc: "x > (- ?N x e) / real c" 
haftmann@29789
  1545
      by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
haftmann@29789
  1546
    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
haftmann@29789
  1547
    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
haftmann@29789
  1548
    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
haftmann@29789
  1549
    moreover {assume y: "y > (-?N x e)/ real c"
haftmann@29789
  1550
      hence "y * real c > - ?N x e"
haftmann@29789
  1551
	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
haftmann@29789
  1552
      hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
haftmann@29789
  1553
      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
haftmann@29789
  1554
    moreover {assume y: "y < (- ?N x e) / real c" 
haftmann@29789
  1555
      with ly have eu: "l < (- ?N x e) / real c" by auto
haftmann@29789
  1556
      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
haftmann@29789
  1557
      with xu pxc have "False" by auto
haftmann@29789
  1558
      hence ?case by simp }
haftmann@29789
  1559
    ultimately show ?case by blast
haftmann@29789
  1560
next
haftmann@29789
  1561
  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
haftmann@29789
  1562
    from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
haftmann@29789
  1563
    hence pxc: "x \<ge> (- ?N x e) / real c" 
haftmann@29789
  1564
      by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
haftmann@29789
  1565
    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
haftmann@29789
  1566
    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
haftmann@29789
  1567
    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
haftmann@29789
  1568
    moreover {assume y: "y > (-?N x e)/ real c"
haftmann@29789
  1569
      hence "y * real c > - ?N x e"
haftmann@29789
  1570
	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
haftmann@29789
  1571
      hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
haftmann@29789
  1572
      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
haftmann@29789
  1573
    moreover {assume y: "y < (- ?N x e) / real c" 
haftmann@29789
  1574
      with ly have eu: "l < (- ?N x e) / real c" by auto
haftmann@29789
  1575
      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
haftmann@29789
  1576
      with xu pxc have "False" by auto
haftmann@29789
  1577
      hence ?case by simp }
haftmann@29789
  1578
    ultimately show ?case by blast
haftmann@29789
  1579
next
haftmann@29789
  1580
  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
haftmann@29789
  1581
    from cp have cnz: "real c \<noteq> 0" by simp
haftmann@29789
  1582
    from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
haftmann@29789
  1583
    hence pxc: "x = (- ?N x e) / real c" 
haftmann@29789
  1584
      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
haftmann@29789
  1585
    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
haftmann@29789
  1586
    with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
haftmann@29789
  1587
    with pxc show ?case by simp
haftmann@29789
  1588
next
haftmann@29789
  1589
  case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
haftmann@29789
  1590
    from cp have cnz: "real c \<noteq> 0" by simp
haftmann@29789
  1591
    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
haftmann@29789
  1592
    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
haftmann@29789
  1593
    hence "y* real c \<noteq> -?N x e"      
haftmann@29789
  1594
      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
haftmann@29789
  1595
    hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
haftmann@29789
  1596
    thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
haftmann@29789
  1597
      by (simp add: algebra_simps)
haftmann@29789
  1598
qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
haftmann@29789
  1599
haftmann@29789
  1600
lemma finite_set_intervals:
haftmann@29789
  1601
  assumes px: "P (x::real)" 
haftmann@29789
  1602
  and lx: "l \<le> x" and xu: "x \<le> u"
haftmann@29789
  1603
  and linS: "l\<in> S" and uinS: "u \<in> S"
haftmann@29789
  1604
  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
haftmann@29789
  1605
  shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x"
haftmann@29789
  1606
proof-
haftmann@29789
  1607
  let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
haftmann@29789
  1608
  let ?xM = "{y. y\<in> S \<and> x \<le> y}"
haftmann@29789
  1609
  let ?a = "Max ?Mx"
haftmann@29789
  1610
  let ?b = "Min ?xM"
haftmann@29789
  1611
  have MxS: "?Mx \<subseteq> S" by blast
haftmann@29789
  1612
  hence fMx: "finite ?Mx" using fS finite_subset by auto
haftmann@29789
  1613
  from lx linS have linMx: "l \<in> ?Mx" by blast
haftmann@29789
  1614
  hence Mxne: "?Mx \<noteq> {}" by blast
haftmann@29789
  1615
  have xMS: "?xM \<subseteq> S" by blast
haftmann@29789
  1616
  hence fxM: "finite ?xM" using fS finite_subset by auto
haftmann@29789
  1617
  from xu uinS have linxM: "u \<in> ?xM" by blast
haftmann@29789
  1618
  hence xMne: "?xM \<noteq> {}" by blast
haftmann@29789
  1619
  have ax:"?a \<le> x" using Mxne fMx by auto
haftmann@29789
  1620
  have xb:"x \<le> ?b" using xMne fxM by auto
haftmann@29789
  1621
  have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
haftmann@29789
  1622
  have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
haftmann@29789
  1623
  have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
haftmann@29789
  1624
  proof(clarsimp)
haftmann@29789
  1625
    fix y
haftmann@29789
  1626
    assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
haftmann@29789
  1627
    from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
haftmann@29789
  1628
    moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
haftmann@29789
  1629
    moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
haftmann@29789
  1630
    ultimately show "False" by blast
haftmann@29789
  1631
  qed
haftmann@29789
  1632
  from ainS binS noy ax xb px show ?thesis by blast
haftmann@29789
  1633
qed
haftmann@29789
  1634
haftmann@29789
  1635
lemma finite_set_intervals2:
haftmann@29789
  1636
  assumes px: "P (x::real)" 
haftmann@29789
  1637
  and lx: "l \<le> x" and xu: "x \<le> u"
haftmann@29789
  1638
  and linS: "l\<in> S" and uinS: "u \<in> S"
haftmann@29789
  1639
  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
haftmann@29789
  1640
  shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)"
haftmann@29789
  1641
proof-
haftmann@29789
  1642
  from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
haftmann@29789
  1643
  obtain a and b where 
haftmann@29789
  1644
    as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
haftmann@29789
  1645
  from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto
haftmann@29789
  1646
  thus ?thesis using px as bs noS by blast 
haftmann@29789
  1647
qed
haftmann@29789
  1648
haftmann@29789
  1649
lemma rinf_uset:
haftmann@29789
  1650
  assumes lp: "isrlfm p"
haftmann@29789
  1651
  and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
haftmann@29789
  1652
  and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
haftmann@29789
  1653
  and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
haftmann@29789
  1654
  shows "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" 
haftmann@29789
  1655
proof-
haftmann@29789
  1656
  let ?N = "\<lambda> x t. Inum (x#bs) t"
haftmann@29789
  1657
  let ?U = "set (uset p)"
haftmann@29789
  1658
  from ex obtain a where pa: "?I a p" by blast
haftmann@29789
  1659
  from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
haftmann@29789
  1660
  have nmi': "\<not> (?I a (?M p))" by simp
haftmann@29789
  1661
  from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
haftmann@29789
  1662
  have npi': "\<not> (?I a (?P p))" by simp
haftmann@29789
  1663
  have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p"
haftmann@29789
  1664
  proof-
haftmann@29789
  1665
    let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
haftmann@29789
  1666
    have fM: "finite ?M" by auto
haftmann@29789
  1667
    from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa] 
haftmann@29789
  1668
    have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast
haftmann@29789
  1669
    then obtain "t" "n" "s" "m" where 
haftmann@29789
  1670
      tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" 
haftmann@29789
  1671
      and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
haftmann@29789
  1672
    from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto
haftmann@29789
  1673
    from tnU have Mne: "?M \<noteq> {}" by auto
haftmann@29789
  1674
    hence Une: "?U \<noteq> {}" by simp
haftmann@29789
  1675
    let ?l = "Min ?M"
haftmann@29789
  1676
    let ?u = "Max ?M"
haftmann@29789
  1677
    have linM: "?l \<in> ?M" using fM Mne by simp
haftmann@29789
  1678
    have uinM: "?u \<in> ?M" using fM Mne by simp
haftmann@29789
  1679
    have tnM: "?N a t / real n \<in> ?M" using tnU by auto
haftmann@29789
  1680
    have smM: "?N a s / real m \<in> ?M" using smU by auto 
haftmann@29789
  1681
    have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
haftmann@29789
  1682
    have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
haftmann@29789
  1683
    have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
haftmann@29789
  1684
    have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
haftmann@29789
  1685
    from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
haftmann@29789
  1686
    have "(\<exists> s\<in> ?M. ?I s p) \<or> 
haftmann@29789
  1687
      (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" .
haftmann@29789
  1688
    moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
haftmann@29789
  1689
      hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
haftmann@29789
  1690
      then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
haftmann@29789
  1691
      have "(u + u) / 2 = u" by auto with pu tuu 
haftmann@29789
  1692
      have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
haftmann@29789
  1693
      with tuU have ?thesis by blast}
haftmann@29789
  1694
    moreover{
haftmann@29789
  1695
      assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p"
haftmann@29789
  1696
      then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
haftmann@29789
  1697
	and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p"
haftmann@29789
  1698
	by blast
haftmann@29789
  1699
      from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
haftmann@29789
  1700
      then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
haftmann@29789
  1701
      from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
haftmann@29789
  1702
      then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
haftmann@29789
  1703
      from t1x xt2 have t1t2: "t1 < t2" by simp
haftmann@29789
  1704
      let ?u = "(t1 + t2) / 2"
haftmann@29789
  1705
      from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
haftmann@29789
  1706
      from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
haftmann@29789
  1707
      with t1uU t2uU t1u t2u have ?thesis by blast}
haftmann@29789
  1708
    ultimately show ?thesis by blast
haftmann@29789
  1709
  qed
haftmann@29789
  1710
  then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" 
haftmann@29789
  1711
    and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
haftmann@29789
  1712
  from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
haftmann@29789
  1713
  from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
haftmann@29789
  1714
    numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
haftmann@29789
  1715
  have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
haftmann@29789
  1716
  with lnU smU
haftmann@29789
  1717
  show ?thesis by auto
haftmann@29789
  1718
qed
haftmann@29789
  1719
    (* The Ferrante - Rackoff Theorem *)
haftmann@29789
  1720
haftmann@29789
  1721
theorem fr_eq: 
haftmann@29789
  1722
  assumes lp: "isrlfm p"
haftmann@29789
  1723
  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). Ifm ((((Inum (x#bs) t)/  real n + (Inum (x#bs) s) / real m) /2)#bs) p))"
haftmann@29789
  1724
  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
haftmann@29789
  1725
proof
haftmann@29789
  1726
  assume px: "\<exists> x. ?I x p"
haftmann@29789
  1727
  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
haftmann@29789
  1728
  moreover {assume "?M \<or> ?P" hence "?D" by blast}
haftmann@29789
  1729
  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
haftmann@29789
  1730
    from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
haftmann@29789
  1731
  ultimately show "?D" by blast
haftmann@29789
  1732
next
haftmann@29789
  1733
  assume "?D" 
haftmann@29789
  1734
  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
haftmann@29789
  1735
  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
haftmann@29789
  1736
  moreover {assume f:"?F" hence "?E" by blast}
haftmann@29789
  1737
  ultimately show "?E" by blast
haftmann@29789
  1738
qed
haftmann@29789
  1739
haftmann@29789
  1740
haftmann@29789
  1741
lemma fr_equsubst: 
haftmann@29789
  1742
  assumes lp: "isrlfm p"
haftmann@29789
  1743
  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (uset p). \<exists> (s,l) \<in> set (uset p). Ifm (x#bs) (usubst p (Add(Mul l t) (Mul k s) , 2*k*l))))"
haftmann@29789
  1744
  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
haftmann@29789
  1745
proof
haftmann@29789
  1746
  assume px: "\<exists> x. ?I x p"
haftmann@29789
  1747
  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
haftmann@29789
  1748
  moreover {assume "?M \<or> ?P" hence "?D" by blast}
haftmann@29789
  1749
  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
haftmann@29789
  1750
    let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
haftmann@29789
  1751
    let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1752
    {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
haftmann@29789
  1753
      with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
haftmann@29789
  1754
	by auto
haftmann@29789
  1755
      let ?st = "Add (Mul m t) (Mul n s)"
haftmann@29789
  1756
      from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
haftmann@29789
  1757
	by (simp add: mult_commute)
haftmann@29789
  1758
      from tnb snb have st_nb: "numbound0 ?st" by simp
haftmann@29789
  1759
      have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1760
	using mnp mp np by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1761
      from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
haftmann@29789
  1762
      have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
haftmann@29789
  1763
    with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
haftmann@29789
  1764
  ultimately show "?D" by blast
haftmann@29789
  1765
next
haftmann@29789
  1766
  assume "?D" 
haftmann@29789
  1767
  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
haftmann@29789
  1768
  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
haftmann@29789
  1769
  moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)" 
haftmann@29789
  1770
    and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
haftmann@29789
  1771
    with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
haftmann@29789
  1772
    let ?st = "Add (Mul l t) (Mul k s)"
haftmann@29789
  1773
    from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" 
haftmann@29789
  1774
      by (simp add: mult_commute)
haftmann@29789
  1775
    from tnb snb have st_nb: "numbound0 ?st" by simp
haftmann@29789
  1776
    from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
haftmann@29789
  1777
  ultimately show "?E" by blast
haftmann@29789
  1778
qed
haftmann@29789
  1779
haftmann@29789
  1780
haftmann@29789
  1781
    (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
haftmann@29789
  1782
constdefs ferrack:: "fm \<Rightarrow> fm"
haftmann@29789
  1783
  "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
haftmann@29789
  1784
                in if (mp = T \<or> pp = T) then T else 
haftmann@29789
  1785
                   (let U = remdps(map simp_num_pair 
haftmann@29789
  1786
                     (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
haftmann@29789
  1787
                           (alluopairs (uset p')))) 
haftmann@29789
  1788
                    in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
haftmann@29789
  1789
haftmann@29789
  1790
lemma uset_cong_aux:
haftmann@29789
  1791
  assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
haftmann@29789
  1792
  shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))"
haftmann@29789
  1793
  (is "?lhs = ?rhs")
haftmann@29789
  1794
proof(auto)
haftmann@29789
  1795
  fix t n s m
haftmann@29789
  1796
  assume "((t,n),(s,m)) \<in> set (alluopairs U)"
haftmann@29789
  1797
  hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
haftmann@29789
  1798
    using alluopairs_set1[where xs="U"] by blast
haftmann@29789
  1799
  let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1800
  let ?st= "Add (Mul m t) (Mul n s)"
haftmann@29789
  1801
  from Ul th have mnz: "m \<noteq> 0" by auto
haftmann@29789
  1802
  from Ul th have  nnz: "n \<noteq> 0" by auto  
haftmann@29789
  1803
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1804
   using mnz nnz by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1805
 
haftmann@29789
  1806
  thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
haftmann@29789
  1807
       (2 * real n * real m)
haftmann@29789
  1808
       \<in> (\<lambda>((t, n), s, m).
haftmann@29789
  1809
             (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
haftmann@29789
  1810
         (set U \<times> set U)"using mnz nnz th  
haftmann@29789
  1811
    apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
haftmann@29789
  1812
    by (rule_tac x="(s,m)" in bexI,simp_all) 
haftmann@29789
  1813
  (rule_tac x="(t,n)" in bexI,simp_all)
haftmann@29789
  1814
next
haftmann@29789
  1815
  fix t n s m
haftmann@29789
  1816
  assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
haftmann@29789
  1817
  let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1818
  let ?st= "Add (Mul m t) (Mul n s)"
haftmann@29789
  1819
  from Ul smU have mnz: "m \<noteq> 0" by auto
haftmann@29789
  1820
  from Ul tnU have  nnz: "n \<noteq> 0" by auto  
haftmann@29789
  1821
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1822
   using mnz nnz by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1823
 let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
haftmann@29789
  1824
 have Pc:"\<forall> a b. ?P a b = ?P b a"
haftmann@29789
  1825
   by auto
haftmann@29789
  1826
 from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
haftmann@29789
  1827
 from alluopairs_ex[OF Pc, where xs="U"] tnU smU
haftmann@29789
  1828
 have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
haftmann@29789
  1829
   by blast
haftmann@29789
  1830
 then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" 
haftmann@29789
  1831
   and Pts': "?P (t',n') (s',m')" by blast
haftmann@29789
  1832
 from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
haftmann@29789
  1833
 let ?st' = "Add (Mul m' t') (Mul n' s')"
haftmann@29789
  1834
   have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
haftmann@29789
  1835
   using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1836
 from Pts' have 
haftmann@29789
  1837
   "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
haftmann@29789
  1838
 also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
haftmann@29789
  1839
 finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
haftmann@29789
  1840
          \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
haftmann@29789
  1841
            (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
haftmann@29789
  1842
            set (alluopairs U)"
haftmann@29789
  1843
   using ts'_U by blast
haftmann@29789
  1844
qed
haftmann@29789
  1845
haftmann@29789
  1846
lemma uset_cong:
haftmann@29789
  1847
  assumes lp: "isrlfm p"
haftmann@29789
  1848
  and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)")
haftmann@29789
  1849
  and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
haftmann@29789
  1850
  and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
haftmann@29789
  1851
  shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (usubst p (t,n)))"
haftmann@29789
  1852
  (is "?lhs = ?rhs")
haftmann@29789
  1853
proof
haftmann@29789
  1854
  assume ?lhs
haftmann@29789
  1855
  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
haftmann@29789
  1856
    Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
haftmann@29789
  1857
  let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1858
  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
haftmann@29789
  1859
    and snb: "numbound0 s" and mp:"m > 0"  by auto
haftmann@29789
  1860
  let ?st= "Add (Mul m t) (Mul n s)"
haftmann@29789
  1861
  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
haftmann@29789
  1862
      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
haftmann@29789
  1863
    from tnb snb have stnb: "numbound0 ?st" by simp
haftmann@29789
  1864
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1865
   using mp np by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1866
  from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
haftmann@29789
  1867
  hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
haftmann@29789
  1868
    by auto (rule_tac x="(a,b)" in bexI, auto)
haftmann@29789
  1869
  then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
haftmann@29789
  1870
  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
haftmann@29789
  1871
  from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst 
haftmann@29789
  1872
  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
haftmann@29789
  1873
  from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
haftmann@29789
  1874
  have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st) 
haftmann@29789
  1875
  then show ?rhs using tnU' by auto 
haftmann@29789
  1876
next
haftmann@29789
  1877
  assume ?rhs
haftmann@29789
  1878
  then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))" 
haftmann@29789
  1879
    by blast
haftmann@29789
  1880
  from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
haftmann@29789
  1881
  hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" 
haftmann@29789
  1882
    by auto (rule_tac x="(a,b)" in bexI, auto)
haftmann@29789
  1883
  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
haftmann@29789
  1884
    th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
haftmann@29789
  1885
    let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1886
  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
haftmann@29789
  1887
    and snb: "numbound0 s" and mp:"m > 0"  by auto
haftmann@29789
  1888
  let ?st= "Add (Mul m t) (Mul n s)"
haftmann@29789
  1889
  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
haftmann@29789
  1890
      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
haftmann@29789
  1891
    from tnb snb have stnb: "numbound0 ?st" by simp
haftmann@29789
  1892
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1893
   using mp np by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1894
  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
haftmann@29789
  1895
  from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
haftmann@29789
  1896
  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
haftmann@29789
  1897
  with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
haftmann@29789
  1898
qed
haftmann@29789
  1899
haftmann@29789
  1900
lemma ferrack: 
haftmann@29789
  1901
  assumes qf: "qfree p"
haftmann@29789
  1902
  shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
haftmann@29789
  1903
  (is "_ \<and> (?rhs = ?lhs)")
haftmann@29789
  1904
proof-
haftmann@29789
  1905
  let ?I = "\<lambda> x p. Ifm (x#bs) p"
haftmann@29789
  1906
  fix x
haftmann@29789
  1907
  let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1908
  let ?q = "rlfm (simpfm p)" 
haftmann@29789
  1909
  let ?U = "uset ?q"
haftmann@29789
  1910
  let ?Up = "alluopairs ?U"
haftmann@29789
  1911
  let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
haftmann@29789
  1912
  let ?S = "map ?g ?Up"
haftmann@29789
  1913
  let ?SS = "map simp_num_pair ?S"
haftmann@29789
  1914
  let ?Y = "remdps ?SS"
haftmann@29789
  1915
  let ?f= "(\<lambda> (t,n). ?N t / real n)"
haftmann@29789
  1916
  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
haftmann@29789
  1917
  let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
haftmann@29789
  1918
  let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
haftmann@29789
  1919
  from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
haftmann@29789
  1920
  from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
haftmann@29789
  1921
  from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
haftmann@29789
  1922
  from U_l UpU 
haftmann@29789
  1923
  have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
haftmann@29789
  1924
  hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
haftmann@29789
  1925
    by (auto simp add: mult_pos_pos)
haftmann@29789
  1926
  have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" 
haftmann@29789
  1927
  proof-
haftmann@29789
  1928
    { fix t n assume tnY: "(t,n) \<in> set ?Y" 
haftmann@29789
  1929
      hence "(t,n) \<in> set ?SS" by simp
haftmann@29789
  1930
      hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
haftmann@29789
  1931
	by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
haftmann@29789
  1932
      then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
haftmann@29789
  1933
      from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
haftmann@29789
  1934
      from simp_num_pair_l[OF tnb np tns]
haftmann@29789
  1935
      have "numbound0 t \<and> n > 0" . }
haftmann@29789
  1936
    thus ?thesis by blast
haftmann@29789
  1937
  qed
haftmann@29789
  1938
haftmann@29789
  1939
  have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
haftmann@29789
  1940
  proof-
haftmann@29789
  1941
     from simp_num_pair_ci[where bs="x#bs"] have 
haftmann@29789
  1942
    "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
haftmann@29789
  1943
     hence th: "?f o simp_num_pair = ?f" using ext by blast
haftmann@29789
  1944
    have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
haftmann@29789
  1945
    also have "\<dots> = (?f ` set ?S)" by (simp add: th)
haftmann@29789
  1946
    also have "\<dots> = ((?f o ?g) ` set ?Up)" 
haftmann@29789
  1947
      by (simp only: set_map o_def image_compose[symmetric])
haftmann@29789
  1948
    also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
haftmann@29789
  1949
      using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
haftmann@29789
  1950
    finally show ?thesis .
haftmann@29789
  1951
  qed
haftmann@29789
  1952
  have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
haftmann@29789
  1953
  proof-
haftmann@29789
  1954
    { fix t n assume tnY: "(t,n) \<in> set ?Y"
haftmann@29789
  1955
      with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
haftmann@29789
  1956
      from usubst_I[OF lq np tnb]
haftmann@29789
  1957
    have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))" 
haftmann@29789
  1958
      using simpfm_bound0 by simp}
haftmann@29789
  1959
    thus ?thesis by blast
haftmann@29789
  1960
  qed
haftmann@29789
  1961
  hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
haftmann@29789
  1962
  let ?mp = "minusinf ?q"
haftmann@29789
  1963
  let ?pp = "plusinf ?q"
haftmann@29789
  1964
  let ?M = "?I x ?mp"
haftmann@29789
  1965
  let ?P = "?I x ?pp"
haftmann@29789
  1966
  let ?res = "disj ?mp (disj ?pp ?ep)"
haftmann@29789
  1967
  from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
haftmann@29789
  1968
  have nbth: "bound0 ?res" by auto
haftmann@29789
  1969
haftmann@29789
  1970
  from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm  
haftmann@29789
  1971
haftmann@29789
  1972
  have th: "?lhs = (\<exists> x. ?I x ?q)" by auto 
haftmann@29789
  1973
  from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
haftmann@29789
  1974
    by (simp only: split_def fst_conv snd_conv)
haftmann@29789
  1975
  also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" 
haftmann@29789
  1976
    using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm) 
haftmann@29789
  1977
  also have "\<dots> = (Ifm (x#bs) ?res)"
haftmann@29789
  1978
    using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
haftmann@29789
  1979
    by (simp add: split_def pair_collapse)
haftmann@29789
  1980
  finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
haftmann@29789
  1981
  hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
haftmann@29789
  1982
    by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
haftmann@29789
  1983
  from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
haftmann@29789
  1984
  with lr show ?thesis by blast
haftmann@29789
  1985
qed
haftmann@29789
  1986
haftmann@29789
  1987
definition linrqe:: "fm \<Rightarrow> fm" where
haftmann@29789
  1988
  "linrqe p = qelim (prep p) ferrack"
haftmann@29789
  1989
haftmann@29789
  1990
theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)"
haftmann@29789
  1991
using ferrack qelim_ci prep
haftmann@29789
  1992
unfolding linrqe_def by auto
haftmann@29789
  1993
haftmann@29789
  1994
definition ferrack_test :: "unit \<Rightarrow> fm" where
haftmann@29789
  1995
  "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
haftmann@29789
  1996
    (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
haftmann@29789
  1997
haftmann@29789
  1998
ML {* @{code ferrack_test} () *}
haftmann@29789
  1999
haftmann@29789
  2000
oracle linr_oracle = {*
haftmann@29789
  2001
let
haftmann@29789
  2002
haftmann@29789
  2003
fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
haftmann@29789
  2004
     of NONE => error "Variable not found in the list!"
haftmann@29789
  2005
      | SOME n => @{code Bound} n)
haftmann@29789
  2006
  | num_of_term vs @{term "real (0::int)"} = @{code C} 0
haftmann@29789
  2007
  | num_of_term vs @{term "real (1::int)"} = @{code C} 1
haftmann@29789
  2008
  | num_of_term vs @{term "0::real"} = @{code C} 0
haftmann@29789
  2009
  | num_of_term vs @{term "1::real"} = @{code C} 1
haftmann@29789
  2010
  | num_of_term vs (Bound i) = @{code Bound} i
haftmann@29789
  2011
  | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
haftmann@29789
  2012
  | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Add} (num_of_term vs t1, num_of_term vs t2)
haftmann@29789
  2013
  | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Sub} (num_of_term vs t1, num_of_term vs t2)
haftmann@29789
  2014
  | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (case (num_of_term vs t1)
haftmann@29789
  2015
     of @{code C} i => @{code Mul} (i, num_of_term vs t2)
haftmann@29789
  2016
      | _ => error "num_of_term: unsupported Multiplication")
haftmann@29789
  2017
  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t')) = @{code C} (HOLogic.dest_numeral t')
haftmann@29789
  2018
  | num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} $ t') = @{code C} (HOLogic.dest_numeral t')
haftmann@29789
  2019
  | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);
haftmann@29789
  2020
haftmann@29789
  2021
fun fm_of_term vs @{term True} = @{code T}
haftmann@29789
  2022
  | fm_of_term vs @{term False} = @{code F}
haftmann@29789
  2023
  | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
haftmann@29789
  2024
  | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
haftmann@29789
  2025
  | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) 
haftmann@29789
  2026
  | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
haftmann@29789
  2027
  | fm_of_term vs (@{term "op &"} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
haftmann@29789
  2028
  | fm_of_term vs (@{term "op |"} $ t1 $ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
haftmann@29789
  2029
  | fm_of_term vs (@{term "op -->"} $ t1 $ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
haftmann@29789
  2030
  | fm_of_term vs (@{term "Not"} $ t') = @{code NOT} (fm_of_term vs t')
haftmann@29789
  2031
  | fm_of_term vs (Const ("Ex", _) $ Abs (xn, xT, p)) =
haftmann@29789
  2032
      @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
haftmann@29789
  2033
  | fm_of_term vs (Const ("All", _) $ Abs (xn, xT, p)) =
haftmann@29789
  2034
      @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
haftmann@29789
  2035
  | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
haftmann@29789
  2036
haftmann@29789
  2037
fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} $ HOLogic.mk_number HOLogic.intT i
haftmann@29789
  2038
  | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs))
haftmann@29789
  2039
  | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
haftmann@29789
  2040
  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
haftmann@29789
  2041
      term_of_num vs t1 $ term_of_num vs t2
haftmann@29789
  2042
  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $
haftmann@29789
  2043
      term_of_num vs t1 $ term_of_num vs t2
haftmann@29789
  2044
  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
haftmann@29789
  2045
      term_of_num vs (@{code C} i) $ term_of_num vs t2
haftmann@29789
  2046
  | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
haftmann@29789
  2047
haftmann@29789
  2048
fun term_of_fm vs @{code T} = HOLogic.true_const 
haftmann@29789
  2049
  | term_of_fm vs @{code F} = HOLogic.false_const
haftmann@29789
  2050
  | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  2051
      term_of_num vs t $ @{term "0::real"}
haftmann@29789
  2052
  | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  2053
      term_of_num vs t $ @{term "0::real"}
haftmann@29789
  2054
  | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  2055
      @{term "0::real"} $ term_of_num vs t
haftmann@29789
  2056
  | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  2057
      @{term "0::real"} $ term_of_num vs t
haftmann@29789
  2058
  | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  2059
      term_of_num vs t $ @{term "0::real"}
haftmann@29789
  2060
  | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
haftmann@29789
  2061
  | term_of_fm vs (@{code NOT} t') = HOLogic.Not $ term_of_fm vs t'
haftmann@29789
  2062
  | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2
haftmann@29789
  2063
  | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2
haftmann@29789
  2064
  | term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
haftmann@29789
  2065
  | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $
haftmann@29789
  2066
      term_of_fm vs t1 $ term_of_fm vs t2
haftmann@29789
  2067
  | term_of_fm vs _ = error "If this is raised, Isabelle/HOL or generate_code is inconsistent.";
haftmann@29789
  2068
haftmann@29789
  2069
in fn ct =>
haftmann@29789
  2070
  let 
haftmann@29789
  2071
    val thy = Thm.theory_of_cterm ct;
haftmann@29789
  2072
    val t = Thm.term_of ct;
haftmann@29789
  2073
    val fs = OldTerm.term_frees t;
haftmann@29789
  2074
    val vs = fs ~~ (0 upto (length fs - 1));
haftmann@29789
  2075
    val res = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_fm vs (@{code linrqe} (fm_of_term vs t))));
haftmann@29789
  2076
  in Thm.cterm_of thy res end
haftmann@29789
  2077
end;
haftmann@29789
  2078
*}
haftmann@29789
  2079
haftmann@29789
  2080
use "ferrack_tac.ML"
haftmann@29789
  2081
setup Ferrack_Tac.setup
haftmann@29789
  2082
haftmann@29789
  2083
lemma
haftmann@29789
  2084
  fixes x :: real
haftmann@29789
  2085
  shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1"
haftmann@29789
  2086
apply rferrack
haftmann@29789
  2087
done
haftmann@29789
  2088
haftmann@29789
  2089
lemma
haftmann@29789
  2090
  fixes x :: real
haftmann@29789
  2091
  shows "\<exists>y \<le> x. x = y + 1"
haftmann@29789
  2092
apply rferrack
haftmann@29789
  2093
done
haftmann@29789
  2094
haftmann@29789
  2095
lemma
haftmann@29789
  2096
  fixes x :: real
haftmann@29789
  2097
  shows "\<not> (\<exists>z. x + z = x + z + 1)"
haftmann@29789
  2098
apply rferrack
haftmann@29789
  2099
done
haftmann@29789
  2100
haftmann@29789
  2101
end