--- a/src/HOL/Complex/ex/ReflectedFerrack.thy Wed Dec 03 09:53:58 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2101 +0,0 @@
-(* Title: Complex/ex/ReflectedFerrack.thy
- Author: Amine Chaieb
-*)
-
-theory ReflectedFerrack
-imports Main GCD Real Efficient_Nat
-uses ("linrtac.ML")
-begin
-
-section {* Quantifier elimination for @{text "\<real> (0, 1, +, <)"} *}
-
- (*********************************************************************************)
- (* SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB *)
- (*********************************************************************************)
-
-consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
-primrec
- "alluopairs [] = []"
- "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
-
-lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
-by (induct xs, auto)
-
-lemma alluopairs_set:
- "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
-by (induct xs, auto)
-
-lemma alluopairs_ex:
- assumes Pc: "\<forall> x y. P x y = P y x"
- shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
-proof
- assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
- then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y" by blast
- from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y"
- by auto
-next
- assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
- then obtain "x" and "y" where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
- from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
- with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
-qed
-
-lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
-using Nat.gr0_conv_Suc
-by clarsimp
-
-lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
- apply (induct xs, auto) done
-
-consts remdps:: "'a list \<Rightarrow> 'a list"
-
-recdef remdps "measure size"
- "remdps [] = []"
- "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
-(hints simp add: filter_length[rule_format])
-
-lemma remdps_set[simp]: "set (remdps xs) = set xs"
- by (induct xs rule: remdps.induct, auto)
-
-
-
- (*********************************************************************************)
- (**** SHADOW SYNTAX AND SEMANTICS ****)
- (*********************************************************************************)
-
-datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num
- | Mul int num
-
- (* A size for num to make inductive proofs simpler*)
-consts num_size :: "num \<Rightarrow> nat"
-primrec
- "num_size (C c) = 1"
- "num_size (Bound n) = 1"
- "num_size (Neg a) = 1 + num_size a"
- "num_size (Add a b) = 1 + num_size a + num_size b"
- "num_size (Sub a b) = 3 + num_size a + num_size b"
- "num_size (Mul c a) = 1 + num_size a"
- "num_size (CN n c a) = 3 + num_size a "
-
- (* Semantics of numeral terms (num) *)
-consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
-primrec
- "Inum bs (C c) = (real c)"
- "Inum bs (Bound n) = bs!n"
- "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
- "Inum bs (Neg a) = -(Inum bs a)"
- "Inum bs (Add a b) = Inum bs a + Inum bs b"
- "Inum bs (Sub a b) = Inum bs a - Inum bs b"
- "Inum bs (Mul c a) = (real c) * Inum bs a"
- (* FORMULAE *)
-datatype fm =
- T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
- NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
-
-
- (* A size for fm *)
-consts fmsize :: "fm \<Rightarrow> nat"
-recdef fmsize "measure size"
- "fmsize (NOT p) = 1 + fmsize p"
- "fmsize (And p q) = 1 + fmsize p + fmsize q"
- "fmsize (Or p q) = 1 + fmsize p + fmsize q"
- "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
- "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
- "fmsize (E p) = 1 + fmsize p"
- "fmsize (A p) = 4+ fmsize p"
- "fmsize p = 1"
- (* several lemmas about fmsize *)
-lemma fmsize_pos: "fmsize p > 0"
-by (induct p rule: fmsize.induct) simp_all
-
- (* Semantics of formulae (fm) *)
-consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
-primrec
- "Ifm bs T = True"
- "Ifm bs F = False"
- "Ifm bs (Lt a) = (Inum bs a < 0)"
- "Ifm bs (Gt a) = (Inum bs a > 0)"
- "Ifm bs (Le a) = (Inum bs a \<le> 0)"
- "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
- "Ifm bs (Eq a) = (Inum bs a = 0)"
- "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
- "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
- "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
- "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
- "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
- "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
- "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
- "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
-
-lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
-apply simp
-done
-
-lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
-apply simp
-done
-lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
-apply simp
-done
-lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
-apply simp
-done
-lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
-apply simp
-done
-lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
-apply simp
-done
-lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
-apply simp
-done
-lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
-apply simp
-done
-
-lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
-apply simp
-done
-lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
-apply simp
-done
-
-consts not:: "fm \<Rightarrow> fm"
-recdef not "measure size"
- "not (NOT p) = p"
- "not T = F"
- "not F = T"
- "not p = NOT p"
-lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
-by (cases p) auto
-
-constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
- "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else
- if p = q then p else And p q)"
-lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
-by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
-
-constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
- "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p
- else if p=q then p else Or p q)"
-
-lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
-by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
-
-constdefs imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
- "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p
- else Imp p q)"
-lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
-by (cases "p=F \<or> q=T",simp_all add: imp_def)
-
-constdefs iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
- "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else
- 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
- Iff p q)"
-lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
- by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
-
-lemma conj_simps:
- "conj F Q = F"
- "conj P F = F"
- "conj T Q = Q"
- "conj P T = P"
- "conj P P = P"
- "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"
- by (simp_all add: conj_def)
-
-lemma disj_simps:
- "disj T Q = T"
- "disj P T = T"
- "disj F Q = Q"
- "disj P F = P"
- "disj P P = P"
- "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"
- by (simp_all add: disj_def)
-lemma imp_simps:
- "imp F Q = T"
- "imp P T = T"
- "imp T Q = Q"
- "imp P F = not P"
- "imp P P = T"
- "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"
- by (simp_all add: imp_def)
-lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
-apply (induct p, auto)
-done
-
-lemma iff_simps:
- "iff p p = T"
- "iff p (NOT p) = F"
- "iff (NOT p) p = F"
- "iff p F = not p"
- "iff F p = not p"
- "p \<noteq> NOT T \<Longrightarrow> iff T p = p"
- "p\<noteq> NOT T \<Longrightarrow> iff p T = p"
- "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"
- using trivNOT
- by (simp_all add: iff_def, cases p, auto)
- (* Quantifier freeness *)
-consts qfree:: "fm \<Rightarrow> bool"
-recdef qfree "measure size"
- "qfree (E p) = False"
- "qfree (A p) = False"
- "qfree (NOT p) = qfree p"
- "qfree (And p q) = (qfree p \<and> qfree q)"
- "qfree (Or p q) = (qfree p \<and> qfree q)"
- "qfree (Imp p q) = (qfree p \<and> qfree q)"
- "qfree (Iff p q) = (qfree p \<and> qfree q)"
- "qfree p = True"
-
- (* Boundedness and substitution *)
-consts
- numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
- bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
-primrec
- "numbound0 (C c) = True"
- "numbound0 (Bound n) = (n>0)"
- "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
- "numbound0 (Neg a) = numbound0 a"
- "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
- "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)"
- "numbound0 (Mul i a) = numbound0 a"
-lemma numbound0_I:
- assumes nb: "numbound0 a"
- shows "Inum (b#bs) a = Inum (b'#bs) a"
-using nb
-by (induct a rule: numbound0.induct,auto simp add: nth_pos2)
-
-primrec
- "bound0 T = True"
- "bound0 F = True"
- "bound0 (Lt a) = numbound0 a"
- "bound0 (Le a) = numbound0 a"
- "bound0 (Gt a) = numbound0 a"
- "bound0 (Ge a) = numbound0 a"
- "bound0 (Eq a) = numbound0 a"
- "bound0 (NEq a) = numbound0 a"
- "bound0 (NOT p) = bound0 p"
- "bound0 (And p q) = (bound0 p \<and> bound0 q)"
- "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
- "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
- "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
- "bound0 (E p) = False"
- "bound0 (A p) = False"
-
-lemma bound0_I:
- assumes bp: "bound0 p"
- shows "Ifm (b#bs) p = Ifm (b'#bs) p"
-using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
-by (induct p rule: bound0.induct) (auto simp add: nth_pos2)
-
-lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
-by (cases p, auto)
-lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
-by (cases p, auto)
-
-
-lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
-using conj_def by auto
-lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
-using conj_def by auto
-
-lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
-using disj_def by auto
-lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
-using disj_def by auto
-
-lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
-using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
-lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
-using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
-
-lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
- by (unfold iff_def,cases "p=q", auto)
-lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
-using iff_def by (unfold iff_def,cases "p=q", auto)
-
-consts
- decrnum:: "num \<Rightarrow> num"
- decr :: "fm \<Rightarrow> fm"
-
-recdef decrnum "measure size"
- "decrnum (Bound n) = Bound (n - 1)"
- "decrnum (Neg a) = Neg (decrnum a)"
- "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
- "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
- "decrnum (Mul c a) = Mul c (decrnum a)"
- "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
- "decrnum a = a"
-
-recdef decr "measure size"
- "decr (Lt a) = Lt (decrnum a)"
- "decr (Le a) = Le (decrnum a)"
- "decr (Gt a) = Gt (decrnum a)"
- "decr (Ge a) = Ge (decrnum a)"
- "decr (Eq a) = Eq (decrnum a)"
- "decr (NEq a) = NEq (decrnum a)"
- "decr (NOT p) = NOT (decr p)"
- "decr (And p q) = conj (decr p) (decr q)"
- "decr (Or p q) = disj (decr p) (decr q)"
- "decr (Imp p q) = imp (decr p) (decr q)"
- "decr (Iff p q) = iff (decr p) (decr q)"
- "decr p = p"
-
-lemma decrnum: assumes nb: "numbound0 t"
- shows "Inum (x#bs) t = Inum bs (decrnum t)"
- using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)
-
-lemma decr: assumes nb: "bound0 p"
- shows "Ifm (x#bs) p = Ifm bs (decr p)"
- using nb
- by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)
-
-lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
-by (induct p, simp_all)
-
-consts
- isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
-recdef isatom "measure size"
- "isatom T = True"
- "isatom F = True"
- "isatom (Lt a) = True"
- "isatom (Le a) = True"
- "isatom (Gt a) = True"
- "isatom (Ge a) = True"
- "isatom (Eq a) = True"
- "isatom (NEq a) = True"
- "isatom p = False"
-
-lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
-by (induct p, simp_all)
-
-constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
- "djf f p q \<equiv> (if q=T then T else if q=F then f p else
- (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
-constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
- "evaldjf f ps \<equiv> foldr (djf f) ps F"
-
-lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
-by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def)
-(cases "f p", simp_all add: Let_def djf_def)
-
-
-lemma djf_simps:
- "djf f p T = T"
- "djf f p F = f p"
- "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)"
- by (simp_all add: djf_def)
-
-lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
- by(induct ps, simp_all add: evaldjf_def djf_Or)
-
-lemma evaldjf_bound0:
- assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
- shows "bound0 (evaldjf f xs)"
- using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
-
-lemma evaldjf_qf:
- assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
- shows "qfree (evaldjf f xs)"
- using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
-
-consts disjuncts :: "fm \<Rightarrow> fm list"
-recdef disjuncts "measure size"
- "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
- "disjuncts F = []"
- "disjuncts p = [p]"
-
-lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
-by(induct p rule: disjuncts.induct, auto)
-
-lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
-proof-
- assume nb: "bound0 p"
- hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
- thus ?thesis by (simp only: list_all_iff)
-qed
-
-lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
-proof-
- assume qf: "qfree p"
- hence "list_all qfree (disjuncts p)"
- by (induct p rule: disjuncts.induct, auto)
- thus ?thesis by (simp only: list_all_iff)
-qed
-
-constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
- "DJ f p \<equiv> evaldjf f (disjuncts p)"
-
-lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
- and fF: "f F = F"
- shows "Ifm bs (DJ f p) = Ifm bs (f p)"
-proof-
- have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
- by (simp add: DJ_def evaldjf_ex)
- also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
- finally show ?thesis .
-qed
-
-lemma DJ_qf: assumes
- fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
- shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
-proof(clarify)
- fix p assume qf: "qfree p"
- have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
- from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
- with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
-
- from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
-qed
-
-lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
- shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
-proof(clarify)
- fix p::fm and bs
- assume qf: "qfree p"
- from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
- from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
- have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
- by (simp add: DJ_def evaldjf_ex)
- also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
- also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
- finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
-qed
- (* Simplification *)
-consts
- numgcd :: "num \<Rightarrow> int"
- numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
- reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
- reducecoeff :: "num \<Rightarrow> num"
- dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
-consts maxcoeff:: "num \<Rightarrow> int"
-recdef maxcoeff "measure size"
- "maxcoeff (C i) = abs i"
- "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
- "maxcoeff t = 1"
-
-lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
- by (induct t rule: maxcoeff.induct, auto)
-
-recdef numgcdh "measure size"
- "numgcdh (C i) = (\<lambda>g. zgcd i g)"
- "numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))"
- "numgcdh t = (\<lambda>g. 1)"
-defs numgcd_def [code]: "numgcd t \<equiv> numgcdh t (maxcoeff t)"
-
-recdef reducecoeffh "measure size"
- "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
- "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
- "reducecoeffh t = (\<lambda>g. t)"
-
-defs reducecoeff_def: "reducecoeff t \<equiv>
- (let g = numgcd t in
- if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
-
-recdef dvdnumcoeff "measure size"
- "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
- "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
- "dvdnumcoeff t = (\<lambda>g. False)"
-
-lemma dvdnumcoeff_trans:
- assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
- shows "dvdnumcoeff t g"
- using dgt' gdg
- by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg])
-
-declare zdvd_trans [trans add]
-
-lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
-by arith
-
-lemma numgcd0:
- assumes g0: "numgcd t = 0"
- shows "Inum bs t = 0"
- using g0[simplified numgcd_def]
- by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)
-
-lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
- using gp
- by (induct t rule: numgcdh.induct, auto simp add: zgcd_def)
-
-lemma numgcd_pos: "numgcd t \<ge>0"
- by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
-
-lemma reducecoeffh:
- assumes gt: "dvdnumcoeff t g" and gp: "g > 0"
- shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
- using gt
-proof(induct t rule: reducecoeffh.induct)
- case (1 i) hence gd: "g dvd i" by simp
- from gp have gnz: "g \<noteq> 0" by simp
- from prems show ?case by (simp add: real_of_int_div[OF gnz gd])
-next
- case (2 n c t) hence gd: "g dvd c" by simp
- from gp have gnz: "g \<noteq> 0" by simp
- from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_simps)
-qed (auto simp add: numgcd_def gp)
-consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
-recdef ismaxcoeff "measure size"
- "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
- "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
- "ismaxcoeff t = (\<lambda>x. True)"
-
-lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
-by (induct t rule: ismaxcoeff.induct, auto)
-
-lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
-proof (induct t rule: maxcoeff.induct)
- case (2 n c t)
- hence H:"ismaxcoeff t (maxcoeff t)" .
- have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2)
- from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
-qed simp_all
-
-lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
- apply (cases "abs i = 0", simp_all add: zgcd_def)
- apply (cases "abs j = 0", simp_all)
- apply (cases "abs i = 1", simp_all)
- apply (cases "abs j = 1", simp_all)
- apply auto
- done
-lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow> m =0"
- by (induct t rule: numgcdh.induct, auto simp add:zgcd0)
-
-lemma dvdnumcoeff_aux:
- assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
- shows "dvdnumcoeff t (numgcdh t m)"
-using prems
-proof(induct t rule: numgcdh.induct)
- case (2 n c t)
- let ?g = "numgcdh t m"
- from prems have th:"zgcd c ?g > 1" by simp
- from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
- have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
- moreover {assume "abs c > 1" and gp: "?g > 1" with prems
- have th: "dvdnumcoeff t ?g" by simp
- have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
- from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
- moreover {assume "abs c = 0 \<and> ?g > 1"
- with prems have th: "dvdnumcoeff t ?g" by simp
- have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
- from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
- hence ?case by simp }
- moreover {assume "abs c > 1" and g0:"?g = 0"
- from numgcdh0[OF g0] have "m=0". with prems have ?case by simp }
- ultimately show ?case by blast
-qed(auto simp add: zgcd_zdvd1)
-
-lemma dvdnumcoeff_aux2:
- assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
- using prems
-proof (simp add: numgcd_def)
- let ?mc = "maxcoeff t"
- let ?g = "numgcdh t ?mc"
- have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
- have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
- assume H: "numgcdh t ?mc > 1"
- from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" .
-qed
-
-lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
-proof-
- let ?g = "numgcd t"
- have "?g \<ge> 0" by (simp add: numgcd_pos)
- hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
- moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)}
- moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)}
- moreover { assume g1:"?g > 1"
- from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
- from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis
- by (simp add: reducecoeff_def Let_def)}
- ultimately show ?thesis by blast
-qed
-
-lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
-by (induct t rule: reducecoeffh.induct, auto)
-
-lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
-using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
-
-consts
- simpnum:: "num \<Rightarrow> num"
- numadd:: "num \<times> num \<Rightarrow> num"
- nummul:: "num \<Rightarrow> int \<Rightarrow> num"
-recdef numadd "measure (\<lambda> (t,s). size t + size s)"
- "numadd (CN n1 c1 r1,CN n2 c2 r2) =
- (if n1=n2 then
- (let c = c1 + c2
- in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
- else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2)))
- else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
- "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"
- "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))"
- "numadd (C b1, C b2) = C (b1+b2)"
- "numadd (a,b) = Add a b"
-
-lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
-apply (induct t s rule: numadd.induct, simp_all add: Let_def)
-apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
-apply (case_tac "n1 = n2", simp_all add: ring_simps)
-by (simp only: left_distrib[symmetric],simp)
-
-lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
-by (induct t s rule: numadd.induct, auto simp add: Let_def)
-
-recdef nummul "measure size"
- "nummul (C j) = (\<lambda> i. C (i*j))"
- "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
- "nummul t = (\<lambda> i. Mul i t)"
-
-lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
-by (induct t rule: nummul.induct, auto simp add: ring_simps)
-
-lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
-by (induct t rule: nummul.induct, auto )
-
-constdefs numneg :: "num \<Rightarrow> num"
- "numneg t \<equiv> nummul t (- 1)"
-
-constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
- "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
-
-lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
-using numneg_def by simp
-
-lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
-using numneg_def by simp
-
-lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
-using numsub_def by simp
-
-lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
-using numsub_def by simp
-
-recdef simpnum "measure size"
- "simpnum (C j) = C j"
- "simpnum (Bound n) = CN n 1 (C 0)"
- "simpnum (Neg t) = numneg (simpnum t)"
- "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
- "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
- "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
- "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
-
-lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
-by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)
-
-lemma simpnum_numbound0[simp]:
- "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
-by (induct t rule: simpnum.induct, auto)
-
-consts nozerocoeff:: "num \<Rightarrow> bool"
-recdef nozerocoeff "measure size"
- "nozerocoeff (C c) = True"
- "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
- "nozerocoeff t = True"
-
-lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
-by (induct a b rule: numadd.induct,auto simp add: Let_def)
-
-lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
-by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
-
-lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
-by (simp add: numneg_def nummul_nz)
-
-lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
-by (simp add: numsub_def numneg_nz numadd_nz)
-
-lemma simpnum_nz: "nozerocoeff (simpnum t)"
-by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz)
-
-lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
-proof (induct t rule: maxcoeff.induct)
- case (2 n c t)
- hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
- have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
- with cnz have "max (abs c) (maxcoeff t) > 0" by arith
- with prems show ?case by simp
-qed auto
-
-lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
-proof-
- from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
- from numgcdh0[OF th] have th:"maxcoeff t = 0" .
- from maxcoeff_nz[OF nz th] show ?thesis .
-qed
-
-constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
- "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
- (let t' = simpnum t ; g = numgcd t' in
- if g > 1 then (let g' = zgcd n g in
- if g' = 1 then (t',n)
- else (reducecoeffh t' g', n div g'))
- else (t',n))))"
-
-lemma simp_num_pair_ci:
- shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
- (is "?lhs = ?rhs")
-proof-
- let ?t' = "simpnum t"
- let ?g = "numgcd ?t'"
- let ?g' = "zgcd n ?g"
- {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
- moreover
- { assume nnz: "n \<noteq> 0"
- {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
- moreover
- {assume g1:"?g>1" hence g0: "?g > 0" by simp
- from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
- hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
- hence "?g'= 1 \<or> ?g' > 1" by arith
- moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
- moreover {assume g'1:"?g'>1"
- from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
- let ?tt = "reducecoeffh ?t' ?g'"
- let ?t = "Inum bs ?tt"
- have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
- have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
- have gpdgp: "?g' dvd ?g'" by simp
- from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
- have th2:"real ?g' * ?t = Inum bs ?t'" by simp
- from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
- also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
- also have "\<dots> = (Inum bs ?t' / real n)"
- using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
- finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci)
- then have ?thesis using prems by (simp add: simp_num_pair_def)}
- ultimately have ?thesis by blast}
- ultimately have ?thesis by blast}
- ultimately show ?thesis by blast
-qed
-
-lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
- shows "numbound0 t' \<and> n' >0"
-proof-
- let ?t' = "simpnum t"
- let ?g = "numgcd ?t'"
- let ?g' = "zgcd n ?g"
- {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
- moreover
- { assume nnz: "n \<noteq> 0"
- {assume "\<not> ?g > 1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
- moreover
- {assume g1:"?g>1" hence g0: "?g > 0" by simp
- from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
- hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
- hence "?g'= 1 \<or> ?g' > 1" by arith
- moreover {assume "?g'=1" hence ?thesis using prems
- by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
- moreover {assume g'1:"?g'>1"
- have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
- have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
- have gpdgp: "?g' dvd ?g'" by simp
- from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
- from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
- have "n div ?g' >0" by simp
- hence ?thesis using prems
- by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)}
- ultimately have ?thesis by blast}
- ultimately have ?thesis by blast}
- ultimately show ?thesis by blast
-qed
-
-consts simpfm :: "fm \<Rightarrow> fm"
-recdef simpfm "measure fmsize"
- "simpfm (And p q) = conj (simpfm p) (simpfm q)"
- "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
- "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
- "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
- "simpfm (NOT p) = not (simpfm p)"
- "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F
- | _ \<Rightarrow> Lt a')"
- "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')"
- "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0) then T else F | _ \<Rightarrow> Gt a')"
- "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')"
- "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0) then T else F | _ \<Rightarrow> Eq a')"
- "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')"
- "simpfm p = p"
-lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p"
-proof(induct p rule: simpfm.induct)
- case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
- {fix v assume "?sa = C v" hence ?case using sa by simp }
- moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
- by (cases ?sa, simp_all add: Let_def)}
- ultimately show ?case by blast
-next
- case (7 a) let ?sa = "simpnum a"
- from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
- {fix v assume "?sa = C v" hence ?case using sa by simp }
- moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
- by (cases ?sa, simp_all add: Let_def)}
- ultimately show ?case by blast
-next
- case (8 a) let ?sa = "simpnum a"
- from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
- {fix v assume "?sa = C v" hence ?case using sa by simp }
- moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
- by (cases ?sa, simp_all add: Let_def)}
- ultimately show ?case by blast
-next
- case (9 a) let ?sa = "simpnum a"
- from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
- {fix v assume "?sa = C v" hence ?case using sa by simp }
- moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
- by (cases ?sa, simp_all add: Let_def)}
- ultimately show ?case by blast
-next
- case (10 a) let ?sa = "simpnum a"
- from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
- {fix v assume "?sa = C v" hence ?case using sa by simp }
- moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
- by (cases ?sa, simp_all add: Let_def)}
- ultimately show ?case by blast
-next
- case (11 a) let ?sa = "simpnum a"
- from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
- {fix v assume "?sa = C v" hence ?case using sa by simp }
- moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
- by (cases ?sa, simp_all add: Let_def)}
- ultimately show ?case by blast
-qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
-
-
-lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
-proof(induct p rule: simpfm.induct)
- case (6 a) hence nb: "numbound0 a" by simp
- hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
- case (7 a) hence nb: "numbound0 a" by simp
- hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
- case (8 a) hence nb: "numbound0 a" by simp
- hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
- case (9 a) hence nb: "numbound0 a" by simp
- hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
- case (10 a) hence nb: "numbound0 a" by simp
- hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
-next
- case (11 a) hence nb: "numbound0 a" by simp
- hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
-qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
-
-lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
-by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
- (case_tac "simpnum a",auto)+
-
-consts prep :: "fm \<Rightarrow> fm"
-recdef prep "measure fmsize"
- "prep (E T) = T"
- "prep (E F) = F"
- "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
- "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
- "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))"
- "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
- "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
- "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
- "prep (E p) = E (prep p)"
- "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
- "prep (A p) = prep (NOT (E (NOT p)))"
- "prep (NOT (NOT p)) = prep p"
- "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
- "prep (NOT (A p)) = prep (E (NOT p))"
- "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
- "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
- "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
- "prep (NOT p) = not (prep p)"
- "prep (Or p q) = disj (prep p) (prep q)"
- "prep (And p q) = conj (prep p) (prep q)"
- "prep (Imp p q) = prep (Or (NOT p) q)"
- "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
- "prep p = p"
-(hints simp add: fmsize_pos)
-lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
-by (induct p rule: prep.induct, auto)
-
- (* Generic quantifier elimination *)
-consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
-recdef qelim "measure fmsize"
- "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
- "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
- "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
- "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))"
- "qelim (Or p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))"
- "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
- "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
- "qelim p = (\<lambda> y. simpfm p)"
-
-lemma qelim_ci:
- assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
- shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
-using qe_inv DJ_qe[OF qe_inv]
-by(induct p rule: qelim.induct)
-(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf
- simpfm simpfm_qf simp del: simpfm.simps)
-
-consts
- plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
- minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
-recdef minusinf "measure size"
- "minusinf (And p q) = conj (minusinf p) (minusinf q)"
- "minusinf (Or p q) = disj (minusinf p) (minusinf q)"
- "minusinf (Eq (CN 0 c e)) = F"
- "minusinf (NEq (CN 0 c e)) = T"
- "minusinf (Lt (CN 0 c e)) = T"
- "minusinf (Le (CN 0 c e)) = T"
- "minusinf (Gt (CN 0 c e)) = F"
- "minusinf (Ge (CN 0 c e)) = F"
- "minusinf p = p"
-
-recdef plusinf "measure size"
- "plusinf (And p q) = conj (plusinf p) (plusinf q)"
- "plusinf (Or p q) = disj (plusinf p) (plusinf q)"
- "plusinf (Eq (CN 0 c e)) = F"
- "plusinf (NEq (CN 0 c e)) = T"
- "plusinf (Lt (CN 0 c e)) = F"
- "plusinf (Le (CN 0 c e)) = F"
- "plusinf (Gt (CN 0 c e)) = T"
- "plusinf (Ge (CN 0 c e)) = T"
- "plusinf p = p"
-
-consts
- isrlfm :: "fm \<Rightarrow> bool" (* Linearity test for fm *)
-recdef isrlfm "measure size"
- "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)"
- "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)"
- "isrlfm (Eq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
- "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
- "isrlfm (Lt (CN 0 c e)) = (c>0 \<and> numbound0 e)"
- "isrlfm (Le (CN 0 c e)) = (c>0 \<and> numbound0 e)"
- "isrlfm (Gt (CN 0 c e)) = (c>0 \<and> numbound0 e)"
- "isrlfm (Ge (CN 0 c e)) = (c>0 \<and> numbound0 e)"
- "isrlfm p = (isatom p \<and> (bound0 p))"
-
- (* splits the bounded from the unbounded part*)
-consts rsplit0 :: "num \<Rightarrow> int \<times> num"
-recdef rsplit0 "measure num_size"
- "rsplit0 (Bound 0) = (1,C 0)"
- "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b
- in (ca+cb, Add ta tb))"
- "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
- "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
- "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
- "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
- "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
- "rsplit0 t = (0,t)"
-lemma rsplit0:
- shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
-proof (induct t rule: rsplit0.induct)
- case (2 a b)
- let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
- let ?ca = "fst ?sa" let ?cb = "fst ?sb"
- let ?ta = "snd ?sa" let ?tb = "snd ?sb"
- from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))"
- by(cases "rsplit0 a",auto simp add: Let_def split_def)
- have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) =
- Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
- by (simp add: Let_def split_def ring_simps)
- also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all)
- finally show ?case using nb by simp
-qed(auto simp add: Let_def split_def ring_simps , simp add: right_distrib[symmetric])
-
- (* Linearize a formula*)
-definition
- lt :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
- "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t))
- else (Gt (CN 0 (-c) (Neg t))))"
-
-definition
- le :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
- "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t))
- else (Ge (CN 0 (-c) (Neg t))))"
-
-definition
- gt :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
- "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t))
- else (Lt (CN 0 (-c) (Neg t))))"
-
-definition
- ge :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
- "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t))
- else (Le (CN 0 (-c) (Neg t))))"
-
-definition
- eq :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
- "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t))
- else (Eq (CN 0 (-c) (Neg t))))"
-
-definition
- neq :: "int \<Rightarrow> num \<Rightarrow> fm"
-where
- "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t))
- else (NEq (CN 0 (-c) (Neg t))))"
-
-lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
-using rsplit0[where bs = "bs" and t="t"]
-by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
-
-lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
-by (auto simp add: conj_def)
-lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
-by (auto simp add: disj_def)
-
-consts rlfm :: "fm \<Rightarrow> fm"
-recdef rlfm "measure fmsize"
- "rlfm (And p q) = conj (rlfm p) (rlfm q)"
- "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
- "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
- "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
- "rlfm (Lt a) = split lt (rsplit0 a)"
- "rlfm (Le a) = split le (rsplit0 a)"
- "rlfm (Gt a) = split gt (rsplit0 a)"
- "rlfm (Ge a) = split ge (rsplit0 a)"
- "rlfm (Eq a) = split eq (rsplit0 a)"
- "rlfm (NEq a) = split neq (rsplit0 a)"
- "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
- "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
- "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
- "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
- "rlfm (NOT (NOT p)) = rlfm p"
- "rlfm (NOT T) = F"
- "rlfm (NOT F) = T"
- "rlfm (NOT (Lt a)) = rlfm (Ge a)"
- "rlfm (NOT (Le a)) = rlfm (Gt a)"
- "rlfm (NOT (Gt a)) = rlfm (Le a)"
- "rlfm (NOT (Ge a)) = rlfm (Lt a)"
- "rlfm (NOT (Eq a)) = rlfm (NEq a)"
- "rlfm (NOT (NEq a)) = rlfm (Eq a)"
- "rlfm p = p" (hints simp add: fmsize_pos)
-
-lemma rlfm_I:
- assumes qfp: "qfree p"
- shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
- using qfp
-by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
-
- (* Operations needed for Ferrante and Rackoff *)
-lemma rminusinf_inf:
- assumes lp: "isrlfm p"
- shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
-using lp
-proof (induct p rule: minusinf.induct)
- case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
-next
- case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
-next
- case (3 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x < ?z"
- hence "(real c * x < - ?e)"
- by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
- hence "real c * x + ?e < 0" by arith
- hence "real c * x + ?e \<noteq> 0" by simp
- with xz have "?P ?z x (Eq (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (4 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x < ?z"
- hence "(real c * x < - ?e)"
- by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
- hence "real c * x + ?e < 0" by arith
- hence "real c * x + ?e \<noteq> 0" by simp
- with xz have "?P ?z x (NEq (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (5 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x < ?z"
- hence "(real c * x < - ?e)"
- by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
- hence "real c * x + ?e < 0" by arith
- with xz have "?P ?z x (Lt (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (6 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x < ?z"
- hence "(real c * x < - ?e)"
- by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
- hence "real c * x + ?e < 0" by arith
- with xz have "?P ?z x (Le (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (7 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x < ?z"
- hence "(real c * x < - ?e)"
- by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
- hence "real c * x + ?e < 0" by arith
- with xz have "?P ?z x (Gt (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (8 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x < ?z"
- hence "(real c * x < - ?e)"
- by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
- hence "real c * x + ?e < 0" by arith
- with xz have "?P ?z x (Ge (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
- thus ?case by blast
-qed simp_all
-
-lemma rplusinf_inf:
- assumes lp: "isrlfm p"
- shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
-using lp
-proof (induct p rule: isrlfm.induct)
- case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
-next
- case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
-next
- case (3 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x > ?z"
- with mult_strict_right_mono [OF xz cp] cp
- have "(real c * x > - ?e)" by (simp add: mult_ac)
- hence "real c * x + ?e > 0" by arith
- hence "real c * x + ?e \<noteq> 0" by simp
- with xz have "?P ?z x (Eq (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (4 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x > ?z"
- with mult_strict_right_mono [OF xz cp] cp
- have "(real c * x > - ?e)" by (simp add: mult_ac)
- hence "real c * x + ?e > 0" by arith
- hence "real c * x + ?e \<noteq> 0" by simp
- with xz have "?P ?z x (NEq (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (5 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x > ?z"
- with mult_strict_right_mono [OF xz cp] cp
- have "(real c * x > - ?e)" by (simp add: mult_ac)
- hence "real c * x + ?e > 0" by arith
- with xz have "?P ?z x (Lt (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (6 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x > ?z"
- with mult_strict_right_mono [OF xz cp] cp
- have "(real c * x > - ?e)" by (simp add: mult_ac)
- hence "real c * x + ?e > 0" by arith
- with xz have "?P ?z x (Le (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (7 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x > ?z"
- with mult_strict_right_mono [OF xz cp] cp
- have "(real c * x > - ?e)" by (simp add: mult_ac)
- hence "real c * x + ?e > 0" by arith
- with xz have "?P ?z x (Gt (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
- thus ?case by blast
-next
- case (8 c e)
- from prems have nb: "numbound0 e" by simp
- from prems have cp: "real c > 0" by simp
- fix a
- let ?e="Inum (a#bs) e"
- let ?z = "(- ?e) / real c"
- {fix x
- assume xz: "x > ?z"
- with mult_strict_right_mono [OF xz cp] cp
- have "(real c * x > - ?e)" by (simp add: mult_ac)
- hence "real c * x + ?e > 0" by arith
- with xz have "?P ?z x (Ge (CN 0 c e))"
- using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
- hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
- thus ?case by blast
-qed simp_all
-
-lemma rminusinf_bound0:
- assumes lp: "isrlfm p"
- shows "bound0 (minusinf p)"
- using lp
- by (induct p rule: minusinf.induct) simp_all
-
-lemma rplusinf_bound0:
- assumes lp: "isrlfm p"
- shows "bound0 (plusinf p)"
- using lp
- by (induct p rule: plusinf.induct) simp_all
-
-lemma rminusinf_ex:
- assumes lp: "isrlfm p"
- and ex: "Ifm (a#bs) (minusinf p)"
- shows "\<exists> x. Ifm (x#bs) p"
-proof-
- from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
- have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
- from rminusinf_inf[OF lp, where bs="bs"]
- obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
- from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
- moreover have "z - 1 < z" by simp
- ultimately show ?thesis using z_def by auto
-qed
-
-lemma rplusinf_ex:
- assumes lp: "isrlfm p"
- and ex: "Ifm (a#bs) (plusinf p)"
- shows "\<exists> x. Ifm (x#bs) p"
-proof-
- from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
- have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
- from rplusinf_inf[OF lp, where bs="bs"]
- obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
- from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
- moreover have "z + 1 > z" by simp
- ultimately show ?thesis using z_def by auto
-qed
-
-consts
- uset:: "fm \<Rightarrow> (num \<times> int) list"
- usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
-recdef uset "measure size"
- "uset (And p q) = (uset p @ uset q)"
- "uset (Or p q) = (uset p @ uset q)"
- "uset (Eq (CN 0 c e)) = [(Neg e,c)]"
- "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
- "uset (Lt (CN 0 c e)) = [(Neg e,c)]"
- "uset (Le (CN 0 c e)) = [(Neg e,c)]"
- "uset (Gt (CN 0 c e)) = [(Neg e,c)]"
- "uset (Ge (CN 0 c e)) = [(Neg e,c)]"
- "uset p = []"
-recdef usubst "measure size"
- "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
- "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
- "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
- "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
- "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
- "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
- "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
- "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
- "usubst p = (\<lambda> (t,n). p)"
-
-lemma usubst_I: assumes lp: "isrlfm p"
- and np: "real n > 0" and nbt: "numbound0 t"
- 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> _")
- using lp
-proof(induct p rule: usubst.induct)
- case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
- have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
- using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
- also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
- by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
- also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
- using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
-next
- case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
- have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
- using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
- also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
- by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
- also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
- using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
-next
- case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
- have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
- using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
- also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
- by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
- also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
- using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
-next
- case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
- have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
- using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
- also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
- by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
- also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
- using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
-next
- case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
- from np have np: "real n \<noteq> 0" by simp
- have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
- using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
- also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
- by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
- also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
- using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
-next
- case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
- from np have np: "real n \<noteq> 0" by simp
- have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
- using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
- also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
- by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
- also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
- using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
-qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
-
-lemma uset_l:
- assumes lp: "isrlfm p"
- shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
-using lp
-by(induct p rule: uset.induct,auto)
-
-lemma rminusinf_uset:
- assumes lp: "isrlfm p"
- and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
- and ex: "Ifm (x#bs) p" (is "?I x p")
- 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")
-proof-
- 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")
- using lp nmi ex
- by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
- then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
- from uset_l[OF lp] smU have mp: "real m > 0" by auto
- 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"
- by (auto simp add: mult_commute)
- thus ?thesis using smU by auto
-qed
-
-lemma rplusinf_uset:
- assumes lp: "isrlfm p"
- and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
- and ex: "Ifm (x#bs) p" (is "?I x p")
- 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")
-proof-
- 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")
- using lp nmi ex
- by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
- then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
- from uset_l[OF lp] smU have mp: "real m > 0" by auto
- 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"
- by (auto simp add: mult_commute)
- thus ?thesis using smU by auto
-qed
-
-lemma lin_dense:
- assumes lp: "isrlfm p"
- and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)"
- (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
- and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
- and ly: "l < y" and yu: "y < u"
- shows "Ifm (y#bs) p"
-using lp px noS
-proof (induct p rule: isrlfm.induct)
- case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e < 0" by (simp add: ring_simps)
- hence pxc: "x < (- ?N x e) / real c"
- by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
- from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
- with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
- hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
- moreover {assume y: "y < (-?N x e)/ real c"
- hence "y * real c < - ?N x e"
- by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e < 0" by (simp add: ring_simps)
- hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
- moreover {assume y: "y > (- ?N x e) / real c"
- with yu have eu: "u > (- ?N x e) / real c" by auto
- with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
- with lx pxc have "False" by auto
- hence ?case by simp }
- ultimately show ?case by blast
-next
- case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
- from prems have "x * real c + ?N x e \<le> 0" by (simp add: ring_simps)
- hence pxc: "x \<le> (- ?N x e) / real c"
- by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
- from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
- with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
- hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
- moreover {assume y: "y < (-?N x e)/ real c"
- hence "y * real c < - ?N x e"
- by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e < 0" by (simp add: ring_simps)
- hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
- moreover {assume y: "y > (- ?N x e) / real c"
- with yu have eu: "u > (- ?N x e) / real c" by auto
- with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
- with lx pxc have "False" by auto
- hence ?case by simp }
- ultimately show ?case by blast
-next
- case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e > 0" by (simp add: ring_simps)
- hence pxc: "x > (- ?N x e) / real c"
- by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
- from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
- with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
- hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
- moreover {assume y: "y > (-?N x e)/ real c"
- hence "y * real c > - ?N x e"
- by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e > 0" by (simp add: ring_simps)
- hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
- moreover {assume y: "y < (- ?N x e) / real c"
- with ly have eu: "l < (- ?N x e) / real c" by auto
- with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
- with xu pxc have "False" by auto
- hence ?case by simp }
- ultimately show ?case by blast
-next
- case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e \<ge> 0" by (simp add: ring_simps)
- hence pxc: "x \<ge> (- ?N x e) / real c"
- by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
- from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
- with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
- hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
- moreover {assume y: "y > (-?N x e)/ real c"
- hence "y * real c > - ?N x e"
- by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e > 0" by (simp add: ring_simps)
- hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
- moreover {assume y: "y < (- ?N x e) / real c"
- with ly have eu: "l < (- ?N x e) / real c" by auto
- with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
- with xu pxc have "False" by auto
- hence ?case by simp }
- ultimately show ?case by blast
-next
- case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from cp have cnz: "real c \<noteq> 0" by simp
- from prems have "x * real c + ?N x e = 0" by (simp add: ring_simps)
- hence pxc: "x = (- ?N x e) / real c"
- by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
- from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
- with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
- with pxc show ?case by simp
-next
- case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from cp have cnz: "real c \<noteq> 0" by simp
- from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
- with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
- hence "y* real c \<noteq> -?N x e"
- by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
- hence "y* real c + ?N x e \<noteq> 0" by (simp add: ring_simps)
- thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
- by (simp add: ring_simps)
-qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
-
-lemma finite_set_intervals:
- assumes px: "P (x::real)"
- and lx: "l \<le> x" and xu: "x \<le> u"
- and linS: "l\<in> S" and uinS: "u \<in> S"
- and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
- 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"
-proof-
- let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
- let ?xM = "{y. y\<in> S \<and> x \<le> y}"
- let ?a = "Max ?Mx"
- let ?b = "Min ?xM"
- have MxS: "?Mx \<subseteq> S" by blast
- hence fMx: "finite ?Mx" using fS finite_subset by auto
- from lx linS have linMx: "l \<in> ?Mx" by blast
- hence Mxne: "?Mx \<noteq> {}" by blast
- have xMS: "?xM \<subseteq> S" by blast
- hence fxM: "finite ?xM" using fS finite_subset by auto
- from xu uinS have linxM: "u \<in> ?xM" by blast
- hence xMne: "?xM \<noteq> {}" by blast
- have ax:"?a \<le> x" using Mxne fMx by auto
- have xb:"x \<le> ?b" using xMne fxM by auto
- have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
- have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
- have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
- proof(clarsimp)
- fix y
- assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
- from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
- moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
- moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
- ultimately show "False" by blast
- qed
- from ainS binS noy ax xb px show ?thesis by blast
-qed
-
-lemma finite_set_intervals2:
- assumes px: "P (x::real)"
- and lx: "l \<le> x" and xu: "x \<le> u"
- and linS: "l\<in> S" and uinS: "u \<in> S"
- and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
- shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)"
-proof-
- from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
- obtain a and b where
- as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x" by auto
- from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto
- thus ?thesis using px as bs noS by blast
-qed
-
-lemma rinf_uset:
- assumes lp: "isrlfm p"
- and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
- and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
- and ex: "\<exists> x. Ifm (x#bs) p" (is "\<exists> x. ?I x p")
- 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"
-proof-
- let ?N = "\<lambda> x t. Inum (x#bs) t"
- let ?U = "set (uset p)"
- from ex obtain a where pa: "?I a p" by blast
- from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
- have nmi': "\<not> (?I a (?M p))" by simp
- from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
- have npi': "\<not> (?I a (?P p))" by simp
- 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"
- proof-
- let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
- have fM: "finite ?M" by auto
- from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa]
- 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
- then obtain "t" "n" "s" "m" where
- tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U"
- and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
- 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
- from tnU have Mne: "?M \<noteq> {}" by auto
- hence Une: "?U \<noteq> {}" by simp
- let ?l = "Min ?M"
- let ?u = "Max ?M"
- have linM: "?l \<in> ?M" using fM Mne by simp
- have uinM: "?u \<in> ?M" using fM Mne by simp
- have tnM: "?N a t / real n \<in> ?M" using tnU by auto
- have smM: "?N a s / real m \<in> ?M" using smU by auto
- have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
- have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
- have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
- have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
- from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
- have "(\<exists> s\<in> ?M. ?I s p) \<or>
- (\<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)" .
- moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
- hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
- then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
- have "(u + u) / 2 = u" by auto with pu tuu
- have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
- with tuU have ?thesis by blast}
- moreover{
- 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"
- then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M"
- 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"
- by blast
- from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
- then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
- from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
- then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
- from t1x xt2 have t1t2: "t1 < t2" by simp
- let ?u = "(t1 + t2) / 2"
- from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
- from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
- with t1uU t2uU t1u t2u have ?thesis by blast}
- ultimately show ?thesis by blast
- qed
- then obtain "l" "n" "s" "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U"
- and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
- from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
- from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
- numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
- have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
- with lnU smU
- show ?thesis by auto
-qed
- (* The Ferrante - Rackoff Theorem *)
-
-theorem fr_eq:
- assumes lp: "isrlfm p"
- 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))"
- (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
-proof
- assume px: "\<exists> x. ?I x p"
- have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
- moreover {assume "?M \<or> ?P" hence "?D" by blast}
- moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
- from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
- ultimately show "?D" by blast
-next
- assume "?D"
- moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
- moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
- moreover {assume f:"?F" hence "?E" by blast}
- ultimately show "?E" by blast
-qed
-
-
-lemma fr_equsubst:
- assumes lp: "isrlfm p"
- 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))))"
- (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
-proof
- assume px: "\<exists> x. ?I x p"
- have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
- moreover {assume "?M \<or> ?P" hence "?D" by blast}
- moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
- let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
- let ?N = "\<lambda> t. Inum (x#bs) t"
- {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
- with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
- by auto
- let ?st = "Add (Mul m t) (Mul n s)"
- from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
- by (simp add: mult_commute)
- from tnb snb have st_nb: "numbound0 ?st" by simp
- have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnp mp np by (simp add: ring_simps add_divide_distrib)
- from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"]
- 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])}
- with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
- ultimately show "?D" by blast
-next
- assume "?D"
- moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
- moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
- moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)"
- and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
- 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
- let ?st = "Add (Mul l t) (Mul k s)"
- from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0"
- by (simp add: mult_commute)
- from tnb snb have st_nb: "numbound0 ?st" by simp
- from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
- ultimately show "?E" by blast
-qed
-
-
- (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
-constdefs ferrack:: "fm \<Rightarrow> fm"
- "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
- in if (mp = T \<or> pp = T) then T else
- (let U = remdps(map simp_num_pair
- (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
- (alluopairs (uset p'))))
- in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
-
-lemma uset_cong_aux:
- assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
- 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))"
- (is "?lhs = ?rhs")
-proof(auto)
- fix t n s m
- assume "((t,n),(s,m)) \<in> set (alluopairs U)"
- hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
- using alluopairs_set1[where xs="U"] by blast
- let ?N = "\<lambda> t. Inum (x#bs) t"
- let ?st= "Add (Mul m t) (Mul n s)"
- from Ul th have mnz: "m \<noteq> 0" by auto
- from Ul th have nnz: "n \<noteq> 0" by auto
- have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnz nnz by (simp add: ring_simps add_divide_distrib)
-
- thus "(real m * Inum (x # bs) t + real n * Inum (x # bs) s) /
- (2 * real n * real m)
- \<in> (\<lambda>((t, n), s, m).
- (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
- (set U \<times> set U)"using mnz nnz th
- apply (auto simp add: th add_divide_distrib ring_simps split_def image_def)
- by (rule_tac x="(s,m)" in bexI,simp_all)
- (rule_tac x="(t,n)" in bexI,simp_all)
-next
- fix t n s m
- assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U"
- let ?N = "\<lambda> t. Inum (x#bs) t"
- let ?st= "Add (Mul m t) (Mul n s)"
- from Ul smU have mnz: "m \<noteq> 0" by auto
- from Ul tnU have nnz: "n \<noteq> 0" by auto
- have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnz nnz by (simp add: ring_simps add_divide_distrib)
- 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"
- have Pc:"\<forall> a b. ?P a b = ?P b a"
- by auto
- from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
- from alluopairs_ex[OF Pc, where xs="U"] tnU smU
- have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
- by blast
- then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)"
- and Pts': "?P (t',n') (s',m')" by blast
- from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
- let ?st' = "Add (Mul m' t') (Mul n' s')"
- have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
- using mnz' nnz' by (simp add: ring_simps add_divide_distrib)
- from Pts' have
- "(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
- 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')
- finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
- \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
- (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
- set (alluopairs U)"
- using ts'_U by blast
-qed
-
-lemma uset_cong:
- assumes lp: "isrlfm p"
- 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)")
- and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
- and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
- 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)))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
- Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
- let ?N = "\<lambda> t. Inum (x#bs) t"
- from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
- and snb: "numbound0 s" and mp:"m > 0" by auto
- let ?st= "Add (Mul m t) (Mul n s)"
- from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
- by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
- from tnb snb have stnb: "numbound0 ?st" by simp
- have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mp np by (simp add: ring_simps add_divide_distrib)
- from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
- hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
- by auto (rule_tac x="(a,b)" in bexI, auto)
- then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
- from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
- from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst
- have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
- 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]]
- have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st)
- then show ?rhs using tnU' by auto
-next
- assume ?rhs
- then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))"
- by blast
- from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
- hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))"
- by auto (rule_tac x="(a,b)" in bexI, auto)
- then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
- th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
- let ?N = "\<lambda> t. Inum (x#bs) t"
- from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
- and snb: "numbound0 s" and mp:"m > 0" by auto
- let ?st= "Add (Mul m t) (Mul n s)"
- from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
- by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
- from tnb snb have stnb: "numbound0 ?st" by simp
- have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mp np by (simp add: ring_simps add_divide_distrib)
- from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
- from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
- have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
- with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
-qed
-
-lemma ferrack:
- assumes qf: "qfree p"
- shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
- (is "_ \<and> (?rhs = ?lhs)")
-proof-
- let ?I = "\<lambda> x p. Ifm (x#bs) p"
- fix x
- let ?N = "\<lambda> t. Inum (x#bs) t"
- let ?q = "rlfm (simpfm p)"
- let ?U = "uset ?q"
- let ?Up = "alluopairs ?U"
- let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
- let ?S = "map ?g ?Up"
- let ?SS = "map simp_num_pair ?S"
- let ?Y = "remdps ?SS"
- let ?f= "(\<lambda> (t,n). ?N t / real n)"
- let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
- let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
- let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
- from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
- from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
- from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
- from U_l UpU
- have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
- hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
- by (auto simp add: mult_pos_pos)
- have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0"
- proof-
- { fix t n assume tnY: "(t,n) \<in> set ?Y"
- hence "(t,n) \<in> set ?SS" by simp
- hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
- by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
- then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
- from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
- from simp_num_pair_l[OF tnb np tns]
- have "numbound0 t \<and> n > 0" . }
- thus ?thesis by blast
- qed
-
- have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
- proof-
- from simp_num_pair_ci[where bs="x#bs"] have
- "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
- hence th: "?f o simp_num_pair = ?f" using ext by blast
- have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
- also have "\<dots> = (?f ` set ?S)" by (simp add: th)
- also have "\<dots> = ((?f o ?g) ` set ?Up)"
- by (simp only: set_map o_def image_compose[symmetric])
- also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
- using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
- finally show ?thesis .
- qed
- have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
- proof-
- { fix t n assume tnY: "(t,n) \<in> set ?Y"
- with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
- from usubst_I[OF lq np tnb]
- have "bound0 (usubst ?q (t,n))" by simp hence "bound0 (simpfm (usubst ?q (t,n)))"
- using simpfm_bound0 by simp}
- thus ?thesis by blast
- qed
- hence ep_nb: "bound0 ?ep" using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
- let ?mp = "minusinf ?q"
- let ?pp = "plusinf ?q"
- let ?M = "?I x ?mp"
- let ?P = "?I x ?pp"
- let ?res = "disj ?mp (disj ?pp ?ep)"
- from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
- have nbth: "bound0 ?res" by auto
-
- from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm
-
- have th: "?lhs = (\<exists> x. ?I x ?q)" by auto
- from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
- by (simp only: split_def fst_conv snd_conv)
- also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))"
- using uset_cong[OF lq YU U_l Y_l] by (simp only: split_def fst_conv snd_conv simpfm)
- also have "\<dots> = (Ifm (x#bs) ?res)"
- using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
- by (simp add: split_def pair_collapse)
- finally have lheq: "?lhs = (Ifm bs (decr ?res))" using decr[OF nbth] by blast
- hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
- by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
- from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
- with lr show ?thesis by blast
-qed
-
-definition linrqe:: "fm \<Rightarrow> fm" where
- "linrqe p = qelim (prep p) ferrack"
-
-theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)"
-using ferrack qelim_ci prep
-unfolding linrqe_def by auto
-
-definition ferrack_test :: "unit \<Rightarrow> fm" where
- "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
- (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
-
-ML {* @{code ferrack_test} () *}
-
-oracle linr_oracle = {*
-let
-
-fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
- of NONE => error "Variable not found in the list!"
- | SOME n => @{code Bound} n)
- | num_of_term vs @{term "real (0::int)"} = @{code C} 0
- | num_of_term vs @{term "real (1::int)"} = @{code C} 1
- | num_of_term vs @{term "0::real"} = @{code C} 0
- | num_of_term vs @{term "1::real"} = @{code C} 1
- | num_of_term vs (Bound i) = @{code Bound} i
- | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
- | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Add} (num_of_term vs t1, num_of_term vs t2)
- | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Sub} (num_of_term vs t1, num_of_term vs t2)
- | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (case (num_of_term vs t1)
- of @{code C} i => @{code Mul} (i, num_of_term vs t2)
- | _ => error "num_of_term: unsupported Multiplication")
- | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t')) = @{code C} (HOLogic.dest_numeral t')
- | num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} $ t') = @{code C} (HOLogic.dest_numeral t')
- | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);
-
-fun fm_of_term vs @{term True} = @{code T}
- | fm_of_term vs @{term False} = @{code F}
- | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
- | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
- | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
- | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
- | fm_of_term vs (@{term "op &"} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
- | fm_of_term vs (@{term "op |"} $ t1 $ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
- | fm_of_term vs (@{term "op -->"} $ t1 $ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
- | fm_of_term vs (@{term "Not"} $ t') = @{code NOT} (fm_of_term vs t')
- | fm_of_term vs (Const ("Ex", _) $ Abs (xn, xT, p)) =
- @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
- | fm_of_term vs (Const ("All", _) $ Abs (xn, xT, p)) =
- @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
- | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
-
-fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} $ HOLogic.mk_number HOLogic.intT i
- | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs))
- | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
- | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
- term_of_num vs t1 $ term_of_num vs t2
- | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $
- term_of_num vs t1 $ term_of_num vs t2
- | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
- term_of_num vs (@{code C} i) $ term_of_num vs t2
- | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
-
-fun term_of_fm vs @{code T} = HOLogic.true_const
- | term_of_fm vs @{code F} = HOLogic.false_const
- | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
- term_of_num vs t $ @{term "0::real"}
- | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
- term_of_num vs t $ @{term "0::real"}
- | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
- @{term "0::real"} $ term_of_num vs t
- | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
- @{term "0::real"} $ term_of_num vs t
- | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $
- term_of_num vs t $ @{term "0::real"}
- | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
- | term_of_fm vs (@{code NOT} t') = HOLogic.Not $ term_of_fm vs t'
- | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2
- | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2
- | term_of_fm vs (@{code Imp} (t1, t2)) = HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
- | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $
- term_of_fm vs t1 $ term_of_fm vs t2
- | term_of_fm vs _ = error "If this is raised, Isabelle/HOL or generate_code is inconsistent.";
-
-in fn ct =>
- let
- val thy = Thm.theory_of_cterm ct;
- val t = Thm.term_of ct;
- val fs = term_frees t;
- val vs = fs ~~ (0 upto (length fs - 1));
- val res = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_fm vs (@{code linrqe} (fm_of_term vs t))));
- in Thm.cterm_of thy res end
-end;
-*}
-
-use "linrtac.ML"
-setup LinrTac.setup
-
-lemma
- fixes x :: real
- shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1"
-apply rferrack
-done
-
-lemma
- fixes x :: real
- shows "\<exists>y \<le> x. x = y + 1"
-apply rferrack
-done
-
-lemma
- fixes x :: real
- shows "\<not> (\<exists>z. x + z = x + z + 1)"
-apply rferrack
-done
-
-end