src/HOL/Decision_Procs/Ferrack.thy
author wenzelm
Fri Dec 02 14:54:25 2011 +0100 (2011-12-02)
changeset 45740 132a3e1c0fe5
parent 44779 98d597c4193d
child 46670 e9aa6d151329
permissions -rw-r--r--
more antiquotations;
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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 DP_Library
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  "~~/src/HOL/Library/Efficient_Nat" "~~/src/HOL/Library/Old_Recdef"
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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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  (****                            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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primrec num_size :: "num \<Rightarrow> nat" where
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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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primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where
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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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fun fmsize :: "fm \<Rightarrow> nat" where
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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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primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where
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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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fun not:: "fm \<Rightarrow> fm" where
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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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definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
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  "conj p q = (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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definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
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  "disj p q = (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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definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
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  "imp p q = (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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definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
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  "iff p q = (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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fun qfree:: "fm \<Rightarrow> bool" where
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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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primrec numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) where
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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) simp_all
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primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
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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"
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| "bound0 (A p) = False"
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lemma bound0_I:
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  assumes bp: "bound0 p"
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  shows "Ifm (b#bs) p = Ifm (b'#bs) p"
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using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
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by (induct p) auto
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lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
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by (cases p, auto)
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lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
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by (cases p, auto)
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lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
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using conj_def by auto 
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lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
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using conj_def by auto 
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lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
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using disj_def by auto 
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lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
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using disj_def by auto 
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lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
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using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
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lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
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using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
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lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
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  by (unfold iff_def,cases "p=q", auto)
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lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
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using iff_def by (unfold iff_def,cases "p=q", auto)
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fun decrnum:: "num \<Rightarrow> num"  where
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  "decrnum (Bound n) = Bound (n - 1)"
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| "decrnum (Neg a) = Neg (decrnum a)"
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| "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
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| "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
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| "decrnum (Mul c a) = Mul c (decrnum a)"
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| "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
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| "decrnum a = a"
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fun decr :: "fm \<Rightarrow> fm" where
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  "decr (Lt a) = Lt (decrnum a)"
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| "decr (Le a) = Le (decrnum a)"
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| "decr (Gt a) = Gt (decrnum a)"
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| "decr (Ge a) = Ge (decrnum a)"
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| "decr (Eq a) = Eq (decrnum a)"
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| "decr (NEq a) = NEq (decrnum a)"
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| "decr (NOT p) = NOT (decr p)" 
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| "decr (And p q) = conj (decr p) (decr q)"
haftmann@36853
   278
| "decr (Or p q) = disj (decr p) (decr q)"
haftmann@36853
   279
| "decr (Imp p q) = imp (decr p) (decr q)"
haftmann@36853
   280
| "decr (Iff p q) = iff (decr p) (decr q)"
haftmann@36853
   281
| "decr p = p"
haftmann@29789
   282
haftmann@29789
   283
lemma decrnum: assumes nb: "numbound0 t"
haftmann@29789
   284
  shows "Inum (x#bs) t = Inum bs (decrnum t)"
nipkow@41842
   285
  using nb by (induct t rule: decrnum.induct, simp_all)
haftmann@29789
   286
haftmann@29789
   287
lemma decr: assumes nb: "bound0 p"
haftmann@29789
   288
  shows "Ifm (x#bs) p = Ifm bs (decr p)"
haftmann@29789
   289
  using nb 
nipkow@41842
   290
  by (induct p rule: decr.induct, simp_all add: decrnum)
haftmann@29789
   291
haftmann@29789
   292
lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
haftmann@29789
   293
by (induct p, simp_all)
haftmann@29789
   294
haftmann@36853
   295
fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
haftmann@29789
   296
  "isatom T = True"
haftmann@36853
   297
| "isatom F = True"
haftmann@36853
   298
| "isatom (Lt a) = True"
haftmann@36853
   299
| "isatom (Le a) = True"
haftmann@36853
   300
| "isatom (Gt a) = True"
haftmann@36853
   301
| "isatom (Ge a) = True"
haftmann@36853
   302
| "isatom (Eq a) = True"
haftmann@36853
   303
| "isatom (NEq a) = True"
haftmann@36853
   304
| "isatom p = False"
haftmann@29789
   305
haftmann@29789
   306
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
haftmann@29789
   307
by (induct p, simp_all)
haftmann@29789
   308
haftmann@35416
   309
definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
haftmann@36853
   310
  "djf f p q = (if q=T then T else if q=F then f p else 
haftmann@29789
   311
  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
haftmann@35416
   312
definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
haftmann@36853
   313
  "evaldjf f ps = foldr (djf f) ps F"
haftmann@29789
   314
haftmann@29789
   315
lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
haftmann@29789
   316
by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
haftmann@29789
   317
(cases "f p", simp_all add: Let_def djf_def) 
haftmann@29789
   318
haftmann@29789
   319
haftmann@29789
   320
lemma djf_simps:
haftmann@29789
   321
  "djf f p T = T"
haftmann@29789
   322
  "djf f p F = f p"
haftmann@29789
   323
  "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
   324
  by (simp_all add: djf_def)
haftmann@29789
   325
haftmann@29789
   326
lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
haftmann@29789
   327
  by(induct ps, simp_all add: evaldjf_def djf_Or)
haftmann@29789
   328
haftmann@29789
   329
lemma evaldjf_bound0: 
haftmann@29789
   330
  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
haftmann@29789
   331
  shows "bound0 (evaldjf f xs)"
haftmann@29789
   332
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
haftmann@29789
   333
haftmann@29789
   334
lemma evaldjf_qf: 
haftmann@29789
   335
  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
haftmann@29789
   336
  shows "qfree (evaldjf f xs)"
haftmann@29789
   337
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
haftmann@29789
   338
haftmann@36853
   339
fun disjuncts :: "fm \<Rightarrow> fm list" where
haftmann@36853
   340
  "disjuncts (Or p q) = disjuncts p @ disjuncts q"
haftmann@36853
   341
| "disjuncts F = []"
haftmann@36853
   342
| "disjuncts p = [p]"
haftmann@29789
   343
haftmann@29789
   344
lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
haftmann@29789
   345
by(induct p rule: disjuncts.induct, auto)
haftmann@29789
   346
haftmann@29789
   347
lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
haftmann@29789
   348
proof-
haftmann@29789
   349
  assume nb: "bound0 p"
haftmann@29789
   350
  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
haftmann@29789
   351
  thus ?thesis by (simp only: list_all_iff)
haftmann@29789
   352
qed
haftmann@29789
   353
haftmann@29789
   354
lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
haftmann@29789
   355
proof-
haftmann@29789
   356
  assume qf: "qfree p"
haftmann@29789
   357
  hence "list_all qfree (disjuncts p)"
haftmann@29789
   358
    by (induct p rule: disjuncts.induct, auto)
haftmann@29789
   359
  thus ?thesis by (simp only: list_all_iff)
haftmann@29789
   360
qed
haftmann@29789
   361
haftmann@35416
   362
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
haftmann@36853
   363
  "DJ f p = evaldjf f (disjuncts p)"
haftmann@29789
   364
haftmann@29789
   365
lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
haftmann@29789
   366
  and fF: "f F = F"
haftmann@29789
   367
  shows "Ifm bs (DJ f p) = Ifm bs (f p)"
haftmann@29789
   368
proof-
haftmann@29789
   369
  have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
haftmann@29789
   370
    by (simp add: DJ_def evaldjf_ex) 
haftmann@29789
   371
  also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
haftmann@29789
   372
  finally show ?thesis .
haftmann@29789
   373
qed
haftmann@29789
   374
haftmann@29789
   375
lemma DJ_qf: assumes 
haftmann@29789
   376
  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
haftmann@29789
   377
  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
haftmann@29789
   378
proof(clarify)
haftmann@29789
   379
  fix  p assume qf: "qfree p"
haftmann@29789
   380
  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
haftmann@29789
   381
  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
haftmann@29789
   382
  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
haftmann@29789
   383
  
haftmann@29789
   384
  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
haftmann@29789
   385
qed
haftmann@29789
   386
haftmann@29789
   387
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
   388
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
haftmann@29789
   389
proof(clarify)
haftmann@29789
   390
  fix p::fm and bs
haftmann@29789
   391
  assume qf: "qfree p"
haftmann@29789
   392
  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
haftmann@29789
   393
  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
haftmann@29789
   394
  have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
haftmann@29789
   395
    by (simp add: DJ_def evaldjf_ex)
haftmann@29789
   396
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
haftmann@29789
   397
  also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
haftmann@29789
   398
  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
haftmann@29789
   399
qed
haftmann@29789
   400
  (* Simplification *)
haftmann@36853
   401
haftmann@36853
   402
fun maxcoeff:: "num \<Rightarrow> int" where
haftmann@29789
   403
  "maxcoeff (C i) = abs i"
haftmann@36853
   404
| "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
haftmann@36853
   405
| "maxcoeff t = 1"
haftmann@29789
   406
haftmann@29789
   407
lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
haftmann@29789
   408
  by (induct t rule: maxcoeff.induct, auto)
haftmann@29789
   409
haftmann@36853
   410
fun numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" where
huffman@31706
   411
  "numgcdh (C i) = (\<lambda>g. gcd i g)"
haftmann@36853
   412
| "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
haftmann@36853
   413
| "numgcdh t = (\<lambda>g. 1)"
haftmann@36853
   414
haftmann@36853
   415
definition numgcd :: "num \<Rightarrow> int" where
haftmann@36853
   416
  "numgcd t = numgcdh t (maxcoeff t)"
haftmann@29789
   417
haftmann@36853
   418
fun reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" where
haftmann@29789
   419
  "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
haftmann@36853
   420
| "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
haftmann@36853
   421
| "reducecoeffh t = (\<lambda>g. t)"
haftmann@29789
   422
haftmann@36853
   423
definition reducecoeff :: "num \<Rightarrow> num" where
haftmann@36853
   424
  "reducecoeff t =
haftmann@29789
   425
  (let g = numgcd t in 
haftmann@29789
   426
  if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
haftmann@29789
   427
haftmann@36853
   428
fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
haftmann@29789
   429
  "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
haftmann@36853
   430
| "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
haftmann@36853
   431
| "dvdnumcoeff t = (\<lambda>g. False)"
haftmann@29789
   432
haftmann@29789
   433
lemma dvdnumcoeff_trans: 
haftmann@29789
   434
  assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
haftmann@29789
   435
  shows "dvdnumcoeff t g"
haftmann@29789
   436
  using dgt' gdg 
nipkow@30042
   437
  by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg dvd_trans[OF gdg])
haftmann@29789
   438
nipkow@30042
   439
declare dvd_trans [trans add]
haftmann@29789
   440
haftmann@29789
   441
lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
haftmann@29789
   442
by arith
haftmann@29789
   443
haftmann@29789
   444
lemma numgcd0:
haftmann@29789
   445
  assumes g0: "numgcd t = 0"
haftmann@29789
   446
  shows "Inum bs t = 0"
haftmann@29789
   447
  using g0[simplified numgcd_def] 
haftmann@32642
   448
  by (induct t rule: numgcdh.induct, auto simp add: natabs0 maxcoeff_pos min_max.sup_absorb2)
haftmann@29789
   449
haftmann@29789
   450
lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
haftmann@29789
   451
  using gp
huffman@31706
   452
  by (induct t rule: numgcdh.induct, auto)
haftmann@29789
   453
haftmann@29789
   454
lemma numgcd_pos: "numgcd t \<ge>0"
haftmann@29789
   455
  by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
haftmann@29789
   456
haftmann@29789
   457
lemma reducecoeffh:
haftmann@29789
   458
  assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
haftmann@29789
   459
  shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
haftmann@29789
   460
  using gt
wenzelm@41807
   461
proof (induct t rule: reducecoeffh.induct) 
wenzelm@41807
   462
  case (1 i)
wenzelm@41807
   463
  hence gd: "g dvd i" by simp
haftmann@29789
   464
  from gp have gnz: "g \<noteq> 0" by simp
wenzelm@41807
   465
  with assms show ?case by (simp add: real_of_int_div[OF gnz gd])
haftmann@29789
   466
next
wenzelm@41807
   467
  case (2 n c t)
wenzelm@41807
   468
  hence gd: "g dvd c" by simp
haftmann@29789
   469
  from gp have gnz: "g \<noteq> 0" by simp
wenzelm@41807
   470
  from assms 2 show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
haftmann@29789
   471
qed (auto simp add: numgcd_def gp)
haftmann@36853
   472
haftmann@36853
   473
fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
haftmann@29789
   474
  "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
haftmann@36853
   475
| "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
haftmann@36853
   476
| "ismaxcoeff t = (\<lambda>x. True)"
haftmann@29789
   477
haftmann@29789
   478
lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
wenzelm@41807
   479
  by (induct t rule: ismaxcoeff.induct) auto
haftmann@29789
   480
haftmann@29789
   481
lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
haftmann@29789
   482
proof (induct t rule: maxcoeff.induct)
haftmann@29789
   483
  case (2 n c t)
haftmann@29789
   484
  hence H:"ismaxcoeff t (maxcoeff t)" .
wenzelm@41807
   485
  have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by simp
wenzelm@41807
   486
  from ismaxcoeff_mono[OF H thh] show ?case by simp
haftmann@29789
   487
qed simp_all
haftmann@29789
   488
huffman@31706
   489
lemma zgcd_gt1: "gcd i j > (1::int) \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
huffman@31706
   490
  apply (cases "abs i = 0", simp_all add: gcd_int_def)
haftmann@29789
   491
  apply (cases "abs j = 0", simp_all)
haftmann@29789
   492
  apply (cases "abs i = 1", simp_all)
haftmann@29789
   493
  apply (cases "abs j = 1", simp_all)
haftmann@29789
   494
  apply auto
haftmann@29789
   495
  done
haftmann@29789
   496
lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
huffman@31706
   497
  by (induct t rule: numgcdh.induct, auto)
haftmann@29789
   498
haftmann@29789
   499
lemma dvdnumcoeff_aux:
haftmann@29789
   500
  assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
haftmann@29789
   501
  shows "dvdnumcoeff t (numgcdh t m)"
wenzelm@41807
   502
using assms
haftmann@29789
   503
proof(induct t rule: numgcdh.induct)
haftmann@29789
   504
  case (2 n c t) 
haftmann@29789
   505
  let ?g = "numgcdh t m"
wenzelm@41807
   506
  from 2 have th:"gcd c ?g > 1" by simp
haftmann@29789
   507
  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
haftmann@29789
   508
  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
wenzelm@41807
   509
  moreover {assume "abs c > 1" and gp: "?g > 1" with 2
haftmann@29789
   510
    have th: "dvdnumcoeff t ?g" by simp
huffman@31706
   511
    have th': "gcd c ?g dvd ?g" by simp
huffman@31706
   512
    from dvdnumcoeff_trans[OF th' th] have ?case by simp }
haftmann@29789
   513
  moreover {assume "abs c = 0 \<and> ?g > 1"
wenzelm@41807
   514
    with 2 have th: "dvdnumcoeff t ?g" by simp
huffman@31706
   515
    have th': "gcd c ?g dvd ?g" by simp
huffman@31706
   516
    from dvdnumcoeff_trans[OF th' th] have ?case by simp
haftmann@29789
   517
    hence ?case by simp }
haftmann@29789
   518
  moreover {assume "abs c > 1" and g0:"?g = 0" 
wenzelm@41807
   519
    from numgcdh0[OF g0] have "m=0". with 2 g0 have ?case by simp }
haftmann@29789
   520
  ultimately show ?case by blast
huffman@31706
   521
qed auto
haftmann@29789
   522
haftmann@29789
   523
lemma dvdnumcoeff_aux2:
wenzelm@41807
   524
  assumes "numgcd t > 1"
wenzelm@41807
   525
  shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
wenzelm@41807
   526
  using assms
haftmann@29789
   527
proof (simp add: numgcd_def)
haftmann@29789
   528
  let ?mc = "maxcoeff t"
haftmann@29789
   529
  let ?g = "numgcdh t ?mc"
haftmann@29789
   530
  have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
haftmann@29789
   531
  have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
haftmann@29789
   532
  assume H: "numgcdh t ?mc > 1"
haftmann@29789
   533
  from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
haftmann@29789
   534
qed
haftmann@29789
   535
haftmann@29789
   536
lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
haftmann@29789
   537
proof-
haftmann@29789
   538
  let ?g = "numgcd t"
haftmann@29789
   539
  have "?g \<ge> 0"  by (simp add: numgcd_pos)
wenzelm@32960
   540
  hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
haftmann@29789
   541
  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} 
haftmann@29789
   542
  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} 
haftmann@29789
   543
  moreover { assume g1:"?g > 1"
haftmann@29789
   544
    from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
haftmann@29789
   545
    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis 
haftmann@29789
   546
      by (simp add: reducecoeff_def Let_def)} 
haftmann@29789
   547
  ultimately show ?thesis by blast
haftmann@29789
   548
qed
haftmann@29789
   549
haftmann@29789
   550
lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
haftmann@29789
   551
by (induct t rule: reducecoeffh.induct, auto)
haftmann@29789
   552
haftmann@29789
   553
lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
haftmann@29789
   554
using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
haftmann@29789
   555
haftmann@29789
   556
consts
haftmann@29789
   557
  numadd:: "num \<times> num \<Rightarrow> num"
haftmann@36853
   558
haftmann@29789
   559
recdef numadd "measure (\<lambda> (t,s). size t + size s)"
haftmann@29789
   560
  "numadd (CN n1 c1 r1,CN n2 c2 r2) =
haftmann@29789
   561
  (if n1=n2 then 
haftmann@29789
   562
  (let c = c1 + c2
haftmann@29789
   563
  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
haftmann@29789
   564
  else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) 
haftmann@29789
   565
  else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
haftmann@29789
   566
  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
haftmann@29789
   567
  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
haftmann@29789
   568
  "numadd (C b1, C b2) = C (b1+b2)"
haftmann@29789
   569
  "numadd (a,b) = Add a b"
haftmann@29789
   570
haftmann@29789
   571
lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
haftmann@29789
   572
apply (induct t s rule: numadd.induct, simp_all add: Let_def)
haftmann@29789
   573
apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
haftmann@29789
   574
apply (case_tac "n1 = n2", simp_all add: algebra_simps)
haftmann@29789
   575
by (simp only: left_distrib[symmetric],simp)
haftmann@29789
   576
haftmann@29789
   577
lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
haftmann@29789
   578
by (induct t s rule: numadd.induct, auto simp add: Let_def)
haftmann@29789
   579
haftmann@36853
   580
fun nummul:: "num \<Rightarrow> int \<Rightarrow> num" where
haftmann@29789
   581
  "nummul (C j) = (\<lambda> i. C (i*j))"
haftmann@36853
   582
| "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
haftmann@36853
   583
| "nummul t = (\<lambda> i. Mul i t)"
haftmann@29789
   584
haftmann@29789
   585
lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
haftmann@29789
   586
by (induct t rule: nummul.induct, auto simp add: algebra_simps)
haftmann@29789
   587
haftmann@29789
   588
lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
haftmann@29789
   589
by (induct t rule: nummul.induct, auto )
haftmann@29789
   590
haftmann@35416
   591
definition numneg :: "num \<Rightarrow> num" where
haftmann@36853
   592
  "numneg t = nummul t (- 1)"
haftmann@29789
   593
haftmann@35416
   594
definition numsub :: "num \<Rightarrow> num \<Rightarrow> num" where
haftmann@36853
   595
  "numsub s t = (if s = t then C 0 else numadd (s,numneg t))"
haftmann@29789
   596
haftmann@29789
   597
lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
haftmann@29789
   598
using numneg_def by simp
haftmann@29789
   599
haftmann@29789
   600
lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
haftmann@29789
   601
using numneg_def by simp
haftmann@29789
   602
haftmann@29789
   603
lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
haftmann@29789
   604
using numsub_def by simp
haftmann@29789
   605
haftmann@29789
   606
lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
haftmann@29789
   607
using numsub_def by simp
haftmann@29789
   608
haftmann@36853
   609
primrec simpnum:: "num \<Rightarrow> num" where
haftmann@29789
   610
  "simpnum (C j) = C j"
haftmann@36853
   611
| "simpnum (Bound n) = CN n 1 (C 0)"
haftmann@36853
   612
| "simpnum (Neg t) = numneg (simpnum t)"
haftmann@36853
   613
| "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
haftmann@36853
   614
| "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
haftmann@36853
   615
| "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
haftmann@36853
   616
| "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
haftmann@29789
   617
haftmann@29789
   618
lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
haftmann@36853
   619
by (induct t) simp_all
haftmann@29789
   620
haftmann@29789
   621
lemma simpnum_numbound0[simp]: 
haftmann@29789
   622
  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
haftmann@36853
   623
by (induct t) simp_all
haftmann@29789
   624
haftmann@36853
   625
fun nozerocoeff:: "num \<Rightarrow> bool" where
haftmann@29789
   626
  "nozerocoeff (C c) = True"
haftmann@36853
   627
| "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
haftmann@36853
   628
| "nozerocoeff t = True"
haftmann@29789
   629
haftmann@29789
   630
lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
haftmann@29789
   631
by (induct a b rule: numadd.induct,auto simp add: Let_def)
haftmann@29789
   632
haftmann@29789
   633
lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
haftmann@29789
   634
by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
haftmann@29789
   635
haftmann@29789
   636
lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
haftmann@29789
   637
by (simp add: numneg_def nummul_nz)
haftmann@29789
   638
haftmann@29789
   639
lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
haftmann@29789
   640
by (simp add: numsub_def numneg_nz numadd_nz)
haftmann@29789
   641
haftmann@29789
   642
lemma simpnum_nz: "nozerocoeff (simpnum t)"
haftmann@36853
   643
by(induct t) (simp_all add: numadd_nz numneg_nz numsub_nz nummul_nz)
haftmann@29789
   644
haftmann@29789
   645
lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
haftmann@29789
   646
proof (induct t rule: maxcoeff.induct)
haftmann@29789
   647
  case (2 n c t)
wenzelm@41807
   648
  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp_all
wenzelm@41807
   649
  have "max (abs c) (maxcoeff t) \<ge> abs c" by simp
haftmann@29789
   650
  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
wenzelm@41807
   651
  with 2 show ?case by simp
haftmann@29789
   652
qed auto
haftmann@29789
   653
haftmann@29789
   654
lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
haftmann@29789
   655
proof-
haftmann@29789
   656
  from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
haftmann@29789
   657
  from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
haftmann@29789
   658
  from maxcoeff_nz[OF nz th] show ?thesis .
haftmann@29789
   659
qed
haftmann@29789
   660
haftmann@35416
   661
definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where
haftmann@36853
   662
  "simp_num_pair = (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
haftmann@29789
   663
   (let t' = simpnum t ; g = numgcd t' in 
huffman@31706
   664
      if g > 1 then (let g' = gcd n g in 
haftmann@29789
   665
        if g' = 1 then (t',n) 
haftmann@29789
   666
        else (reducecoeffh t' g', n div g')) 
haftmann@29789
   667
      else (t',n))))"
haftmann@29789
   668
haftmann@29789
   669
lemma simp_num_pair_ci:
haftmann@29789
   670
  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
   671
  (is "?lhs = ?rhs")
haftmann@29789
   672
proof-
haftmann@29789
   673
  let ?t' = "simpnum t"
haftmann@29789
   674
  let ?g = "numgcd ?t'"
huffman@31706
   675
  let ?g' = "gcd n ?g"
haftmann@29789
   676
  {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
haftmann@29789
   677
  moreover
haftmann@29789
   678
  { assume nnz: "n \<noteq> 0"
wenzelm@44779
   679
    {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def) }
haftmann@29789
   680
    moreover
haftmann@29789
   681
    {assume g1:"?g>1" hence g0: "?g > 0" by simp
huffman@31706
   682
      from g1 nnz have gp0: "?g' \<noteq> 0" by simp
nipkow@31952
   683
      hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith 
haftmann@29789
   684
      hence "?g'= 1 \<or> ?g' > 1" by arith
wenzelm@44779
   685
      moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
haftmann@29789
   686
      moreover {assume g'1:"?g'>1"
wenzelm@32960
   687
        from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
wenzelm@32960
   688
        let ?tt = "reducecoeffh ?t' ?g'"
wenzelm@32960
   689
        let ?t = "Inum bs ?tt"
wenzelm@32960
   690
        have gpdg: "?g' dvd ?g" by simp
wenzelm@32960
   691
        have gpdd: "?g' dvd n" by simp 
wenzelm@32960
   692
        have gpdgp: "?g' dvd ?g'" by simp
wenzelm@32960
   693
        from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
wenzelm@32960
   694
        have th2:"real ?g' * ?t = Inum bs ?t'" by simp
wenzelm@41807
   695
        from g1 g'1 have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
wenzelm@32960
   696
        also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
wenzelm@32960
   697
        also have "\<dots> = (Inum bs ?t' / real n)"
wenzelm@32960
   698
          using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
wenzelm@41807
   699
        finally have "?lhs = Inum bs t / real n" by simp
wenzelm@41807
   700
        then have ?thesis by (simp add: simp_num_pair_def) }
wenzelm@41807
   701
      ultimately have ?thesis by blast }
wenzelm@41807
   702
    ultimately have ?thesis by blast }
haftmann@29789
   703
  ultimately show ?thesis by blast
haftmann@29789
   704
qed
haftmann@29789
   705
haftmann@29789
   706
lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
haftmann@29789
   707
  shows "numbound0 t' \<and> n' >0"
haftmann@29789
   708
proof-
wenzelm@41807
   709
  let ?t' = "simpnum t"
haftmann@29789
   710
  let ?g = "numgcd ?t'"
huffman@31706
   711
  let ?g' = "gcd n ?g"
wenzelm@41807
   712
  { assume nz: "n = 0" hence ?thesis using assms by (simp add: Let_def simp_num_pair_def) }
haftmann@29789
   713
  moreover
haftmann@29789
   714
  { assume nnz: "n \<noteq> 0"
wenzelm@41807
   715
    { assume "\<not> ?g > 1" hence ?thesis using assms
wenzelm@41807
   716
        by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0) }
haftmann@29789
   717
    moreover
wenzelm@41807
   718
    { assume g1:"?g>1" hence g0: "?g > 0" by simp
huffman@31706
   719
      from g1 nnz have gp0: "?g' \<noteq> 0" by simp
nipkow@31952
   720
      hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
haftmann@29789
   721
      hence "?g'= 1 \<or> ?g' > 1" by arith
wenzelm@41807
   722
      moreover {
wenzelm@41807
   723
        assume "?g' = 1" hence ?thesis using assms g1
wenzelm@41807
   724
          by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0) }
wenzelm@41807
   725
      moreover {
wenzelm@41807
   726
        assume g'1: "?g' > 1"
wenzelm@32960
   727
        have gpdg: "?g' dvd ?g" by simp
wenzelm@41807
   728
        have gpdd: "?g' dvd n" by simp
wenzelm@32960
   729
        have gpdgp: "?g' dvd ?g'" by simp
wenzelm@32960
   730
        from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
wenzelm@32960
   731
        from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
wenzelm@32960
   732
        have "n div ?g' >0" by simp
wenzelm@41807
   733
        hence ?thesis using assms g1 g'1
wenzelm@41807
   734
          by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0) }
wenzelm@41807
   735
      ultimately have ?thesis by blast }
wenzelm@41807
   736
    ultimately have ?thesis by blast }
haftmann@29789
   737
  ultimately show ?thesis by blast
haftmann@29789
   738
qed
haftmann@29789
   739
haftmann@36853
   740
fun simpfm :: "fm \<Rightarrow> fm" where
haftmann@29789
   741
  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
haftmann@36853
   742
| "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
haftmann@36853
   743
| "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
haftmann@36853
   744
| "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
haftmann@36853
   745
| "simpfm (NOT p) = not (simpfm p)"
haftmann@36853
   746
| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
haftmann@29789
   747
  | _ \<Rightarrow> Lt a')"
haftmann@36853
   748
| "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@36853
   749
| "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@36853
   750
| "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@36853
   751
| "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@36853
   752
| "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@36853
   753
| "simpfm p = p"
haftmann@29789
   754
lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p"
haftmann@29789
   755
proof(induct p rule: simpfm.induct)
haftmann@29789
   756
  case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   757
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   758
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   759
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   760
  ultimately show ?case by blast
haftmann@29789
   761
next
haftmann@29789
   762
  case (7 a)  let ?sa = "simpnum a" 
haftmann@29789
   763
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   764
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   765
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   766
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   767
  ultimately show ?case by blast
haftmann@29789
   768
next
haftmann@29789
   769
  case (8 a)  let ?sa = "simpnum a" 
haftmann@29789
   770
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   771
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   772
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   773
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   774
  ultimately show ?case by blast
haftmann@29789
   775
next
haftmann@29789
   776
  case (9 a)  let ?sa = "simpnum a" 
haftmann@29789
   777
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   778
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   779
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   780
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   781
  ultimately show ?case by blast
haftmann@29789
   782
next
haftmann@29789
   783
  case (10 a)  let ?sa = "simpnum a" 
haftmann@29789
   784
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   785
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   786
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   787
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   788
  ultimately show ?case by blast
haftmann@29789
   789
next
haftmann@29789
   790
  case (11 a)  let ?sa = "simpnum a" 
haftmann@29789
   791
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
haftmann@29789
   792
  {fix v assume "?sa = C v" hence ?case using sa by simp }
haftmann@29789
   793
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
haftmann@29789
   794
      by (cases ?sa, simp_all add: Let_def)}
haftmann@29789
   795
  ultimately show ?case by blast
haftmann@29789
   796
qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
haftmann@29789
   797
haftmann@29789
   798
haftmann@29789
   799
lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
haftmann@29789
   800
proof(induct p rule: simpfm.induct)
haftmann@29789
   801
  case (6 a) hence nb: "numbound0 a" by simp
haftmann@29789
   802
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
wenzelm@44779
   803
  thus ?case by (cases "simpnum a") (auto simp add: Let_def)
haftmann@29789
   804
next
haftmann@29789
   805
  case (7 a) hence nb: "numbound0 a" by simp
haftmann@29789
   806
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
wenzelm@44779
   807
  thus ?case by (cases "simpnum a") (auto simp add: Let_def)
haftmann@29789
   808
next
haftmann@29789
   809
  case (8 a) hence nb: "numbound0 a" by simp
haftmann@29789
   810
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
wenzelm@44779
   811
  thus ?case by (cases "simpnum a") (auto simp add: Let_def)
haftmann@29789
   812
next
haftmann@29789
   813
  case (9 a) hence nb: "numbound0 a" by simp
haftmann@29789
   814
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
wenzelm@44779
   815
  thus ?case by (cases "simpnum a") (auto simp add: Let_def)
haftmann@29789
   816
next
haftmann@29789
   817
  case (10 a) hence nb: "numbound0 a" by simp
haftmann@29789
   818
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
wenzelm@44779
   819
  thus ?case by (cases "simpnum a") (auto simp add: Let_def)
haftmann@29789
   820
next
haftmann@29789
   821
  case (11 a) hence nb: "numbound0 a" by simp
haftmann@29789
   822
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
wenzelm@44779
   823
  thus ?case by (cases "simpnum a") (auto simp add: Let_def)
haftmann@29789
   824
qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
haftmann@29789
   825
haftmann@29789
   826
lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
wenzelm@44779
   827
  apply (induct p rule: simpfm.induct)
wenzelm@44779
   828
  apply (auto simp add: Let_def)
wenzelm@44779
   829
  apply (case_tac "simpnum a", auto)+
wenzelm@44779
   830
  done
haftmann@29789
   831
haftmann@29789
   832
consts prep :: "fm \<Rightarrow> fm"
haftmann@29789
   833
recdef prep "measure fmsize"
haftmann@29789
   834
  "prep (E T) = T"
haftmann@29789
   835
  "prep (E F) = F"
haftmann@29789
   836
  "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
haftmann@29789
   837
  "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
haftmann@29789
   838
  "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
haftmann@29789
   839
  "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
haftmann@29789
   840
  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
haftmann@29789
   841
  "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
haftmann@29789
   842
  "prep (E p) = E (prep p)"
haftmann@29789
   843
  "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
haftmann@29789
   844
  "prep (A p) = prep (NOT (E (NOT p)))"
haftmann@29789
   845
  "prep (NOT (NOT p)) = prep p"
haftmann@29789
   846
  "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
haftmann@29789
   847
  "prep (NOT (A p)) = prep (E (NOT p))"
haftmann@29789
   848
  "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
haftmann@29789
   849
  "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
haftmann@29789
   850
  "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
haftmann@29789
   851
  "prep (NOT p) = not (prep p)"
haftmann@29789
   852
  "prep (Or p q) = disj (prep p) (prep q)"
haftmann@29789
   853
  "prep (And p q) = conj (prep p) (prep q)"
haftmann@29789
   854
  "prep (Imp p q) = prep (Or (NOT p) q)"
haftmann@29789
   855
  "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
haftmann@29789
   856
  "prep p = p"
haftmann@29789
   857
(hints simp add: fmsize_pos)
haftmann@29789
   858
lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
wenzelm@44779
   859
  by (induct p rule: prep.induct) auto
haftmann@29789
   860
haftmann@29789
   861
  (* Generic quantifier elimination *)
haftmann@36853
   862
function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where
haftmann@29789
   863
  "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
haftmann@36853
   864
| "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
haftmann@36853
   865
| "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
haftmann@36853
   866
| "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
haftmann@36853
   867
| "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
haftmann@36853
   868
| "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
haftmann@36853
   869
| "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
haftmann@36853
   870
| "qelim p = (\<lambda> y. simpfm p)"
haftmann@36853
   871
by pat_completeness auto
haftmann@36853
   872
termination qelim by (relation "measure fmsize") simp_all
haftmann@29789
   873
haftmann@29789
   874
lemma qelim_ci:
haftmann@29789
   875
  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
haftmann@29789
   876
  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
haftmann@29789
   877
using qe_inv DJ_qe[OF qe_inv] 
haftmann@29789
   878
by(induct p rule: qelim.induct) 
haftmann@29789
   879
(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
haftmann@29789
   880
  simpfm simpfm_qf simp del: simpfm.simps)
haftmann@29789
   881
haftmann@36853
   882
fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) where
haftmann@29789
   883
  "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
haftmann@36853
   884
| "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
haftmann@36853
   885
| "minusinf (Eq  (CN 0 c e)) = F"
haftmann@36853
   886
| "minusinf (NEq (CN 0 c e)) = T"
haftmann@36853
   887
| "minusinf (Lt  (CN 0 c e)) = T"
haftmann@36853
   888
| "minusinf (Le  (CN 0 c e)) = T"
haftmann@36853
   889
| "minusinf (Gt  (CN 0 c e)) = F"
haftmann@36853
   890
| "minusinf (Ge  (CN 0 c e)) = F"
haftmann@36853
   891
| "minusinf p = p"
haftmann@29789
   892
haftmann@36853
   893
fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) where
haftmann@29789
   894
  "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
haftmann@36853
   895
| "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
haftmann@36853
   896
| "plusinf (Eq  (CN 0 c e)) = F"
haftmann@36853
   897
| "plusinf (NEq (CN 0 c e)) = T"
haftmann@36853
   898
| "plusinf (Lt  (CN 0 c e)) = F"
haftmann@36853
   899
| "plusinf (Le  (CN 0 c e)) = F"
haftmann@36853
   900
| "plusinf (Gt  (CN 0 c e)) = T"
haftmann@36853
   901
| "plusinf (Ge  (CN 0 c e)) = T"
haftmann@36853
   902
| "plusinf p = p"
haftmann@29789
   903
haftmann@36853
   904
fun isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *) where
haftmann@29789
   905
  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" 
haftmann@36853
   906
| "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" 
haftmann@36853
   907
| "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@36853
   908
| "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@36853
   909
| "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@36853
   910
| "isrlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@36853
   911
| "isrlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@36853
   912
| "isrlfm (Ge  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
haftmann@36853
   913
| "isrlfm p = (isatom p \<and> (bound0 p))"
haftmann@29789
   914
haftmann@29789
   915
  (* splits the bounded from the unbounded part*)
haftmann@36853
   916
function (sequential) rsplit0 :: "num \<Rightarrow> int \<times> num" where
haftmann@29789
   917
  "rsplit0 (Bound 0) = (1,C 0)"
haftmann@36853
   918
| "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b 
haftmann@29789
   919
              in (ca+cb, Add ta tb))"
haftmann@36853
   920
| "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
haftmann@36853
   921
| "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
haftmann@36853
   922
| "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
haftmann@36853
   923
| "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
haftmann@36853
   924
| "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
haftmann@36853
   925
| "rsplit0 t = (0,t)"
haftmann@36853
   926
by pat_completeness auto
haftmann@36853
   927
termination rsplit0 by (relation "measure num_size") simp_all
haftmann@36853
   928
haftmann@29789
   929
lemma rsplit0: 
haftmann@29789
   930
  shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
haftmann@29789
   931
proof (induct t rule: rsplit0.induct)
haftmann@29789
   932
  case (2 a b) 
haftmann@29789
   933
  let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
haftmann@29789
   934
  let ?ca = "fst ?sa" let ?cb = "fst ?sb"
haftmann@29789
   935
  let ?ta = "snd ?sa" let ?tb = "snd ?sb"
wenzelm@41807
   936
  from 2 have nb: "numbound0 (snd(rsplit0 (Add a b)))" 
haftmann@36853
   937
    by (cases "rsplit0 a") (auto simp add: Let_def split_def)
haftmann@29789
   938
  have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) = 
haftmann@29789
   939
    Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
haftmann@29789
   940
    by (simp add: Let_def split_def algebra_simps)
wenzelm@41807
   941
  also have "\<dots> = Inum bs a + Inum bs b" using 2 by (cases "rsplit0 a") auto
haftmann@29789
   942
  finally show ?case using nb by simp 
wenzelm@41807
   943
qed (auto simp add: Let_def split_def algebra_simps, simp add: right_distrib[symmetric])
haftmann@29789
   944
haftmann@29789
   945
    (* Linearize a formula*)
haftmann@29789
   946
definition
haftmann@29789
   947
  lt :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
   948
where
haftmann@29789
   949
  "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) 
haftmann@29789
   950
    else (Gt (CN 0 (-c) (Neg t))))"
haftmann@29789
   951
haftmann@29789
   952
definition
haftmann@29789
   953
  le :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
   954
where
haftmann@29789
   955
  "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) 
haftmann@29789
   956
    else (Ge (CN 0 (-c) (Neg t))))"
haftmann@29789
   957
haftmann@29789
   958
definition
haftmann@29789
   959
  gt :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
   960
where
haftmann@29789
   961
  "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) 
haftmann@29789
   962
    else (Lt (CN 0 (-c) (Neg t))))"
haftmann@29789
   963
haftmann@29789
   964
definition
haftmann@29789
   965
  ge :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
   966
where
haftmann@29789
   967
  "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) 
haftmann@29789
   968
    else (Le (CN 0 (-c) (Neg t))))"
haftmann@29789
   969
haftmann@29789
   970
definition
haftmann@29789
   971
  eq :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
   972
where
haftmann@29789
   973
  "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) 
haftmann@29789
   974
    else (Eq (CN 0 (-c) (Neg t))))"
haftmann@29789
   975
haftmann@29789
   976
definition
haftmann@29789
   977
  neq :: "int \<Rightarrow> num \<Rightarrow> fm"
haftmann@29789
   978
where
haftmann@29789
   979
  "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) 
haftmann@29789
   980
    else (NEq (CN 0 (-c) (Neg t))))"
haftmann@29789
   981
haftmann@29789
   982
lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
haftmann@29789
   983
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
   984
by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
   985
haftmann@29789
   986
lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
haftmann@29789
   987
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
   988
by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
   989
haftmann@29789
   990
lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
haftmann@29789
   991
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
   992
by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
   993
haftmann@29789
   994
lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
haftmann@29789
   995
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
   996
by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
   997
haftmann@29789
   998
lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
haftmann@29789
   999
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
  1000
by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
  1001
haftmann@29789
  1002
lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
haftmann@29789
  1003
using rsplit0[where bs = "bs" and t="t"]
haftmann@29789
  1004
by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
haftmann@29789
  1005
haftmann@29789
  1006
lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
haftmann@29789
  1007
by (auto simp add: conj_def)
haftmann@29789
  1008
lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
haftmann@29789
  1009
by (auto simp add: disj_def)
haftmann@29789
  1010
haftmann@29789
  1011
consts rlfm :: "fm \<Rightarrow> fm"
haftmann@29789
  1012
recdef rlfm "measure fmsize"
haftmann@29789
  1013
  "rlfm (And p q) = conj (rlfm p) (rlfm q)"
haftmann@29789
  1014
  "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
haftmann@29789
  1015
  "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
haftmann@29789
  1016
  "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
haftmann@29789
  1017
  "rlfm (Lt a) = split lt (rsplit0 a)"
haftmann@29789
  1018
  "rlfm (Le a) = split le (rsplit0 a)"
haftmann@29789
  1019
  "rlfm (Gt a) = split gt (rsplit0 a)"
haftmann@29789
  1020
  "rlfm (Ge a) = split ge (rsplit0 a)"
haftmann@29789
  1021
  "rlfm (Eq a) = split eq (rsplit0 a)"
haftmann@29789
  1022
  "rlfm (NEq a) = split neq (rsplit0 a)"
haftmann@29789
  1023
  "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
haftmann@29789
  1024
  "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
haftmann@29789
  1025
  "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
haftmann@29789
  1026
  "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
haftmann@29789
  1027
  "rlfm (NOT (NOT p)) = rlfm p"
haftmann@29789
  1028
  "rlfm (NOT T) = F"
haftmann@29789
  1029
  "rlfm (NOT F) = T"
haftmann@29789
  1030
  "rlfm (NOT (Lt a)) = rlfm (Ge a)"
haftmann@29789
  1031
  "rlfm (NOT (Le a)) = rlfm (Gt a)"
haftmann@29789
  1032
  "rlfm (NOT (Gt a)) = rlfm (Le a)"
haftmann@29789
  1033
  "rlfm (NOT (Ge a)) = rlfm (Lt a)"
haftmann@29789
  1034
  "rlfm (NOT (Eq a)) = rlfm (NEq a)"
haftmann@29789
  1035
  "rlfm (NOT (NEq a)) = rlfm (Eq a)"
haftmann@29789
  1036
  "rlfm p = p" (hints simp add: fmsize_pos)
haftmann@29789
  1037
haftmann@29789
  1038
lemma rlfm_I:
haftmann@29789
  1039
  assumes qfp: "qfree p"
haftmann@29789
  1040
  shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
haftmann@29789
  1041
  using qfp 
wenzelm@44779
  1042
by (induct p rule: rlfm.induct) (auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
haftmann@29789
  1043
haftmann@29789
  1044
    (* Operations needed for Ferrante and Rackoff *)
haftmann@29789
  1045
lemma rminusinf_inf:
haftmann@29789
  1046
  assumes lp: "isrlfm p"
haftmann@29789
  1047
  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
  1048
using lp
haftmann@29789
  1049
proof (induct p rule: minusinf.induct)
wenzelm@44779
  1050
  case (1 p q)
wenzelm@44779
  1051
  thus ?case apply auto apply (rule_tac x= "min z za" in exI) apply auto done
haftmann@29789
  1052
next
wenzelm@44779
  1053
  case (2 p q)
wenzelm@44779
  1054
  thus ?case apply auto apply (rule_tac x= "min z za" in exI) apply auto done
haftmann@29789
  1055
next
haftmann@29789
  1056
  case (3 c e) 
wenzelm@41807
  1057
  from 3 have nb: "numbound0 e" by simp
wenzelm@41807
  1058
  from 3 have cp: "real c > 0" by simp
haftmann@29789
  1059
  fix a
haftmann@29789
  1060
  let ?e="Inum (a#bs) e"
haftmann@29789
  1061
  let ?z = "(- ?e) / real c"
haftmann@29789
  1062
  {fix x
haftmann@29789
  1063
    assume xz: "x < ?z"
haftmann@29789
  1064
    hence "(real c * x < - ?e)" 
haftmann@29789
  1065
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1066
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1067
    hence "real c * x + ?e \<noteq> 0" by simp
haftmann@29789
  1068
    with xz have "?P ?z x (Eq (CN 0 c e))"
haftmann@29789
  1069
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
haftmann@29789
  1070
  hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
haftmann@29789
  1071
  thus ?case by blast
haftmann@29789
  1072
next
haftmann@29789
  1073
  case (4 c e)   
wenzelm@41807
  1074
  from 4 have nb: "numbound0 e" by simp
wenzelm@41807
  1075
  from 4 have cp: "real c > 0" by simp
haftmann@29789
  1076
  fix a
haftmann@29789
  1077
  let ?e="Inum (a#bs) e"
haftmann@29789
  1078
  let ?z = "(- ?e) / real c"
haftmann@29789
  1079
  {fix x
haftmann@29789
  1080
    assume xz: "x < ?z"
haftmann@29789
  1081
    hence "(real c * x < - ?e)" 
haftmann@29789
  1082
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1083
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1084
    hence "real c * x + ?e \<noteq> 0" by simp
haftmann@29789
  1085
    with xz have "?P ?z x (NEq (CN 0 c e))"
haftmann@29789
  1086
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1087
  hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
haftmann@29789
  1088
  thus ?case by blast
haftmann@29789
  1089
next
haftmann@29789
  1090
  case (5 c e) 
wenzelm@41807
  1091
  from 5 have nb: "numbound0 e" by simp
wenzelm@41807
  1092
  from 5 have cp: "real c > 0" by simp
haftmann@29789
  1093
  fix a
haftmann@29789
  1094
  let ?e="Inum (a#bs) e"
haftmann@29789
  1095
  let ?z = "(- ?e) / real c"
haftmann@29789
  1096
  {fix x
haftmann@29789
  1097
    assume xz: "x < ?z"
haftmann@29789
  1098
    hence "(real c * x < - ?e)" 
haftmann@29789
  1099
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1100
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1101
    with xz have "?P ?z x (Lt (CN 0 c e))"
haftmann@29789
  1102
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
haftmann@29789
  1103
  hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
haftmann@29789
  1104
  thus ?case by blast
haftmann@29789
  1105
next
haftmann@29789
  1106
  case (6 c e)  
wenzelm@41807
  1107
  from 6 have nb: "numbound0 e" by simp
wenzelm@41807
  1108
  from lp 6 have cp: "real c > 0" by simp
haftmann@29789
  1109
  fix a
haftmann@29789
  1110
  let ?e="Inum (a#bs) e"
haftmann@29789
  1111
  let ?z = "(- ?e) / real c"
haftmann@29789
  1112
  {fix x
haftmann@29789
  1113
    assume xz: "x < ?z"
haftmann@29789
  1114
    hence "(real c * x < - ?e)" 
haftmann@29789
  1115
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1116
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1117
    with xz have "?P ?z x (Le (CN 0 c e))"
haftmann@29789
  1118
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1119
  hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
haftmann@29789
  1120
  thus ?case by blast
haftmann@29789
  1121
next
haftmann@29789
  1122
  case (7 c e)  
wenzelm@41807
  1123
  from 7 have nb: "numbound0 e" by simp
wenzelm@41807
  1124
  from 7 have cp: "real c > 0" by simp
haftmann@29789
  1125
  fix a
haftmann@29789
  1126
  let ?e="Inum (a#bs) e"
haftmann@29789
  1127
  let ?z = "(- ?e) / real c"
haftmann@29789
  1128
  {fix x
haftmann@29789
  1129
    assume xz: "x < ?z"
haftmann@29789
  1130
    hence "(real c * x < - ?e)" 
haftmann@29789
  1131
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1132
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1133
    with xz have "?P ?z x (Gt (CN 0 c e))"
haftmann@29789
  1134
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1135
  hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
haftmann@29789
  1136
  thus ?case by blast
haftmann@29789
  1137
next
haftmann@29789
  1138
  case (8 c e)  
wenzelm@41807
  1139
  from 8 have nb: "numbound0 e" by simp
wenzelm@41807
  1140
  from 8 have cp: "real c > 0" by simp
haftmann@29789
  1141
  fix a
haftmann@29789
  1142
  let ?e="Inum (a#bs) e"
haftmann@29789
  1143
  let ?z = "(- ?e) / real c"
haftmann@29789
  1144
  {fix x
haftmann@29789
  1145
    assume xz: "x < ?z"
haftmann@29789
  1146
    hence "(real c * x < - ?e)" 
haftmann@29789
  1147
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
haftmann@29789
  1148
    hence "real c * x + ?e < 0" by arith
haftmann@29789
  1149
    with xz have "?P ?z x (Ge (CN 0 c e))"
haftmann@29789
  1150
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1151
  hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
haftmann@29789
  1152
  thus ?case by blast
haftmann@29789
  1153
qed simp_all
haftmann@29789
  1154
haftmann@29789
  1155
lemma rplusinf_inf:
haftmann@29789
  1156
  assumes lp: "isrlfm p"
haftmann@29789
  1157
  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
  1158
using lp
haftmann@29789
  1159
proof (induct p rule: isrlfm.induct)
haftmann@29789
  1160
  case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
haftmann@29789
  1161
next
haftmann@29789
  1162
  case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
haftmann@29789
  1163
next
haftmann@29789
  1164
  case (3 c e) 
wenzelm@41807
  1165
  from 3 have nb: "numbound0 e" by simp
wenzelm@41807
  1166
  from 3 have cp: "real c > 0" by simp
haftmann@29789
  1167
  fix a
haftmann@29789
  1168
  let ?e="Inum (a#bs) e"
haftmann@29789
  1169
  let ?z = "(- ?e) / real c"
haftmann@29789
  1170
  {fix x
haftmann@29789
  1171
    assume xz: "x > ?z"
haftmann@29789
  1172
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1173
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1174
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1175
    hence "real c * x + ?e \<noteq> 0" by simp
haftmann@29789
  1176
    with xz have "?P ?z x (Eq (CN 0 c e))"
haftmann@29789
  1177
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1178
  hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
haftmann@29789
  1179
  thus ?case by blast
haftmann@29789
  1180
next
haftmann@29789
  1181
  case (4 c e) 
wenzelm@41807
  1182
  from 4 have nb: "numbound0 e" by simp
wenzelm@41807
  1183
  from 4 have cp: "real c > 0" by simp
haftmann@29789
  1184
  fix a
haftmann@29789
  1185
  let ?e="Inum (a#bs) e"
haftmann@29789
  1186
  let ?z = "(- ?e) / real c"
haftmann@29789
  1187
  {fix x
haftmann@29789
  1188
    assume xz: "x > ?z"
haftmann@29789
  1189
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1190
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1191
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1192
    hence "real c * x + ?e \<noteq> 0" by simp
haftmann@29789
  1193
    with xz have "?P ?z x (NEq (CN 0 c e))"
haftmann@29789
  1194
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1195
  hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
haftmann@29789
  1196
  thus ?case by blast
haftmann@29789
  1197
next
haftmann@29789
  1198
  case (5 c e) 
wenzelm@41807
  1199
  from 5 have nb: "numbound0 e" by simp
wenzelm@41807
  1200
  from 5 have cp: "real c > 0" by simp
haftmann@29789
  1201
  fix a
haftmann@29789
  1202
  let ?e="Inum (a#bs) e"
haftmann@29789
  1203
  let ?z = "(- ?e) / real c"
haftmann@29789
  1204
  {fix x
haftmann@29789
  1205
    assume xz: "x > ?z"
haftmann@29789
  1206
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1207
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1208
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1209
    with xz have "?P ?z x (Lt (CN 0 c e))"
haftmann@29789
  1210
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1211
  hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
haftmann@29789
  1212
  thus ?case by blast
haftmann@29789
  1213
next
haftmann@29789
  1214
  case (6 c e) 
wenzelm@41807
  1215
  from 6 have nb: "numbound0 e" by simp
wenzelm@41807
  1216
  from 6 have cp: "real c > 0" by simp
haftmann@29789
  1217
  fix a
haftmann@29789
  1218
  let ?e="Inum (a#bs) e"
haftmann@29789
  1219
  let ?z = "(- ?e) / real c"
haftmann@29789
  1220
  {fix x
haftmann@29789
  1221
    assume xz: "x > ?z"
haftmann@29789
  1222
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1223
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1224
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1225
    with xz have "?P ?z x (Le (CN 0 c e))"
haftmann@29789
  1226
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1227
  hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
haftmann@29789
  1228
  thus ?case by blast
haftmann@29789
  1229
next
haftmann@29789
  1230
  case (7 c e) 
wenzelm@41807
  1231
  from 7 have nb: "numbound0 e" by simp
wenzelm@41807
  1232
  from 7 have cp: "real c > 0" by simp
haftmann@29789
  1233
  fix a
haftmann@29789
  1234
  let ?e="Inum (a#bs) e"
haftmann@29789
  1235
  let ?z = "(- ?e) / real c"
haftmann@29789
  1236
  {fix x
haftmann@29789
  1237
    assume xz: "x > ?z"
haftmann@29789
  1238
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1239
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1240
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1241
    with xz have "?P ?z x (Gt (CN 0 c e))"
haftmann@29789
  1242
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
haftmann@29789
  1243
  hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
haftmann@29789
  1244
  thus ?case by blast
haftmann@29789
  1245
next
haftmann@29789
  1246
  case (8 c e) 
wenzelm@41807
  1247
  from 8 have nb: "numbound0 e" by simp
wenzelm@41807
  1248
  from 8 have cp: "real c > 0" by simp
haftmann@29789
  1249
  fix a
haftmann@29789
  1250
  let ?e="Inum (a#bs) e"
haftmann@29789
  1251
  let ?z = "(- ?e) / real c"
haftmann@29789
  1252
  {fix x
haftmann@29789
  1253
    assume xz: "x > ?z"
haftmann@29789
  1254
    with mult_strict_right_mono [OF xz cp] cp
haftmann@29789
  1255
    have "(real c * x > - ?e)" by (simp add: mult_ac)
haftmann@29789
  1256
    hence "real c * x + ?e > 0" by arith
haftmann@29789
  1257
    with xz have "?P ?z x (Ge (CN 0 c e))"
haftmann@29789
  1258
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
haftmann@29789
  1259
  hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
haftmann@29789
  1260
  thus ?case by blast
haftmann@29789
  1261
qed simp_all
haftmann@29789
  1262
haftmann@29789
  1263
lemma rminusinf_bound0:
haftmann@29789
  1264
  assumes lp: "isrlfm p"
haftmann@29789
  1265
  shows "bound0 (minusinf p)"
haftmann@29789
  1266
  using lp
haftmann@29789
  1267
  by (induct p rule: minusinf.induct) simp_all
haftmann@29789
  1268
haftmann@29789
  1269
lemma rplusinf_bound0:
haftmann@29789
  1270
  assumes lp: "isrlfm p"
haftmann@29789
  1271
  shows "bound0 (plusinf p)"
haftmann@29789
  1272
  using lp
haftmann@29789
  1273
  by (induct p rule: plusinf.induct) simp_all
haftmann@29789
  1274
haftmann@29789
  1275
lemma rminusinf_ex:
haftmann@29789
  1276
  assumes lp: "isrlfm p"
haftmann@29789
  1277
  and ex: "Ifm (a#bs) (minusinf p)"
haftmann@29789
  1278
  shows "\<exists> x. Ifm (x#bs) p"
haftmann@29789
  1279
proof-
haftmann@29789
  1280
  from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
haftmann@29789
  1281
  have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
haftmann@29789
  1282
  from rminusinf_inf[OF lp, where bs="bs"] 
haftmann@29789
  1283
  obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
haftmann@29789
  1284
  from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
haftmann@29789
  1285
  moreover have "z - 1 < z" by simp
haftmann@29789
  1286
  ultimately show ?thesis using z_def by auto
haftmann@29789
  1287
qed
haftmann@29789
  1288
haftmann@29789
  1289
lemma rplusinf_ex:
haftmann@29789
  1290
  assumes lp: "isrlfm p"
haftmann@29789
  1291
  and ex: "Ifm (a#bs) (plusinf p)"
haftmann@29789
  1292
  shows "\<exists> x. Ifm (x#bs) p"
haftmann@29789
  1293
proof-
haftmann@29789
  1294
  from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
haftmann@29789
  1295
  have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
haftmann@29789
  1296
  from rplusinf_inf[OF lp, where bs="bs"] 
haftmann@29789
  1297
  obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
haftmann@29789
  1298
  from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
haftmann@29789
  1299
  moreover have "z + 1 > z" by simp
haftmann@29789
  1300
  ultimately show ?thesis using z_def by auto
haftmann@29789
  1301
qed
haftmann@29789
  1302
haftmann@29789
  1303
consts 
haftmann@29789
  1304
  uset:: "fm \<Rightarrow> (num \<times> int) list"
haftmann@29789
  1305
  usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
haftmann@29789
  1306
recdef uset "measure size"
haftmann@29789
  1307
  "uset (And p q) = (uset p @ uset q)" 
haftmann@29789
  1308
  "uset (Or p q) = (uset p @ uset q)" 
haftmann@29789
  1309
  "uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1310
  "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1311
  "uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1312
  "uset (Le  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1313
  "uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1314
  "uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
haftmann@29789
  1315
  "uset p = []"
haftmann@29789
  1316
recdef usubst "measure size"
haftmann@29789
  1317
  "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
haftmann@29789
  1318
  "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
haftmann@29789
  1319
  "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1320
  "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1321
  "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1322
  "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1323
  "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1324
  "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
haftmann@29789
  1325
  "usubst p = (\<lambda> (t,n). p)"
haftmann@29789
  1326
haftmann@29789
  1327
lemma usubst_I: assumes lp: "isrlfm p"
haftmann@29789
  1328
  and np: "real n > 0" and nbt: "numbound0 t"
haftmann@29789
  1329
  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
  1330
  using lp
haftmann@29789
  1331
proof(induct p rule: usubst.induct)
wenzelm@41807
  1332
  case (5 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
haftmann@29789
  1333
  have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
haftmann@29789
  1334
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1335
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
haftmann@29789
  1336
    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1337
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1338
  also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
haftmann@29789
  1339
    using np by simp 
haftmann@29789
  1340
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1341
next
wenzelm@41807
  1342
  case (6 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
haftmann@29789
  1343
  have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
haftmann@29789
  1344
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1345
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
haftmann@29789
  1346
    by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1347
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1348
  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
haftmann@29789
  1349
    using np by simp 
haftmann@29789
  1350
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1351
next
wenzelm@41807
  1352
  case (7 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
haftmann@29789
  1353
  have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
haftmann@29789
  1354
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1355
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
haftmann@29789
  1356
    by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1357
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1358
  also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
haftmann@29789
  1359
    using np by simp 
haftmann@29789
  1360
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1361
next
wenzelm@41807
  1362
  case (8 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
haftmann@29789
  1363
  have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
haftmann@29789
  1364
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1365
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
haftmann@29789
  1366
    by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1367
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1368
  also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
haftmann@29789
  1369
    using np by simp 
haftmann@29789
  1370
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1371
next
wenzelm@41807
  1372
  case (3 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
haftmann@29789
  1373
  from np have np: "real n \<noteq> 0" by simp
haftmann@29789
  1374
  have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
haftmann@29789
  1375
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1376
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
haftmann@29789
  1377
    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1378
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1379
  also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
haftmann@29789
  1380
    using np by simp 
haftmann@29789
  1381
  finally show ?case using nbt nb by (simp add: algebra_simps)
haftmann@29789
  1382
next
wenzelm@41807
  1383
  case (4 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
haftmann@29789
  1384
  from np have np: "real n \<noteq> 0" by simp
haftmann@29789
  1385
  have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
haftmann@29789
  1386
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
haftmann@29789
  1387
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
haftmann@29789
  1388
    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
haftmann@29789
  1389
      and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
haftmann@29789
  1390
  also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
haftmann@29789
  1391
    using np by simp 
haftmann@29789
  1392
  finally show ?case using nbt nb by (simp add: algebra_simps)
nipkow@41842
  1393
qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"])
haftmann@29789
  1394
haftmann@29789
  1395
lemma uset_l:
haftmann@29789
  1396
  assumes lp: "isrlfm p"
haftmann@29789
  1397
  shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
haftmann@29789
  1398
using lp
haftmann@29789
  1399
by(induct p rule: uset.induct,auto)
haftmann@29789
  1400
haftmann@29789
  1401
lemma rminusinf_uset:
haftmann@29789
  1402
  assumes lp: "isrlfm p"
haftmann@29789
  1403
  and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
haftmann@29789
  1404
  and ex: "Ifm (x#bs) p" (is "?I x p")
haftmann@29789
  1405
  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
  1406
proof-
haftmann@29789
  1407
  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
  1408
    using lp nmi ex
nipkow@41842
  1409
    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
haftmann@29789
  1410
  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
  1411
  from uset_l[OF lp] smU have mp: "real m > 0" by auto
haftmann@29789
  1412
  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
  1413
    by (auto simp add: mult_commute)
haftmann@29789
  1414
  thus ?thesis using smU by auto
haftmann@29789
  1415
qed
haftmann@29789
  1416
haftmann@29789
  1417
lemma rplusinf_uset:
haftmann@29789
  1418
  assumes lp: "isrlfm p"
haftmann@29789
  1419
  and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
haftmann@29789
  1420
  and ex: "Ifm (x#bs) p" (is "?I x p")
haftmann@29789
  1421
  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
  1422
proof-
haftmann@29789
  1423
  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
  1424
    using lp nmi ex
nipkow@41842
  1425
    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
haftmann@29789
  1426
  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
  1427
  from uset_l[OF lp] smU have mp: "real m > 0" by auto
haftmann@29789
  1428
  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
  1429
    by (auto simp add: mult_commute)
haftmann@29789
  1430
  thus ?thesis using smU by auto
haftmann@29789
  1431
qed
haftmann@29789
  1432
haftmann@29789
  1433
lemma lin_dense: 
haftmann@29789
  1434
  assumes lp: "isrlfm p"
haftmann@29789
  1435
  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
  1436
  (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
haftmann@29789
  1437
  and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
haftmann@29789
  1438
  and ly: "l < y" and yu: "y < u"
haftmann@29789
  1439
  shows "Ifm (y#bs) p"
haftmann@29789
  1440
using lp px noS
haftmann@29789
  1441
proof (induct p rule: isrlfm.induct)
haftmann@29789
  1442
  case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
wenzelm@41807
  1443
  from 5 have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
wenzelm@41807
  1444
  hence pxc: "x < (- ?N x e) / real c" 
wenzelm@41807
  1445
    by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
wenzelm@41807
  1446
  from 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
wenzelm@41807
  1447
  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
wenzelm@41807
  1448
  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
wenzelm@41807
  1449
  moreover {assume y: "y < (-?N x e)/ real c"
wenzelm@41807
  1450
    hence "y * real c < - ?N x e"
wenzelm@41807
  1451
      by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
wenzelm@41807
  1452
    hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
wenzelm@41807
  1453
    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
wenzelm@41807
  1454
  moreover {assume y: "y > (- ?N x e) / real c" 
wenzelm@41807
  1455
    with yu have eu: "u > (- ?N x e) / real c" by auto
wenzelm@41807
  1456
    with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
wenzelm@41807
  1457
    with lx pxc have "False" by auto
wenzelm@41807
  1458
    hence ?case by simp }
wenzelm@41807
  1459
  ultimately show ?case by blast
haftmann@29789
  1460
next
haftmann@29789
  1461
  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
wenzelm@41807
  1462
  from 6 have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
wenzelm@41807
  1463
  hence pxc: "x \<le> (- ?N x e) / real c" 
wenzelm@41807
  1464
    by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
wenzelm@41807
  1465
  from 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
wenzelm@41807
  1466
  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
wenzelm@41807
  1467
  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
wenzelm@41807
  1468
  moreover {assume y: "y < (-?N x e)/ real c"
wenzelm@41807
  1469
    hence "y * real c < - ?N x e"
wenzelm@41807
  1470
      by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
wenzelm@41807
  1471
    hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
wenzelm@41807
  1472
    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
wenzelm@41807
  1473
  moreover {assume y: "y > (- ?N x e) / real c" 
wenzelm@41807
  1474
    with yu have eu: "u > (- ?N x e) / real c" by auto
wenzelm@41807
  1475
    with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
wenzelm@41807
  1476
    with lx pxc have "False" by auto
wenzelm@41807
  1477
    hence ?case by simp }
wenzelm@41807
  1478
  ultimately show ?case by blast
haftmann@29789
  1479
next
haftmann@29789
  1480
  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
wenzelm@41807
  1481
  from 7 have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
wenzelm@41807
  1482
  hence pxc: "x > (- ?N x e) / real c" 
wenzelm@41807
  1483
    by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
wenzelm@41807
  1484
  from 7 have noSc: "\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
wenzelm@41807
  1485
  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
wenzelm@41807
  1486
  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
wenzelm@41807
  1487
  moreover {assume y: "y > (-?N x e)/ real c"
wenzelm@41807
  1488
    hence "y * real c > - ?N x e"
wenzelm@41807
  1489
      by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
wenzelm@41807
  1490
    hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
wenzelm@41807
  1491
    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
wenzelm@41807
  1492
  moreover {assume y: "y < (- ?N x e) / real c" 
wenzelm@41807
  1493
    with ly have eu: "l < (- ?N x e) / real c" by auto
wenzelm@41807
  1494
    with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
wenzelm@41807
  1495
    with xu pxc have "False" by auto
wenzelm@41807
  1496
    hence ?case by simp }
wenzelm@41807
  1497
  ultimately show ?case by blast
haftmann@29789
  1498
next
haftmann@29789
  1499
  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
wenzelm@41807
  1500
  from 8 have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
wenzelm@41807
  1501
  hence pxc: "x \<ge> (- ?N x e) / real c" 
wenzelm@41807
  1502
    by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
wenzelm@41807
  1503
  from 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
wenzelm@41807
  1504
  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
wenzelm@41807
  1505
  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
wenzelm@41807
  1506
  moreover {assume y: "y > (-?N x e)/ real c"
wenzelm@41807
  1507
    hence "y * real c > - ?N x e"
wenzelm@41807
  1508
      by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
wenzelm@41807
  1509
    hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
wenzelm@41807
  1510
    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
wenzelm@41807
  1511
  moreover {assume y: "y < (- ?N x e) / real c" 
wenzelm@41807
  1512
    with ly have eu: "l < (- ?N x e) / real c" by auto
wenzelm@41807
  1513
    with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
wenzelm@41807
  1514
    with xu pxc have "False" by auto
wenzelm@41807
  1515
    hence ?case by simp }
wenzelm@41807
  1516
  ultimately show ?case by blast
haftmann@29789
  1517
next
haftmann@29789
  1518
  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
wenzelm@41807
  1519
  from cp have cnz: "real c \<noteq> 0" by simp
wenzelm@41807
  1520
  from 3 have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
wenzelm@41807
  1521
  hence pxc: "x = (- ?N x e) / real c" 
wenzelm@41807
  1522
    by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
wenzelm@41807
  1523
  from 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
wenzelm@41807
  1524
  with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
wenzelm@41807
  1525
  with pxc show ?case by simp
haftmann@29789
  1526
next
haftmann@29789
  1527
  case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
wenzelm@41807
  1528
  from cp have cnz: "real c \<noteq> 0" by simp
wenzelm@41807
  1529
  from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
wenzelm@41807
  1530
  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
wenzelm@41807
  1531
  hence "y* real c \<noteq> -?N x e"      
wenzelm@41807
  1532
    by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
wenzelm@41807
  1533
  hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
wenzelm@41807
  1534
  thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
wenzelm@41807
  1535
    by (simp add: algebra_simps)
nipkow@41842
  1536
qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"])
haftmann@29789
  1537
haftmann@29789
  1538
lemma finite_set_intervals:
haftmann@29789
  1539
  assumes px: "P (x::real)" 
haftmann@29789
  1540
  and lx: "l \<le> x" and xu: "x \<le> u"
haftmann@29789
  1541
  and linS: "l\<in> S" and uinS: "u \<in> S"
haftmann@29789
  1542
  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
haftmann@29789
  1543
  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
  1544
proof-
haftmann@29789
  1545
  let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
haftmann@29789
  1546
  let ?xM = "{y. y\<in> S \<and> x \<le> y}"
haftmann@29789
  1547
  let ?a = "Max ?Mx"
haftmann@29789
  1548
  let ?b = "Min ?xM"
haftmann@29789
  1549
  have MxS: "?Mx \<subseteq> S" by blast
haftmann@29789
  1550
  hence fMx: "finite ?Mx" using fS finite_subset by auto
haftmann@29789
  1551
  from lx linS have linMx: "l \<in> ?Mx" by blast
haftmann@29789
  1552
  hence Mxne: "?Mx \<noteq> {}" by blast
haftmann@29789
  1553
  have xMS: "?xM \<subseteq> S" by blast
haftmann@29789
  1554
  hence fxM: "finite ?xM" using fS finite_subset by auto
haftmann@29789
  1555
  from xu uinS have linxM: "u \<in> ?xM" by blast
haftmann@29789
  1556
  hence xMne: "?xM \<noteq> {}" by blast
haftmann@29789
  1557
  have ax:"?a \<le> x" using Mxne fMx by auto
haftmann@29789
  1558
  have xb:"x \<le> ?b" using xMne fxM by auto
haftmann@29789
  1559
  have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
haftmann@29789
  1560
  have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
haftmann@29789
  1561
  have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
haftmann@29789
  1562
  proof(clarsimp)
haftmann@29789
  1563
    fix y
haftmann@29789
  1564
    assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
haftmann@29789
  1565
    from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
haftmann@29789
  1566
    moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
haftmann@29789
  1567
    moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
haftmann@29789
  1568
    ultimately show "False" by blast
haftmann@29789
  1569
  qed
haftmann@29789
  1570
  from ainS binS noy ax xb px show ?thesis by blast
haftmann@29789
  1571
qed
haftmann@29789
  1572
haftmann@29789
  1573
lemma rinf_uset:
haftmann@29789
  1574
  assumes lp: "isrlfm p"
haftmann@29789
  1575
  and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
haftmann@29789
  1576
  and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
haftmann@29789
  1577
  and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
haftmann@29789
  1578
  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
  1579
proof-
haftmann@29789
  1580
  let ?N = "\<lambda> x t. Inum (x#bs) t"
haftmann@29789
  1581
  let ?U = "set (uset p)"
haftmann@29789
  1582
  from ex obtain a where pa: "?I a p" by blast
haftmann@29789
  1583
  from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
haftmann@29789
  1584
  have nmi': "\<not> (?I a (?M p))" by simp
haftmann@29789
  1585
  from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
haftmann@29789
  1586
  have npi': "\<not> (?I a (?P p))" by simp
haftmann@29789
  1587
  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
  1588
  proof-
haftmann@29789
  1589
    let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
haftmann@29789
  1590
    have fM: "finite ?M" by auto
haftmann@29789
  1591
    from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa] 
haftmann@29789
  1592
    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
  1593
    then obtain "t" "n" "s" "m" where 
haftmann@29789
  1594
      tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" 
haftmann@29789
  1595
      and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
haftmann@29789
  1596
    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
  1597
    from tnU have Mne: "?M \<noteq> {}" by auto
haftmann@29789
  1598
    hence Une: "?U \<noteq> {}" by simp
haftmann@29789
  1599
    let ?l = "Min ?M"
haftmann@29789
  1600
    let ?u = "Max ?M"
haftmann@29789
  1601
    have linM: "?l \<in> ?M" using fM Mne by simp
haftmann@29789
  1602
    have uinM: "?u \<in> ?M" using fM Mne by simp
haftmann@29789
  1603
    have tnM: "?N a t / real n \<in> ?M" using tnU by auto
haftmann@29789
  1604
    have smM: "?N a s / real m \<in> ?M" using smU by auto 
haftmann@29789
  1605
    have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
haftmann@29789
  1606
    have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
haftmann@29789
  1607
    have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
haftmann@29789
  1608
    have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
haftmann@29789
  1609
    from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
haftmann@29789
  1610
    have "(\<exists> s\<in> ?M. ?I s p) \<or> 
haftmann@29789
  1611
      (\<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
  1612
    moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
haftmann@29789
  1613
      hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
haftmann@29789
  1614
      then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
haftmann@29789
  1615
      have "(u + u) / 2 = u" by auto with pu tuu 
haftmann@29789
  1616
      have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
haftmann@29789
  1617
      with tuU have ?thesis by blast}
haftmann@29789
  1618
    moreover{
haftmann@29789
  1619
      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
  1620
      then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
wenzelm@32960
  1621
        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"
wenzelm@32960
  1622
        by blast
haftmann@29789
  1623
      from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
haftmann@29789
  1624
      then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
haftmann@29789
  1625
      from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
haftmann@29789
  1626
      then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
haftmann@29789
  1627
      from t1x xt2 have t1t2: "t1 < t2" by simp
haftmann@29789
  1628
      let ?u = "(t1 + t2) / 2"
haftmann@29789
  1629
      from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
haftmann@29789
  1630
      from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
haftmann@29789
  1631
      with t1uU t2uU t1u t2u have ?thesis by blast}
haftmann@29789
  1632
    ultimately show ?thesis by blast
haftmann@29789
  1633
  qed
haftmann@29789
  1634
  then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" 
haftmann@29789
  1635
    and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
haftmann@29789
  1636
  from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
haftmann@29789
  1637
  from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
haftmann@29789
  1638
    numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
haftmann@29789
  1639
  have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
haftmann@29789
  1640
  with lnU smU
haftmann@29789
  1641
  show ?thesis by auto
haftmann@29789
  1642
qed
haftmann@29789
  1643
    (* The Ferrante - Rackoff Theorem *)
haftmann@29789
  1644
haftmann@29789
  1645
theorem fr_eq: 
haftmann@29789
  1646
  assumes lp: "isrlfm p"
haftmann@29789
  1647
  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
  1648
  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
haftmann@29789
  1649
proof
haftmann@29789
  1650
  assume px: "\<exists> x. ?I x p"
haftmann@29789
  1651
  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
haftmann@29789
  1652
  moreover {assume "?M \<or> ?P" hence "?D" by blast}
haftmann@29789
  1653
  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
haftmann@29789
  1654
    from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
haftmann@29789
  1655
  ultimately show "?D" by blast
haftmann@29789
  1656
next
haftmann@29789
  1657
  assume "?D" 
haftmann@29789
  1658
  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
haftmann@29789
  1659
  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
haftmann@29789
  1660
  moreover {assume f:"?F" hence "?E" by blast}
haftmann@29789
  1661
  ultimately show "?E" by blast
haftmann@29789
  1662
qed
haftmann@29789
  1663
haftmann@29789
  1664
haftmann@29789
  1665
lemma fr_equsubst: 
haftmann@29789
  1666
  assumes lp: "isrlfm p"
haftmann@29789
  1667
  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
  1668
  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
haftmann@29789
  1669
proof
haftmann@29789
  1670
  assume px: "\<exists> x. ?I x p"
haftmann@29789
  1671
  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
haftmann@29789
  1672
  moreover {assume "?M \<or> ?P" hence "?D" by blast}
haftmann@29789
  1673
  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
haftmann@29789
  1674
    let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
haftmann@29789
  1675
    let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1676
    {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
haftmann@29789
  1677
      with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
wenzelm@32960
  1678
        by auto
haftmann@29789
  1679
      let ?st = "Add (Mul m t) (Mul n s)"
haftmann@29789
  1680
      from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
wenzelm@32960
  1681
        by (simp add: mult_commute)
haftmann@29789
  1682
      from tnb snb have st_nb: "numbound0 ?st" by simp
haftmann@29789
  1683
      have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
wenzelm@32960
  1684
        using mnp mp np by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1685
      from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
haftmann@29789
  1686
      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
  1687
    with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
haftmann@29789
  1688
  ultimately show "?D" by blast
haftmann@29789
  1689
next
haftmann@29789
  1690
  assume "?D" 
haftmann@29789
  1691
  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
haftmann@29789
  1692
  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
haftmann@29789
  1693
  moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)" 
haftmann@29789
  1694
    and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
haftmann@29789
  1695
    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
  1696
    let ?st = "Add (Mul l t) (Mul k s)"
haftmann@29789
  1697
    from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" 
haftmann@29789
  1698
      by (simp add: mult_commute)
haftmann@29789
  1699
    from tnb snb have st_nb: "numbound0 ?st" by simp
haftmann@29789
  1700
    from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
haftmann@29789
  1701
  ultimately show "?E" by blast
haftmann@29789
  1702
qed
haftmann@29789
  1703
haftmann@29789
  1704
haftmann@29789
  1705
    (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
haftmann@35416
  1706
definition ferrack :: "fm \<Rightarrow> fm" where
haftmann@36853
  1707
  "ferrack p = (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
haftmann@29789
  1708
                in if (mp = T \<or> pp = T) then T else 
haftmann@36853
  1709
                   (let U = remdups(map simp_num_pair 
haftmann@29789
  1710
                     (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
haftmann@29789
  1711
                           (alluopairs (uset p')))) 
haftmann@29789
  1712
                    in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
haftmann@29789
  1713
haftmann@29789
  1714
lemma uset_cong_aux:
haftmann@29789
  1715
  assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
haftmann@29789
  1716
  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
  1717
  (is "?lhs = ?rhs")
haftmann@29789
  1718
proof(auto)
haftmann@29789
  1719
  fix t n s m
haftmann@29789
  1720
  assume "((t,n),(s,m)) \<in> set (alluopairs U)"
haftmann@29789
  1721
  hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
haftmann@29789
  1722
    using alluopairs_set1[where xs="U"] by blast
haftmann@29789
  1723
  let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1724
  let ?st= "Add (Mul m t) (Mul n s)"
haftmann@29789
  1725
  from Ul th have mnz: "m \<noteq> 0" by auto
haftmann@29789
  1726
  from Ul th have  nnz: "n \<noteq> 0" by auto  
haftmann@29789
  1727
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1728
   using mnz nnz by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1729
 
haftmann@29789
  1730
  thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
haftmann@29789
  1731
       (2 * real n * real m)
haftmann@29789
  1732
       \<in> (\<lambda>((t, n), s, m).
haftmann@29789
  1733
             (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
haftmann@29789
  1734
         (set U \<times> set U)"using mnz nnz th  
haftmann@29789
  1735
    apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
haftmann@29789
  1736
    by (rule_tac x="(s,m)" in bexI,simp_all) 
haftmann@29789
  1737
  (rule_tac x="(t,n)" in bexI,simp_all)
haftmann@29789
  1738
next
haftmann@29789
  1739
  fix t n s m
haftmann@29789
  1740
  assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
haftmann@29789
  1741
  let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1742
  let ?st= "Add (Mul m t) (Mul n s)"
haftmann@29789
  1743
  from Ul smU have mnz: "m \<noteq> 0" by auto
haftmann@29789
  1744
  from Ul tnU have  nnz: "n \<noteq> 0" by auto  
haftmann@29789
  1745
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1746
   using mnz nnz by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1747
 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
  1748
 have Pc:"\<forall> a b. ?P a b = ?P b a"
haftmann@29789
  1749
   by auto
haftmann@29789
  1750
 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
  1751
 from alluopairs_ex[OF Pc, where xs="U"] tnU smU
haftmann@29789
  1752
 have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
haftmann@29789
  1753
   by blast
haftmann@29789
  1754
 then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" 
haftmann@29789
  1755
   and Pts': "?P (t',n') (s',m')" by blast
haftmann@29789
  1756
 from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
haftmann@29789
  1757
 let ?st' = "Add (Mul m' t') (Mul n' s')"
haftmann@29789
  1758
   have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
haftmann@29789
  1759
   using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1760
 from Pts' have 
haftmann@29789
  1761
   "(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
  1762
 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
  1763
 finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
haftmann@29789
  1764
          \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
haftmann@29789
  1765
            (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
haftmann@29789
  1766
            set (alluopairs U)"
haftmann@29789
  1767
   using ts'_U by blast
haftmann@29789
  1768
qed
haftmann@29789
  1769
haftmann@29789
  1770
lemma uset_cong:
haftmann@29789
  1771
  assumes lp: "isrlfm p"
haftmann@29789
  1772
  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
  1773
  and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
haftmann@29789
  1774
  and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
haftmann@29789
  1775
  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
  1776
  (is "?lhs = ?rhs")
haftmann@29789
  1777
proof
haftmann@29789
  1778
  assume ?lhs
haftmann@29789
  1779
  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
haftmann@29789
  1780
    Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
haftmann@29789
  1781
  let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1782
  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
haftmann@29789
  1783
    and snb: "numbound0 s" and mp:"m > 0"  by auto
haftmann@29789
  1784
  let ?st= "Add (Mul m t) (Mul n s)"
haftmann@29789
  1785
  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
haftmann@29789
  1786
      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
haftmann@29789
  1787
    from tnb snb have stnb: "numbound0 ?st" by simp
haftmann@29789
  1788
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1789
   using mp np by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1790
  from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
haftmann@29789
  1791
  hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
haftmann@29789
  1792
    by auto (rule_tac x="(a,b)" in bexI, auto)
haftmann@29789
  1793
  then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
haftmann@29789
  1794
  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
haftmann@29789
  1795
  from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst 
haftmann@29789
  1796
  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
haftmann@29789
  1797
  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
  1798
  have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st) 
haftmann@29789
  1799
  then show ?rhs using tnU' by auto 
haftmann@29789
  1800
next
haftmann@29789
  1801
  assume ?rhs
haftmann@29789
  1802
  then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))" 
haftmann@29789
  1803
    by blast
haftmann@29789
  1804
  from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
haftmann@29789
  1805
  hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" 
haftmann@29789
  1806
    by auto (rule_tac x="(a,b)" in bexI, auto)
haftmann@29789
  1807
  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
haftmann@29789
  1808
    th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
haftmann@29789
  1809
    let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1810
  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
haftmann@29789
  1811
    and snb: "numbound0 s" and mp:"m > 0"  by auto
haftmann@29789
  1812
  let ?st= "Add (Mul m t) (Mul n s)"
haftmann@29789
  1813
  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
haftmann@29789
  1814
      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
haftmann@29789
  1815
    from tnb snb have stnb: "numbound0 ?st" by simp
haftmann@29789
  1816
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
haftmann@29789
  1817
   using mp np by (simp add: algebra_simps add_divide_distrib)
haftmann@29789
  1818
  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
haftmann@29789
  1819
  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
  1820
  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
haftmann@29789
  1821
  with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
haftmann@29789
  1822
qed
haftmann@29789
  1823
haftmann@29789
  1824
lemma ferrack: 
haftmann@29789
  1825
  assumes qf: "qfree p"
haftmann@29789
  1826
  shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
haftmann@29789
  1827
  (is "_ \<and> (?rhs = ?lhs)")
haftmann@29789
  1828
proof-
haftmann@29789
  1829
  let ?I = "\<lambda> x p. Ifm (x#bs) p"
haftmann@29789
  1830
  fix x
haftmann@29789
  1831
  let ?N = "\<lambda> t. Inum (x#bs) t"
haftmann@29789
  1832
  let ?q = "rlfm (simpfm p)" 
haftmann@29789
  1833
  let ?U = "uset ?q"
haftmann@29789
  1834
  let ?Up = "alluopairs ?U"
haftmann@29789
  1835
  let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
haftmann@29789
  1836
  let ?S = "map ?g ?Up"
haftmann@29789
  1837
  let ?SS = "map simp_num_pair ?S"
haftmann@36853
  1838
  let ?Y = "remdups ?SS"
haftmann@29789
  1839
  let ?f= "(\<lambda> (t,n). ?N t / real n)"
haftmann@29789
  1840
  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
haftmann@29789
  1841
  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
  1842
  let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
haftmann@29789
  1843
  from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
haftmann@29789
  1844
  from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
haftmann@29789
  1845
  from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
haftmann@29789
  1846
  from U_l UpU 
haftmann@29789
  1847
  have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
haftmann@29789
  1848
  hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
haftmann@29789
  1849
    by (auto simp add: mult_pos_pos)
haftmann@29789
  1850
  have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" 
haftmann@29789
  1851
  proof-
haftmann@29789
  1852
    { fix t n assume tnY: "(t,n) \<in> set ?Y" 
haftmann@29789
  1853
      hence "(t,n) \<in> set ?SS" by simp
haftmann@29789
  1854
      hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
hoelzl@33639
  1855
        by (auto simp add: split_def simp del: map_map)
hoelzl@33639
  1856
           (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
haftmann@29789
  1857
      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
  1858
      from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
haftmann@29789
  1859
      from simp_num_pair_l[OF tnb np tns]
haftmann@29789
  1860
      have "numbound0 t \<and> n > 0" . }
haftmann@29789
  1861
    thus ?thesis by blast
haftmann@29789
  1862
  qed
haftmann@29789
  1863
haftmann@29789
  1864
  have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
haftmann@29789
  1865
  proof-
haftmann@29789
  1866
     from simp_num_pair_ci[where bs="x#bs"] have 
haftmann@29789
  1867
    "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
haftmann@29789
  1868
     hence th: "?f o simp_num_pair = ?f" using ext by blast
haftmann@29789
  1869
    have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
haftmann@29789
  1870
    also have "\<dots> = (?f ` set ?S)" by (simp add: th)
haftmann@29789
  1871
    also have "\<dots> = ((?f o ?g) ` set ?Up)" 
haftmann@29789
  1872
      by (simp only: set_map o_def image_compose[symmetric])
haftmann@29789
  1873
    also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
haftmann@29789
  1874
      using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
haftmann@29789
  1875
    finally show ?thesis .
haftmann@29789
  1876
  qed
haftmann@29789
  1877
  have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
haftmann@29789
  1878
  proof-
haftmann@29789
  1879
    { fix t n assume tnY: "(t,n) \<in> set ?Y"
haftmann@29789
  1880
      with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
haftmann@29789
  1881
      from usubst_I[OF lq np tnb]
haftmann@29789
  1882
    have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))" 
haftmann@29789
  1883
      using simpfm_bound0 by simp}
haftmann@29789
  1884
    thus ?thesis by blast
haftmann@29789
  1885
  qed
haftmann@29789
  1886
  hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
haftmann@29789
  1887
  let ?mp = "minusinf ?q"
haftmann@29789
  1888
  let ?pp = "plusinf ?q"
haftmann@29789
  1889
  let ?M = "?I x ?mp"
haftmann@29789
  1890
  let ?P = "?I x ?pp"
haftmann@29789
  1891
  let ?res = "disj ?mp (disj ?pp ?ep)"
haftmann@29789
  1892
  from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
haftmann@29789
  1893
  have nbth: "bound0 ?res" by auto
haftmann@29789
  1894
haftmann@29789
  1895
  from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm  
haftmann@29789
  1896
haftmann@29789
  1897
  have th: "?lhs = (\<exists> x. ?I x ?q)" by auto 
haftmann@29789
  1898
  from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
haftmann@29789
  1899
    by (simp only: split_def fst_conv snd_conv)
haftmann@29789
  1900
  also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" 
haftmann@29789
  1901
    using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm) 
haftmann@29789
  1902
  also have "\<dots> = (Ifm (x#bs) ?res)"
haftmann@29789
  1903
    using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
haftmann@29789
  1904
    by (simp add: split_def pair_collapse)
haftmann@29789
  1905
  finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
haftmann@29789
  1906
  hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
haftmann@29789
  1907
    by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
haftmann@29789
  1908
  from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
haftmann@29789
  1909
  with lr show ?thesis by blast
haftmann@29789
  1910
qed
haftmann@29789
  1911
haftmann@29789
  1912
definition linrqe:: "fm \<Rightarrow> fm" where
haftmann@29789
  1913
  "linrqe p = qelim (prep p) ferrack"
haftmann@29789
  1914
haftmann@29789
  1915
theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)"
haftmann@29789
  1916
using ferrack qelim_ci prep
haftmann@29789
  1917
unfolding linrqe_def by auto
haftmann@29789
  1918
haftmann@29789
  1919
definition ferrack_test :: "unit \<Rightarrow> fm" where
haftmann@29789
  1920
  "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
haftmann@29789
  1921
    (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
haftmann@29789
  1922
haftmann@29789
  1923
ML {* @{code ferrack_test} () *}
haftmann@29789
  1924
haftmann@29789
  1925
oracle linr_oracle = {*
haftmann@29789
  1926
let
haftmann@29789
  1927
haftmann@36853
  1928
fun num_of_term vs (Free vT) = @{code Bound} (find_index (fn vT' => vT = vT') vs)
haftmann@29789
  1929
  | num_of_term vs @{term "real (0::int)"} = @{code C} 0
haftmann@29789
  1930
  | num_of_term vs @{term "real (1::int)"} = @{code C} 1
haftmann@29789
  1931
  | num_of_term vs @{term "0::real"} = @{code C} 0
haftmann@29789
  1932
  | num_of_term vs @{term "1::real"} = @{code C} 1
haftmann@29789
  1933
  | num_of_term vs (Bound i) = @{code Bound} i
haftmann@29789
  1934
  | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
haftmann@36853
  1935
  | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
haftmann@36853
  1936
     @{code Add} (num_of_term vs t1, num_of_term vs t2)
haftmann@36853
  1937
  | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) =
haftmann@36853
  1938
     @{code Sub} (num_of_term vs t1, num_of_term vs t2)
haftmann@36853
  1939
  | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (case num_of_term vs t1
haftmann@29789
  1940
     of @{code C} i => @{code Mul} (i, num_of_term vs t2)
haftmann@36853
  1941
      | _ => error "num_of_term: unsupported multiplication")
haftmann@36853
  1942
  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t')) =
haftmann@36853
  1943
     @{code C} (HOLogic.dest_numeral t')
haftmann@36853
  1944
  | num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} $ t') =
haftmann@36853
  1945
     @{code C} (HOLogic.dest_numeral t')
haftmann@36853
  1946
  | num_of_term vs t = error ("num_of_term: unknown term");
haftmann@29789
  1947
haftmann@29789
  1948
fun fm_of_term vs @{term True} = @{code T}
haftmann@29789
  1949
  | fm_of_term vs @{term False} = @{code F}
haftmann@36853
  1950
  | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
haftmann@36853
  1951
      @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
haftmann@36853
  1952
  | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
haftmann@36853
  1953
      @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
haftmann@36853
  1954
  | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
haftmann@36853
  1955
      @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) 
haftmann@36853
  1956
  | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
haftmann@36853
  1957
      @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
haftmann@38795
  1958
  | fm_of_term vs (@{term HOL.conj} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
haftmann@38795
  1959
  | fm_of_term vs (@{term HOL.disj} $ t1 $ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
haftmann@38786
  1960
  | fm_of_term vs (@{term HOL.implies} $ t1 $ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
haftmann@29789
  1961
  | fm_of_term vs (@{term "Not"} $ t') = @{code NOT} (fm_of_term vs t')
haftmann@38558
  1962
  | fm_of_term vs (Const (@{const_name Ex}, _) $ Abs (xn, xT, p)) =
haftmann@36853
  1963
      @{code E} (fm_of_term (("", dummyT) :: vs) p)
haftmann@38558
  1964
  | fm_of_term vs (Const (@{const_name All}, _) $ Abs (xn, xT, p)) =
haftmann@36853
  1965
      @{code A} (fm_of_term (("", dummyT) ::  vs) p)
haftmann@29789
  1966
  | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
haftmann@29789
  1967
haftmann@29789
  1968
fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} $ HOLogic.mk_number HOLogic.intT i
haftmann@36853
  1969
  | term_of_num vs (@{code Bound} n) = Free (nth vs n)
haftmann@29789
  1970
  | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
haftmann@29789
  1971
  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
haftmann@29789
  1972
      term_of_num vs t1 $ term_of_num vs t2
haftmann@29789
  1973
  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $
haftmann@29789
  1974
      term_of_num vs t1 $ term_of_num vs t2
haftmann@29789
  1975
  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
haftmann@29789
  1976
      term_of_num vs (@{code C} i) $ term_of_num vs t2
haftmann@29789
  1977
  | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
haftmann@29789
  1978
wenzelm@45740
  1979
fun term_of_fm vs @{code T} = @{term True} 
wenzelm@45740
  1980
  | term_of_fm vs @{code F} = @{term False}
haftmann@29789
  1981
  | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  1982
      term_of_num vs t $ @{term "0::real"}
haftmann@29789
  1983
  | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  1984
      term_of_num vs t $ @{term "0::real"}
haftmann@29789
  1985
  | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  1986
      @{term "0::real"} $ term_of_num vs t
haftmann@29789
  1987
  | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  1988
      @{term "0::real"} $ term_of_num vs t
haftmann@29789
  1989
  | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $
haftmann@29789
  1990
      term_of_num vs t $ @{term "0::real"}
haftmann@29789
  1991
  | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
haftmann@29789
  1992
  | term_of_fm vs (@{code NOT} t') = HOLogic.Not $ term_of_fm vs t'
haftmann@29789
  1993
  | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2
haftmann@29789
  1994
  | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2
haftmann@29789
  1995
  | term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
haftmann@29789
  1996
  | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $
haftmann@36853
  1997
      term_of_fm vs t1 $ term_of_fm vs t2;
haftmann@29789
  1998
haftmann@36853
  1999
in fn (ctxt, t) =>
haftmann@29789
  2000
  let 
haftmann@36853
  2001
    val vs = Term.add_frees t [];
haftmann@36853
  2002
    val t' = (term_of_fm vs o @{code linrqe} o fm_of_term vs) t;
wenzelm@42361
  2003
  in (Thm.cterm_of (Proof_Context.theory_of ctxt) o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end
haftmann@29789
  2004
end;
haftmann@29789
  2005
*}
haftmann@29789
  2006
haftmann@29789
  2007
use "ferrack_tac.ML"
haftmann@29789
  2008
setup Ferrack_Tac.setup
haftmann@29789
  2009
haftmann@29789
  2010
lemma
haftmann@29789
  2011
  fixes x :: real
haftmann@29789
  2012
  shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1"
haftmann@29789
  2013
apply rferrack
haftmann@29789
  2014
done
haftmann@29789
  2015
haftmann@29789
  2016
lemma
haftmann@29789
  2017
  fixes x :: real
haftmann@29789
  2018
  shows "\<exists>y \<le> x. x = y + 1"
haftmann@29789
  2019
apply rferrack
haftmann@29789
  2020
done
haftmann@29789
  2021
haftmann@29789
  2022
lemma
haftmann@29789
  2023
  fixes x :: real
haftmann@29789
  2024
  shows "\<not> (\<exists>z. x + z = x + z + 1)"
haftmann@29789
  2025
apply rferrack
haftmann@29789
  2026
done
haftmann@29789
  2027
haftmann@29789
  2028
end