1. New cancellation simprocs for common factors in inequations
2. Updated the documentation
(* Title: HOL/Decision_Procs/MIR.thy
Author: Amine Chaieb
*)
theory MIR
imports Complex_Main Dense_Linear_Order Efficient_Nat
uses ("mir_tac.ML")
begin
section {* Quantifier elimination for @{text "\<real> (0, 1, +, floor, <)"} *}
declare real_of_int_floor_cancel [simp del]
primrec alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" where
"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
(* generate a list from i to j*)
consts iupt :: "int \<times> int \<Rightarrow> int list"
recdef iupt "measure (\<lambda> (i,j). nat (j-i +1))"
"iupt (i,j) = (if j <i then [] else (i# iupt(i+1, j)))"
lemma iupt_set: "set (iupt(i,j)) = {i .. j}"
proof(induct rule: iupt.induct)
case (1 a b)
show ?case
using prems by (simp add: simp_from_to)
qed
lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
using Nat.gr0_conv_Suc
by clarsimp
lemma myl: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a \<le> b) = (0 \<le> b - a)"
proof(clarify)
fix x y ::"'a"
have "(x \<le> y) = (x - y \<le> 0)" by (simp only: le_iff_diff_le_0[where a="x" and b="y"])
also have "\<dots> = (- (y - x) \<le> 0)" by simp
also have "\<dots> = (0 \<le> y - x)" by (simp only: neg_le_0_iff_le[where a="y-x"])
finally show "(x \<le> y) = (0 \<le> y - x)" .
qed
lemma myless: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a < b) = (0 < b - a)"
proof(clarify)
fix x y ::"'a"
have "(x < y) = (x - y < 0)" by (simp only: less_iff_diff_less_0[where a="x" and b="y"])
also have "\<dots> = (- (y - x) < 0)" by simp
also have "\<dots> = (0 < y - x)" by (simp only: neg_less_0_iff_less[where a="y-x"])
finally show "(x < y) = (0 < y - x)" .
qed
lemma myeq: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a = b) = (0 = b - a)"
by auto
(* Maybe should be added to the library \<dots> *)
lemma floor_int_eq: "(real n\<le> x \<and> x < real (n+1)) = (floor x = n)"
proof( auto)
assume lb: "real n \<le> x"
and ub: "x < real n + 1"
have "real (floor x) \<le> x" by simp
hence "real (floor x) < real (n + 1) " using ub by arith
hence "floor x < n+1" by simp
moreover from lb have "n \<le> floor x" using floor_mono[where x="real n" and y="x"]
by simp ultimately show "floor x = n" by simp
qed
(* Periodicity of dvd *)
lemma dvd_period:
assumes advdd: "(a::int) dvd d"
shows "(a dvd (x + t)) = (a dvd ((x+ c*d) + t))"
using advdd
proof-
{fix x k
from inf_period(3)[OF advdd, rule_format, where x=x and k="-k"]
have " ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp}
hence "\<forall>x.\<forall>k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp
then show ?thesis by simp
qed
(* The Divisibility relation between reals *)
definition
rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl "rdvd" 50)
where
rdvd_def: "x rdvd y \<longleftrightarrow> (\<exists>k\<Colon>int. y = x * real k)"
lemma int_rdvd_real:
shows "real (i::int) rdvd x = (i dvd (floor x) \<and> real (floor x) = x)" (is "?l = ?r")
proof
assume "?l"
hence th: "\<exists> k. x=real (i*k)" by (simp add: rdvd_def)
hence th': "real (floor x) = x" by (auto simp del: real_of_int_mult)
with th have "\<exists> k. real (floor x) = real (i*k)" by simp
hence "\<exists> k. floor x = i*k" by (simp only: real_of_int_inject)
thus ?r using th' by (simp add: dvd_def)
next
assume "?r" hence "(i\<Colon>int) dvd \<lfloor>x\<Colon>real\<rfloor>" ..
hence "\<exists> k. real (floor x) = real (i*k)"
by (simp only: real_of_int_inject) (simp add: dvd_def)
thus ?l using prems by (simp add: rdvd_def)
qed
lemma int_rdvd_iff: "(real (i::int) rdvd real t) = (i dvd t)"
by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only :real_of_int_mult[symmetric])
lemma rdvd_abs1:
"(abs (real d) rdvd t) = (real (d ::int) rdvd t)"
proof
assume d: "real d rdvd t"
from d int_rdvd_real have d2: "d dvd (floor t)" and ti: "real (floor t) = t" by auto
from iffD2[OF abs_dvd_iff] d2 have "(abs d) dvd (floor t)" by blast
with ti int_rdvd_real[symmetric] have "real (abs d) rdvd t" by blast
thus "abs (real d) rdvd t" by simp
next
assume "abs (real d) rdvd t" hence "real (abs d) rdvd t" by simp
with int_rdvd_real[where i="abs d" and x="t"] have d2: "abs d dvd floor t" and ti: "real (floor t) =t" by auto
from iffD1[OF abs_dvd_iff] d2 have "d dvd floor t" by blast
with ti int_rdvd_real[symmetric] show "real d rdvd t" by blast
qed
lemma rdvd_minus: "(real (d::int) rdvd t) = (real d rdvd -t)"
apply (auto simp add: rdvd_def)
apply (rule_tac x="-k" in exI, simp)
apply (rule_tac x="-k" in exI, simp)
done
lemma rdvd_left_0_eq: "(0 rdvd t) = (t=0)"
by (auto simp add: rdvd_def)
lemma rdvd_mult:
assumes knz: "k\<noteq>0"
shows "(real (n::int) * real (k::int) rdvd x * real k) = (real n rdvd x)"
using knz by (simp add:rdvd_def)
lemma rdvd_trans: assumes mn:"m rdvd n" and nk:"n rdvd k"
shows "m rdvd k"
proof-
from rdvd_def mn obtain c where nmc:"n = m * real (c::int)" by auto
from rdvd_def nk obtain c' where nkc:"k = n * real (c'::int)" by auto
hence "k = m * real (c * c')" using nmc by simp
thus ?thesis using rdvd_def by blast
qed
(*********************************************************************************)
(**** 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 | Floor num| CF int num num
(* A size for num to make inductive proofs simpler*)
primrec num_size :: "num \<Rightarrow> nat" where
"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 (CN n c a) = 4 + num_size a "
| "num_size (CF c a b) = 4 + num_size a + num_size b"
| "num_size (Mul c a) = 1 + num_size a"
| "num_size (Floor a) = 1 + num_size a"
(* Semantics of numeral terms (num) *)
primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where
"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"
| "Inum bs (Floor a) = real (floor (Inum bs a))"
| "Inum bs (CF c a b) = real c * real (floor (Inum bs a)) + Inum bs b"
definition "isint t bs \<equiv> real (floor (Inum bs t)) = Inum bs t"
lemma isint_iff: "isint n bs = (real (floor (Inum bs n)) = Inum bs n)"
by (simp add: isint_def)
lemma isint_Floor: "isint (Floor n) bs"
by (simp add: isint_iff)
lemma isint_Mul: "isint e bs \<Longrightarrow> isint (Mul c e) bs"
proof-
let ?e = "Inum bs e"
let ?fe = "floor ?e"
assume be: "isint e bs" hence efe:"real ?fe = ?e" by (simp add: isint_iff)
have "real ((floor (Inum bs (Mul c e)))) = real (floor (real (c * ?fe)))" using efe by simp
also have "\<dots> = real (c* ?fe)" by (simp only: floor_real_of_int)
also have "\<dots> = real c * ?e" using efe by simp
finally show ?thesis using isint_iff by simp
qed
lemma isint_neg: "isint e bs \<Longrightarrow> isint (Neg e) bs"
proof-
let ?I = "\<lambda> t. Inum bs t"
assume ie: "isint e bs"
hence th: "real (floor (?I e)) = ?I e" by (simp add: isint_def)
have "real (floor (?I (Neg e))) = real (floor (- (real (floor (?I e)))))" by (simp add: th)
also have "\<dots> = - real (floor (?I e))" by(simp add: floor_minus_real_of_int)
finally show "isint (Neg e) bs" by (simp add: isint_def th)
qed
lemma isint_sub:
assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs"
proof-
let ?I = "\<lambda> t. Inum bs t"
from ie have th: "real (floor (?I e)) = ?I e" by (simp add: isint_def)
have "real (floor (?I (Sub (C c) e))) = real (floor ((real (c -floor (?I e)))))" by (simp add: th)
also have "\<dots> = real (c- floor (?I e))" by(simp add: floor_minus_real_of_int)
finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th)
qed
lemma isint_add: assumes
ai:"isint a bs" and bi: "isint b bs" shows "isint (Add a b) bs"
proof-
let ?a = "Inum bs a"
let ?b = "Inum bs b"
from ai bi isint_iff have "real (floor (?a + ?b)) = real (floor (real (floor ?a) + real (floor ?b)))" by simp
also have "\<dots> = real (floor ?a) + real (floor ?b)" by simp
also have "\<dots> = ?a + ?b" using ai bi isint_iff by simp
finally show "isint (Add a b) bs" by (simp add: isint_iff)
qed
lemma isint_c: "isint (C j) bs"
by (simp add: isint_iff)
(* FORMULAE *)
datatype fm =
T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num|
NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
(* A size for fm *)
fun fmsize :: "fm \<Rightarrow> nat" where
"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 (Dvd i t) = 2"
| "fmsize (NDvd i t) = 2"
| "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) *)
primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where
"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 (Dvd i b) = (real i rdvd Inum bs b)"
| "Ifm bs (NDvd i b) = (\<not>(real i rdvd Inum bs b))"
| "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)"
consts prep :: "fm \<Rightarrow> fm"
recdef prep "measure fmsize"
"prep (E T) = T"
"prep (E F) = F"
"prep (E (Or p q)) = Or (prep (E p)) (prep (E q))"
"prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))"
"prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))"
"prep (E (NOT (And p q))) = Or (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))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
"prep (E p) = E (prep p)"
"prep (A (And p q)) = And (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)) = Or (prep (NOT p)) (prep (NOT q))"
"prep (NOT (A p)) = prep (E (NOT p))"
"prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))"
"prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))"
"prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))"
"prep (NOT p) = NOT (prep p)"
"prep (Or p q) = Or (prep p) (prep q)"
"prep (And p q) = And (prep p) (prep q)"
"prep (Imp p q) = prep (Or (NOT p) q)"
"prep (Iff p q) = Or (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)
(* Quantifier freeness *)
fun qfree:: "fm \<Rightarrow> bool" where
"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 *)
primrec numbound0 :: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) where
"numbound0 (C c) = True"
| "numbound0 (Bound n) = (n>0)"
| "numbound0 (CN n i a) = (n > 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"
| "numbound0 (Floor a) = numbound0 a"
| "numbound0 (CF c a b) = (numbound0 a \<and> numbound0 b)"
lemma numbound0_I:
assumes nb: "numbound0 a"
shows "Inum (b#bs) a = Inum (b'#bs) a"
using nb by (induct a) (auto simp add: nth_pos2)
lemma numbound0_gen:
assumes nb: "numbound0 t" and ti: "isint t (x#bs)"
shows "\<forall> y. isint t (y#bs)"
using nb ti
proof(clarify)
fix y
from numbound0_I[OF nb, where bs="bs" and b="y" and b'="x"] ti[simplified isint_def]
show "isint t (y#bs)"
by (simp add: isint_def)
qed
primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
"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 (Dvd i a) = numbound0 a"
| "bound0 (NDvd i 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) (auto simp add: nth_pos2)
primrec numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" (* substitute a num into a num for Bound 0 *) where
"numsubst0 t (C c) = (C c)"
| "numsubst0 t (Bound n) = (if n=0 then t else Bound n)"
| "numsubst0 t (CN n i a) = (if n=0 then Add (Mul i t) (numsubst0 t a) else CN n i (numsubst0 t a))"
| "numsubst0 t (CF i a b) = CF i (numsubst0 t a) (numsubst0 t b)"
| "numsubst0 t (Neg a) = Neg (numsubst0 t a)"
| "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)"
| "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)"
| "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)"
| "numsubst0 t (Floor a) = Floor (numsubst0 t a)"
lemma numsubst0_I:
shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t"
by (induct t) (simp_all add: nth_pos2)
lemma numsubst0_I':
assumes nb: "numbound0 a"
shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t"
by (induct t) (simp_all add: nth_pos2 numbound0_I[OF nb, where b="b" and b'="b'"])
primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *) where
"subst0 t T = T"
| "subst0 t F = F"
| "subst0 t (Lt a) = Lt (numsubst0 t a)"
| "subst0 t (Le a) = Le (numsubst0 t a)"
| "subst0 t (Gt a) = Gt (numsubst0 t a)"
| "subst0 t (Ge a) = Ge (numsubst0 t a)"
| "subst0 t (Eq a) = Eq (numsubst0 t a)"
| "subst0 t (NEq a) = NEq (numsubst0 t a)"
| "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)"
| "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)"
| "subst0 t (NOT p) = NOT (subst0 t p)"
| "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
| "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
| "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)"
| "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
lemma subst0_I: assumes qfp: "qfree p"
shows "Ifm (b#bs) (subst0 a p) = Ifm ((Inum (b#bs) a)#bs) p"
using qfp numsubst0_I[where b="b" and bs="bs" and a="a"]
by (induct p) (simp_all add: nth_pos2 )
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 (Floor a) = Floor (decrnum a)"
"decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
"decrnum (CF c a b) = CF c (decrnum a) (decrnum b)"
"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 (Dvd i a) = Dvd i (decrnum a)"
"decr (NDvd i a) = NDvd i (decrnum a)"
"decr (NOT p) = NOT (decr p)"
"decr (And p q) = And (decr p) (decr q)"
"decr (Or p q) = Or (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 (Dvd i b) = True"
"isatom (NDvd i b) = True"
"isatom p = False"
lemma numsubst0_numbound0: assumes nb: "numbound0 t"
shows "numbound0 (numsubst0 t a)"
using nb by (induct a, auto)
lemma subst0_bound0: assumes qf: "qfree p" and nb: "numbound0 t"
shows "bound0 (subst0 t p)"
using qf numsubst0_numbound0[OF nb] by (induct p, auto)
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
by (induct p, simp_all)
definition djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
"djf f p q = (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 fp q))"
definition evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
"evaldjf f ps = 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 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"
conjuncts :: "fm \<Rightarrow> fm list"
recdef disjuncts "measure size"
"disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
"disjuncts F = []"
"disjuncts p = [p]"
recdef conjuncts "measure size"
"conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
"conjuncts T = []"
"conjuncts p = [p]"
lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
by(induct p rule: disjuncts.induct, auto)
lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm bs q) = Ifm bs p"
by(induct p rule: conjuncts.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 conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
proof-
assume nb: "bound0 p"
hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.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
lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
proof-
assume qf: "qfree p"
hence "list_all qfree (conjuncts p)"
by (induct p rule: conjuncts.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. f (Or p q) = 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 *)
(* Algebraic simplifications for nums *)
consts bnds:: "num \<Rightarrow> nat list"
lex_ns:: "nat list \<times> nat list \<Rightarrow> bool"
recdef bnds "measure size"
"bnds (Bound n) = [n]"
"bnds (CN n c a) = n#(bnds a)"
"bnds (Neg a) = bnds a"
"bnds (Add a b) = (bnds a)@(bnds b)"
"bnds (Sub a b) = (bnds a)@(bnds b)"
"bnds (Mul i a) = bnds a"
"bnds (Floor a) = bnds a"
"bnds (CF c a b) = (bnds a)@(bnds b)"
"bnds a = []"
recdef lex_ns "measure (\<lambda> (xs,ys). length xs + length ys)"
"lex_ns ([], ms) = True"
"lex_ns (ns, []) = False"
"lex_ns (n#ns, m#ms) = (n<m \<or> ((n = m) \<and> lex_ns (ns,ms))) "
constdefs lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool"
"lex_bnd t s \<equiv> lex_ns (bnds t, bnds s)"
consts
numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
reducecoeffh:: "num \<Rightarrow> int \<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 (CF c t s) = max (abs c) (maxcoeff s)"
"maxcoeff t = 1"
lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
apply (induct t rule: maxcoeff.induct, auto)
done
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 (CF c s t) = (\<lambda>g. zgcd c (numgcdh t g))"
"numgcdh t = (\<lambda>g. 1)"
definition
numgcd :: "num \<Rightarrow> int"
where
numgcd_def: "numgcd t = 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 (CF c s t) = (\<lambda> g. CF (c div g) s (reducecoeffh t g))"
"reducecoeffh t = (\<lambda>g. t)"
definition
reducecoeff :: "num \<Rightarrow> num"
where
reducecoeff_def: "reducecoeff t =
(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 (CF c s 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 dvd_trans[OF gdg])
declare dvd_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"
proof-
have "\<And>x. numgcdh t x= 0 \<Longrightarrow> Inum bs t = 0"
by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)
thus ?thesis using g0[simplified numgcd_def] by blast
qed
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] algebra_simps)
next
case (3 c s 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] algebra_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 (CF c s 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)
next
case (3 c t s)
hence H1:"ismaxcoeff s (maxcoeff s)" by auto
have thh1: "maxcoeff s \<le> max \<bar>c\<bar> (maxcoeff s)" by (simp add: max_def)
from ismaxcoeff_mono[OF H1 thh1] 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 (unfold zgcd_def)
apply (cases "i = 0", simp_all)
apply (cases "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
next
case (3 c s 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 (CF c1 t1 r1,CF c2 t2 r2) =
(if t1 = t2 then
(let c=c1+c2; s= numadd(r1,r2) in (if c=0 then s else CF c t1 s))
else if lex_bnd t1 t2 then CF c1 t1 (numadd(r1,CF c2 t2 r2))
else CF c2 t2 (numadd(CF c1 t1 r1,r2)))"
"numadd (CF c1 t1 r1,C c) = CF c1 t1 (numadd (r1, C c))"
"numadd (C c,CF c1 t1 r1) = CF c1 t1 (numadd (r1, C c))"
"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: algebra_simps)
apply (simp only: left_distrib[symmetric])
apply simp
apply (case_tac "lex_bnd t1 t2", simp_all)
apply (case_tac "c1+c2 = 0")
by (case_tac "t1 = t2", simp_all add: algebra_simps left_distrib[symmetric] real_of_int_mult[symmetric] real_of_int_add[symmetric]del: real_of_int_mult real_of_int_add left_distrib)
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 t) = (\<lambda> i. CN n (c*i) (nummul t i))"
"nummul (CF c t s) = (\<lambda> i. CF (c*i) t (nummul s i))"
"nummul (Mul c t) = (\<lambda> i. nummul t (i*c))"
"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: algebra_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 nummul 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
lemma isint_CF: assumes si: "isint s bs" shows "isint (CF c t s) bs"
proof-
have cti: "isint (Mul c (Floor t)) bs" by (simp add: isint_Mul isint_Floor)
have "?thesis = isint (Add (Mul c (Floor t)) s) bs" by (simp add: isint_def)
also have "\<dots>" by (simp add: isint_add cti si)
finally show ?thesis .
qed
consts split_int:: "num \<Rightarrow> num\<times>num"
recdef split_int "measure num_size"
"split_int (C c) = (C 0, C c)"
"split_int (CN n c b) =
(let (bv,bi) = split_int b
in (CN n c bv, bi))"
"split_int (CF c a b) =
(let (bv,bi) = split_int b
in (bv, CF c a bi))"
"split_int a = (a,C 0)"
lemma split_int:"\<And> tv ti. split_int t = (tv,ti) \<Longrightarrow> (Inum bs (Add tv ti) = Inum bs t) \<and> isint ti bs"
proof (induct t rule: split_int.induct)
case (2 c n b tv ti)
let ?bv = "fst (split_int b)"
let ?bi = "snd (split_int b)"
have "split_int b = (?bv,?bi)" by simp
with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
from prems(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def)
from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def)
next
case (3 c a b tv ti)
let ?bv = "fst (split_int b)"
let ?bi = "snd (split_int b)"
have "split_int b = (?bv,?bi)" by simp
with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
from prems(2) have tibi: "ti = CF c a ?bi" by (simp add: Let_def split_def)
from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def isint_Floor isint_add isint_Mul isint_CF)
qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def algebra_simps)
lemma split_int_nb: "numbound0 t \<Longrightarrow> numbound0 (fst (split_int t)) \<and> numbound0 (snd (split_int t)) "
by (induct t rule: split_int.induct, auto simp add: Let_def split_def)
definition
numfloor:: "num \<Rightarrow> num"
where
numfloor_def: "numfloor t = (let (tv,ti) = split_int t in
(case tv of C i \<Rightarrow> numadd (tv,ti)
| _ \<Rightarrow> numadd(CF 1 tv (C 0),ti)))"
lemma numfloor[simp]: "Inum bs (numfloor t) = Inum bs (Floor t)" (is "?n t = ?N (Floor t)")
proof-
let ?tv = "fst (split_int t)"
let ?ti = "snd (split_int t)"
have tvti:"split_int t = (?tv,?ti)" by simp
{assume H: "\<forall> v. ?tv \<noteq> C v"
hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)"
by (cases ?tv, auto simp add: numfloor_def Let_def split_def numadd)
from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp
also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))"
by (simp,subst tii[simplified isint_iff, symmetric]) simp
also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
finally have ?thesis using th1 by simp}
moreover {fix v assume H:"?tv = C v"
from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp
also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))"
by (simp,subst tii[simplified isint_iff, symmetric]) simp
also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
finally have ?thesis by (simp add: H numfloor_def Let_def split_def numadd) }
ultimately show ?thesis by auto
qed
lemma numfloor_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numfloor t)"
using split_int_nb[where t="t"]
by (cases "fst(split_int t)" , auto simp add: numfloor_def Let_def split_def numadd_nb)
recdef simpnum "measure num_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 (Floor t) = numfloor (simpnum t)"
"simpnum (CN n c t) = (if c=0 then simpnum t else CN n c (simpnum t))"
"simpnum (CF c t s) = simpnum(Add (Mul c (Floor t)) s)"
lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
by (induct t rule: simpnum.induct, auto)
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 (CF c s t) = (c \<noteq> 0 \<and> nozerocoeff t)"
"nozerocoeff (Mul 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 split_int_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (fst (split_int t)) \<and> nozerocoeff (snd (split_int t))"
by (induct t rule: split_int.induct,auto simp add: Let_def split_def)
lemma numfloor_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (numfloor t)"
by (simp add: numfloor_def Let_def split_def)
(cases "fst (split_int t)", simp_all add: split_int_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 numfloor_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
next
case (3 c s 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)}
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)}
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
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)}
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)}
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)}
ultimately have ?thesis by blast}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed
consts not:: "fm \<Rightarrow> fm"
recdef not "measure size"
"not (NOT p) = p"
"not T = F"
"not F = T"
"not (Lt t) = Ge t"
"not (Le t) = Gt t"
"not (Gt t) = Le t"
"not (Ge t) = Lt t"
"not (Eq t) = NEq t"
"not (NEq t) = Eq t"
"not (Dvd i t) = NDvd i t"
"not (NDvd i t) = Dvd i t"
"not (And p q) = Or (not p) (not q)"
"not (Or p q) = And (not p) (not q)"
"not p = NOT p"
lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
by (induct p) auto
lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
by (induct p, auto)
lemma not_nb[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
by (induct 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)
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
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)
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
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)
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)
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 add:not)
(cases "not p= q", auto simp add:not)
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 check_int:: "num \<Rightarrow> bool"
recdef check_int "measure size"
"check_int (C i) = True"
"check_int (Floor t) = True"
"check_int (Mul i t) = check_int t"
"check_int (Add t s) = (check_int t \<and> check_int s)"
"check_int (Neg t) = check_int t"
"check_int (CF c t s) = check_int s"
"check_int t = False"
lemma check_int: "check_int t \<Longrightarrow> isint t bs"
by (induct t, auto simp add: isint_add isint_Floor isint_Mul isint_neg isint_c isint_CF)
lemma rdvd_left1_int: "real \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t"
by (simp add: rdvd_def,rule_tac x="\<lfloor>t\<rfloor>" in exI) simp
lemma rdvd_reduce:
assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0"
shows "real (d::int) rdvd real (c::int)*t = (real (d div g) rdvd real (c div g)*t)"
proof
assume d: "real d rdvd real c * t"
from d rdvd_def obtain k where k_def: "real c * t = real d* real (k::int)" by auto
from gd dvd_def obtain kd where kd_def: "d = g * kd" by auto
from gc dvd_def obtain kc where kc_def: "c = g * kc" by auto
from k_def kd_def kc_def have "real g * real kc * t = real g * real kd * real k" by simp
hence "real kc * t = real kd * real k" using gp by simp
hence th:"real kd rdvd real kc * t" using rdvd_def by blast
from kd_def gp have th':"kd = d div g" by simp
from kc_def gp have "kc = c div g" by simp
with th th' show "real (d div g) rdvd real (c div g) * t" by simp
next
assume d: "real (d div g) rdvd real (c div g) * t"
from gp have gnz: "g \<noteq> 0" by simp
thus "real d rdvd real c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real (c div g) * t"] real_of_int_div[OF gnz gd] real_of_int_div[OF gnz gc] by simp
qed
constdefs simpdvd:: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)"
"simpdvd d t \<equiv>
(let g = numgcd t in
if g > 1 then (let g' = zgcd d g in
if g' = 1 then (d, t)
else (d div g',reducecoeffh t g'))
else (d, t))"
lemma simpdvd:
assumes tnz: "nozerocoeff t" and dnz: "d \<noteq> 0"
shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)"
proof-
let ?g = "numgcd t"
let ?g' = "zgcd d ?g"
{assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simpdvd_def)}
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
from zgcd0 g1 dnz have gp0: "?g' \<noteq> 0" by simp
hence g'p: "?g' > 0" using zgcd_pos[where i="d" 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 simpdvd_def)}
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 d" 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 "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)"
by (simp add: simpdvd_def Let_def)
also have "\<dots> = (real d rdvd (Inum bs t))"
using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified zdiv_self[OF gp0]]
th2[symmetric] by simp
finally have ?thesis by simp }
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 (reducecoeff 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 (reducecoeff a'))"
"simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0) then T else F | _ \<Rightarrow> Gt (reducecoeff 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 (reducecoeff a'))"
"simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0) then T else F | _ \<Rightarrow> Eq (reducecoeff 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 (reducecoeff a'))"
"simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
else if (abs i = 1) \<and> check_int a then T
else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in Dvd d t))"
"simpfm (NDvd i a) = (if i=0 then simpfm (NEq a)
else if (abs i = 1) \<and> check_int a then F
else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in NDvd d t))"
"simpfm p = p"
lemma simpfm[simp]: "Ifm bs (simpfm p) = Ifm bs p"
proof(induct p rule: simpfm.induct)
case (6 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
{fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
let ?g = "numgcd ?sa"
let ?rsa = "reducecoeff ?sa"
let ?r = "Inum bs ?rsa"
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
hence gp: "real ?g > 0" by simp
have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
with sa have "Inum bs a < 0 = (real ?g * ?r < real ?g * 0)" by simp
also have "\<dots> = (?r < 0)" using gp
by (simp only: mult_less_cancel_left) simp
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
ultimately show ?case by blast
next
case (7 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
{fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
let ?g = "numgcd ?sa"
let ?rsa = "reducecoeff ?sa"
let ?r = "Inum bs ?rsa"
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
hence gp: "real ?g > 0" by simp
have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
with sa have "Inum bs a \<le> 0 = (real ?g * ?r \<le> real ?g * 0)" by simp
also have "\<dots> = (?r \<le> 0)" using gp
by (simp only: mult_le_cancel_left) simp
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
ultimately show ?case by blast
next
case (8 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
{fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
let ?g = "numgcd ?sa"
let ?rsa = "reducecoeff ?sa"
let ?r = "Inum bs ?rsa"
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
hence gp: "real ?g > 0" by simp
have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
with sa have "Inum bs a > 0 = (real ?g * ?r > real ?g * 0)" by simp
also have "\<dots> = (?r > 0)" using gp
by (simp only: mult_less_cancel_left) simp
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
ultimately show ?case by blast
next
case (9 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
{fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
let ?g = "numgcd ?sa"
let ?rsa = "reducecoeff ?sa"
let ?r = "Inum bs ?rsa"
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
hence gp: "real ?g > 0" by simp
have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
with sa have "Inum bs a \<ge> 0 = (real ?g * ?r \<ge> real ?g * 0)" by simp
also have "\<dots> = (?r \<ge> 0)" using gp
by (simp only: mult_le_cancel_left) simp
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
ultimately show ?case by blast
next
case (10 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
{fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
let ?g = "numgcd ?sa"
let ?rsa = "reducecoeff ?sa"
let ?r = "Inum bs ?rsa"
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
hence gp: "real ?g > 0" by simp
have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
with sa have "Inum bs a = 0 = (real ?g * ?r = 0)" by simp
also have "\<dots> = (?r = 0)" using gp
by (simp add: mult_eq_0_iff)
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
ultimately show ?case by blast
next
case (11 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
{fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
let ?g = "numgcd ?sa"
let ?rsa = "reducecoeff ?sa"
let ?r = "Inum bs ?rsa"
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
hence gp: "real ?g > 0" by simp
have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
with sa have "Inum bs a \<noteq> 0 = (real ?g * ?r \<noteq> 0)" by simp
also have "\<dots> = (?r \<noteq> 0)" using gp
by (simp add: mult_eq_0_iff)
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
ultimately show ?case by blast
next
case (12 i a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
have "i=0 \<or> (abs i = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((abs i \<noteq> 1) \<or> (\<not> check_int a)))" by auto
{assume "i=0" hence ?case using "12.hyps" by (simp add: rdvd_left_0_eq Let_def)}
moreover
{assume ai1: "abs i = 1" and ai: "check_int a"
hence "i=1 \<or> i= - 1" by arith
moreover {assume i1: "i = 1"
from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]]
have ?case using i1 ai by simp }
moreover {assume i1: "i = - 1"
from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]]
rdvd_abs1[where d="- 1" and t="Inum bs a"]
have ?case using i1 ai by simp }
ultimately have ?case by blast}
moreover
{assume inz: "i\<noteq>0" and cond: "(abs i \<noteq> 1) \<or> (\<not> check_int a)"
{fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
by (cases "abs i = 1", auto simp add: int_rdvd_iff) }
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
hence th: "simpfm (Dvd i a) = Dvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond by (cases ?sa, auto simp add: Let_def split_def)
from simpnum_nz have nz:"nozerocoeff ?sa" by simp
from simpdvd [OF nz inz] th have ?case using sa by simp}
ultimately have ?case by blast}
ultimately show ?case by blast
next
case (13 i a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp
have "i=0 \<or> (abs i = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((abs i \<noteq> 1) \<or> (\<not> check_int a)))" by auto
{assume "i=0" hence ?case using "13.hyps" by (simp add: rdvd_left_0_eq Let_def)}
moreover
{assume ai1: "abs i = 1" and ai: "check_int a"
hence "i=1 \<or> i= - 1" by arith
moreover {assume i1: "i = 1"
from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]]
have ?case using i1 ai by simp }
moreover {assume i1: "i = - 1"
from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]]
rdvd_abs1[where d="- 1" and t="Inum bs a"]
have ?case using i1 ai by simp }
ultimately have ?case by blast}
moreover
{assume inz: "i\<noteq>0" and cond: "(abs i \<noteq> 1) \<or> (\<not> check_int a)"
{fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
by (cases "abs i = 1", auto simp add: int_rdvd_iff) }
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
hence th: "simpfm (NDvd i a) = NDvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond
by (cases ?sa, auto simp add: Let_def split_def)
from simpnum_nz have nz:"nozerocoeff ?sa" by simp
from simpdvd [OF nz inz] th have ?case using sa by simp}
ultimately have ?case by blast}
ultimately show ?case by blast
qed (induct p rule: simpfm.induct, simp_all)
lemma simpdvd_numbound0: "numbound0 t \<Longrightarrow> numbound0 (snd (simpdvd d t))"
by (simp add: simpdvd_def Let_def split_def reducecoeffh_numbound0)
lemma simpfm_bound0[simp]: "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 reducecoeff_numbound0)
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 reducecoeff_numbound0)
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 reducecoeff_numbound0)
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 reducecoeff_numbound0)
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 reducecoeff_numbound0)
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 reducecoeff_numbound0)
next
case (12 i 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 reducecoeff_numbound0 simpdvd_numbound0 split_def)
next
case (13 i 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 reducecoeff_numbound0 simpdvd_numbound0 split_def)
qed(auto simp add: disj_def imp_def iff_def conj_def)
lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
by (induct p rule: simpfm.induct, auto simp add: Let_def)
(case_tac "simpnum a",auto simp add: split_def Let_def)+
(* Generic quantifier elimination *)
constdefs list_conj :: "fm list \<Rightarrow> fm"
"list_conj ps \<equiv> foldr conj ps T"
lemma list_conj: "Ifm bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm bs p)"
by (induct ps, auto simp add: list_conj_def)
lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
by (induct ps, auto simp add: list_conj_def)
lemma list_conj_nb: " \<forall>p\<in> set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
by (induct ps, auto simp add: list_conj_def)
constdefs CJNB:: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
"CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = List.partition bound0 cjs
in conj (decr (list_conj yes)) (f (list_conj no)))"
lemma CJNB_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 (CJNB qe p) \<and> (Ifm bs ((CJNB qe p)) = Ifm bs (E p))"
proof(clarify)
fix bs p
assume qfp: "qfree p"
let ?cjs = "conjuncts p"
let ?yes = "fst (List.partition bound0 ?cjs)"
let ?no = "snd (List.partition bound0 ?cjs)"
let ?cno = "list_conj ?no"
let ?cyes = "list_conj ?yes"
have part: "List.partition bound0 ?cjs = (?yes,?no)" by simp
from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast
hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb)
hence yes_qf: "qfree (decr ?cyes )" by (simp add: decr_qf)
from conjuncts_qf[OF qfp] partition_set[OF part]
have " \<forall>q\<in> set ?no. qfree q" by auto
hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
with qe have cno_qf:"qfree (qe ?cno )"
and noE: "Ifm bs (qe ?cno) = Ifm bs (E ?cno)" by blast+
from cno_qf yes_qf have qf: "qfree (CJNB qe p)"
by (simp add: CJNB_def Let_def conj_qf split_def)
{fix bs
from conjuncts have "Ifm bs p = (\<forall>q\<in> set ?cjs. Ifm bs q)" by blast
also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm bs q) \<and> (\<forall>q\<in> set ?no. Ifm bs q))"
using partition_set[OF part] by auto
finally have "Ifm bs p = ((Ifm bs ?cyes) \<and> (Ifm bs ?cno))" using list_conj by simp}
hence "Ifm bs (E p) = (\<exists>x. (Ifm (x#bs) ?cyes) \<and> (Ifm (x#bs) ?cno))" by simp
also fix y have "\<dots> = (\<exists>x. (Ifm (y#bs) ?cyes) \<and> (Ifm (x#bs) ?cno))"
using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
also have "\<dots> = (Ifm bs (decr ?cyes) \<and> Ifm bs (E ?cno))"
by (auto simp add: decr[OF yes_nb])
also have "\<dots> = (Ifm bs (conj (decr ?cyes) (qe ?cno)))"
using qe[rule_format, OF no_qf] by auto
finally have "Ifm bs (E p) = Ifm bs (CJNB qe p)"
by (simp add: Let_def CJNB_def split_def)
with qf show "qfree (CJNB qe p) \<and> Ifm bs (CJNB qe p) = Ifm bs (E p)" by blast
qed
consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
recdef qelim "measure fmsize"
"qelim (E p) = (\<lambda> qe. DJ (CJNB 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. disj (qelim (NOT 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 CJNB_qe[OF qe_inv]]
by(induct p rule: qelim.induct)
(auto simp del: simpfm.simps)
text {* The @{text "\<int>"} Part *}
text{* Linearity for fm where Bound 0 ranges over @{text "\<int>"} *}
consts
zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*)
recdef zsplit0 "measure num_size"
"zsplit0 (C c) = (0,C c)"
"zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))"
"zsplit0 (CN n c a) = zsplit0 (Add (Mul c (Bound n)) a)"
"zsplit0 (CF c a b) = zsplit0 (Add (Mul c (Floor a)) b)"
"zsplit0 (Neg a) = (let (i',a') = zsplit0 a in (-i', Neg a'))"
"zsplit0 (Add a b) = (let (ia,a') = zsplit0 a ;
(ib,b') = zsplit0 b
in (ia+ib, Add a' b'))"
"zsplit0 (Sub a b) = (let (ia,a') = zsplit0 a ;
(ib,b') = zsplit0 b
in (ia-ib, Sub a' b'))"
"zsplit0 (Mul i a) = (let (i',a') = zsplit0 a in (i*i', Mul i a'))"
"zsplit0 (Floor a) = (let (i',a') = zsplit0 a in (i',Floor a'))"
(hints simp add: Let_def)
lemma zsplit0_I:
shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((real (x::int)) #bs) (CN 0 n a) = Inum (real x #bs) t) \<and> numbound0 a"
(is "\<And> n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a")
proof(induct t rule: zsplit0.induct)
case (1 c n a) thus ?case by auto
next
case (2 m n a) thus ?case by (cases "m=0") auto
next
case (3 n i a n a') thus ?case by auto
next
case (4 c a b n a') thus ?case by auto
next
case (5 t n a)
let ?nt = "fst (zsplit0 t)"
let ?at = "snd (zsplit0 t)"
have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using prems
by (simp add: Let_def split_def)
from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
from th2[simplified] th[simplified] show ?case by simp
next
case (6 s t n a)
let ?ns = "fst (zsplit0 s)"
let ?as = "snd (zsplit0 s)"
let ?nt = "fst (zsplit0 t)"
let ?at = "snd (zsplit0 t)"
have abjs: "zsplit0 s = (?ns,?as)" by simp
moreover have abjt: "zsplit0 t = (?nt,?at)" by simp
ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using prems
by (simp add: Let_def split_def)
from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by simp
with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
from abjs prems have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
from th3[simplified] th2[simplified] th[simplified] show ?case
by (simp add: left_distrib)
next
case (7 s t n a)
let ?ns = "fst (zsplit0 s)"
let ?as = "snd (zsplit0 s)"
let ?nt = "fst (zsplit0 t)"
let ?at = "snd (zsplit0 t)"
have abjs: "zsplit0 s = (?ns,?as)" by simp
moreover have abjt: "zsplit0 t = (?nt,?at)" by simp
ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using prems
by (simp add: Let_def split_def)
from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by simp
with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
from abjs prems have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
from th3[simplified] th2[simplified] th[simplified] show ?case
by (simp add: left_diff_distrib)
next
case (8 i t n a)
let ?nt = "fst (zsplit0 t)"
let ?at = "snd (zsplit0 t)"
have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using prems
by (simp add: Let_def split_def)
from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
hence " ?I x (Mul i t) = (real i) * ?I x (CN 0 ?nt ?at)" by simp
also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: right_distrib)
finally show ?case using th th2 by simp
next
case (9 t n a)
let ?nt = "fst (zsplit0 t)"
let ?at = "snd (zsplit0 t)"
have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a= Floor ?at \<and> n=?nt" using prems
by (simp add: Let_def split_def)
from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
hence na: "?N a" using th by simp
have th': "(real ?nt)*(real x) = real (?nt * x)" by simp
have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp
also have "\<dots> = real (floor ((real ?nt)* real(x) + ?I x ?at))" by simp
also have "\<dots> = real (floor (?I x ?at + real (?nt* x)))" by (simp add: add_ac)
also have "\<dots> = real (floor (?I x ?at) + (?nt* x))"
using floor_add[where x="?I x ?at" and a="?nt* x"] by simp
also have "\<dots> = real (?nt)*(real x) + real (floor (?I x ?at))" by (simp add: add_ac)
finally have "?I x (Floor t) = ?I x (CN 0 n a)" using th by simp
with na show ?case by simp
qed
consts
iszlfm :: "fm \<Rightarrow> real list \<Rightarrow> bool" (* Linearity test for fm *)
zlfm :: "fm \<Rightarrow> fm" (* Linearity transformation for fm *)
recdef iszlfm "measure size"
"iszlfm (And p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)"
"iszlfm (Or p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)"
"iszlfm (Eq (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
"iszlfm (NEq (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
"iszlfm (Lt (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
"iszlfm (Le (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
"iszlfm (Gt (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
"iszlfm (Ge (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
"iszlfm (Dvd i (CN 0 c e)) =
(\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)"
"iszlfm (NDvd i (CN 0 c e))=
(\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)"
"iszlfm p = (\<lambda> bs. isatom p \<and> (bound0 p))"
lemma zlin_qfree: "iszlfm p bs \<Longrightarrow> qfree p"
by (induct p rule: iszlfm.induct) auto
lemma iszlfm_gen:
assumes lp: "iszlfm p (x#bs)"
shows "\<forall> y. iszlfm p (y#bs)"
proof
fix y
show "iszlfm p (y#bs)"
using lp
by(induct p rule: iszlfm.induct, simp_all add: numbound0_gen[rule_format, where x="x" and y="y"])
qed
lemma conj_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm q bs \<Longrightarrow> iszlfm (conj p q) bs"
using conj_def by (cases p,auto)
lemma disj_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm q bs \<Longrightarrow> iszlfm (disj p q) bs"
using disj_def by (cases p,auto)
lemma not_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm (not p) bs"
by (induct p rule:iszlfm.induct ,auto)
recdef zlfm "measure fmsize"
"zlfm (And p q) = conj (zlfm p) (zlfm q)"
"zlfm (Or p q) = disj (zlfm p) (zlfm q)"
"zlfm (Imp p q) = disj (zlfm (NOT p)) (zlfm q)"
"zlfm (Iff p q) = disj (conj (zlfm p) (zlfm q)) (conj (zlfm (NOT p)) (zlfm (NOT q)))"
"zlfm (Lt a) = (let (c,r) = zsplit0 a in
if c=0 then Lt r else
if c>0 then Or (Lt (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r)))
else Or (Gt (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))"
"zlfm (Le a) = (let (c,r) = zsplit0 a in
if c=0 then Le r else
if c>0 then Or (Le (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r)))
else Or (Ge (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))"
"zlfm (Gt a) = (let (c,r) = zsplit0 a in
if c=0 then Gt r else
if c>0 then Or (Gt (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r)))
else Or (Lt (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))"
"zlfm (Ge a) = (let (c,r) = zsplit0 a in
if c=0 then Ge r else
if c>0 then Or (Ge (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r)))
else Or (Le (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))"
"zlfm (Eq a) = (let (c,r) = zsplit0 a in
if c=0 then Eq r else
if c>0 then (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Eq (Add (Floor (Neg r)) r)))
else (And (Eq (CN 0 (-c) (Floor (Neg r)))) (Eq (Add (Floor (Neg r)) r))))"
"zlfm (NEq a) = (let (c,r) = zsplit0 a in
if c=0 then NEq r else
if c>0 then (Or (NEq (CN 0 c (Neg (Floor (Neg r))))) (NEq (Add (Floor (Neg r)) r)))
else (Or (NEq (CN 0 (-c) (Floor (Neg r)))) (NEq (Add (Floor (Neg r)) r))))"
"zlfm (Dvd i a) = (if i=0 then zlfm (Eq a)
else (let (c,r) = zsplit0 a in
if c=0 then Dvd (abs i) r else
if c>0 then And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 c (Floor r)))
else And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 (-c) (Neg (Floor r))))))"
"zlfm (NDvd i a) = (if i=0 then zlfm (NEq a)
else (let (c,r) = zsplit0 a in
if c=0 then NDvd (abs i) r else
if c>0 then Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 c (Floor r)))
else Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 (-c) (Neg (Floor r))))))"
"zlfm (NOT (And p q)) = disj (zlfm (NOT p)) (zlfm (NOT q))"
"zlfm (NOT (Or p q)) = conj (zlfm (NOT p)) (zlfm (NOT q))"
"zlfm (NOT (Imp p q)) = conj (zlfm p) (zlfm (NOT q))"
"zlfm (NOT (Iff p q)) = disj (conj(zlfm p) (zlfm(NOT q))) (conj (zlfm(NOT p)) (zlfm q))"
"zlfm (NOT (NOT p)) = zlfm p"
"zlfm (NOT T) = F"
"zlfm (NOT F) = T"
"zlfm (NOT (Lt a)) = zlfm (Ge a)"
"zlfm (NOT (Le a)) = zlfm (Gt a)"
"zlfm (NOT (Gt a)) = zlfm (Le a)"
"zlfm (NOT (Ge a)) = zlfm (Lt a)"
"zlfm (NOT (Eq a)) = zlfm (NEq a)"
"zlfm (NOT (NEq a)) = zlfm (Eq a)"
"zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)"
"zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)"
"zlfm p = p" (hints simp add: fmsize_pos)
lemma split_int_less_real:
"(real (a::int) < b) = (a < floor b \<or> (a = floor b \<and> real (floor b) < b))"
proof( auto)
assume alb: "real a < b" and agb: "\<not> a < floor b"
from agb have "floor b \<le> a" by simp hence th: "b < real a + 1" by (simp only: floor_le_eq)
from floor_eq[OF alb th] show "a= floor b" by simp
next
assume alb: "a < floor b"
hence "real a < real (floor b)" by simp
moreover have "real (floor b) \<le> b" by simp ultimately show "real a < b" by arith
qed
lemma split_int_less_real':
"(real (a::int) + b < 0) = (real a - real (floor(-b)) < 0 \<or> (real a - real (floor (-b)) = 0 \<and> real (floor (-b)) + b < 0))"
proof-
have "(real a + b <0) = (real a < -b)" by arith
with split_int_less_real[where a="a" and b="-b"] show ?thesis by arith
qed
lemma split_int_gt_real':
"(real (a::int) + b > 0) = (real a + real (floor b) > 0 \<or> (real a + real (floor b) = 0 \<and> real (floor b) - b < 0))"
proof-
have th: "(real a + b >0) = (real (-a) + (-b)< 0)" by arith
show ?thesis using myless[rule_format, where b="real (floor b)"]
by (simp only:th split_int_less_real'[where a="-a" and b="-b"])
(simp add: algebra_simps diff_def[symmetric],arith)
qed
lemma split_int_le_real:
"(real (a::int) \<le> b) = (a \<le> floor b \<or> (a = floor b \<and> real (floor b) < b))"
proof( auto)
assume alb: "real a \<le> b" and agb: "\<not> a \<le> floor b"
from alb have "floor (real a) \<le> floor b " by (simp only: floor_mono)
hence "a \<le> floor b" by simp with agb show "False" by simp
next
assume alb: "a \<le> floor b"
hence "real a \<le> real (floor b)" by (simp only: floor_mono)
also have "\<dots>\<le> b" by simp finally show "real a \<le> b" .
qed
lemma split_int_le_real':
"(real (a::int) + b \<le> 0) = (real a - real (floor(-b)) \<le> 0 \<or> (real a - real (floor (-b)) = 0 \<and> real (floor (-b)) + b < 0))"
proof-
have "(real a + b \<le>0) = (real a \<le> -b)" by arith
with split_int_le_real[where a="a" and b="-b"] show ?thesis by arith
qed
lemma split_int_ge_real':
"(real (a::int) + b \<ge> 0) = (real a + real (floor b) \<ge> 0 \<or> (real a + real (floor b) = 0 \<and> real (floor b) - b < 0))"
proof-
have th: "(real a + b \<ge>0) = (real (-a) + (-b) \<le> 0)" by arith
show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"])
(simp add: algebra_simps diff_def[symmetric],arith)
qed
lemma split_int_eq_real: "(real (a::int) = b) = ( a = floor b \<and> b = real (floor b))" (is "?l = ?r")
by auto
lemma split_int_eq_real': "(real (a::int) + b = 0) = ( a - floor (-b) = 0 \<and> real (floor (-b)) + b = 0)" (is "?l = ?r")
proof-
have "?l = (real a = -b)" by arith
with split_int_eq_real[where a="a" and b="-b"] show ?thesis by simp arith
qed
lemma zlfm_I:
assumes qfp: "qfree p"
shows "(Ifm (real i #bs) (zlfm p) = Ifm (real i# bs) p) \<and> iszlfm (zlfm p) (real (i::int) #bs)"
(is "(?I (?l p) = ?I p) \<and> ?L (?l p)")
using qfp
proof(induct p rule: zlfm.induct)
case (5 a)
let ?c = "fst (zsplit0 a)"
let ?r = "snd (zsplit0 a)"
have spl: "zsplit0 a = (?c,?r)" by simp
from zsplit0_I[OF spl, where x="i" and bs="bs"]
have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (real i#bs) t"
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
moreover
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
by (cases "?r", simp_all add: Let_def split_def,case_tac "nat", simp_all)}
moreover
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
also have "\<dots> = (?I (?l (Lt a)))" apply (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) by (simp add: Ia cp cnz Let_def split_def diff_def)
finally have ?case using l by simp}
moreover
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
also from cn cnz have "\<dots> = (?I (?l (Lt a)))" by (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith)
finally have ?case using l by simp}
ultimately show ?case by blast
next
case (6 a)
let ?c = "fst (zsplit0 a)"
let ?r = "snd (zsplit0 a)"
have spl: "zsplit0 a = (?c,?r)" by simp
from zsplit0_I[OF spl, where x="i" and bs="bs"]
have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (real i#bs) t"
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
moreover
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
by (cases "?r", simp_all add: Let_def split_def, case_tac "nat",simp_all)}
moreover
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Le a) = (real (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
also have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def)
finally have ?case using l by simp}
moreover
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Le a) = (real (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
also from cn cnz have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac ,arith)
finally have ?case using l by simp}
ultimately show ?case by blast
next
case (7 a)
let ?c = "fst (zsplit0 a)"
let ?r = "snd (zsplit0 a)"
have spl: "zsplit0 a = (?c,?r)" by simp
from zsplit0_I[OF spl, where x="i" and bs="bs"]
have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (real i#bs) t"
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
moreover
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
moreover
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
also have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def)
finally have ?case using l by simp}
moreover
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
also from cn cnz have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith)
finally have ?case using l by simp}
ultimately show ?case by blast
next
case (8 a)
let ?c = "fst (zsplit0 a)"
let ?r = "snd (zsplit0 a)"
have spl: "zsplit0 a = (?c,?r)" by simp
from zsplit0_I[OF spl, where x="i" and bs="bs"]
have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (real i#bs) t"
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
moreover
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
moreover
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Ge a) = (real (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
also have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def)
finally have ?case using l by simp}
moreover
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Ge a) = (real (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
also from cn cnz have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith)
finally have ?case using l by simp}
ultimately show ?case by blast
next
case (9 a)
let ?c = "fst (zsplit0 a)"
let ?r = "snd (zsplit0 a)"
have spl: "zsplit0 a = (?c,?r)" by simp
from zsplit0_I[OF spl, where x="i" and bs="bs"]
have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (real i#bs) t"
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
moreover
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
moreover
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
also have "\<dots> = (?I (?l (Eq a)))" using cp cnz by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult)
finally have ?case using l by simp}
moreover
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
also from cn cnz have "\<dots> = (?I (?l (Eq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith)
finally have ?case using l by simp}
ultimately show ?case by blast
next
case (10 a)
let ?c = "fst (zsplit0 a)"
let ?r = "snd (zsplit0 a)"
have spl: "zsplit0 a = (?c,?r)" by simp
from zsplit0_I[OF spl, where x="i" and bs="bs"]
have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (real i#bs) t"
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
moreover
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
moreover
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (NEq a) = (real (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
also have "\<dots> = (?I (?l (NEq a)))" using cp cnz by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult)
finally have ?case using l by simp}
moreover
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (NEq a) = (real (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
also from cn cnz have "\<dots> = (?I (?l (NEq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith)
finally have ?case using l by simp}
ultimately show ?case by blast
next
case (11 j a)
let ?c = "fst (zsplit0 a)"
let ?r = "snd (zsplit0 a)"
have spl: "zsplit0 a = (?c,?r)" by simp
from zsplit0_I[OF spl, where x="i" and bs="bs"]
have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (real i#bs) t"
have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
moreover
{assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def)
hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)}
moreover
{assume "?c=0" and "j\<noteq>0" hence ?case
using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"]
by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
moreover
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))"
using Ia by (simp add: Let_def split_def)
also have "\<dots> = (real (abs j) rdvd real (?c*i) + (?N ?r))"
by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and>
(real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))"
by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac)
also have "\<dots> = (?I (?l (Dvd j a)))" using cp cnz jnz
by (simp add: Let_def split_def int_rdvd_iff[symmetric]
del: real_of_int_mult) (auto simp add: add_ac)
finally have ?case using l jnz by simp }
moreover
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))"
using Ia by (simp add: Let_def split_def)
also have "\<dots> = (real (abs j) rdvd real (?c*i) + (?N ?r))"
by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and>
(real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))"
by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac)
also have "\<dots> = (?I (?l (Dvd j a)))" using cn cnz jnz
using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric]
by (simp add: Let_def split_def int_rdvd_iff[symmetric]
del: real_of_int_mult) (auto simp add: add_ac)
finally have ?case using l jnz by blast }
ultimately show ?case by blast
next
case (12 j a)
let ?c = "fst (zsplit0 a)"
let ?r = "snd (zsplit0 a)"
have spl: "zsplit0 a = (?c,?r)" by simp
from zsplit0_I[OF spl, where x="i" and bs="bs"]
have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (real i#bs) t"
have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
moreover
{assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def)
hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)}
moreover
{assume "?c=0" and "j\<noteq>0" hence ?case
using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"]
by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)}
moreover
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (NDvd j a) = (\<not> (real j rdvd (real (?c * i) + (?N ?r))))"
using Ia by (simp add: Let_def split_def)
also have "\<dots> = (\<not> (real (abs j) rdvd real (?c*i) + (?N ?r)))"
by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and>
(real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))"
by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac)
also have "\<dots> = (?I (?l (NDvd j a)))" using cp cnz jnz
by (simp add: Let_def split_def int_rdvd_iff[symmetric]
del: real_of_int_mult) (auto simp add: add_ac)
finally have ?case using l jnz by simp }
moreover
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))"
by (simp add: nb Let_def split_def isint_Floor isint_neg)
have "?I (NDvd j a) = (\<not> (real j rdvd (real (?c * i) + (?N ?r))))"
using Ia by (simp add: Let_def split_def)
also have "\<dots> = (\<not> (real (abs j) rdvd real (?c*i) + (?N ?r)))"
by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and>
(real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))"
by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac)
also have "\<dots> = (?I (?l (NDvd j a)))" using cn cnz jnz
using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric]
by (simp add: Let_def split_def int_rdvd_iff[symmetric]
del: real_of_int_mult) (auto simp add: add_ac)
finally have ?case using l jnz by blast }
ultimately show ?case by blast
qed auto
text{* plusinf : Virtual substitution of @{text "+\<infinity>"}
minusinf: Virtual substitution of @{text "-\<infinity>"}
@{text "\<delta>"} Compute lcm @{text "d| Dvd d c*x+t \<in> p"}
@{text "d\<delta>"} checks if a given l divides all the ds above*}
consts
plusinf:: "fm \<Rightarrow> fm"
minusinf:: "fm \<Rightarrow> fm"
\<delta> :: "fm \<Rightarrow> int"
d\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool"
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"
lemma minusinf_qfree: "qfree p \<Longrightarrow> qfree (minusinf p)"
by (induct p rule: minusinf.induct, auto)
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"
recdef \<delta> "measure size"
"\<delta> (And p q) = zlcm (\<delta> p) (\<delta> q)"
"\<delta> (Or p q) = zlcm (\<delta> p) (\<delta> q)"
"\<delta> (Dvd i (CN 0 c e)) = i"
"\<delta> (NDvd i (CN 0 c e)) = i"
"\<delta> p = 1"
recdef d\<delta> "measure size"
"d\<delta> (And p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)"
"d\<delta> (Or p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)"
"d\<delta> (Dvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
"d\<delta> (NDvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
"d\<delta> p = (\<lambda> d. True)"
lemma delta_mono:
assumes lin: "iszlfm p bs"
and d: "d dvd d'"
and ad: "d\<delta> p d"
shows "d\<delta> p d'"
using lin ad d
proof(induct p rule: iszlfm.induct)
case (9 i c e) thus ?case using d
by (simp add: dvd_trans[of "i" "d" "d'"])
next
case (10 i c e) thus ?case using d
by (simp add: dvd_trans[of "i" "d" "d'"])
qed simp_all
lemma \<delta> : assumes lin:"iszlfm p bs"
shows "d\<delta> p (\<delta> p) \<and> \<delta> p >0"
using lin
proof (induct p rule: iszlfm.induct)
case (1 p q)
let ?d = "\<delta> (And p q)"
from prems zlcm_pos have dp: "?d >0" by simp
have d1: "\<delta> p dvd \<delta> (And p q)" using prems by simp
hence th: "d\<delta> p ?d"
using delta_mono prems by (auto simp del: dvd_zlcm_self1)
have "\<delta> q dvd \<delta> (And p q)" using prems by simp
hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: dvd_zlcm_self2)
from th th' dp show ?case by simp
next
case (2 p q)
let ?d = "\<delta> (And p q)"
from prems zlcm_pos have dp: "?d >0" by simp
have "\<delta> p dvd \<delta> (And p q)" using prems by simp hence th: "d\<delta> p ?d" using delta_mono prems
by (auto simp del: dvd_zlcm_self1)
have "\<delta> q dvd \<delta> (And p q)" using prems by simp hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: dvd_zlcm_self2)
from th th' dp show ?case by simp
qed simp_all
lemma minusinf_inf:
assumes linp: "iszlfm p (a # bs)"
shows "\<exists> (z::int). \<forall> x < z. Ifm ((real x)#bs) (minusinf p) = Ifm ((real x)#bs) p"
(is "?P p" is "\<exists> (z::int). \<forall> x < z. ?I x (?M p) = ?I x p")
using linp
proof (induct p rule: minusinf.induct)
case (1 f g)
from prems have "?P f" by simp
then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast
from prems have "?P g" by simp
then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast
let ?z = "min z1 z2"
from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (And f g)) = ?I x (And f g)" by simp
thus ?case by blast
next
case (2 f g) from prems have "?P f" by simp
then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast
from prems have "?P g" by simp
then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast
let ?z = "min z1 z2"
from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (Or f g)) = ?I x (Or f g)" by simp
thus ?case by blast
next
case (3 c e)
from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
from prems have nbe: "numbound0 e" by simp
fix y
have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))"
proof (simp add: less_floor_eq , rule allI, rule impI)
fix x
assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))"
by (simp only: real_mult_less_mono2[OF rcpos th1])
hence "real c * real x + Inum (y # bs) e \<noteq> 0"using rcpos by simp
thus "real c * real x + Inum (real x # bs) e \<noteq> 0"
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] by simp
qed
thus ?case by blast
next
case (4 c e)
from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
from prems have nbe: "numbound0 e" by simp
fix y
have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))"
proof (simp add: less_floor_eq , rule allI, rule impI)
fix x
assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))"
by (simp only: real_mult_less_mono2[OF rcpos th1])
hence "real c * real x + Inum (y # bs) e \<noteq> 0"using rcpos by simp
thus "real c * real x + Inum (real x # bs) e \<noteq> 0"
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] by simp
qed
thus ?case by blast
next
case (5 c e)
from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
from prems have nbe: "numbound0 e" by simp
fix y
have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))"
proof (simp add: less_floor_eq , rule allI, rule impI)
fix x
assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))"
by (simp only: real_mult_less_mono2[OF rcpos th1])
thus "real c * real x + Inum (real x # bs) e < 0"
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
qed
thus ?case by blast
next
case (6 c e)
from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
from prems have nbe: "numbound0 e" by simp
fix y
have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))"
proof (simp add: less_floor_eq , rule allI, rule impI)
fix x
assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))"
by (simp only: real_mult_less_mono2[OF rcpos th1])
thus "real c * real x + Inum (real x # bs) e \<le> 0"
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
qed
thus ?case by blast
next
case (7 c e)
from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
from prems have nbe: "numbound0 e" by simp
fix y
have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))"
proof (simp add: less_floor_eq , rule allI, rule impI)
fix x
assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))"
by (simp only: real_mult_less_mono2[OF rcpos th1])
thus "\<not> (real c * real x + Inum (real x # bs) e>0)"
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
qed
thus ?case by blast
next
case (8 c e)
from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp
from prems have nbe: "numbound0 e" by simp
fix y
have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))"
proof (simp add: less_floor_eq , rule allI, rule impI)
fix x
assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)"
hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))"
by (simp only: real_mult_less_mono2[OF rcpos th1])
thus "\<not> real c * real x + Inum (real x # bs) e \<ge> 0"
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
qed
thus ?case by blast
qed simp_all
lemma minusinf_repeats:
assumes d: "d\<delta> p d" and linp: "iszlfm p (a # bs)"
shows "Ifm ((real(x - k*d))#bs) (minusinf p) = Ifm (real x #bs) (minusinf p)"
using linp d
proof(induct p rule: iszlfm.induct)
case (9 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+
hence "\<exists> k. d=i*k" by (simp add: dvd_def)
then obtain "di" where di_def: "d=i*di" by blast
show ?case
proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI)
assume
"real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e"
(is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))"
by (simp add: algebra_simps di_def)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))"
by (simp add: algebra_simps)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast
thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp
next
assume
"real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)"
by blast
thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp
qed
next
case (10 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+
hence "\<exists> k. d=i*k" by (simp add: dvd_def)
then obtain "di" where di_def: "d=i*di" by blast
show ?case
proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI)
assume
"real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e"
(is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))"
by (simp add: algebra_simps di_def)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))"
by (simp add: algebra_simps)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast
thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp
next
assume
"real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)"
by blast
thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp
qed
qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="real(x - k*d)" and b'="real x"] simp del: real_of_int_mult real_of_int_diff)
lemma minusinf_ex:
assumes lin: "iszlfm p (real (a::int) #bs)"
and exmi: "\<exists> (x::int). Ifm (real x#bs) (minusinf p)" (is "\<exists> x. ?P1 x")
shows "\<exists> (x::int). Ifm (real x#bs) p" (is "\<exists> x. ?P x")
proof-
let ?d = "\<delta> p"
from \<delta> [OF lin] have dpos: "?d >0" by simp
from \<delta> [OF lin] have alld: "d\<delta> p ?d" by simp
from minusinf_repeats[OF alld lin] have th1:"\<forall> x k. ?P1 x = ?P1 (x - (k * ?d))" by simp
from minusinf_inf[OF lin] have th2:"\<exists> z. \<forall> x. x<z \<longrightarrow> (?P x = ?P1 x)" by blast
from minusinfinity [OF dpos th1 th2] exmi show ?thesis by blast
qed
lemma minusinf_bex:
assumes lin: "iszlfm p (real (a::int) #bs)"
shows "(\<exists> (x::int). Ifm (real x#bs) (minusinf p)) =
(\<exists> (x::int)\<in> {1..\<delta> p}. Ifm (real x#bs) (minusinf p))"
(is "(\<exists> x. ?P x) = _")
proof-
let ?d = "\<delta> p"
from \<delta> [OF lin] have dpos: "?d >0" by simp
from \<delta> [OF lin] have alld: "d\<delta> p ?d" by simp
from minusinf_repeats[OF alld lin] have th1:"\<forall> x k. ?P x = ?P (x - (k * ?d))" by simp
from periodic_finite_ex[OF dpos th1] show ?thesis by blast
qed
lemma dvd1_eq1: "x >0 \<Longrightarrow> (x::int) dvd 1 = (x = 1)" by auto
consts
a\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" (* adjusts the coeffitients of a formula *)
d\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* tests if all coeffs c of c divide a given l*)
\<zeta> :: "fm \<Rightarrow> int" (* computes the lcm of all coefficients of x*)
\<beta> :: "fm \<Rightarrow> num list"
\<alpha> :: "fm \<Rightarrow> num list"
recdef a\<beta> "measure size"
"a\<beta> (And p q) = (\<lambda> k. And (a\<beta> p k) (a\<beta> q k))"
"a\<beta> (Or p q) = (\<lambda> k. Or (a\<beta> p k) (a\<beta> q k))"
"a\<beta> (Eq (CN 0 c e)) = (\<lambda> k. Eq (CN 0 1 (Mul (k div c) e)))"
"a\<beta> (NEq (CN 0 c e)) = (\<lambda> k. NEq (CN 0 1 (Mul (k div c) e)))"
"a\<beta> (Lt (CN 0 c e)) = (\<lambda> k. Lt (CN 0 1 (Mul (k div c) e)))"
"a\<beta> (Le (CN 0 c e)) = (\<lambda> k. Le (CN 0 1 (Mul (k div c) e)))"
"a\<beta> (Gt (CN 0 c e)) = (\<lambda> k. Gt (CN 0 1 (Mul (k div c) e)))"
"a\<beta> (Ge (CN 0 c e)) = (\<lambda> k. Ge (CN 0 1 (Mul (k div c) e)))"
"a\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
"a\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
"a\<beta> p = (\<lambda> k. p)"
recdef d\<beta> "measure size"
"d\<beta> (And p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))"
"d\<beta> (Or p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))"
"d\<beta> (Eq (CN 0 c e)) = (\<lambda> k. c dvd k)"
"d\<beta> (NEq (CN 0 c e)) = (\<lambda> k. c dvd k)"
"d\<beta> (Lt (CN 0 c e)) = (\<lambda> k. c dvd k)"
"d\<beta> (Le (CN 0 c e)) = (\<lambda> k. c dvd k)"
"d\<beta> (Gt (CN 0 c e)) = (\<lambda> k. c dvd k)"
"d\<beta> (Ge (CN 0 c e)) = (\<lambda> k. c dvd k)"
"d\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. c dvd k)"
"d\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. c dvd k)"
"d\<beta> p = (\<lambda> k. True)"
recdef \<zeta> "measure size"
"\<zeta> (And p q) = zlcm (\<zeta> p) (\<zeta> q)"
"\<zeta> (Or p q) = zlcm (\<zeta> p) (\<zeta> q)"
"\<zeta> (Eq (CN 0 c e)) = c"
"\<zeta> (NEq (CN 0 c e)) = c"
"\<zeta> (Lt (CN 0 c e)) = c"
"\<zeta> (Le (CN 0 c e)) = c"
"\<zeta> (Gt (CN 0 c e)) = c"
"\<zeta> (Ge (CN 0 c e)) = c"
"\<zeta> (Dvd i (CN 0 c e)) = c"
"\<zeta> (NDvd i (CN 0 c e))= c"
"\<zeta> p = 1"
recdef \<beta> "measure size"
"\<beta> (And p q) = (\<beta> p @ \<beta> q)"
"\<beta> (Or p q) = (\<beta> p @ \<beta> q)"
"\<beta> (Eq (CN 0 c e)) = [Sub (C -1) e]"
"\<beta> (NEq (CN 0 c e)) = [Neg e]"
"\<beta> (Lt (CN 0 c e)) = []"
"\<beta> (Le (CN 0 c e)) = []"
"\<beta> (Gt (CN 0 c e)) = [Neg e]"
"\<beta> (Ge (CN 0 c e)) = [Sub (C -1) e]"
"\<beta> p = []"
recdef \<alpha> "measure size"
"\<alpha> (And p q) = (\<alpha> p @ \<alpha> q)"
"\<alpha> (Or p q) = (\<alpha> p @ \<alpha> q)"
"\<alpha> (Eq (CN 0 c e)) = [Add (C -1) e]"
"\<alpha> (NEq (CN 0 c e)) = [e]"
"\<alpha> (Lt (CN 0 c e)) = [e]"
"\<alpha> (Le (CN 0 c e)) = [Add (C -1) e]"
"\<alpha> (Gt (CN 0 c e)) = []"
"\<alpha> (Ge (CN 0 c e)) = []"
"\<alpha> p = []"
consts mirror :: "fm \<Rightarrow> fm"
recdef mirror "measure size"
"mirror (And p q) = And (mirror p) (mirror q)"
"mirror (Or p q) = Or (mirror p) (mirror q)"
"mirror (Eq (CN 0 c e)) = Eq (CN 0 c (Neg e))"
"mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))"
"mirror (Lt (CN 0 c e)) = Gt (CN 0 c (Neg e))"
"mirror (Le (CN 0 c e)) = Ge (CN 0 c (Neg e))"
"mirror (Gt (CN 0 c e)) = Lt (CN 0 c (Neg e))"
"mirror (Ge (CN 0 c e)) = Le (CN 0 c (Neg e))"
"mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))"
"mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))"
"mirror p = p"
lemma mirror\<alpha>\<beta>:
assumes lp: "iszlfm p (a#bs)"
shows "(Inum (real (i::int)#bs)) ` set (\<alpha> p) = (Inum (real i#bs)) ` set (\<beta> (mirror p))"
using lp
by (induct p rule: mirror.induct, auto)
lemma mirror:
assumes lp: "iszlfm p (a#bs)"
shows "Ifm (real (x::int)#bs) (mirror p) = Ifm (real (- x)#bs) p"
using lp
proof(induct p rule: iszlfm.induct)
case (9 j c e)
have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
(real j rdvd - (real c * real x - Inum (real x # bs) e))"
by (simp only: rdvd_minus[symmetric])
from prems th show ?case
by (simp add: algebra_simps
numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
next
case (10 j c e)
have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
(real j rdvd - (real c * real x - Inum (real x # bs) e))"
by (simp only: rdvd_minus[symmetric])
from prems th show ?case
by (simp add: algebra_simps
numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
qed (auto simp add: numbound0_I[where bs="bs" and b="real x" and b'="- real x"] nth_pos2)
lemma mirror_l: "iszlfm p (a#bs) \<Longrightarrow> iszlfm (mirror p) (a#bs)"
by (induct p rule: mirror.induct, auto simp add: isint_neg)
lemma mirror_d\<beta>: "iszlfm p (a#bs) \<and> d\<beta> p 1
\<Longrightarrow> iszlfm (mirror p) (a#bs) \<and> d\<beta> (mirror p) 1"
by (induct p rule: mirror.induct, auto simp add: isint_neg)
lemma mirror_\<delta>: "iszlfm p (a#bs) \<Longrightarrow> \<delta> (mirror p) = \<delta> p"
by (induct p rule: mirror.induct,auto)
lemma mirror_ex:
assumes lp: "iszlfm p (real (i::int)#bs)"
shows "(\<exists> (x::int). Ifm (real x#bs) (mirror p)) = (\<exists> (x::int). Ifm (real x#bs) p)"
(is "(\<exists> x. ?I x ?mp) = (\<exists> x. ?I x p)")
proof(auto)
fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast
thus "\<exists> x. ?I x p" by blast
next
fix x assume "?I x p" hence "?I (- x) ?mp"
using mirror[OF lp, where x="- x", symmetric] by auto
thus "\<exists> x. ?I x ?mp" by blast
qed
lemma \<beta>_numbound0: assumes lp: "iszlfm p bs"
shows "\<forall> b\<in> set (\<beta> p). numbound0 b"
using lp by (induct p rule: \<beta>.induct,auto)
lemma d\<beta>_mono:
assumes linp: "iszlfm p (a #bs)"
and dr: "d\<beta> p l"
and d: "l dvd l'"
shows "d\<beta> p l'"
using dr linp dvd_trans[of _ "l" "l'", simplified d]
by (induct p rule: iszlfm.induct) simp_all
lemma \<alpha>_l: assumes lp: "iszlfm p (a#bs)"
shows "\<forall> b\<in> set (\<alpha> p). numbound0 b \<and> isint b (a#bs)"
using lp
by(induct p rule: \<alpha>.induct, auto simp add: isint_add isint_c)
lemma \<zeta>:
assumes linp: "iszlfm p (a #bs)"
shows "\<zeta> p > 0 \<and> d\<beta> p (\<zeta> p)"
using linp
proof(induct p rule: iszlfm.induct)
case (1 p q)
from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
dl1 dl2 show ?case by (auto simp add: zlcm_pos)
next
case (2 p q)
from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
dl1 dl2 show ?case by (auto simp add: zlcm_pos)
qed (auto simp add: zlcm_pos)
lemma a\<beta>: assumes linp: "iszlfm p (a #bs)" and d: "d\<beta> p l" and lp: "l > 0"
shows "iszlfm (a\<beta> p l) (a #bs) \<and> d\<beta> (a\<beta> p l) 1 \<and> (Ifm (real (l * x) #bs) (a\<beta> p l) = Ifm ((real x)#bs) p)"
using linp d
proof (induct p rule: iszlfm.induct)
case (5 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
by (simp add: zdiv_self[OF cnz])
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(real l * real x + real (l div c) * Inum (real x # bs) e < (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e < 0)"
by simp
also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) < (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e < 0)"
using mult_less_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
next
case (6 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
by (simp add: zdiv_self[OF cnz])
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<le> (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<le> 0)"
by simp
also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<le> (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e \<le> 0)"
using mult_le_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
next
case (7 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
by (simp add: zdiv_self[OF cnz])
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(real l * real x + real (l div c) * Inum (real x # bs) e > (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e > 0)"
by simp
also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) > (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e > 0)"
using zero_less_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
next
case (8 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
by (simp add: zdiv_self[OF cnz])
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<ge> (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<ge> 0)"
by simp
also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<ge> (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e \<ge> 0)"
using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
next
case (3 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
by (simp add: zdiv_self[OF cnz])
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(real l * real x + real (l div c) * Inum (real x # bs) e = (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = 0)"
by simp
also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) = (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e = 0)"
using mult_eq_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
next
case (4 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
by (simp add: zdiv_self[OF cnz])
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<noteq> (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<noteq> 0)"
by simp
also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<noteq> (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e \<noteq> 0)"
using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
next
case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
by (simp add: zdiv_self[OF cnz])
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)" by simp
also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)"
using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp
also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp
next
case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
by (simp add: zdiv_self[OF cnz])
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)" by simp
also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)"
using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp
also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp
qed (simp_all add: nth_pos2 numbound0_I[where bs="bs" and b="real (l * x)" and b'="real x"] isint_Mul del: real_of_int_mult)
lemma a\<beta>_ex: assumes linp: "iszlfm p (a#bs)" and d: "d\<beta> p l" and lp: "l>0"
shows "(\<exists> x. l dvd x \<and> Ifm (real x #bs) (a\<beta> p l)) = (\<exists> (x::int). Ifm (real x#bs) p)"
(is "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> x. ?P' x)")
proof-
have "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> (x::int). ?P (l*x))"
using unity_coeff_ex[where l="l" and P="?P", simplified] by simp
also have "\<dots> = (\<exists> (x::int). ?P' x)" using a\<beta>[OF linp d lp] by simp
finally show ?thesis .
qed
lemma \<beta>:
assumes lp: "iszlfm p (a#bs)"
and u: "d\<beta> p 1"
and d: "d\<delta> p d"
and dp: "d > 0"
and nob: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real x = b + real j)"
and p: "Ifm (real x#bs) p" (is "?P x")
shows "?P (x - d)"
using lp u d dp nob p
proof(induct p rule: iszlfm.induct)
case (5 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems
show ?case by (simp del: real_of_int_minus)
next
case (6 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems
show ?case by (simp del: real_of_int_minus)
next
case (7 c e) hence p: "Ifm (real x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" and ie1:"isint e (a#bs)" using dvd1_eq1[where x="c"] by simp+
let ?e = "Inum (real x # bs) e"
from ie1 have ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"]
numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]
by (simp add: isint_iff)
{assume "real (x-d) +?e > 0" hence ?case using c1
numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"]
by (simp del: real_of_int_minus)}
moreover
{assume H: "\<not> real (x-d) + ?e > 0"
let ?v="Neg e"
have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp
from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]]
have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x = - ?e + real j)" by auto
from H p have "real x + ?e > 0 \<and> real x + ?e \<le> real d" by (simp add: c1)
hence "real (x + floor ?e) > real (0::int) \<and> real (x + floor ?e) \<le> real d"
using ie by simp
hence "x + floor ?e \<ge> 1 \<and> x + floor ?e \<le> d" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. real x = real (- floor ?e + j)"
by (simp only: real_of_int_inject) (simp add: algebra_simps)
hence "\<exists> (j::int) \<in> {1 .. d}. real x = - ?e + real j"
by (simp add: ie[simplified isint_iff])
with nob have ?case by auto}
ultimately show ?case by blast
next
case (8 c e) hence p: "Ifm (real x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e"
and ie1:"isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
let ?e = "Inum (real x # bs) e"
from ie1 have ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"]
by (simp add: isint_iff)
{assume "real (x-d) +?e \<ge> 0" hence ?case using c1
numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"]
by (simp del: real_of_int_minus)}
moreover
{assume H: "\<not> real (x-d) + ?e \<ge> 0"
let ?v="Sub (C -1) e"
have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp
from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]]
have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x = - ?e - 1 + real j)" by auto
from H p have "real x + ?e \<ge> 0 \<and> real x + ?e < real d" by (simp add: c1)
hence "real (x + floor ?e) \<ge> real (0::int) \<and> real (x + floor ?e) < real d"
using ie by simp
hence "x + floor ?e +1 \<ge> 1 \<and> x + floor ?e + 1 \<le> d" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e + 1" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. x= - floor ?e - 1 + j" by (simp add: algebra_simps)
hence "\<exists> (j::int) \<in> {1 .. d}. real x= real (- floor ?e - 1 + j)"
by (simp only: real_of_int_inject)
hence "\<exists> (j::int) \<in> {1 .. d}. real x= - ?e - 1 + real j"
by (simp add: ie[simplified isint_iff])
with nob have ?case by simp }
ultimately show ?case by blast
next
case (3 c e) hence p: "Ifm (real x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1"
and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
let ?e = "Inum (real x # bs) e"
let ?v="(Sub (C -1) e)"
have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp
from p have "real x= - ?e" by (simp add: c1) with prems(11) show ?case using dp
by simp (erule ballE[where x="1"],
simp_all add:algebra_simps numbound0_I[OF bn,where b="real x"and b'="a"and bs="bs"])
next
case (4 c e)hence p: "Ifm (real x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1"
and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
let ?e = "Inum (real x # bs) e"
let ?v="Neg e"
have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp
{assume "real x - real d + Inum ((real (x -d)) # bs) e \<noteq> 0"
hence ?case by (simp add: c1)}
moreover
{assume H: "real x - real d + Inum ((real (x -d)) # bs) e = 0"
hence "real x = - Inum ((real (x -d)) # bs) e + real d" by simp
hence "real x = - Inum (a # bs) e + real d"
by (simp add: numbound0_I[OF bn,where b="real x - real d"and b'="a"and bs="bs"])
with prems(11) have ?case using dp by simp}
ultimately show ?case by blast
next
case (9 j c e) hence p: "Ifm (real x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1"
and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
let ?e = "Inum (real x # bs) e"
from prems have "isint e (a #bs)" by simp
hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real x)#bs"] numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"]
by (simp add: isint_iff)
from prems have id: "j dvd d" by simp
from c1 ie[symmetric] have "?p x = (real j rdvd real (x+ floor ?e))" by simp
also have "\<dots> = (j dvd x + floor ?e)"
using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
also have "\<dots> = (j dvd x - d + floor ?e)"
using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
also have "\<dots> = (real j rdvd real (x - d + floor ?e))"
using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
ie by simp
also have "\<dots> = (real j rdvd real x - real d + ?e)"
using ie by simp
finally show ?case
using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
next
case (10 j c e) hence p: "Ifm (real x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
let ?e = "Inum (real x # bs) e"
from prems have "isint e (a#bs)" by simp
hence ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"]
by (simp add: isint_iff)
from prems have id: "j dvd d" by simp
from c1 ie[symmetric] have "?p x = (\<not> real j rdvd real (x+ floor ?e))" by simp
also have "\<dots> = (\<not> j dvd x + floor ?e)"
using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
also have "\<dots> = (\<not> j dvd x - d + floor ?e)"
using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
also have "\<dots> = (\<not> real j rdvd real (x - d + floor ?e))"
using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
ie by simp
also have "\<dots> = (\<not> real j rdvd real x - real d + ?e)"
using ie by simp
finally show ?case using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
qed (auto simp add: numbound0_I[where bs="bs" and b="real (x - d)" and b'="real x"] nth_pos2 simp del: real_of_int_diff)
lemma \<beta>':
assumes lp: "iszlfm p (a #bs)"
and u: "d\<beta> p 1"
and d: "d\<delta> p d"
and dp: "d > 0"
shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm ((Inum (a#bs) b + real j) #bs) p) \<longrightarrow> Ifm (real x#bs) p \<longrightarrow> Ifm (real (x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
proof(clarify)
fix x
assume nb:"?b" and px: "?P x"
hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real x = b + real j)"
by auto
from \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" .
qed
lemma \<beta>_int: assumes lp: "iszlfm p bs"
shows "\<forall> b\<in> set (\<beta> p). isint b bs"
using lp by (induct p rule: iszlfm.induct) (auto simp add: isint_neg isint_sub)
lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
apply(rule iffI)
prefer 2
apply(drule minusinfinity)
apply assumption+
apply(fastsimp)
apply clarsimp
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
apply(frule_tac x = x and z=z in decr_lemma)
apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
prefer 2
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
prefer 2 apply arith
apply fastsimp
apply(drule (1) periodic_finite_ex)
apply blast
apply(blast dest:decr_mult_lemma)
done
theorem cp_thm:
assumes lp: "iszlfm p (a #bs)"
and u: "d\<beta> p 1"
and d: "d\<delta> p d"
and dp: "d > 0"
shows "(\<exists> (x::int). Ifm (real x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm (real j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm ((Inum (a#bs) b + real j) #bs) p))"
(is "(\<exists> (x::int). ?P (real x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + real j)))")
proof-
from minusinf_inf[OF lp]
have th: "\<exists>(z::int). \<forall>x<z. ?P (real x) = ?M x" by blast
let ?B' = "{floor (?I b) | b. b\<in> ?B}"
from \<beta>_int[OF lp] isint_iff[where bs="a # bs"] have B: "\<forall> b\<in> ?B. real (floor (?I b)) = ?I b" by simp
from B[rule_format]
have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real (floor (?I b)) + real j))"
by simp
also have "\<dots> = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real (floor (?I b) + j)))" by simp
also have"\<dots> = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j)))" by blast
finally have BB':
"(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j)))"
by blast
hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j))) \<longrightarrow> ?P (real x) \<longrightarrow> ?P (real (x - d))" using \<beta>'[OF lp u d dp] by blast
from minusinf_repeats[OF d lp]
have th3: "\<forall> x k. ?M x = ?M (x-k*d)" by simp
from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast
qed
(* Reddy and Loveland *)
consts
\<rho> :: "fm \<Rightarrow> (num \<times> int) list" (* Compute the Reddy and Loveland Bset*)
\<sigma>\<rho>:: "fm \<Rightarrow> num \<times> int \<Rightarrow> fm" (* Performs the modified substitution of Reddy and Loveland*)
\<alpha>\<rho> :: "fm \<Rightarrow> (num\<times>int) list"
a\<rho> :: "fm \<Rightarrow> int \<Rightarrow> fm"
recdef \<rho> "measure size"
"\<rho> (And p q) = (\<rho> p @ \<rho> q)"
"\<rho> (Or p q) = (\<rho> p @ \<rho> q)"
"\<rho> (Eq (CN 0 c e)) = [(Sub (C -1) e,c)]"
"\<rho> (NEq (CN 0 c e)) = [(Neg e,c)]"
"\<rho> (Lt (CN 0 c e)) = []"
"\<rho> (Le (CN 0 c e)) = []"
"\<rho> (Gt (CN 0 c e)) = [(Neg e, c)]"
"\<rho> (Ge (CN 0 c e)) = [(Sub (C (-1)) e, c)]"
"\<rho> p = []"
recdef \<sigma>\<rho> "measure size"
"\<sigma>\<rho> (And p q) = (\<lambda> (t,k). And (\<sigma>\<rho> p (t,k)) (\<sigma>\<rho> q (t,k)))"
"\<sigma>\<rho> (Or p q) = (\<lambda> (t,k). Or (\<sigma>\<rho> p (t,k)) (\<sigma>\<rho> q (t,k)))"
"\<sigma>\<rho> (Eq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Eq (Add (Mul (c div k) t) e))
else (Eq (Add (Mul c t) (Mul k e))))"
"\<sigma>\<rho> (NEq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (NEq (Add (Mul (c div k) t) e))
else (NEq (Add (Mul c t) (Mul k e))))"
"\<sigma>\<rho> (Lt (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Lt (Add (Mul (c div k) t) e))
else (Lt (Add (Mul c t) (Mul k e))))"
"\<sigma>\<rho> (Le (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Le (Add (Mul (c div k) t) e))
else (Le (Add (Mul c t) (Mul k e))))"
"\<sigma>\<rho> (Gt (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Gt (Add (Mul (c div k) t) e))
else (Gt (Add (Mul c t) (Mul k e))))"
"\<sigma>\<rho> (Ge (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Ge (Add (Mul (c div k) t) e))
else (Ge (Add (Mul c t) (Mul k e))))"
"\<sigma>\<rho> (Dvd i (CN 0 c e)) =(\<lambda> (t,k). if k dvd c then (Dvd i (Add (Mul (c div k) t) e))
else (Dvd (i*k) (Add (Mul c t) (Mul k e))))"
"\<sigma>\<rho> (NDvd i (CN 0 c e))=(\<lambda> (t,k). if k dvd c then (NDvd i (Add (Mul (c div k) t) e))
else (NDvd (i*k) (Add (Mul c t) (Mul k e))))"
"\<sigma>\<rho> p = (\<lambda> (t,k). p)"
recdef \<alpha>\<rho> "measure size"
"\<alpha>\<rho> (And p q) = (\<alpha>\<rho> p @ \<alpha>\<rho> q)"
"\<alpha>\<rho> (Or p q) = (\<alpha>\<rho> p @ \<alpha>\<rho> q)"
"\<alpha>\<rho> (Eq (CN 0 c e)) = [(Add (C -1) e,c)]"
"\<alpha>\<rho> (NEq (CN 0 c e)) = [(e,c)]"
"\<alpha>\<rho> (Lt (CN 0 c e)) = [(e,c)]"
"\<alpha>\<rho> (Le (CN 0 c e)) = [(Add (C -1) e,c)]"
"\<alpha>\<rho> p = []"
(* Simulates normal substituion by modifying the formula see correctness theorem *)
recdef a\<rho> "measure size"
"a\<rho> (And p q) = (\<lambda> k. And (a\<rho> p k) (a\<rho> q k))"
"a\<rho> (Or p q) = (\<lambda> k. Or (a\<rho> p k) (a\<rho> q k))"
"a\<rho> (Eq (CN 0 c e)) = (\<lambda> k. if k dvd c then (Eq (CN 0 (c div k) e))
else (Eq (CN 0 c (Mul k e))))"
"a\<rho> (NEq (CN 0 c e)) = (\<lambda> k. if k dvd c then (NEq (CN 0 (c div k) e))
else (NEq (CN 0 c (Mul k e))))"
"a\<rho> (Lt (CN 0 c e)) = (\<lambda> k. if k dvd c then (Lt (CN 0 (c div k) e))
else (Lt (CN 0 c (Mul k e))))"
"a\<rho> (Le (CN 0 c e)) = (\<lambda> k. if k dvd c then (Le (CN 0 (c div k) e))
else (Le (CN 0 c (Mul k e))))"
"a\<rho> (Gt (CN 0 c e)) = (\<lambda> k. if k dvd c then (Gt (CN 0 (c div k) e))
else (Gt (CN 0 c (Mul k e))))"
"a\<rho> (Ge (CN 0 c e)) = (\<lambda> k. if k dvd c then (Ge (CN 0 (c div k) e))
else (Ge (CN 0 c (Mul k e))))"
"a\<rho> (Dvd i (CN 0 c e)) = (\<lambda> k. if k dvd c then (Dvd i (CN 0 (c div k) e))
else (Dvd (i*k) (CN 0 c (Mul k e))))"
"a\<rho> (NDvd i (CN 0 c e)) = (\<lambda> k. if k dvd c then (NDvd i (CN 0 (c div k) e))
else (NDvd (i*k) (CN 0 c (Mul k e))))"
"a\<rho> p = (\<lambda> k. p)"
constdefs \<sigma> :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm"
"\<sigma> p k t \<equiv> And (Dvd k t) (\<sigma>\<rho> p (t,k))"
lemma \<sigma>\<rho>:
assumes linp: "iszlfm p (real (x::int)#bs)"
and kpos: "real k > 0"
and tnb: "numbound0 t"
and tint: "isint t (real x#bs)"
and kdt: "k dvd floor (Inum (b'#bs) t)"
shows "Ifm (real x#bs) (\<sigma>\<rho> p (t,k)) =
(Ifm ((real ((floor (Inum (b'#bs) t)) div k))#bs) p)"
(is "?I (real x) (?s p) = (?I (real ((floor (?N b' t)) div k)) p)" is "_ = (?I ?tk p)")
using linp kpos tnb
proof(induct p rule: \<sigma>\<rho>.induct)
case (3 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
{assume kdc: "k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
moreover
{assume "\<not> k dvd c"
from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have "?I (real x) (?s (Eq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k = 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
finally have ?case . }
ultimately show ?case by blast
next
case (4 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
{assume kdc: "k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
moreover
{assume "\<not> k dvd c"
from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have "?I (real x) (?s (NEq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<noteq> 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
finally have ?case . }
ultimately show ?case by blast
next
case (5 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
{assume kdc: "k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
moreover
{assume "\<not> k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have "?I (real x) (?s (Lt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k < 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))" using pos_less_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
finally have ?case . }
ultimately show ?case by blast
next
case (6 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
{assume kdc: "k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
moreover
{assume "\<not> k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have "?I (real x) (?s (Le (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<le> 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Le (CN 0 c e)))" using pos_le_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
finally have ?case . }
ultimately show ?case by blast
next
case (7 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
{assume kdc: "k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
moreover
{assume "\<not> k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have "?I (real x) (?s (Gt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k > 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))" using pos_divide_less_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
finally have ?case . }
ultimately show ?case by blast
next
case (8 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
{assume kdc: "k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
moreover
{assume "\<not> k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have "?I (real x) (?s (Ge (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<ge> 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))" using pos_divide_le_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
finally have ?case . }
ultimately show ?case by blast
next
case (9 i c e) from prems have cp: "c > 0" and nb: "numbound0 e" by auto
{assume kdc: "k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
moreover
{assume "\<not> k dvd c"
from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have "?I (real x) (?s (Dvd i (CN 0 c e))) = (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Dvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
finally have ?case . }
ultimately show ?case by blast
next
case (10 i c e) from prems have cp: "c > 0" and nb: "numbound0 e" by auto
{assume kdc: "k dvd c"
from kpos have knz: "k\<noteq>0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) }
moreover
{assume "\<not> k dvd c"
from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
from prems have "?I (real x) (?s (NDvd i (CN 0 c e))) = (\<not> (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k))"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (NDvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
finally have ?case . }
ultimately show ?case by blast
qed (simp_all add: nth_pos2 bound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"] numbound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"])
lemma a\<rho>:
assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "real k > 0"
shows "Ifm (real (x*k)#bs) (a\<rho> p k) = Ifm (real x#bs) p" (is "?I (x*k) (?f p k) = ?I x p")
using lp bound0_I[where bs="bs" and b="real (x*k)" and b'="real x"] numbound0_I[where bs="bs" and b="real (x*k)" and b'="real x"]
proof(induct p rule: a\<rho>.induct)
case (3 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
{assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (4 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
{assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (5 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
{assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_less_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (6 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
{assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_le_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (7 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
{assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_less_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (8 c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
{assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_le_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (9 i c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
{assume "\<not> k dvd c"
hence "Ifm (real (x*k)#bs) (a\<rho> (Dvd i (CN 0 c e)) k) =
(real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k)"
using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"]
by (simp add: algebra_simps)
also have "\<dots> = (Ifm (real x#bs) (Dvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"])
finally have ?case . }
ultimately show ?case by blast
next
case (10 i c e)
from prems have cp: "c > 0" and nb: "numbound0 e" by auto
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
{assume "\<not> k dvd c"
hence "Ifm (real (x*k)#bs) (a\<rho> (NDvd i (CN 0 c e)) k) =
(\<not> (real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k))"
using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"]
by (simp add: algebra_simps)
also have "\<dots> = (Ifm (real x#bs) (NDvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"])
finally have ?case . }
ultimately show ?case by blast
qed (simp_all add: nth_pos2)
lemma a\<rho>_ex:
assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "k > 0"
shows "(\<exists> (x::int). real k rdvd real x \<and> Ifm (real x#bs) (a\<rho> p k)) =
(\<exists> (x::int). Ifm (real x#bs) p)" (is "(\<exists> x. ?D x \<and> ?P' x) = (\<exists> x. ?P x)")
proof-
have "(\<exists> x. ?D x \<and> ?P' x) = (\<exists> x. k dvd x \<and> ?P' x)" using int_rdvd_iff by simp
also have "\<dots> = (\<exists>x. ?P' (x*k))" using unity_coeff_ex[where P="?P'" and l="k", simplified]
by (simp add: algebra_simps)
also have "\<dots> = (\<exists> x. ?P x)" using a\<rho> iszlfm_gen[OF lp] kp by auto
finally show ?thesis .
qed
lemma \<sigma>\<rho>': assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "k > 0" and nb: "numbound0 t"
shows "Ifm (real x#bs) (\<sigma>\<rho> p (t,k)) = Ifm ((Inum (real x#bs) t)#bs) (a\<rho> p k)"
using lp
by(induct p rule: \<sigma>\<rho>.induct, simp_all add:
numbound0_I[OF nb, where bs="bs" and b="Inum (real x#bs) t" and b'="real x"]
numbound0_I[where bs="bs" and b="Inum (real x#bs) t" and b'="real x"]
bound0_I[where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] nth_pos2 cong: imp_cong)
lemma \<sigma>\<rho>_nb: assumes lp:"iszlfm p (a#bs)" and nb: "numbound0 t"
shows "bound0 (\<sigma>\<rho> p (t,k))"
using lp
by (induct p rule: iszlfm.induct, auto simp add: nb)
lemma \<rho>_l:
assumes lp: "iszlfm p (real (i::int)#bs)"
shows "\<forall> (b,k) \<in> set (\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real i#bs)"
using lp by (induct p rule: \<rho>.induct, auto simp add: isint_sub isint_neg)
lemma \<alpha>\<rho>_l:
assumes lp: "iszlfm p (real (i::int)#bs)"
shows "\<forall> (b,k) \<in> set (\<alpha>\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real i#bs)"
using lp isint_add [OF isint_c[where j="- 1"],where bs="real i#bs"]
by (induct p rule: \<alpha>\<rho>.induct, auto)
lemma zminusinf_\<rho>:
assumes lp: "iszlfm p (real (i::int)#bs)"
and nmi: "\<not> (Ifm (real i#bs) (minusinf p))" (is "\<not> (Ifm (real i#bs) (?M p))")
and ex: "Ifm (real i#bs) p" (is "?I i p")
shows "\<exists> (e,c) \<in> set (\<rho> p). real (c*i) > Inum (real i#bs) e" (is "\<exists> (e,c) \<in> ?R p. real (c*i) > ?N i e")
using lp nmi ex
by (induct p rule: minusinf.induct, auto)
lemma \<sigma>_And: "Ifm bs (\<sigma> (And p q) k t) = Ifm bs (And (\<sigma> p k t) (\<sigma> q k t))"
using \<sigma>_def by auto
lemma \<sigma>_Or: "Ifm bs (\<sigma> (Or p q) k t) = Ifm bs (Or (\<sigma> p k t) (\<sigma> q k t))"
using \<sigma>_def by auto
lemma \<rho>: assumes lp: "iszlfm p (real (i::int) #bs)"
and pi: "Ifm (real i#bs) p"
and d: "d\<delta> p d"
and dp: "d > 0"
and nob: "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> Inum (real i#bs) e + real j"
(is "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. _ \<noteq> ?N i e + _")
shows "Ifm (real(i - d)#bs) p"
using lp pi d nob
proof(induct p rule: iszlfm.induct)
case (3 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
and pi: "real (c*i) = - 1 - ?N i e + real (1::int)" and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> -1 - ?N i e + real j"
by simp+
from mult_strict_left_mono[OF dp cp] have one:"1 \<in> {1 .. c*d}" by auto
from nob[rule_format, where j="1", OF one] pi show ?case by simp
next
case (4 c e)
hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j"
by simp+
{assume "real (c*i) \<noteq> - ?N i e + real (c*d)"
with numbound0_I[OF nb, where bs="bs" and b="real i - real d" and b'="real i"]
have ?case by (simp add: algebra_simps)}
moreover
{assume pi: "real (c*i) = - ?N i e + real (c*d)"
from mult_strict_left_mono[OF dp cp] have d: "(c*d) \<in> {1 .. c*d}" by simp
from nob[rule_format, where j="c*d", OF d] pi have ?case by simp }
ultimately show ?case by blast
next
case (5 c e) hence cp: "c > 0" by simp
from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric]
real_of_int_mult]
show ?case using prems dp
by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"]
algebra_simps)
next
case (6 c e) hence cp: "c > 0" by simp
from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric]
real_of_int_mult]
show ?case using prems dp
by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"]
algebra_simps)
next
case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j"
and pi: "real (c*i) + ?N i e > 0" and cp': "real c >0"
by simp+
let ?fe = "floor (?N i e)"
from pi cp have th:"(real i +?N i e / real c)*real c > 0" by (simp add: algebra_simps)
from pi ei[simplified isint_iff] have "real (c*i + ?fe) > real (0::int)" by simp
hence pi': "c*i + ?fe > 0" by (simp only: real_of_int_less_iff[symmetric])
have "real (c*i) + ?N i e > real (c*d) \<or> real (c*i) + ?N i e \<le> real (c*d)" by auto
moreover
{assume "real (c*i) + ?N i e > real (c*d)" hence ?case
by (simp add: algebra_simps
numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])}
moreover
{assume H:"real (c*i) + ?N i e \<le> real (c*d)"
with ei[simplified isint_iff] have "real (c*i + ?fe) \<le> real (c*d)" by simp
hence pid: "c*i + ?fe \<le> c*d" by (simp only: real_of_int_le_iff)
with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + ?fe = j1" by auto
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - ?N i e + real j1"
by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] algebra_simps)
with nob have ?case by blast }
ultimately show ?case by blast
next
case (8 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - 1 - ?N i e + real j"
and pi: "real (c*i) + ?N i e \<ge> 0" and cp': "real c >0"
by simp+
let ?fe = "floor (?N i e)"
from pi cp have th:"(real i +?N i e / real c)*real c \<ge> 0" by (simp add: algebra_simps)
from pi ei[simplified isint_iff] have "real (c*i + ?fe) \<ge> real (0::int)" by simp
hence pi': "c*i + 1 + ?fe \<ge> 1" by (simp only: real_of_int_le_iff[symmetric])
have "real (c*i) + ?N i e \<ge> real (c*d) \<or> real (c*i) + ?N i e < real (c*d)" by auto
moreover
{assume "real (c*i) + ?N i e \<ge> real (c*d)" hence ?case
by (simp add: algebra_simps
numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])}
moreover
{assume H:"real (c*i) + ?N i e < real (c*d)"
with ei[simplified isint_iff] have "real (c*i + ?fe) < real (c*d)" by simp
hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: real_of_int_le_iff)
with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + 1+ ?fe = j1" by auto
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) + 1= - ?N i e + real j1"
by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] algebra_simps real_of_one)
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = (- ?N i e + real j1) - 1"
by (simp only: algebra_simps diff_def[symmetric])
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - 1 - ?N i e + real j1"
by (simp only: add_ac diff_def)
with nob have ?case by blast }
ultimately show ?case by blast
next
case (9 j c e) hence p: "real j rdvd real (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+
let ?e = "Inum (real i # bs) e"
from prems have "isint e (real i #bs)" by simp
hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
by (simp add: isint_iff)
from prems have id: "j dvd d" by simp
from ie[symmetric] have "?p i = (real j rdvd real (c*i+ floor ?e))" by simp
also have "\<dots> = (j dvd c*i + floor ?e)"
using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
also have "\<dots> = (j dvd c*i - c*d + floor ?e)"
using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
also have "\<dots> = (real j rdvd real (c*i - c*d + floor ?e))"
using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
ie by simp
also have "\<dots> = (real j rdvd real (c*(i - d)) + ?e)"
using ie by (simp add:algebra_simps)
finally show ?case
using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p
by (simp add: algebra_simps)
next
case (10 j c e) hence p: "\<not> (real j rdvd real (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+
let ?e = "Inum (real i # bs) e"
from prems have "isint e (real i #bs)" by simp
hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
by (simp add: isint_iff)
from prems have id: "j dvd d" by simp
from ie[symmetric] have "?p i = (\<not> (real j rdvd real (c*i+ floor ?e)))" by simp
also have "\<dots> = Not (j dvd c*i + floor ?e)"
using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
also have "\<dots> = Not (j dvd c*i - c*d + floor ?e)"
using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
also have "\<dots> = Not (real j rdvd real (c*i - c*d + floor ?e))"
using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
ie by simp
also have "\<dots> = Not (real j rdvd real (c*(i - d)) + ?e)"
using ie by (simp add:algebra_simps)
finally show ?case
using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p
by (simp add: algebra_simps)
qed(auto simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] nth_pos2)
lemma \<sigma>_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t"
shows "bound0 (\<sigma> p k t)"
using \<sigma>\<rho>_nb[OF lp nb] nb by (simp add: \<sigma>_def)
lemma \<rho>': assumes lp: "iszlfm p (a #bs)"
and d: "d\<delta> p d"
and dp: "d > 0"
shows "\<forall> x. \<not>(\<exists> (e,c) \<in> set(\<rho> p). \<exists>(j::int) \<in> {1 .. c*d}. Ifm (a #bs) (\<sigma> p c (Add e (C j)))) \<longrightarrow> Ifm (real x#bs) p \<longrightarrow> Ifm (real (x - d)#bs) p" (is "\<forall> x. ?b x \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
proof(clarify)
fix x
assume nob1:"?b x" and px: "?P x"
from iszlfm_gen[OF lp, rule_format, where y="real x"] have lp': "iszlfm p (real x#bs)".
have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real (c * x) \<noteq> Inum (real x # bs) e + real j"
proof(clarify)
fix e c j assume ecR: "(e,c) \<in> set (\<rho> p)" and jD: "j\<in> {1 .. c*d}"
and cx: "real (c*x) = Inum (real x#bs) e + real j"
let ?e = "Inum (real x#bs) e"
let ?fe = "floor ?e"
from \<rho>_l[OF lp'] ecR have ei:"isint e (real x#bs)" and cp:"c>0" and nb:"numbound0 e"
by auto
from numbound0_gen [OF nb ei, rule_format,where y="a"] have "isint e (a#bs)" .
from cx ei[simplified isint_iff] have "real (c*x) = real (?fe + j)" by simp
hence cx: "c*x = ?fe + j" by (simp only: real_of_int_inject)
hence cdej:"c dvd ?fe + j" by (simp add: dvd_def) (rule_tac x="x" in exI, simp)
hence "real c rdvd real (?fe + j)" by (simp only: int_rdvd_iff)
hence rcdej: "real c rdvd ?e + real j" by (simp add: ei[simplified isint_iff])
from cx have "(c*x) div c = (?fe + j) div c" by simp
with cp have "x = (?fe + j) div c" by simp
with px have th: "?P ((?fe + j) div c)" by auto
from cp have cp': "real c > 0" by simp
from cdej have cdej': "c dvd floor (Inum (real x#bs) (Add e (C j)))" by simp
from nb have nb': "numbound0 (Add e (C j))" by simp
have ji: "isint (C j) (real x#bs)" by (simp add: isint_def)
from isint_add[OF ei ji] have ei':"isint (Add e (C j)) (real x#bs)" .
from th \<sigma>\<rho>[where b'="real x", OF lp' cp' nb' ei' cdej',symmetric]
have "Ifm (real x#bs) (\<sigma>\<rho> p (Add e (C j), c))" by simp
with rcdej have th: "Ifm (real x#bs) (\<sigma> p c (Add e (C j)))" by (simp add: \<sigma>_def)
from th bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"],where bs="bs" and b="real x" and b'="a"]
have "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" by blast
with ecR jD nob1 show "False" by blast
qed
from \<rho>[OF lp' px d dp nob] show "?P (x -d )" .
qed
lemma rl_thm:
assumes lp: "iszlfm p (real (i::int)#bs)"
shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1 .. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
(is "(\<exists>(x::int). ?P x) = ((\<exists> j\<in> {1.. \<delta> p}. ?MP j)\<or>(\<exists> (e,c) \<in> ?R. \<exists> j\<in> _. ?SP c e j))"
is "?lhs = (?MD \<or> ?RD)" is "?lhs = ?rhs")
proof-
let ?d= "\<delta> p"
from \<delta>[OF lp] have d:"d\<delta> p ?d" and dp: "?d > 0" by auto
{ assume H:"?MD" hence th:"\<exists> (x::int). ?MP x" by blast
from H minusinf_ex[OF lp th] have ?thesis by blast}
moreover
{ fix e c j assume exR:"(e,c) \<in> ?R" and jD:"j\<in> {1 .. c*?d}" and spx:"?SP c e j"
from exR \<rho>_l[OF lp] have nb: "numbound0 e" and ei:"isint e (real i#bs)" and cp: "c > 0"
by auto
have "isint (C j) (real i#bs)" by (simp add: isint_iff)
with isint_add[OF numbound0_gen[OF nb ei,rule_format, where y="real i"]]
have eji:"isint (Add e (C j)) (real i#bs)" by simp
from nb have nb': "numbound0 (Add e (C j))" by simp
from spx bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"], where bs="bs" and b="a" and b'="real i"]
have spx': "Ifm (real i # bs) (\<sigma> p c (Add e (C j)))" by blast
from spx' have rcdej:"real c rdvd (Inum (real i#bs) (Add e (C j)))"
and sr:"Ifm (real i#bs) (\<sigma>\<rho> p (Add e (C j),c))" by (simp add: \<sigma>_def)+
from rcdej eji[simplified isint_iff]
have "real c rdvd real (floor (Inum (real i#bs) (Add e (C j))))" by simp
hence cdej:"c dvd floor (Inum (real i#bs) (Add e (C j)))" by (simp only: int_rdvd_iff)
from cp have cp': "real c > 0" by simp
from \<sigma>\<rho>[OF lp cp' nb' eji cdej] spx' have "?P (\<lfloor>Inum (real i # bs) (Add e (C j))\<rfloor> div c)"
by (simp add: \<sigma>_def)
hence ?lhs by blast
with exR jD spx have ?thesis by blast}
moreover
{ fix x assume px: "?P x" and nob: "\<not> ?RD"
from iszlfm_gen [OF lp,rule_format, where y="a"] have lp':"iszlfm p (a#bs)" .
from \<rho>'[OF lp' d dp, rule_format, OF nob] have th:"\<forall> x. ?P x \<longrightarrow> ?P (x - ?d)" by blast
from minusinf_inf[OF lp] obtain z where z:"\<forall> x<z. ?MP x = ?P x" by blast
have zp: "abs (x - z) + 1 \<ge> 0" by arith
from decr_lemma[OF dp,where x="x" and z="z"]
decr_mult_lemma[OF dp th zp, rule_format, OF px] z have th:"\<exists> x. ?MP x" by auto
with minusinf_bex[OF lp] px nob have ?thesis by blast}
ultimately show ?thesis by blast
qed
lemma mirror_\<alpha>\<rho>: assumes lp: "iszlfm p (a#bs)"
shows "(\<lambda> (t,k). (Inum (a#bs) t, k)) ` set (\<alpha>\<rho> p) = (\<lambda> (t,k). (Inum (a#bs) t,k)) ` set (\<rho> (mirror p))"
using lp
by (induct p rule: mirror.induct, simp_all add: split_def image_Un )
text {* The @{text "\<real>"} part*}
text{* Linearity for fm where Bound 0 ranges over @{text "\<real>"}*}
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))"
constdefs fp :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> int \<Rightarrow> fm"
"fp p n s j \<equiv> (if n > 0 then
(And p (And (Ge (CN 0 n (Sub s (Add (Floor s) (C j)))))
(Lt (CN 0 n (Sub s (Add (Floor s) (C (j+1))))))))
else
(And p (And (Le (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C j)))))
(Gt (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C (j + 1)))))))))"
(* splits the bounded from the unbounded part*)
consts rsplit0 :: "num \<Rightarrow> (fm \<times> int \<times> num) list"
recdef rsplit0 "measure num_size"
"rsplit0 (Bound 0) = [(T,1,C 0)]"
"rsplit0 (Add a b) = (let acs = rsplit0 a ; bcs = rsplit0 b
in map (\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s)) [(a,b). a\<leftarrow>acs,b\<leftarrow>bcs])"
"rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
"rsplit0 (Neg a) = map (\<lambda> (p,n,s). (p,-n,Neg s)) (rsplit0 a)"
"rsplit0 (Floor a) = foldl (op @) [] (map
(\<lambda> (p,n,s). if n=0 then [(p,0,Floor s)]
else (map (\<lambda> j. (fp p n s j, 0, Add (Floor s) (C j))) (if n > 0 then iupt (0,n) else iupt(n,0))))
(rsplit0 a))"
"rsplit0 (CN 0 c a) = map (\<lambda> (p,n,s). (p,n+c,s)) (rsplit0 a)"
"rsplit0 (CN m c a) = map (\<lambda> (p,n,s). (p,n,CN m c s)) (rsplit0 a)"
"rsplit0 (CF c t s) = rsplit0 (Add (Mul c (Floor t)) s)"
"rsplit0 (Mul c a) = map (\<lambda> (p,n,s). (p,c*n,Mul c s)) (rsplit0 a)"
"rsplit0 t = [(T,0,t)]"
lemma not_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm (not p)"
by (induct p rule: isrlfm.induct, auto)
lemma conj_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
using conj_def by (cases p, auto)
lemma disj_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
using disj_def by (cases p, auto)
lemma rsplit0_cs:
shows "\<forall> (p,n,s) \<in> set (rsplit0 t).
(Ifm (x#bs) p \<longrightarrow> (Inum (x#bs) t = Inum (x#bs) (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p"
(is "\<forall> (p,n,s) \<in> ?SS t. (?I p \<longrightarrow> ?N t = ?N (CN 0 n s)) \<and> _ \<and> _ ")
proof(induct t rule: rsplit0.induct)
case (5 a)
let ?p = "\<lambda> (p,n,s) j. fp p n s j"
let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),Add (Floor s) (C j)))"
let ?J = "\<lambda> n. if n>0 then iupt (0,n) else iupt (n,0)"
let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))"
have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith
have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) =
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto
have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}.
?ff (p,n,s) = map (?f(p,n,s)) (iupt(0,n))" by auto
hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) =
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s).
set (map (?f(p,n,s)) (iupt(0,n)))))"
proof-
fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c"
thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))"
by (auto simp add: split_def)
qed
have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(n,0))"
by auto
hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) =
(UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n<0} (\<lambda>(p,n,s). set (map (?f(p,n,s)) (iupt(n,0)))))"
proof-
fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c"
thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))"
by (auto simp add: split_def)
qed
have "?SS (Floor a) = UNION (?SS a) (\<lambda>x. set (?ff x))"
by (auto simp add: foldl_conv_concat)
also have "\<dots> = UNION (?SS a) (\<lambda> (p,n,s). set (?ff (p,n,s)))" by auto
also have "\<dots> =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))"
using int_cases[rule_format] by blast
also have "\<dots> =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un
(UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n>0} (\<lambda>(p,n,s). set(map(?f(p,n,s)) (iupt(0,n))))) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).
set (map (?f(p,n,s)) (iupt(n,0))))))" by (simp only: U1 U2 U3)
also have "\<dots> =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))"
by (simp only: set_map iupt_set set.simps)
also have "\<dots> =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast
finally
have FS: "?SS (Floor a) =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast
show ?case
proof(simp only: FS, clarsimp simp del: Ifm.simps Inum.simps, -)
fix p n s
let ?ths = "(?I p \<longrightarrow> (?N (Floor a) = ?N (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p"
assume "(\<exists>ba. (p, 0, ba) \<in> set (rsplit0 a) \<and> n = 0 \<and> s = Floor ba) \<or>
(\<exists>ab ac ba.
(ab, ac, ba) \<in> set (rsplit0 a) \<and>
0 < ac \<and>
(\<exists>j. p = fp ab ac ba j \<and>
n = 0 \<and> s = Add (Floor ba) (C j) \<and> 0 \<le> j \<and> j \<le> ac)) \<or>
(\<exists>ab ac ba.
(ab, ac, ba) \<in> set (rsplit0 a) \<and>
ac < 0 \<and>
(\<exists>j. p = fp ab ac ba j \<and>
n = 0 \<and> s = Add (Floor ba) (C j) \<and> ac \<le> j \<and> j \<le> 0))"
moreover
{fix s'
assume "(p, 0, s') \<in> ?SS a" and "n = 0" and "s = Floor s'"
hence ?ths using prems by auto}
moreover
{ fix p' n' s' j
assume pns: "(p', n', s') \<in> ?SS a"
and np: "0 < n'"
and p_def: "p = ?p (p',n',s') j"
and n0: "n = 0"
and s_def: "s = (Add (Floor s') (C j))"
and jp: "0 \<le> j" and jn: "j \<le> n'"
from prems pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow>
Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and>
numbound0 s' \<and> isrlfm p'" by blast
hence nb: "numbound0 s'" by simp
from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numsub_nb)
let ?nxs = "CN 0 n' s'"
let ?l = "floor (?N s') + j"
from H
have "?I (?p (p',n',s') j) \<longrightarrow>
(((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))"
by (simp add: fp_def np algebra_simps numsub numadd numfloor)
also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
moreover
have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp
ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
by blast
with s_def n0 p_def nb nf have ?ths by auto}
moreover
{fix p' n' s' j
assume pns: "(p', n', s') \<in> ?SS a"
and np: "n' < 0"
and p_def: "p = ?p (p',n',s') j"
and n0: "n = 0"
and s_def: "s = (Add (Floor s') (C j))"
and jp: "n' \<le> j" and jn: "j \<le> 0"
from prems pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow>
Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and>
numbound0 s' \<and> isrlfm p'" by blast
hence nb: "numbound0 s'" by simp
from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numneg_nb)
let ?nxs = "CN 0 n' s'"
let ?l = "floor (?N s') + j"
from H
have "?I (?p (p',n',s') j) \<longrightarrow>
(((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))"
by (simp add: np fp_def algebra_simps numneg numfloor numadd numsub)
also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
moreover
have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp
ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
by blast
with s_def n0 p_def nb nf have ?ths by auto}
ultimately show ?ths by auto
qed
next
case (3 a b) then show ?case
apply auto
apply (erule_tac x = "(aa, aaa, ba)" in ballE) apply simp_all
apply (erule_tac x = "(ab, ac, baa)" in ballE) apply simp_all
done
qed (auto simp add: Let_def split_def algebra_simps conj_rl)
lemma real_in_int_intervals:
assumes xb: "real m \<le> x \<and> x < real ((n::int) + 1)"
shows "\<exists> j\<in> {m.. n}. real j \<le> x \<and> x < real (j+1)" (is "\<exists> j\<in> ?N. ?P j")
by (rule bexI[where P="?P" and x="floor x" and A="?N"])
(auto simp add: floor_less_eq[where x="x" and a="n+1", simplified] xb[simplified] floor_mono[where x="real m" and y="x", OF conjunct1[OF xb], simplified floor_real_of_int[where n="m"]])
lemma rsplit0_complete:
assumes xp:"0 \<le> x" and x1:"x < 1"
shows "\<exists> (p,n,s) \<in> set (rsplit0 t). Ifm (x#bs) p" (is "\<exists> (p,n,s) \<in> ?SS t. ?I p")
proof(induct t rule: rsplit0.induct)
case (2 a b)
from prems have "\<exists> (pa,na,sa) \<in> ?SS a. ?I pa" by auto
then obtain "pa" "na" "sa" where pa: "(pa,na,sa)\<in> ?SS a \<and> ?I pa" by blast
from prems have "\<exists> (pb,nb,sb) \<in> ?SS b. ?I pb" by auto
then obtain "pb" "nb" "sb" where pb: "(pb,nb,sb)\<in> ?SS b \<and> ?I pb" by blast
from pa pb have th: "((pa,na,sa),(pb,nb,sb)) \<in> set[(x,y). x\<leftarrow>rsplit0 a, y\<leftarrow>rsplit0 b]"
by (auto)
let ?f="(\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s))"
from imageI[OF th, where f="?f"] have "?f ((pa,na,sa),(pb,nb,sb)) \<in> ?SS (Add a b)"
by (simp add: Let_def)
hence "(And pa pb, na +nb, Add sa sb) \<in> ?SS (Add a b)" by simp
moreover from pa pb have "?I (And pa pb)" by simp
ultimately show ?case by blast
next
case (5 a)
let ?p = "\<lambda> (p,n,s) j. fp p n s j"
let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),(Add (Floor s) (C j))))"
let ?J = "\<lambda> n. if n>0 then iupt (0,n) else iupt (n,0)"
let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))"
have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith
have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto
have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(0,n))"
by auto
hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(0,n)))))"
proof-
fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c"
thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))"
by (auto simp add: split_def)
qed
have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(n,0))"
by auto
hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(n,0)))))"
proof-
fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c"
thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))"
by (auto simp add: split_def)
qed
have "?SS (Floor a) = UNION (?SS a) (\<lambda>x. set (?ff x))" by (auto simp add: foldl_conv_concat)
also have "\<dots> = UNION (?SS a) (\<lambda> (p,n,s). set (?ff (p,n,s)))" by auto
also have "\<dots> =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))"
using int_cases[rule_format] by blast
also have "\<dots> =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(0,n))))) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(n,0))))))" by (simp only: U1 U2 U3)
also have "\<dots> =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))"
by (simp only: set_map iupt_set set.simps)
also have "\<dots> =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast
finally
have FS: "?SS (Floor a) =
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast
from prems have "\<exists> (p,n,s) \<in> ?SS a. ?I p" by auto
then obtain "p" "n" "s" where pns: "(p,n,s) \<in> ?SS a \<and> ?I p" by blast
let ?N = "\<lambda> t. Inum (x#bs) t"
from rsplit0_cs[rule_format] pns have ans:"(?N a = ?N (CN 0 n s)) \<and> numbound0 s \<and> isrlfm p"
by auto
have "n=0 \<or> n >0 \<or> n <0" by arith
moreover {assume "n=0" hence ?case using pns by (simp only: FS) auto }
moreover
{
assume np: "n > 0"
from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) \<le> ?N s" by simp
also from mult_left_mono[OF xp] np have "?N s \<le> real n * x + ?N s" by simp
finally have "?N (Floor s) \<le> real n * x + ?N s" .
moreover
{from x1 np have "real n *x + ?N s < real n + ?N s" by simp
also from real_of_int_floor_add_one_gt[where r="?N s"]
have "\<dots> < real n + ?N (Floor s) + 1" by simp
finally have "real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp}
ultimately have "?N (Floor s) \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp
hence th: "0 \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (n+1)" by simp
from real_in_int_intervals th have "\<exists> j\<in> {0 .. n}. real j \<le> real n *x + ?N s - ?N (Floor s)\<and> real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp
hence "\<exists> j\<in> {0 .. n}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0"
by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"])
hence "\<exists> j\<in> {0.. n}. ?I (?p (p,n,s) j)"
using pns by (simp add: fp_def np algebra_simps numsub numadd)
then obtain "j" where j_def: "j\<in> {0 .. n} \<and> ?I (?p (p,n,s) j)" by blast
hence "\<exists>x \<in> {?p (p,n,s) j |j. 0\<le> j \<and> j \<le> n }. ?I x" by auto
hence ?case using pns
by (simp only: FS,simp add: bex_Un)
(rule disjI2, rule disjI1,rule exI [where x="p"],
rule exI [where x="n"],rule exI [where x="s"],simp_all add: np)
}
moreover
{ assume nn: "n < 0" hence np: "-n >0" by simp
from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) + 1 > ?N s" by simp
moreover from mult_left_mono_neg[OF xp] nn have "?N s \<ge> real n * x + ?N s" by simp
ultimately have "?N (Floor s) + 1 > real n * x + ?N s" by arith
moreover
{from x1 nn have "real n *x + ?N s \<ge> real n + ?N s" by simp
moreover from real_of_int_floor_le[where r="?N s"] have "real n + ?N s \<ge> real n + ?N (Floor s)" by simp
ultimately have "real n *x + ?N s \<ge> ?N (Floor s) + real n"
by (simp only: algebra_simps)}
ultimately have "?N (Floor s) + real n \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (1::int)" by simp
hence th: "real n \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (1::int)" by simp
have th1: "\<forall> (a::real). (- a > 0) = (a < 0)" by auto
have th2: "\<forall> (a::real). (0 \<ge> - a) = (a \<ge> 0)" by auto
from real_in_int_intervals th have "\<exists> j\<in> {n .. 0}. real j \<le> real n *x + ?N s - ?N (Floor s)\<and> real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp
hence "\<exists> j\<in> {n .. 0}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0"
by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"])
hence "\<exists> j\<in> {n .. 0}. 0 \<ge> - (real n *x + ?N s - ?N (Floor s) - real j) \<and> - (real n *x + ?N s - ?N (Floor s) - real (j+1)) > 0" by (simp only: th1[rule_format] th2[rule_format])
hence "\<exists> j\<in> {n.. 0}. ?I (?p (p,n,s) j)"
using pns by (simp add: fp_def nn diff_def add_ac mult_ac numfloor numadd numneg
del: diff_less_0_iff_less diff_le_0_iff_le)
then obtain "j" where j_def: "j\<in> {n .. 0} \<and> ?I (?p (p,n,s) j)" by blast
hence "\<exists>x \<in> {?p (p,n,s) j |j. n\<le> j \<and> j \<le> 0 }. ?I x" by auto
hence ?case using pns
by (simp only: FS,simp add: bex_Un)
(rule disjI2, rule disjI2,rule exI [where x="p"],
rule exI [where x="n"],rule exI [where x="s"],simp_all add: nn)
}
ultimately show ?case by blast
qed (auto simp add: Let_def split_def)
(* Linearize a formula where Bound 0 ranges over [0,1) *)
constdefs rsplit :: "(int \<Rightarrow> num \<Rightarrow> fm) \<Rightarrow> num \<Rightarrow> fm"
"rsplit f a \<equiv> foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) (rsplit0 a)) F"
lemma foldr_disj_map: "Ifm bs (foldr disj (map f xs) F) = (\<exists> x \<in> set xs. Ifm bs (f x))"
by(induct xs, simp_all)
lemma foldr_conj_map: "Ifm bs (foldr conj (map f xs) T) = (\<forall> x \<in> set xs. Ifm bs (f x))"
by(induct xs, simp_all)
lemma foldr_disj_map_rlfm:
assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)"
and \<phi>: "\<forall> (\<phi>,n,s) \<in> set xs. numbound0 s \<and> isrlfm \<phi>"
shows "isrlfm (foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) xs) F)"
using lf \<phi> by (induct xs, auto)
lemma rsplit_ex: "Ifm bs (rsplit f a) = (\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). Ifm bs (conj \<phi> (f n s)))"
using foldr_disj_map[where xs="rsplit0 a"] rsplit_def by (simp add: split_def)
lemma rsplit_l: assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)"
shows "isrlfm (rsplit f a)"
proof-
from rsplit0_cs[where t="a"] have th: "\<forall> (\<phi>,n,s) \<in> set (rsplit0 a). numbound0 s \<and> isrlfm \<phi>" by blast
from foldr_disj_map_rlfm[OF lf th] rsplit_def show ?thesis by simp
qed
lemma rsplit:
assumes xp: "x \<ge> 0" and x1: "x < 1"
and f: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> (Ifm (x#bs) (f n s) = Ifm (x#bs) (g a))"
shows "Ifm (x#bs) (rsplit f a) = Ifm (x#bs) (g a)"
proof(auto)
let ?I = "\<lambda>x p. Ifm (x#bs) p"
let ?N = "\<lambda> x t. Inum (x#bs) t"
assume "?I x (rsplit f a)"
hence "\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). ?I x (And \<phi> (f n s))" using rsplit_ex by simp
then obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and "?I x (And \<phi> (f n s))" by blast
hence \<phi>: "?I x \<phi>" and fns: "?I x (f n s)" by auto
from rsplit0_cs[where t="a" and bs="bs" and x="x", rule_format, OF fnsS] \<phi>
have th: "(?N x a = ?N x (CN 0 n s)) \<and> numbound0 s" by auto
from f[rule_format, OF th] fns show "?I x (g a)" by simp
next
let ?I = "\<lambda>x p. Ifm (x#bs) p"
let ?N = "\<lambda> x t. Inum (x#bs) t"
assume ga: "?I x (g a)"
from rsplit0_complete[OF xp x1, where bs="bs" and t="a"]
obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and fx: "?I x \<phi>" by blast
from rsplit0_cs[where t="a" and x="x" and bs="bs"] fnsS fx
have ans: "?N x a = ?N x (CN 0 n s)" and nb: "numbound0 s" by auto
with ga f have "?I x (f n s)" by auto
with rsplit_ex fnsS fx show "?I x (rsplit f a)" by auto
qed
definition lt :: "int \<Rightarrow> num \<Rightarrow> fm" where
lt_def: "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_def: "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_def: "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_def: "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_def: "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_def: "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_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (lt n s) = Ifm (x#bs) (Lt a)"
(is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _\<longrightarrow> ?I (lt n s) = ?I (Lt a)")
proof(clarify)
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (lt n s) = ?I (Lt a)" using H by (cases "n=0", (simp add: lt_def))
(cases "n > 0", simp_all add: lt_def algebra_simps myless[rule_format, where b="0"])
qed
lemma lt_l: "isrlfm (rsplit lt a)"
by (rule rsplit_l[where f="lt" and a="a"], auto simp add: lt_def,
case_tac s, simp_all, case_tac "nat", simp_all)
lemma le_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (le n s) = Ifm (x#bs) (Le a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (le n s) = ?I (Le a)")
proof(clarify)
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (le n s) = ?I (Le a)" using H by (cases "n=0", (simp add: le_def))
(cases "n > 0", simp_all add: le_def algebra_simps myl[rule_format, where b="0"])
qed
lemma le_l: "isrlfm (rsplit le a)"
by (rule rsplit_l[where f="le" and a="a"], auto simp add: le_def)
(case_tac s, simp_all, case_tac "nat",simp_all)
lemma gt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (gt n s) = Ifm (x#bs) (Gt a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (gt n s) = ?I (Gt a)")
proof(clarify)
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (gt n s) = ?I (Gt a)" using H by (cases "n=0", (simp add: gt_def))
(cases "n > 0", simp_all add: gt_def algebra_simps myless[rule_format, where b="0"])
qed
lemma gt_l: "isrlfm (rsplit gt a)"
by (rule rsplit_l[where f="gt" and a="a"], auto simp add: gt_def)
(case_tac s, simp_all, case_tac "nat", simp_all)
lemma ge_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (ge n s) = Ifm (x#bs) (Ge a)" (is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (ge n s) = ?I (Ge a)")
proof(clarify)
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (ge n s) = ?I (Ge a)" using H by (cases "n=0", (simp add: ge_def))
(cases "n > 0", simp_all add: ge_def algebra_simps myl[rule_format, where b="0"])
qed
lemma ge_l: "isrlfm (rsplit ge a)"
by (rule rsplit_l[where f="ge" and a="a"], auto simp add: ge_def)
(case_tac s, simp_all, case_tac "nat", simp_all)
lemma eq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (eq n s) = Ifm (x#bs) (Eq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (eq n s) = ?I (Eq a)")
proof(clarify)
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (eq n s) = ?I (Eq a)" using H by (auto simp add: eq_def algebra_simps)
qed
lemma eq_l: "isrlfm (rsplit eq a)"
by (rule rsplit_l[where f="eq" and a="a"], auto simp add: eq_def)
(case_tac s, simp_all, case_tac"nat", simp_all)
lemma neq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (neq n s) = Ifm (x#bs) (NEq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (neq n s) = ?I (NEq a)")
proof(clarify)
fix a n s bs
assume H: "?N a = ?N (CN 0 n s)"
show "?I (neq n s) = ?I (NEq a)" using H by (auto simp add: neq_def algebra_simps)
qed
lemma neq_l: "isrlfm (rsplit neq a)"
by (rule rsplit_l[where f="neq" and a="a"], auto simp add: neq_def)
(case_tac s, simp_all, case_tac"nat", simp_all)
lemma small_le:
assumes u0:"0 \<le> u" and u1: "u < 1"
shows "(-u \<le> real (n::int)) = (0 \<le> n)"
using u0 u1 by auto
lemma small_lt:
assumes u0:"0 \<le> u" and u1: "u < 1"
shows "(real (n::int) < real (m::int) - u) = (n < m)"
using u0 u1 by auto
lemma rdvd01_cs:
assumes up: "u \<ge> 0" and u1: "u<1" and np: "real n > 0"
shows "(real (i::int) rdvd real (n::int) * u - s) = (\<exists> j\<in> {0 .. n - 1}. real n * u = s - real (floor s) + real j \<and> real i rdvd real (j - floor s))" (is "?lhs = ?rhs")
proof-
let ?ss = "s - real (floor s)"
from real_of_int_floor_add_one_gt[where r="s", simplified myless[rule_format,where a="s"]]
real_of_int_floor_le[where r="s"] have ss0:"?ss \<ge> 0" and ss1:"?ss < 1"
by (auto simp add: myl[rule_format, where b="s", symmetric] myless[rule_format, where a="?ss"])
from np have n0: "real n \<ge> 0" by simp
from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np]
have nu0:"real n * u - s \<ge> -s" and nun:"real n * u -s < real n - s" by auto
from int_rdvd_real[where i="i" and x="real (n::int) * u - s"]
have "real i rdvd real n * u - s =
(i dvd floor (real n * u -s) \<and> (real (floor (real n * u - s)) = real n * u - s ))"
(is "_ = (?DE)" is "_ = (?D \<and> ?E)") by simp
also have "\<dots> = (?DE \<and> real(floor (real n * u - s) + floor s)\<ge> -?ss
\<and> real(floor (real n * u - s) + floor s)< real n - ?ss)" (is "_=(?DE \<and>real ?a \<ge> _ \<and> real ?a < _)")
using nu0 nun by auto
also have "\<dots> = (?DE \<and> ?a \<ge> 0 \<and> ?a < n)" by(simp only: small_le[OF ss0 ss1] small_lt[OF ss0 ss1])
also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. ?a = j))" by simp
also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. real (\<lfloor>real n * u - s\<rfloor>) = real j - real \<lfloor>s\<rfloor> ))"
by (simp only: algebra_simps real_of_int_diff[symmetric] real_of_int_inject del: real_of_int_diff)
also have "\<dots> = ((\<exists> j\<in> {0 .. (n - 1)}. real n * u - s = real j - real \<lfloor>s\<rfloor> \<and> real i rdvd real n * u - s))" using int_rdvd_iff[where i="i" and t="\<lfloor>real n * u - s\<rfloor>"]
by (auto cong: conj_cong)
also have "\<dots> = ?rhs" by(simp cong: conj_cong) (simp add: algebra_simps )
finally show ?thesis .
qed
definition
DVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm"
where
DVDJ_def: "DVDJ i n s = (foldr disj (map (\<lambda> j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))) (iupt(0,n - 1))) F)"
definition
NDVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm"
where
NDVDJ_def: "NDVDJ i n s = (foldr conj (map (\<lambda> j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s))))) (iupt(0,n - 1))) T)"
lemma DVDJ_DVD:
assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real n > 0"
shows "Ifm (x#bs) (DVDJ i n s) = Ifm (x#bs) (Dvd i (CN 0 n s))"
proof-
let ?f = "\<lambda> j. conj (Eq(CN 0 n (Add s (Sub(Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))"
let ?s= "Inum (x#bs) s"
from foldr_disj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"]
have "Ifm (x#bs) (DVDJ i n s) = (\<exists> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))"
by (simp add: iupt_set np DVDJ_def del: iupt.simps)
also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s)))" by (simp add: algebra_simps diff_def[symmetric])
also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"]
have "\<dots> = (real i rdvd real n * x - (-?s))" by simp
finally show ?thesis by simp
qed
lemma NDVDJ_NDVD:
assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real n > 0"
shows "Ifm (x#bs) (NDVDJ i n s) = Ifm (x#bs) (NDvd i (CN 0 n s))"
proof-
let ?f = "\<lambda> j. disj(NEq(CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor(Neg s))))"
let ?s= "Inum (x#bs) s"
from foldr_conj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"]
have "Ifm (x#bs) (NDVDJ i n s) = (\<forall> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))"
by (simp add: iupt_set np NDVDJ_def del: iupt.simps)
also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s))))" by (simp add: algebra_simps diff_def[symmetric])
also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"]
have "\<dots> = (\<not> (real i rdvd real n * x - (-?s)))" by simp
finally show ?thesis by simp
qed
lemma foldr_disj_map_rlfm2:
assumes lf: "\<forall> n . isrlfm (f n)"
shows "isrlfm (foldr disj (map f xs) F)"
using lf by (induct xs, auto)
lemma foldr_And_map_rlfm2:
assumes lf: "\<forall> n . isrlfm (f n)"
shows "isrlfm (foldr conj (map f xs) T)"
using lf by (induct xs, auto)
lemma DVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s"
shows "isrlfm (DVDJ i n s)"
proof-
let ?f="\<lambda>j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j)))))
(Dvd i (Sub (C j) (Floor (Neg s))))"
have th: "\<forall> j. isrlfm (?f j)" using nb np by auto
from DVDJ_def foldr_disj_map_rlfm2[OF th] show ?thesis by simp
qed
lemma NDVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s"
shows "isrlfm (NDVDJ i n s)"
proof-
let ?f="\<lambda>j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j)))))
(NDvd i (Sub (C j) (Floor (Neg s))))"
have th: "\<forall> j. isrlfm (?f j)" using nb np by auto
from NDVDJ_def foldr_And_map_rlfm2[OF th] show ?thesis by auto
qed
definition DVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where
DVD_def: "DVD i c t =
(if i=0 then eq c t else
if c = 0 then (Dvd i t) else if c >0 then DVDJ (abs i) c t else DVDJ (abs i) (-c) (Neg t))"
definition NDVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where
"NDVD i c t =
(if i=0 then neq c t else
if c = 0 then (NDvd i t) else if c >0 then NDVDJ (abs i) c t else NDVDJ (abs i) (-c) (Neg t))"
lemma DVD_mono:
assumes xp: "0\<le> x" and x1: "x < 1"
shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (DVD i n s) = Ifm (x#bs) (Dvd i a)"
(is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (DVD i n s) = ?I (Dvd i a)")
proof(clarify)
fix a n s
assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s"
let ?th = "?I (DVD i n s) = ?I (Dvd i a)"
have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith
moreover {assume iz: "i=0" hence ?th using eq_mono[rule_format, OF conjI[OF H nb]]
by (simp add: DVD_def rdvd_left_0_eq)}
moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H DVD_def) }
moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th
by (simp add: DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1
rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) }
moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th by (simp add:DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1)}
ultimately show ?th by blast
qed
lemma NDVD_mono: assumes xp: "0\<le> x" and x1: "x < 1"
shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (NDVD i n s) = Ifm (x#bs) (NDvd i a)"
(is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (NDVD i n s) = ?I (NDvd i a)")
proof(clarify)
fix a n s
assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s"
let ?th = "?I (NDVD i n s) = ?I (NDvd i a)"
have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith
moreover {assume iz: "i=0" hence ?th using neq_mono[rule_format, OF conjI[OF H nb]]
by (simp add: NDVD_def rdvd_left_0_eq)}
moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H NDVD_def) }
moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th
by (simp add: NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1
rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) }
moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th
by (simp add:NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1)}
ultimately show ?th by blast
qed
lemma DVD_l: "isrlfm (rsplit (DVD i) a)"
by (rule rsplit_l[where f="DVD i" and a="a"], auto simp add: DVD_def eq_def DVDJ_l)
(case_tac s, simp_all, case_tac "nat", simp_all)
lemma NDVD_l: "isrlfm (rsplit (NDVD i) a)"
by (rule rsplit_l[where f="NDVD i" and a="a"], auto simp add: NDVD_def neq_def NDVDJ_l)
(case_tac s, simp_all, case_tac "nat", simp_all)
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) = rsplit lt a"
"rlfm (Le a) = rsplit le a"
"rlfm (Gt a) = rsplit gt a"
"rlfm (Ge a) = rsplit ge a"
"rlfm (Eq a) = rsplit eq a"
"rlfm (NEq a) = rsplit neq a"
"rlfm (Dvd i a) = rsplit (\<lambda> t. DVD i t) a"
"rlfm (NDvd i a) = rsplit (\<lambda> t. NDVD i t) 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)) = simpfm (rlfm (Ge a))"
"rlfm (NOT (Le a)) = simpfm (rlfm (Gt a))"
"rlfm (NOT (Gt a)) = simpfm (rlfm (Le a))"
"rlfm (NOT (Ge a)) = simpfm (rlfm (Lt a))"
"rlfm (NOT (Eq a)) = simpfm (rlfm (NEq a))"
"rlfm (NOT (NEq a)) = simpfm (rlfm (Eq a))"
"rlfm (NOT (Dvd i a)) = simpfm (rlfm (NDvd i a))"
"rlfm (NOT (NDvd i a)) = simpfm (rlfm (Dvd i a))"
"rlfm p = p" (hints simp add: fmsize_pos)
lemma bound0at_l : "\<lbrakk>isatom p ; bound0 p\<rbrakk> \<Longrightarrow> isrlfm p"
by (induct p rule: isrlfm.induct, auto)
lemma zgcd_le1: assumes ip: "0 < i" shows "zgcd i j \<le> i"
proof-
from zgcd_zdvd1 have th: "zgcd i j dvd i" by blast
from zdvd_imp_le[OF th ip] show ?thesis .
qed
lemma simpfm_rl: "isrlfm p \<Longrightarrow> isrlfm (simpfm p)"
proof (induct p)
case (Lt a)
hence "bound0 (Lt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
by (cases a,simp_all, case_tac "nat", simp_all)
moreover
{assume "bound0 (Lt a)" hence bn:"bound0 (simpfm (Lt a))"
using simpfm_bound0 by blast
have "isatom (simpfm (Lt a))" by (cases "simpnum a", auto simp add: Let_def)
with bn bound0at_l have ?case by blast}
moreover
{fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
{
assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
with numgcd_pos[where t="CN 0 c (simpnum e)"]
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
by (simp add: numgcd_def zgcd_le1)
from prems have th': "c\<noteq>0" by auto
from prems have cp: "c \<ge> 0" by simp
from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
}
with prems have ?case
by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
ultimately show ?case by blast
next
case (Le a)
hence "bound0 (Le a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
by (cases a,simp_all, case_tac "nat", simp_all)
moreover
{assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))"
using simpfm_bound0 by blast
have "isatom (simpfm (Le a))" by (cases "simpnum a", auto simp add: Let_def)
with bn bound0at_l have ?case by blast}
moreover
{fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
{
assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
with numgcd_pos[where t="CN 0 c (simpnum e)"]
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
by (simp add: numgcd_def zgcd_le1)
from prems have th': "c\<noteq>0" by auto
from prems have cp: "c \<ge> 0" by simp
from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
}
with prems have ?case
by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
ultimately show ?case by blast
next
case (Gt a)
hence "bound0 (Gt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
by (cases a,simp_all, case_tac "nat", simp_all)
moreover
{assume "bound0 (Gt a)" hence bn:"bound0 (simpfm (Gt a))"
using simpfm_bound0 by blast
have "isatom (simpfm (Gt a))" by (cases "simpnum a", auto simp add: Let_def)
with bn bound0at_l have ?case by blast}
moreover
{fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
{
assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
with numgcd_pos[where t="CN 0 c (simpnum e)"]
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
by (simp add: numgcd_def zgcd_le1)
from prems have th': "c\<noteq>0" by auto
from prems have cp: "c \<ge> 0" by simp
from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
}
with prems have ?case
by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
ultimately show ?case by blast
next
case (Ge a)
hence "bound0 (Ge a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
by (cases a,simp_all, case_tac "nat", simp_all)
moreover
{assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))"
using simpfm_bound0 by blast
have "isatom (simpfm (Ge a))" by (cases "simpnum a", auto simp add: Let_def)
with bn bound0at_l have ?case by blast}
moreover
{fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
{
assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
with numgcd_pos[where t="CN 0 c (simpnum e)"]
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
by (simp add: numgcd_def zgcd_le1)
from prems have th': "c\<noteq>0" by auto
from prems have cp: "c \<ge> 0" by simp
from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
}
with prems have ?case
by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
ultimately show ?case by blast
next
case (Eq a)
hence "bound0 (Eq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
by (cases a,simp_all, case_tac "nat", simp_all)
moreover
{assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))"
using simpfm_bound0 by blast
have "isatom (simpfm (Eq a))" by (cases "simpnum a", auto simp add: Let_def)
with bn bound0at_l have ?case by blast}
moreover
{fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
{
assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
with numgcd_pos[where t="CN 0 c (simpnum e)"]
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
by (simp add: numgcd_def zgcd_le1)
from prems have th': "c\<noteq>0" by auto
from prems have cp: "c \<ge> 0" by simp
from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
}
with prems have ?case
by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
ultimately show ?case by blast
next
case (NEq a)
hence "bound0 (NEq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
by (cases a,simp_all, case_tac "nat", simp_all)
moreover
{assume "bound0 (NEq a)" hence bn:"bound0 (simpfm (NEq a))"
using simpfm_bound0 by blast
have "isatom (simpfm (NEq a))" by (cases "simpnum a", auto simp add: Let_def)
with bn bound0at_l have ?case by blast}
moreover
{fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e"
{
assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
with numgcd_pos[where t="CN 0 c (simpnum e)"]
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
by (simp add: numgcd_def zgcd_le1)
from prems have th': "c\<noteq>0" by auto
from prems have cp: "c \<ge> 0" by simp
from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']]
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp
}
with prems have ?case
by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)}
ultimately show ?case by blast
next
case (Dvd i a) hence "bound0 (Dvd i a)" by auto hence bn:"bound0 (simpfm (Dvd i a))"
using simpfm_bound0 by blast
have "isatom (simpfm (Dvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def)
with bn bound0at_l show ?case by blast
next
case (NDvd i a) hence "bound0 (NDvd i a)" by auto hence bn:"bound0 (simpfm (NDvd i a))"
using simpfm_bound0 by blast
have "isatom (simpfm (NDvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def)
with bn bound0at_l show ?case by blast
qed(auto simp add: conj_def imp_def disj_def iff_def Let_def simpfm_bound0 numadd_nb numneg_nb)
lemma rlfm_I:
assumes qfp: "qfree p"
and xp: "0 \<le> x" and x1: "x < 1"
shows "(Ifm (x#bs) (rlfm p) = Ifm (x# bs) p) \<and> isrlfm (rlfm p)"
using qfp
by (induct p rule: rlfm.induct)
(auto simp add: rsplit[OF xp x1 lt_mono] lt_l rsplit[OF xp x1 le_mono] le_l rsplit[OF xp x1 gt_mono] gt_l
rsplit[OF xp x1 ge_mono] ge_l rsplit[OF xp x1 eq_mono] eq_l rsplit[OF xp x1 neq_mono] neq_l
rsplit[OF xp x1 DVD_mono[OF xp x1]] DVD_l rsplit[OF xp x1 NDVD_mono[OF xp x1]] NDVD_l simpfm_rl)
lemma rlfm_l:
assumes qfp: "qfree p"
shows "isrlfm (rlfm p)"
using qfp lt_l gt_l ge_l le_l eq_l neq_l DVD_l NDVD_l
by (induct p rule: rlfm.induct,auto simp add: simpfm_rl)
(* 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
\<Upsilon>:: "fm \<Rightarrow> (num \<times> int) list"
\<upsilon> :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
recdef \<Upsilon> "measure size"
"\<Upsilon> (And p q) = (\<Upsilon> p @ \<Upsilon> q)"
"\<Upsilon> (Or p q) = (\<Upsilon> p @ \<Upsilon> q)"
"\<Upsilon> (Eq (CN 0 c e)) = [(Neg e,c)]"
"\<Upsilon> (NEq (CN 0 c e)) = [(Neg e,c)]"
"\<Upsilon> (Lt (CN 0 c e)) = [(Neg e,c)]"
"\<Upsilon> (Le (CN 0 c e)) = [(Neg e,c)]"
"\<Upsilon> (Gt (CN 0 c e)) = [(Neg e,c)]"
"\<Upsilon> (Ge (CN 0 c e)) = [(Neg e,c)]"
"\<Upsilon> p = []"
recdef \<upsilon> "measure size"
"\<upsilon> (And p q) = (\<lambda> (t,n). And (\<upsilon> p (t,n)) (\<upsilon> q (t,n)))"
"\<upsilon> (Or p q) = (\<lambda> (t,n). Or (\<upsilon> p (t,n)) (\<upsilon> q (t,n)))"
"\<upsilon> (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
"\<upsilon> (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
"\<upsilon> (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
"\<upsilon> (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
"\<upsilon> (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
"\<upsilon> (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
"\<upsilon> p = (\<lambda> (t,n). p)"
lemma \<upsilon>_I: assumes lp: "isrlfm p"
and np: "real n > 0" and nbt: "numbound0 t"
shows "(Ifm (x#bs) (\<upsilon> p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (\<upsilon> p (t,n))" (is "(?I x (\<upsilon> 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: \<upsilon>.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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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 \<Upsilon>_l:
assumes lp: "isrlfm p"
shows "\<forall> (t,k) \<in> set (\<Upsilon> p). numbound0 t \<and> k >0"
using lp
by(induct p rule: \<Upsilon>.induct) auto
lemma rminusinf_\<Upsilon>:
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 (\<Upsilon> 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 (\<Upsilon> 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 (\<Upsilon> p)" and mx: "real m * x \<ge> ?N a s" by blast
from \<Upsilon>_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_\<Upsilon>:
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 (\<Upsilon> 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 (\<Upsilon> 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 (\<Upsilon> p)" and mx: "real m * x \<le> ?N a s" by blast
from \<Upsilon>_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 (\<Upsilon> 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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_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: algebra_simps)
thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
by (simp add: algebra_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_\<Upsilon>:
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 (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> 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 (\<Upsilon> 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 (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> 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_\<Upsilon>[OF lp nmi pa] rplusinf_\<Upsilon>[OF lp npi pa]
have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> 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 \<Upsilon>_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 \<Upsilon>_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 (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> 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_\<Upsilon>[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_eq\<upsilon>:
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 (\<Upsilon> p). \<exists> (s,l) \<in> set (\<Upsilon> p). Ifm (x#bs) (\<upsilon> 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 (\<Upsilon> p)" and "(s,m) \<in> set (\<Upsilon> p)"
with \<Upsilon>_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: algebra_simps add_divide_distrib)
from \<upsilon>_I[OF lp mnp st_nb, where x="x" and bs="bs"]
have "?I x (\<upsilon> p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
with rinf_\<Upsilon>[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 (\<Upsilon> p)" and "(s,l) \<in> set (\<Upsilon> p)"
and px:"?I x (\<upsilon> p (Add (Mul l t) (Mul k s), 2*k*l))"
with \<Upsilon>_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 \<upsilon>_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
ultimately show "?E" by blast
qed
text{* The overall Part *}
lemma real_ex_int_real01:
shows "(\<exists> (x::real). P x) = (\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real i + u))"
proof(auto)
fix x
assume Px: "P x"
let ?i = "floor x"
let ?u = "x - real ?i"
have "x = real ?i + ?u" by simp
hence "P (real ?i + ?u)" using Px by simp
moreover have "real ?i \<le> x" using real_of_int_floor_le by simp hence "0 \<le> ?u" by arith
moreover have "?u < 1" using real_of_int_floor_add_one_gt[where r="x"] by arith
ultimately show "(\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real i + u))" by blast
qed
consts exsplitnum :: "num \<Rightarrow> num"
exsplit :: "fm \<Rightarrow> fm"
recdef exsplitnum "measure size"
"exsplitnum (C c) = (C c)"
"exsplitnum (Bound 0) = Add (Bound 0) (Bound 1)"
"exsplitnum (Bound n) = Bound (n+1)"
"exsplitnum (Neg a) = Neg (exsplitnum a)"
"exsplitnum (Add a b) = Add (exsplitnum a) (exsplitnum b) "
"exsplitnum (Sub a b) = Sub (exsplitnum a) (exsplitnum b) "
"exsplitnum (Mul c a) = Mul c (exsplitnum a)"
"exsplitnum (Floor a) = Floor (exsplitnum a)"
"exsplitnum (CN 0 c a) = CN 0 c (Add (Mul c (Bound 1)) (exsplitnum a))"
"exsplitnum (CN n c a) = CN (n+1) c (exsplitnum a)"
"exsplitnum (CF c s t) = CF c (exsplitnum s) (exsplitnum t)"
recdef exsplit "measure size"
"exsplit (Lt a) = Lt (exsplitnum a)"
"exsplit (Le a) = Le (exsplitnum a)"
"exsplit (Gt a) = Gt (exsplitnum a)"
"exsplit (Ge a) = Ge (exsplitnum a)"
"exsplit (Eq a) = Eq (exsplitnum a)"
"exsplit (NEq a) = NEq (exsplitnum a)"
"exsplit (Dvd i a) = Dvd i (exsplitnum a)"
"exsplit (NDvd i a) = NDvd i (exsplitnum a)"
"exsplit (And p q) = And (exsplit p) (exsplit q)"
"exsplit (Or p q) = Or (exsplit p) (exsplit q)"
"exsplit (Imp p q) = Imp (exsplit p) (exsplit q)"
"exsplit (Iff p q) = Iff (exsplit p) (exsplit q)"
"exsplit (NOT p) = NOT (exsplit p)"
"exsplit p = p"
lemma exsplitnum:
"Inum (x#y#bs) (exsplitnum t) = Inum ((x+y) #bs) t"
by(induct t rule: exsplitnum.induct) (simp_all add: algebra_simps)
lemma exsplit:
assumes qfp: "qfree p"
shows "Ifm (x#y#bs) (exsplit p) = Ifm ((x+y)#bs) p"
using qfp exsplitnum[where x="x" and y="y" and bs="bs"]
by(induct p rule: exsplit.induct) simp_all
lemma splitex:
assumes qf: "qfree p"
shows "(Ifm bs (E p)) = (\<exists> (i::int). Ifm (real i#bs) (E (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))" (is "?lhs = ?rhs")
proof-
have "?rhs = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm (x#(real i)#bs) (exsplit p))"
by (simp add: myless[rule_format, where b="1"] myless[rule_format, where b="0"] add_ac diff_def)
also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real i + x) #bs) p)"
by (simp only: exsplit[OF qf] add_ac)
also have "\<dots> = (\<exists> x. Ifm (x#bs) p)"
by (simp only: real_ex_int_real01[where P="\<lambda> x. Ifm (x#bs) p"])
finally show ?thesis by simp
qed
(* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *)
constdefs ferrack01:: "fm \<Rightarrow> fm"
"ferrack01 p \<equiv> (let p' = rlfm(And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p);
U = remdups(map simp_num_pair
(map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
(alluopairs (\<Upsilon> p'))))
in decr (evaldjf (\<upsilon> p') U ))"
lemma fr_eq_01:
assumes qf: "qfree p"
shows "(\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (\<exists> (t,n) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). \<exists> (s,m) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). Ifm (x#bs) (\<upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) (Add (Mul m t) (Mul n s), 2*n*m)))"
(is "(\<exists> x. ?I x ?q) = ?F")
proof-
let ?rq = "rlfm ?q"
let ?M = "?I x (minusinf ?rq)"
let ?P = "?I x (plusinf ?rq)"
have MF: "?M = False"
apply (simp add: Let_def reducecoeff_def numgcd_def zgcd_def rsplit_def ge_def lt_def conj_def disj_def)
by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all)
have PF: "?P = False" apply (simp add: Let_def reducecoeff_def numgcd_def zgcd_def rsplit_def ge_def lt_def conj_def disj_def)
by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all)
have "(\<exists> x. ?I x ?q ) =
((?I x (minusinf ?rq)) \<or> (?I x (plusinf ?rq )) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> ?rq). \<exists> (s,m) \<in> set (\<Upsilon> ?rq ). ?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))))"
(is "(\<exists> x. ?I x ?q) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
proof
assume "\<exists> x. ?I x ?q"
then obtain x where qx: "?I x ?q" by blast
hence xp: "0\<le> x" and x1: "x< 1" and px: "?I x p"
by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf])
from qx have "?I x ?rq "
by (simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1])
hence lqx: "?I x ?rq " using simpfm[where p="?rq" and bs="x#bs"] by auto
from qf have qfq:"isrlfm ?rq"
by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1])
with lqx fr_eq\<upsilon>[OF qfq] show "?M \<or> ?P \<or> ?F" by blast
next
assume D: "?D"
let ?U = "set (\<Upsilon> ?rq )"
from MF PF D have "?F" by auto
then obtain t n s m where aU:"(t,n) \<in> ?U" and bU:"(s,m)\<in> ?U" and rqx: "?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))" by blast
from qf have lrq:"isrlfm ?rq"using rlfm_l[OF qf]
by (auto simp add: rsplit_def lt_def ge_def)
from aU bU \<Upsilon>_l[OF lrq] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" by (auto simp add: split_def)
let ?st = "Add (Mul m t) (Mul n s)"
from tnb snb have stnb: "numbound0 ?st" by simp
from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
by (simp add: mult_commute)
from conjunct1[OF \<upsilon>_I[OF lrq mnp stnb, where bs="bs" and x="x"], symmetric] rqx
have "\<exists> x. ?I x ?rq" by auto
thus "?E"
using rlfm_I[OF qf] by (auto simp add: rsplit_def lt_def ge_def)
qed
with MF PF show ?thesis by blast
qed
lemma \<Upsilon>_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: algebra_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 algebra_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: algebra_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: algebra_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 \<Upsilon>_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) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (\<upsilon> 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) (\<upsilon> 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: algebra_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 \<upsilon>_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 \<upsilon>_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) (\<upsilon> 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) (\<upsilon> 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: algebra_simps add_divide_distrib)
from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
from \<upsilon>_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 \<upsilon>_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
qed
lemma ferrack01:
assumes qf: "qfree p"
shows "((\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (Ifm bs (ferrack01 p))) \<and> qfree (ferrack01 p)" (is "(?lhs = ?rhs) \<and> _")
proof-
let ?I = "\<lambda> x p. Ifm (x#bs) p"
fix x
let ?N = "\<lambda> t. Inum (x#bs) t"
let ?q = "rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)"
let ?U = "\<Upsilon> ?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 = "remdups ?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 (\<Upsilon> p). \<exists> b \<in> set (\<Upsilon> p). ?I x (\<upsilon> p (?g(a,b)))"
let ?ep = "evaldjf (\<upsilon> ?q) ?Y"
from rlfm_l[OF qf] have lq: "isrlfm ?q"
by (simp add: rsplit_def lt_def ge_def conj_def disj_def Let_def reducecoeff_def numgcd_def zgcd_def)
from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
from \<Upsilon>_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
from U_l UpU
have Up_: "\<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 \<Upsilon>_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 (\<upsilon> ?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 \<upsilon>_I[OF lq np tnb]
have "bound0 (\<upsilon> ?q (t,n))" by simp}
thus ?thesis by blast
qed
hence ep_nb: "bound0 ?ep" using evaldjf_bound0[where xs="?Y" and f="\<upsilon> ?q"]
by auto
from fr_eq_01[OF qf, where bs="bs" and x="x"] have "?lhs = ?F ?q"
by (simp only: split_def fst_conv snd_conv)
also have "\<dots> = (\<exists> (t,n) \<in> set ?Y. ?I x (\<upsilon> ?q (t,n)))" using \<Upsilon>_cong[OF lq YU U_l Y_l]
by (simp only: split_def fst_conv snd_conv)
also have "\<dots> = (Ifm (x#bs) ?ep)"
using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="\<upsilon> ?q",symmetric]
by (simp only: split_def pair_collapse)
also have "\<dots> = (Ifm bs (decr ?ep))" using decr[OF ep_nb] by blast
finally have lr: "?lhs = ?rhs" by (simp only: ferrack01_def Let_def)
from decr_qf[OF ep_nb] have "qfree (ferrack01 p)" by (simp only: Let_def ferrack01_def)
with lr show ?thesis by blast
qed
lemma cp_thm':
assumes lp: "iszlfm p (real (i::int)#bs)"
and up: "d\<beta> p 1" and dd: "d\<delta> p d" and dp: "d > 0"
shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (real i#bs)) ` set (\<beta> p). Ifm ((b+real j)#bs) p))"
using cp_thm[OF lp up dd dp] by auto
constdefs unit:: "fm \<Rightarrow> fm \<times> num list \<times> int"
"unit p \<equiv> (let p' = zlfm p ; l = \<zeta> p' ; q = And (Dvd l (CN 0 1 (C 0))) (a\<beta> p' l); d = \<delta> q;
B = remdups (map simpnum (\<beta> q)) ; a = remdups (map simpnum (\<alpha> q))
in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"
lemma unit: assumes qf: "qfree p"
shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow> ((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> (Inum (real i#bs)) ` set B = (Inum (real i#bs)) ` set (\<beta> q) \<and> d\<beta> q 1 \<and> d\<delta> q d \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
proof-
fix q B d
assume qBd: "unit p = (q,B,d)"
let ?thes = "((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and>
Inum (real i#bs) ` set B = Inum (real i#bs) ` set (\<beta> q) \<and>
d\<beta> q 1 \<and> d\<delta> q d \<and> 0 < d \<and> iszlfm q (real i # bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
let ?p' = "zlfm p"
let ?l = "\<zeta> ?p'"
let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a\<beta> ?p' ?l)"
let ?d = "\<delta> ?q"
let ?B = "set (\<beta> ?q)"
let ?B'= "remdups (map simpnum (\<beta> ?q))"
let ?A = "set (\<alpha> ?q)"
let ?A'= "remdups (map simpnum (\<alpha> ?q))"
from conjunct1[OF zlfm_I[OF qf, where bs="bs"]]
have pp': "\<forall> i. ?I i ?p' = ?I i p" by auto
from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]]
have lp': "\<forall> (i::int). iszlfm ?p' (real i#bs)" by simp
hence lp'': "iszlfm ?p' (real (i::int)#bs)" by simp
from lp' \<zeta>[where p="?p'" and bs="bs"] have lp: "?l >0" and dl: "d\<beta> ?p' ?l" by auto
from a\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp'' dl lp] pp'
have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by (simp add: int_rdvd_iff)
from lp'' lp a\<beta>[OF lp'' dl lp] have lq:"iszlfm ?q (real i#bs)" and uq: "d\<beta> ?q 1"
by (auto simp add: isint_def)
from \<delta>[OF lq] have dp:"?d >0" and dd: "d\<delta> ?q ?d" by blast+
let ?N = "\<lambda> t. Inum (real (i::int)#bs) t"
have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_compose)
also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="real i #bs"] by auto
finally have BB': "?N ` set ?B' = ?N ` ?B" .
have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_compose)
also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real i #bs"] by auto
finally have AA': "?N ` set ?A' = ?N ` ?A" .
from \<beta>_numbound0[OF lq] have B_nb:"\<forall> b\<in> set ?B'. numbound0 b"
by (simp add: simpnum_numbound0)
from \<alpha>_l[OF lq] have A_nb: "\<forall> b\<in> set ?A'. numbound0 b"
by (simp add: simpnum_numbound0)
{assume "length ?B' \<le> length ?A'"
hence q:"q=?q" and "B = ?B'" and d:"d = ?d"
using qBd by (auto simp add: Let_def unit_def)
with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<beta> q)"
and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
with pq_ex dp uq dd lq q d have ?thes by simp}
moreover
{assume "\<not> (length ?B' \<le> length ?A')"
hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d"
using qBd by (auto simp add: Let_def unit_def)
with AA' mirror\<alpha>\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)"
and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
from mirror_ex[OF lq] pq_ex q
have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
from lq uq q mirror_d\<beta> [where p="?q" and bs="bs" and a="real i"]
have lq': "iszlfm q (real i#bs)" and uq: "d\<beta> q 1" by auto
from \<delta>[OF lq'] mirror_\<delta>[OF lq] q d have dq:"d\<delta> q d " by auto
from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp
}
ultimately show ?thes by blast
qed
(* Cooper's Algorithm *)
constdefs cooper :: "fm \<Rightarrow> fm"
"cooper p \<equiv>
(let (q,B,d) = unit p; js = iupt (1,d);
mq = simpfm (minusinf q);
md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) js
in if md = T then T else
(let qd = evaldjf (\<lambda> t. simpfm (subst0 t q))
(remdups (map (\<lambda> (b,j). simpnum (Add b (C j)))
[(b,j). b\<leftarrow>B,j\<leftarrow>js]))
in decr (disj md qd)))"
lemma cooper: assumes qf: "qfree p"
shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)"
(is "(?lhs = ?rhs) \<and> _")
proof-
let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
let ?q = "fst (unit p)"
let ?B = "fst (snd(unit p))"
let ?d = "snd (snd (unit p))"
let ?js = "iupt (1,?d)"
let ?mq = "minusinf ?q"
let ?smq = "simpfm ?mq"
let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
fix i
let ?N = "\<lambda> t. Inum (real (i::int)#bs) t"
let ?bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]"
let ?sbjs = "map (\<lambda> (b,j). simpnum (Add b (C j))) ?bjs"
let ?qd = "evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs)"
have qbf:"unit p = (?q,?B,?d)" by simp
from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and
B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and
uq:"d\<beta> ?q 1" and dd: "d\<delta> ?q ?d" and dp: "?d > 0" and
lq: "iszlfm ?q (real i#bs)" and
Bn: "\<forall> b\<in> set ?B. numbound0 b" by auto
from zlin_qfree[OF lq] have qfq: "qfree ?q" .
from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp
hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)"
by (auto simp only: subst0_bound0[OF qfmq])
hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))"
by (auto simp add: simpfm_bound0)
from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp
from Bn jsnb have "\<forall> (b,j) \<in> set ?bjs. numbound0 (Add b (C j))"
by simp
hence "\<forall> (b,j) \<in> set ?bjs. numbound0 (simpnum (Add b (C j)))"
using simpnum_numbound0 by blast
hence "\<forall> t \<in> set ?sbjs. numbound0 t" by simp
hence "\<forall> t \<in> set (remdups ?sbjs). bound0 (subst0 t ?q)"
using subst0_bound0[OF qfq] by auto
hence th': "\<forall> t \<in> set (remdups ?sbjs). bound0 (simpfm (subst0 t ?q))"
using simpfm_bound0 by blast
from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp
from mdb qdb
have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
from trans [OF pq_ex cp_thm'[OF lq uq dd dp]] B
have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm ((b+ real j)#bs) ?q))" by auto
also have "\<dots> = ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> (b,j) \<in> (?N ` set ?B \<times> set ?js). Ifm ((b+ real j)#bs) ?q))" apply (simp only: iupt_set simpfm) by auto
also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (Add b (C j))) ` set ?bjs. Ifm (t #bs) ?q))" by simp
also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (simpnum (Add b (C j)))) ` set ?bjs. Ifm (t #bs) ?q))" by (simp only: simpnum_ci)
also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> set ?sbjs. Ifm (?N t #bs) ?q))"
by (auto simp add: split_def)
also have "\<dots> = ((\<exists> j\<in> set ?js. (\<lambda> j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or> (\<exists> t \<in> set (remdups ?sbjs). (\<lambda> t. ?I i (simpfm (subst0 t ?q))) t))" by (simp only: simpfm subst0_I[OF qfq] simpfm Inum.simps subst0_I[OF qfmq] set_remdups)
also have "\<dots> = ((?I i (evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js)) \<or> (?I i (evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs))))" by (simp only: evaldjf_ex)
finally have mdqd: "?lhs = (?I i (disj ?md ?qd))" by (simp add: disj)
hence mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" using decr [OF mdqdb] by simp
{assume mdT: "?md = T"
hence cT:"cooper p = T"
by (simp only: cooper_def unit_def split_def Let_def if_True) simp
from mdT mdqd have lhs:"?lhs" by (auto simp add: disj)
from mdT have "?rhs" by (simp add: cooper_def unit_def split_def)
with lhs cT have ?thesis by simp }
moreover
{assume mdT: "?md \<noteq> T" hence "cooper p = decr (disj ?md ?qd)"
by (simp only: cooper_def unit_def split_def Let_def if_False)
with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp }
ultimately show ?thesis by blast
qed
lemma DJcooper:
assumes qf: "qfree p"
shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ cooper p))) \<and> qfree (DJ cooper p)"
proof-
from cooper have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (cooper p)" by blast
from DJ_qf[OF cqf] qf have thqf:"qfree (DJ cooper p)" by blast
have "Ifm bs (DJ cooper p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (cooper q))"
by (simp add: DJ_def evaldjf_ex)
also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real x#bs) q)"
using cooper disjuncts_qf[OF qf] by blast
also have "\<dots> = (\<exists> (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto)
finally show ?thesis using thqf by blast
qed
(* Redy and Loveland *)
lemma \<sigma>\<rho>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'"
shows "Ifm (a#bs) (\<sigma>\<rho> p (t,c)) = Ifm (a#bs) (\<sigma>\<rho> p (t',c))"
using lp
by (induct p rule: iszlfm.induct, auto simp add: tt')
lemma \<sigma>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'"
shows "Ifm (a#bs) (\<sigma> p c t) = Ifm (a#bs) (\<sigma> p c t')"
by (simp add: \<sigma>_def tt' \<sigma>\<rho>_cong[OF lp tt'])
lemma \<rho>_cong: assumes lp: "iszlfm p (a#bs)"
and RR: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R = (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)"
shows "(\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))) = (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j))))"
(is "?lhs = ?rhs")
proof
let ?d = "\<delta> p"
assume ?lhs then obtain e c j where ecR: "(e,c) \<in> R" and jD:"j \<in> {1 .. c*?d}"
and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast
from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" by auto
hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" using RR by simp
hence "\<exists> (e',c') \<in> set (\<rho> p). Inum (a#bs) e = Inum (a#bs) e' \<and> c = c'" by auto
then obtain e' c' where ecRo:"(e',c') \<in> set (\<rho> p)" and ee':"Inum (a#bs) e = Inum (a#bs) e'"
and cc':"c = c'" by blast
from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp
from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp
from ecRo jD px' cc' show ?rhs apply auto
by (rule_tac x="(e', c')" in bexI,simp_all)
(rule_tac x="j" in bexI, simp_all add: cc'[symmetric])
next
let ?d = "\<delta> p"
assume ?rhs then obtain e c j where ecR: "(e,c) \<in> set (\<rho> p)" and jD:"j \<in> {1 .. c*?d}"
and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast
from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" by auto
hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" using RR by simp
hence "\<exists> (e',c') \<in> R. Inum (a#bs) e = Inum (a#bs) e' \<and> c = c'" by auto
then obtain e' c' where ecRo:"(e',c') \<in> R" and ee':"Inum (a#bs) e = Inum (a#bs) e'"
and cc':"c = c'" by blast
from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp
from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp
from ecRo jD px' cc' show ?lhs apply auto
by (rule_tac x="(e', c')" in bexI,simp_all)
(rule_tac x="j" in bexI, simp_all add: cc'[symmetric])
qed
lemma rl_thm':
assumes lp: "iszlfm p (real (i::int)#bs)"
and R: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R = (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)"
shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
using rl_thm[OF lp] \<rho>_cong[OF iszlfm_gen[OF lp, rule_format, where y="a"] R] by simp
constdefs chooset:: "fm \<Rightarrow> fm \<times> ((num\<times>int) list) \<times> int"
"chooset p \<equiv> (let q = zlfm p ; d = \<delta> q;
B = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<rho> q)) ;
a = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<alpha>\<rho> q))
in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"
lemma chooset: assumes qf: "qfree p"
shows "\<And> q B d. chooset p = (q,B,d) \<Longrightarrow> ((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)"
proof-
fix q B d
assume qBd: "chooset p = (q,B,d)"
let ?thes = "((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)"
let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
let ?q = "zlfm p"
let ?d = "\<delta> ?q"
let ?B = "set (\<rho> ?q)"
let ?f = "\<lambda> (t,k). (simpnum t,k)"
let ?B'= "remdups (map ?f (\<rho> ?q))"
let ?A = "set (\<alpha>\<rho> ?q)"
let ?A'= "remdups (map ?f (\<alpha>\<rho> ?q))"
from conjunct1[OF zlfm_I[OF qf, where bs="bs"]]
have pp': "\<forall> i. ?I i ?q = ?I i p" by auto
hence pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp
from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]], rule_format, where y="real i"]
have lq: "iszlfm ?q (real (i::int)#bs)" .
from \<delta>[OF lq] have dp:"?d >0" by blast
let ?N = "\<lambda> (t,c). (Inum (real (i::int)#bs) t,c)"
have "?N ` set ?B' = ((?N o ?f) ` ?B)" by (simp add: split_def image_compose)
also have "\<dots> = ?N ` ?B"
by(simp add: split_def image_compose simpnum_ci[where bs="real i #bs"] image_def)
finally have BB': "?N ` set ?B' = ?N ` ?B" .
have "?N ` set ?A' = ((?N o ?f) ` ?A)" by (simp add: split_def image_compose)
also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real i #bs"]
by(simp add: split_def image_compose simpnum_ci[where bs="real i #bs"] image_def)
finally have AA': "?N ` set ?A' = ?N ` ?A" .
from \<rho>_l[OF lq] have B_nb:"\<forall> (e,c)\<in> set ?B'. numbound0 e \<and> c > 0"
by (simp add: simpnum_numbound0 split_def)
from \<alpha>\<rho>_l[OF lq] have A_nb: "\<forall> (e,c)\<in> set ?A'. numbound0 e \<and> c > 0"
by (simp add: simpnum_numbound0 split_def)
{assume "length ?B' \<le> length ?A'"
hence q:"q=?q" and "B = ?B'" and d:"d = ?d"
using qBd by (auto simp add: Let_def chooset_def)
with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<rho> q)"
and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto
with pq_ex dp lq q d have ?thes by simp}
moreover
{assume "\<not> (length ?B' \<le> length ?A')"
hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d"
using qBd by (auto simp add: Let_def chooset_def)
with AA' mirror_\<alpha>\<rho>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<rho> q)"
and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto
from mirror_ex[OF lq] pq_ex q
have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
from lq q mirror_l [where p="?q" and bs="bs" and a="real i"]
have lq': "iszlfm q (real i#bs)" by auto
from mirror_\<delta>[OF lq] pqm_eq b bn lq' dp q dp d have ?thes by simp
}
ultimately show ?thes by blast
qed
constdefs stage:: "fm \<Rightarrow> int \<Rightarrow> (num \<times> int) \<Rightarrow> fm"
"stage p d \<equiv> (\<lambda> (e,c). evaldjf (\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))) (iupt (1,c*d)))"
lemma stage:
shows "Ifm bs (stage p d (e,c)) = (\<exists> j\<in>{1 .. c*d}. Ifm bs (\<sigma> p c (Add e (C j))))"
by (unfold stage_def split_def ,simp only: evaldjf_ex iupt_set simpfm) simp
lemma stage_nb: assumes lp: "iszlfm p (a#bs)" and cp: "c >0" and nb:"numbound0 e"
shows "bound0 (stage p d (e,c))"
proof-
let ?f = "\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))"
have th: "\<forall> j\<in> set (iupt(1,c*d)). bound0 (?f j)"
proof
fix j
from nb have nb':"numbound0 (Add e (C j))" by simp
from simpfm_bound0[OF \<sigma>_nb[OF lp nb', where k="c"]]
show "bound0 (simpfm (\<sigma> p c (Add e (C j))))" .
qed
from evaldjf_bound0[OF th] show ?thesis by (unfold stage_def split_def) simp
qed
constdefs redlove:: "fm \<Rightarrow> fm"
"redlove p \<equiv>
(let (q,B,d) = chooset p;
mq = simpfm (minusinf q);
md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) (iupt (1,d))
in if md = T then T else
(let qd = evaldjf (stage q d) B
in decr (disj md qd)))"
lemma redlove: assumes qf: "qfree p"
shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)"
(is "(?lhs = ?rhs) \<and> _")
proof-
let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
let ?q = "fst (chooset p)"
let ?B = "fst (snd(chooset p))"
let ?d = "snd (snd (chooset p))"
let ?js = "iupt (1,?d)"
let ?mq = "minusinf ?q"
let ?smq = "simpfm ?mq"
let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
fix i
let ?N = "\<lambda> (t,k). (Inum (real (i::int)#bs) t,k)"
let ?qd = "evaldjf (stage ?q ?d) ?B"
have qbf:"chooset p = (?q,?B,?d)" by simp
from chooset[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and
B:"?N ` set ?B = ?N ` set (\<rho> ?q)" and dd: "\<delta> ?q = ?d" and dp: "?d > 0" and
lq: "iszlfm ?q (real i#bs)" and
Bn: "\<forall> (e,c)\<in> set ?B. numbound0 e \<and> c > 0" by auto
from zlin_qfree[OF lq] have qfq: "qfree ?q" .
from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp
hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)"
by (auto simp only: subst0_bound0[OF qfmq])
hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))"
by (auto simp add: simpfm_bound0)
from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp
from Bn stage_nb[OF lq] have th:"\<forall> x \<in> set ?B. bound0 (stage ?q ?d x)" by auto
from evaldjf_bound0[OF th] have qdb: "bound0 ?qd" .
from mdb qdb
have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
from trans [OF pq_ex rl_thm'[OF lq B]] dd
have "?lhs = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq) \<or> (\<exists> (e,c)\<in> set ?B. \<exists> j\<in> {1 .. c*?d}. Ifm (real i#bs) (\<sigma> ?q c (Add e (C j)))))" by auto
also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq) \<or> (\<exists> (e,c)\<in> set ?B. ?I i (stage ?q ?d (e,c) )))"
by (simp add: simpfm stage split_def)
also have "\<dots> = ((\<exists> j\<in> {1 .. ?d}. ?I i (subst0 (C j) ?smq)) \<or> ?I i ?qd)"
by (simp add: evaldjf_ex subst0_I[OF qfmq])
finally have mdqd:"?lhs = (?I i ?md \<or> ?I i ?qd)" by (simp only: evaldjf_ex iupt_set simpfm)
also have "\<dots> = (?I i (disj ?md ?qd))" by (simp add: disj)
also have "\<dots> = (Ifm bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb])
finally have mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" .
{assume mdT: "?md = T"
hence cT:"redlove p = T" by (simp add: redlove_def Let_def chooset_def split_def)
from mdT have lhs:"?lhs" using mdqd by simp
from mdT have "?rhs" by (simp add: redlove_def chooset_def split_def)
with lhs cT have ?thesis by simp }
moreover
{assume mdT: "?md \<noteq> T" hence "redlove p = decr (disj ?md ?qd)"
by (simp add: redlove_def chooset_def split_def Let_def)
with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp }
ultimately show ?thesis by blast
qed
lemma DJredlove:
assumes qf: "qfree p"
shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ redlove p))) \<and> qfree (DJ redlove p)"
proof-
from redlove have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (redlove p)" by blast
from DJ_qf[OF cqf] qf have thqf:"qfree (DJ redlove p)" by blast
have "Ifm bs (DJ redlove p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (redlove q))"
by (simp add: DJ_def evaldjf_ex)
also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real x#bs) q)"
using redlove disjuncts_qf[OF qf] by blast
also have "\<dots> = (\<exists> (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto)
finally show ?thesis using thqf by blast
qed
lemma exsplit_qf: assumes qf: "qfree p"
shows "qfree (exsplit p)"
using qf by (induct p rule: exsplit.induct, auto)
definition mircfr :: "fm \<Rightarrow> fm" where
"mircfr = DJ cooper o ferrack01 o simpfm o exsplit"
definition mirlfr :: "fm \<Rightarrow> fm" where
"mirlfr = DJ redlove o ferrack01 o simpfm o exsplit"
lemma mircfr: "\<forall> bs p. qfree p \<longrightarrow> qfree (mircfr p) \<and> Ifm bs (mircfr p) = Ifm bs (E p)"
proof(clarsimp simp del: Ifm.simps)
fix bs p
assume qf: "qfree p"
show "qfree (mircfr p)\<and>(Ifm bs (mircfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)")
proof-
let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))"
have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real i#bs) ?es)"
using splitex[OF qf] by simp
with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
with DJcooper[OF qf'] show ?thesis by (simp add: mircfr_def)
qed
qed
lemma mirlfr: "\<forall> bs p. qfree p \<longrightarrow> qfree(mirlfr p) \<and> Ifm bs (mirlfr p) = Ifm bs (E p)"
proof(clarsimp simp del: Ifm.simps)
fix bs p
assume qf: "qfree p"
show "qfree (mirlfr p)\<and>(Ifm bs (mirlfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)")
proof-
let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))"
have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real i#bs) ?es)"
using splitex[OF qf] by simp
with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
with DJredlove[OF qf'] show ?thesis by (simp add: mirlfr_def)
qed
qed
definition mircfrqe:: "fm \<Rightarrow> fm" where
"mircfrqe p = qelim (prep p) mircfr"
definition mirlfrqe:: "fm \<Rightarrow> fm" where
"mirlfrqe p = qelim (prep p) mirlfr"
theorem mircfrqe: "(Ifm bs (mircfrqe p) = Ifm bs p) \<and> qfree (mircfrqe p)"
using qelim_ci[OF mircfr] prep by (auto simp add: mircfrqe_def)
theorem mirlfrqe: "(Ifm bs (mirlfrqe p) = Ifm bs p) \<and> qfree (mirlfrqe p)"
using qelim_ci[OF mirlfr] prep by (auto simp add: mirlfrqe_def)
definition
"test1 (u\<Colon>unit) = mircfrqe (A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0)))))))"
definition
"test2 (u\<Colon>unit) = mircfrqe (A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0)))))"
definition
"test3 (u\<Colon>unit) = mirlfrqe (A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0)))))))"
definition
"test4 (u\<Colon>unit) = mirlfrqe (A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0)))))"
definition
"test5 (u\<Colon>unit) = mircfrqe (A(E(And (Ge(Sub (Bound 1) (Bound 0))) (Eq (Add (Floor (Bound 1)) (Floor (Neg(Bound 0))))))))"
ML {* @{code test1} () *}
ML {* @{code test2} () *}
ML {* @{code test3} () *}
ML {* @{code test4} () *}
ML {* @{code test5} () *}
(*export_code mircfrqe mirlfrqe
in SML module_name Mir file "raw_mir.ML"*)
oracle mirfr_oracle = {* fn (proofs, ct) =>
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 "real :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ t')) =
@{code Floor} (num_of_term vs t')
| num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ t')) =
@{code Neg} (@{code Floor} (@{code Neg} (num_of_term vs 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 rdvd"} $ (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t1)) $ t2) =
@{code Dvd} (HOLogic.dest_numeral t1, num_of_term vs t2)
| fm_of_term vs (@{term "op = :: 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} (@{code Floor} (@{code Neg} t'))) =
@{term "real :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ term_of_num vs t')
| 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 Floor} t) = @{term "real :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ term_of_num vs t)
| term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t))
| term_of_num vs (@{code CF} (c, t, s)) = term_of_num vs (@{code Add} (@{code Mul} (c, @{code Floor} t), s));
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 Dvd} (i, t)) =
@{term "op rdvd"} $ term_of_num vs (@{code C} i) $ term_of_num vs t
| term_of_fm vs (@{code NDvd} (i, t)) =
term_of_fm vs (@{code NOT} (@{code Dvd} (i, 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 = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm vs t1 $ term_of_fm vs t2;
in
let
val thy = Thm.theory_of_cterm ct;
val t = Thm.term_of ct;
val fs = OldTerm.term_frees t;
val vs = fs ~~ (0 upto (length fs - 1));
val qe = if proofs then @{code mirlfrqe} else @{code mircfrqe};
val t' = (term_of_fm vs o qe o fm_of_term vs) t;
in (cterm_of thy o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end
end;
*}
use "mir_tac.ML"
setup "Mir_Tac.setup"
lemma "ALL (x::real). (\<lfloor>x\<rfloor> = \<lceil>x\<rceil> = (x = real \<lfloor>x\<rfloor>))"
apply mir
done
lemma "ALL (x::real). real (2::int)*x - (real (1::int)) < real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil> \<and> real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil> \<le> real (2::int)*x + (real (1::int))"
apply mir
done
lemma "ALL (x::real). 2*\<lfloor>x\<rfloor> \<le> \<lfloor>2*x\<rfloor> \<and> \<lfloor>2*x\<rfloor> \<le> 2*\<lfloor>x+1\<rfloor>"
apply mir
done
lemma "ALL (x::real). \<exists>y \<le> x. (\<lfloor>x\<rfloor> = \<lceil>y\<rceil>)"
apply mir
done
lemma "ALL (x::real) (y::real). \<lfloor>x\<rfloor> = \<lfloor>y\<rfloor> \<longrightarrow> 0 \<le> abs (y - x) \<and> abs (y - x) \<le> 1"
apply mir
done
end