two target language numeral types: integer and natural, as replacement for code_numeral;
former theory HOL/Library/Code_Numeral_Types replaces HOL/Code_Numeral;
refined stack of theories implementing int and/or nat by target language numerals;
reduced number of target language numeral types to exactly one
(* Title: HOL/Decision_Procs/Cooper.thy
Author: Amine Chaieb
*)
theory Cooper
imports Complex_Main "~~/src/HOL/Library/Code_Target_Numeral" "~~/src/HOL/Library/Old_Recdef"
begin
(* Periodicity of dvd *)
(*********************************************************************************)
(**** 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
primrec num_size :: "num \<Rightarrow> nat" -- {* A size for num to make inductive proofs simpler *}
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 (Mul c a) = 1 + num_size a"
primrec Inum :: "int list \<Rightarrow> num \<Rightarrow> int" where
"Inum bs (C c) = c"
| "Inum bs (Bound n) = bs!n"
| "Inum bs (CN n c a) = 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) = c* Inum bs a"
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
| Closed nat | NClosed nat
fun fmsize :: "fm \<Rightarrow> nat" -- {* A size for fm *}
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"
lemma fmsize_pos: "fmsize p > 0"
by (induct p rule: fmsize.induct) simp_all
primrec Ifm :: "bool list \<Rightarrow> int list \<Rightarrow> fm \<Rightarrow> bool" -- {* Semantics of formulae (fm) *}
where
"Ifm bbs bs T = True"
| "Ifm bbs bs F = False"
| "Ifm bbs bs (Lt a) = (Inum bs a < 0)"
| "Ifm bbs bs (Gt a) = (Inum bs a > 0)"
| "Ifm bbs bs (Le a) = (Inum bs a \<le> 0)"
| "Ifm bbs bs (Ge a) = (Inum bs a \<ge> 0)"
| "Ifm bbs bs (Eq a) = (Inum bs a = 0)"
| "Ifm bbs bs (NEq a) = (Inum bs a \<noteq> 0)"
| "Ifm bbs bs (Dvd i b) = (i dvd Inum bs b)"
| "Ifm bbs bs (NDvd i b) = (\<not>(i dvd Inum bs b))"
| "Ifm bbs bs (NOT p) = (\<not> (Ifm bbs bs p))"
| "Ifm bbs bs (And p q) = (Ifm bbs bs p \<and> Ifm bbs bs q)"
| "Ifm bbs bs (Or p q) = (Ifm bbs bs p \<or> Ifm bbs bs q)"
| "Ifm bbs bs (Imp p q) = ((Ifm bbs bs p) \<longrightarrow> (Ifm bbs bs q))"
| "Ifm bbs bs (Iff p q) = (Ifm bbs bs p = Ifm bbs bs q)"
| "Ifm bbs bs (E p) = (\<exists>x. Ifm bbs (x#bs) p)"
| "Ifm bbs bs (A p) = (\<forall>x. Ifm bbs (x#bs) p)"
| "Ifm bbs bs (Closed n) = bbs!n"
| "Ifm bbs bs (NClosed n) = (\<not> bbs!n)"
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: "Ifm bbs bs (prep p) = Ifm bbs bs p"
by (induct p arbitrary: bs rule: prep.induct) auto
fun qfree :: "fm \<Rightarrow> bool" -- {* Quantifier freeness *}
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"
text {* 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"
lemma numbound0_I:
assumes nb: "numbound0 a"
shows "Inum (b#bs) a = Inum (b'#bs) a"
using nb by (induct a rule: num.induct) (auto simp add: gr0_conv_Suc)
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"
| "bound0 (Closed P) = True"
| "bound0 (NClosed P) = True"
lemma bound0_I:
assumes bp: "bound0 p"
shows "Ifm bbs (b#bs) p = Ifm bbs (b'#bs) p"
using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
by (induct p rule: fm.induct) (auto simp add: gr0_conv_Suc)
fun numsubst0 :: "num \<Rightarrow> num \<Rightarrow> num"
where
"numsubst0 t (C c) = (C c)"
| "numsubst0 t (Bound n) = (if n=0 then t else Bound n)"
| "numsubst0 t (CN 0 i a) = Add (Mul i t) (numsubst0 t a)"
| "numsubst0 t (CN n i a) = CN n i (numsubst0 t a)"
| "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)"
lemma numsubst0_I: "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t"
by (induct t rule: numsubst0.induct) (auto simp: nth_Cons')
lemma numsubst0_I':
"numbound0 a \<Longrightarrow> Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t"
by (induct t rule: numsubst0.induct) (auto simp: nth_Cons' numbound0_I[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)"
| "subst0 t (Closed P) = (Closed P)"
| "subst0 t (NClosed P) = (NClosed P)"
lemma subst0_I:
assumes qfp: "qfree p"
shows "Ifm bbs (b#bs) (subst0 a p) = Ifm bbs ((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: gr0_conv_Suc)
fun decrnum:: "num \<Rightarrow> num"
where
"decrnum (Bound n) = Bound (n - 1)"
| "decrnum (Neg a) = Neg (decrnum a)"
| "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
| "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
| "decrnum (Mul c a) = Mul c (decrnum a)"
| "decrnum (CN n i a) = (CN (n - 1) i (decrnum a))"
| "decrnum a = a"
fun decr :: "fm \<Rightarrow> fm"
where
"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) (auto simp add: gr0_conv_Suc)
lemma decr:
assumes nb: "bound0 p"
shows "Ifm bbs (x#bs) p = Ifm bbs bs (decr p)"
using nb by (induct p rule: decr.induct) (simp_all add: gr0_conv_Suc decrnum)
lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
by (induct p) simp_all
fun isatom :: "fm \<Rightarrow> bool" -- {* test for atomicity *}
where
"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 (Closed P) = True"
| "isatom (NClosed P) = True"
| "isatom p = False"
lemma numsubst0_numbound0:
assumes nb: "numbound0 t"
shows "numbound0 (numsubst0 t a)"
using nb apply (induct a)
apply simp_all
apply (case_tac nat, simp_all)
done
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 (f p) q))"
definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
where "evaldjf f ps = foldr (djf f) ps F"
lemma djf_Or: "Ifm bbs bs (djf f p q) = Ifm bbs 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 bbs bs (evaldjf f ps) = (\<exists>p \<in> set ps. Ifm bbs 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)
fun disjuncts :: "fm \<Rightarrow> fm list"
where
"disjuncts (Or p q) = disjuncts p @ disjuncts q"
| "disjuncts F = []"
| "disjuncts p = [p]"
lemma disjuncts: "(\<exists>q \<in> set (disjuncts p). Ifm bbs bs q) = Ifm bbs bs p"
by (induct p rule: disjuncts.induct) auto
lemma disjuncts_nb:
assumes nb: "bound0 p"
shows "\<forall>q \<in> set (disjuncts p). bound0 q"
proof -
from nb have "list_all bound0 (disjuncts p)"
by (induct p rule: disjuncts.induct) auto
thus ?thesis by (simp only: list_all_iff)
qed
lemma disjuncts_qf:
assumes qf: "qfree p"
shows "\<forall>q \<in> set (disjuncts p). qfree q"
proof -
from qf have "list_all qfree (disjuncts p)"
by (induct p rule: disjuncts.induct) auto
thus ?thesis by (simp only: list_all_iff)
qed
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
where "DJ f p = 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 bbs bs (DJ f p) = Ifm bbs bs (f p)"
proof -
have "Ifm bbs bs (DJ f p) = (\<exists>q \<in> set (disjuncts p). Ifm bbs bs (f q))"
by (simp add: DJ_def evaldjf_ex)
also have "\<dots> = Ifm bbs 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 bbs bs (qe p) = Ifm bbs bs (E p))"
shows "\<forall>bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bbs bs ((DJ qe p)) = Ifm bbs 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 bbs bs (DJ qe p) = (\<exists>q\<in> set (disjuncts p). Ifm bbs bs (qe q))"
by (simp add: DJ_def evaldjf_ex)
also have "\<dots> = (\<exists>q \<in> set(disjuncts p). Ifm bbs bs (E q))"
using qe disjuncts_qf[OF qf] by auto
also have "\<dots> = Ifm bbs bs (E p)"
by (induct p rule: disjuncts.induct) auto
finally show "qfree (DJ qe p) \<and> Ifm bbs bs (DJ qe p) = Ifm bbs bs (E p)"
using qfth by blast
qed
text {* Simplification *}
text {* Algebraic simplifications for nums *}
fun bnds :: "num \<Rightarrow> nat list"
where
"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 a = []"
fun lex_ns:: "nat list \<Rightarrow> nat list \<Rightarrow> bool"
where
"lex_ns [] ms = True"
| "lex_ns ns [] = False"
| "lex_ns (n#ns) (m#ms) = (n<m \<or> ((n = m) \<and> lex_ns ns ms)) "
definition lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool"
where "lex_bnd t s = lex_ns (bnds t) (bnds s)"
consts numadd:: "num \<times> num \<Rightarrow> num"
recdef numadd "measure (\<lambda>(t,s). num_size t + num_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,Add (Mul c2 (Bound n2)) r2))
else CN n2 c2 (numadd (Add (Mul c1 (Bound n1)) r1, r2)))"
"numadd (CN n1 c1 r1, t) = CN n1 c1 (numadd (r1, t))"
"numadd (t, CN n2 c2 r2) = CN n2 c2 (numadd (t, r2))"
"numadd (C b1, C b2) = C (b1 + b2)"
"numadd (a, b) = Add a b"
(*function (sequential)
numadd :: "num \<Rightarrow> num \<Rightarrow> num"
where
"numadd (Add (Mul c1 (Bound n1)) r1) (Add (Mul c2 (Bound n2)) r2) =
(if n1 = n2 then (let c = c1 + c2
in (if c = 0 then numadd r1 r2 else
Add (Mul c (Bound n1)) (numadd r1 r2)))
else if n1 \<le> n2 then
Add (Mul c1 (Bound n1)) (numadd r1 (Add (Mul c2 (Bound n2)) r2))
else
Add (Mul c2 (Bound n2)) (numadd (Add (Mul c1 (Bound n1)) r1) r2))"
| "numadd (Add (Mul c1 (Bound n1)) r1) t =
Add (Mul c1 (Bound n1)) (numadd r1 t)"
| "numadd t (Add (Mul c2 (Bound n2)) r2) =
Add (Mul c2 (Bound n2)) (numadd t r2)"
| "numadd (C b1) (C b2) = C (b1 + b2)"
| "numadd a b = Add a b"
apply pat_completeness apply auto*)
lemma numadd: "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")
apply(simp_all add: algebra_simps)
apply(simp add: distrib_right[symmetric])
done
lemma numadd_nb: "numbound0 t \<Longrightarrow> numbound0 s \<Longrightarrow> numbound0 (numadd (t, s))"
by (induct t s rule: numadd.induct) (auto simp add: Let_def)
fun nummul :: "int \<Rightarrow> num \<Rightarrow> num"
where
"nummul i (C j) = C (i * j)"
| "nummul i (CN n c t) = CN n (c*i) (nummul i t)"
| "nummul i t = Mul i t"
lemma nummul: "Inum bs (nummul i t) = Inum bs (Mul i t)"
by (induct t arbitrary: i rule: nummul.induct) (auto simp add: algebra_simps numadd)
lemma nummul_nb: "numbound0 t \<Longrightarrow> numbound0 (nummul i t)"
by (induct t arbitrary: i rule: nummul.induct) (auto simp add: numadd_nb)
definition numneg :: "num \<Rightarrow> num"
where "numneg t = nummul (- 1) t"
definition numsub :: "num \<Rightarrow> num \<Rightarrow> num"
where "numsub s t = (if s = t then C 0 else numadd (s, numneg t))"
lemma numneg: "Inum bs (numneg t) = Inum bs (Neg t)"
using numneg_def nummul by simp
lemma numneg_nb: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
using numneg_def nummul_nb by simp
lemma numsub: "Inum bs (numsub a b) = Inum bs (Sub a b)"
using numneg numadd numsub_def by simp
lemma numsub_nb: "numbound0 t \<Longrightarrow> numbound0 s \<Longrightarrow> numbound0 (numsub t s)"
using numsub_def numadd_nb numneg_nb by simp
fun simpnum :: "num \<Rightarrow> num"
where
"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 i (simpnum t))"
| "simpnum t = t"
lemma simpnum_ci: "Inum bs (simpnum t) = Inum bs t"
by (induct t rule: simpnum.induct) (auto simp add: numneg numadd numsub nummul)
lemma simpnum_numbound0: "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
by (induct t rule: simpnum.induct) (auto simp add: numadd_nb numsub_nb nummul_nb numneg_nb)
fun not :: "fm \<Rightarrow> fm"
where
"not (NOT p) = p"
| "not T = F"
| "not F = T"
| "not p = NOT p"
lemma not: "Ifm bbs bs (not p) = Ifm bbs bs (NOT p)"
by (cases p) auto
lemma not_qf: "qfree p \<Longrightarrow> qfree (not p)"
by (cases p) auto
lemma not_bn: "bound0 p \<Longrightarrow> bound0 (not p)"
by (cases p) auto
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
where
"conj p q = (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else And p q)"
lemma conj: "Ifm bbs bs (conj p q) = Ifm bbs bs (And p q)"
by (cases "p=F \<or> q=F", simp_all add: conj_def) (cases p, simp_all)
lemma conj_qf: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (conj p q)"
using conj_def by auto
lemma conj_nb: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (conj p q)"
using conj_def by auto
definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
where
"disj p q =
(if (p = T \<or> q=T) then T else if p=F then q else if q=F then p else Or p q)"
lemma disj: "Ifm bbs bs (disj p q) = Ifm bbs bs (Or p q)"
by (cases "p=T \<or> q=T", simp_all add: disj_def) (cases p, simp_all)
lemma disj_qf: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
using disj_def by auto
lemma disj_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
using disj_def by auto
definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
where
"imp p q = (if (p = F \<or> q=T) then T else if p=T then q else if q=F then not p else Imp p q)"
lemma imp: "Ifm bbs bs (imp p q) = Ifm bbs bs (Imp p q)"
by (cases "p=F \<or> q=T", simp_all add: imp_def, cases p) (simp_all add: not)
lemma imp_qf: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (imp p q)"
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def,cases p) (simp_all add: not_qf)
lemma imp_nb: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (imp p q)"
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def,cases p) simp_all
definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
where
"iff p q =
(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: "Ifm bbs bs (iff p q) = Ifm bbs 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: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
by (unfold iff_def,cases "p=q", auto simp add: not_qf)
lemma iff_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
using iff_def by (unfold iff_def,cases "p=q", auto simp add: not_bn)
function (sequential) simpfm :: "fm \<Rightarrow> fm"
where
"simpfm (And p q) = conj (simpfm p) (simpfm q)"
| "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
| "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
| "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
| "simpfm (NOT p) = not (simpfm p)"
| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F | _ \<Rightarrow> Lt a')"
| "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0) then T else F | _ \<Rightarrow> Le a')"
| "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0) then T else F | _ \<Rightarrow> Gt a')"
| "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0) then T else F | _ \<Rightarrow> Ge a')"
| "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0) then T else F | _ \<Rightarrow> Eq a')"
| "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0) then T else F | _ \<Rightarrow> NEq a')"
| "simpfm (Dvd i a) =
(if i=0 then simpfm (Eq a)
else if (abs i = 1) then T
else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v) then T else F | _ \<Rightarrow> Dvd i a')"
| "simpfm (NDvd i a) =
(if i=0 then simpfm (NEq a)
else if (abs i = 1) then F
else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> NDvd i a')"
| "simpfm p = p"
by pat_completeness auto
termination by (relation "measure fmsize") auto
lemma simpfm: "Ifm bbs bs (simpfm p) = Ifm bbs bs p"
proof(induct p rule: simpfm.induct)
case (6 a)
let ?sa = "simpnum a"
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
{ fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {
assume "\<not> (\<exists>v. ?sa = C v)"
hence ?case using sa by (cases ?sa) (simp_all add: Let_def)
}
ultimately show ?case by blast
next
case (7 a)
let ?sa = "simpnum a"
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
{ fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {
assume "\<not> (\<exists>v. ?sa = C v)"
hence ?case using sa by (cases ?sa) (simp_all add: Let_def)
}
ultimately show ?case by blast
next
case (8 a)
let ?sa = "simpnum a"
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
{ fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {
assume "\<not> (\<exists>v. ?sa = C v)"
hence ?case using sa by (cases ?sa) (simp_all add: Let_def)
}
ultimately show ?case by blast
next
case (9 a)
let ?sa = "simpnum a"
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
{ fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {
assume "\<not> (\<exists>v. ?sa = C v)"
hence ?case using sa by (cases ?sa) (simp_all add: Let_def)
}
ultimately show ?case by blast
next
case (10 a)
let ?sa = "simpnum a"
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
{ fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {
assume "\<not> (\<exists>v. ?sa = C v)"
hence ?case using sa by (cases ?sa) (simp_all add: Let_def)
}
ultimately show ?case by blast
next
case (11 a)
let ?sa = "simpnum a"
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
{ fix v assume "?sa = C v" hence ?case using sa by simp }
moreover {
assume "\<not> (\<exists>v. ?sa = C v)"
hence ?case using sa by (cases ?sa) (simp_all add: Let_def)
}
ultimately show ?case by blast
next
case (12 i a)
let ?sa = "simpnum a"
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
{ assume "i=0" hence ?case using "12.hyps" by (simp add: dvd_def Let_def) }
moreover
{ assume i1: "abs i = 1"
from one_dvd[of "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"]
have ?case using i1
apply (cases "i=0", simp_all add: Let_def)
apply (cases "i > 0", simp_all)
done
}
moreover
{ assume inz: "i\<noteq>0" and cond: "abs i \<noteq> 1"
{ fix v assume "?sa = C v"
hence ?case using sa[symmetric] inz cond
by (cases "abs i = 1") auto }
moreover {
assume "\<not> (\<exists>v. ?sa = C v)"
hence "simpfm (Dvd i a) = Dvd i ?sa" using inz cond
by (cases ?sa) (auto simp add: Let_def)
hence ?case using sa by simp }
ultimately have ?case by blast }
ultimately show ?case by blast
next
case (13 i a)
let ?sa = "simpnum a" from simpnum_ci
have sa: "Inum bs ?sa = Inum bs a" by simp
{ assume "i=0" hence ?case using "13.hyps" by (simp add: dvd_def Let_def) }
moreover
{ assume i1: "abs i = 1"
from one_dvd[of "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"]
have ?case using i1
apply (cases "i=0", simp_all add: Let_def)
apply (cases "i > 0", simp_all)
done
}
moreover
{ assume inz: "i\<noteq>0" and cond: "abs i \<noteq> 1"
{ fix v assume "?sa = C v"
hence ?case using sa[symmetric] inz cond
by (cases "abs i = 1") auto }
moreover {
assume "\<not> (\<exists>v. ?sa = C v)"
hence "simpfm (NDvd i a) = NDvd i ?sa" using inz cond
by (cases ?sa) (auto simp add: Let_def)
hence ?case using sa by simp }
ultimately have ?case by blast }
ultimately show ?case by blast
qed (simp_all add: conj disj imp iff not)
lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
proof (induct p rule: simpfm.induct)
case (6 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (7 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (8 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (9 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (10 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (11 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
thus ?case by (cases "simpnum a") (auto simp add: Let_def)
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)
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)
qed (auto simp add: disj_def imp_def iff_def conj_def not_bn)
lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
(case_tac "simpnum a", auto)+
text {* Generic quantifier elimination *}
function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
where
"qelim (E p) = (\<lambda>qe. DJ qe (qelim p qe))"
| "qelim (A p) = (\<lambda>qe. not (qe ((qelim (NOT p) qe))))"
| "qelim (NOT p) = (\<lambda>qe. not (qelim p qe))"
| "qelim (And p q) = (\<lambda>qe. conj (qelim p qe) (qelim q qe))"
| "qelim (Or p q) = (\<lambda>qe. disj (qelim p qe) (qelim q qe))"
| "qelim (Imp p q) = (\<lambda>qe. imp (qelim p qe) (qelim q qe))"
| "qelim (Iff p q) = (\<lambda>qe. iff (qelim p qe) (qelim q qe))"
| "qelim p = (\<lambda>y. simpfm p)"
by pat_completeness auto
termination by (relation "measure fmsize") auto
lemma qelim_ci:
assumes qe_inv: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bbs bs (qe p) = Ifm bbs bs (E p))"
shows "\<And>bs. qfree (qelim p qe) \<and> (Ifm bbs bs (qelim p qe) = Ifm bbs bs p)"
using qe_inv DJ_qe[OF qe_inv]
by(induct p rule: qelim.induct)
(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf
simpfm simpfm_qf simp del: simpfm.simps)
text {* Linearity for fm where Bound 0 ranges over @{text "\<int>"} *}
fun zsplit0 :: "num \<Rightarrow> int \<times> num" -- {* splits the bounded from the unbounded part *}
where
"zsplit0 (C c) = (0,C c)"
| "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))"
| "zsplit0 (CN n i a) =
(let (i',a') = zsplit0 a
in if n=0 then (i+i', a') else (i',CN n i a'))"
| "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'))"
lemma zsplit0_I:
shows "\<And>n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((x::int) #bs) (CN 0 n a) = Inum (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 m i a n a')
let ?j = "fst (zsplit0 a)"
let ?b = "snd (zsplit0 a)"
have abj: "zsplit0 a = (?j,?b)" by simp
{assume "m\<noteq>0"
with 3(1)[OF abj] 3(2) have ?case by (auto simp add: Let_def split_def)}
moreover
{assume m0: "m =0"
with abj have th: "a'=?b \<and> n=i+?j" using 3
by (simp add: Let_def split_def)
from abj 3 m0 have th2: "(?I x (CN 0 ?j ?b) = ?I x a) \<and> ?N ?b" by blast
from th have "?I x (CN 0 n a') = ?I x (CN 0 (i+?j) ?b)" by simp
also from th2 have "\<dots> = ?I x (CN 0 i (CN 0 ?j ?b))" by (simp add: distrib_right)
finally have "?I x (CN 0 n a') = ?I x (CN 0 i a)" using th2 by simp
with th2 th have ?case using m0 by blast}
ultimately show ?case by blast
next
case (4 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 4
by (simp add: Let_def split_def)
from abj 4 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 (5 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 5
by (simp add: Let_def split_def)
from abjs[symmetric] have bluddy: "\<exists>x y. (x,y) = zsplit0 s" by blast
from 5 have "(\<exists>x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by auto
with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
from abjs 5 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: distrib_right)
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=Sub ?as ?at \<and> n=?ns - ?nt" using 6
by (simp add: Let_def split_def)
from abjs[symmetric] have bluddy: "\<exists>x y. (x,y) = zsplit0 s" by blast
from 6 have "(\<exists>x y. (x,y) = zsplit0 s) \<longrightarrow>
(\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \<and> numbound0 xb)"
by auto
with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
from abjs 6 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 (7 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 7
by (simp add: Let_def split_def)
from abj 7 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
hence "?I x (Mul i t) = i * ?I x (CN 0 ?nt ?at)" by simp
also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: distrib_left)
finally show ?case using th th2 by simp
qed
consts iszlfm :: "fm \<Rightarrow> bool" -- {* Linearity test for fm *}
recdef iszlfm "measure size"
"iszlfm (And p q) = (iszlfm p \<and> iszlfm q)"
"iszlfm (Or p q) = (iszlfm p \<and> iszlfm q)"
"iszlfm (Eq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
"iszlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
"iszlfm (Lt (CN 0 c e)) = (c>0 \<and> numbound0 e)"
"iszlfm (Le (CN 0 c e)) = (c>0 \<and> numbound0 e)"
"iszlfm (Gt (CN 0 c e)) = (c>0 \<and> numbound0 e)"
"iszlfm (Ge (CN 0 c e)) = ( c>0 \<and> numbound0 e)"
"iszlfm (Dvd i (CN 0 c e)) =
(c>0 \<and> i>0 \<and> numbound0 e)"
"iszlfm (NDvd i (CN 0 c e))=
(c>0 \<and> i>0 \<and> numbound0 e)"
"iszlfm p = (isatom p \<and> (bound0 p))"
lemma zlin_qfree: "iszlfm p \<Longrightarrow> qfree p"
by (induct p rule: iszlfm.induct) auto
consts zlfm :: "fm \<Rightarrow> fm" -- {* Linearity transformation for fm *}
recdef zlfm "measure fmsize"
"zlfm (And p q) = And (zlfm p) (zlfm q)"
"zlfm (Or p q) = Or (zlfm p) (zlfm q)"
"zlfm (Imp p q) = Or (zlfm (NOT p)) (zlfm q)"
"zlfm (Iff p q) = Or (And (zlfm p) (zlfm q)) (And (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 (Lt (CN 0 c r)) else (Gt (CN 0 (- c) (Neg r))))"
"zlfm (Le a) = (let (c,r) = zsplit0 a in
if c=0 then Le r else
if c>0 then (Le (CN 0 c r)) else (Ge (CN 0 (- c) (Neg r))))"
"zlfm (Gt a) = (let (c,r) = zsplit0 a in
if c=0 then Gt r else
if c>0 then (Gt (CN 0 c r)) else (Lt (CN 0 (- c) (Neg r))))"
"zlfm (Ge a) = (let (c,r) = zsplit0 a in
if c=0 then Ge r else
if c>0 then (Ge (CN 0 c r)) else (Le (CN 0 (- c) (Neg r))))"
"zlfm (Eq a) = (let (c,r) = zsplit0 a in
if c=0 then Eq r else
if c>0 then (Eq (CN 0 c r)) else (Eq (CN 0 (- c) (Neg r))))"
"zlfm (NEq a) = (let (c,r) = zsplit0 a in
if c=0 then NEq r else
if c>0 then (NEq (CN 0 c r)) else (NEq (CN 0 (- c) (Neg 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 (Dvd (abs i) (CN 0 c r))
else (Dvd (abs i) (CN 0 (- c) (Neg 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 (NDvd (abs i) (CN 0 c r))
else (NDvd (abs i) (CN 0 (- c) (Neg r)))))"
"zlfm (NOT (And p q)) = Or (zlfm (NOT p)) (zlfm (NOT q))"
"zlfm (NOT (Or p q)) = And (zlfm (NOT p)) (zlfm (NOT q))"
"zlfm (NOT (Imp p q)) = And (zlfm p) (zlfm (NOT q))"
"zlfm (NOT (Iff p q)) = Or (And(zlfm p) (zlfm(NOT q))) (And (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 (NOT (Closed P)) = NClosed P"
"zlfm (NOT (NClosed P)) = Closed P"
"zlfm p = p" (hints simp add: fmsize_pos)
lemma zlfm_I:
assumes qfp: "qfree p"
shows "(Ifm bbs (i#bs) (zlfm p) = Ifm bbs (i# bs) p) \<and> iszlfm (zlfm p)"
(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 (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda>t. Inum (i#bs) t"
from 5 Ia nb show ?case
apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r", auto)
apply (case_tac nat, auto)
done
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 (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda>t. Inum (i#bs) t"
from 6 Ia nb show ?case
apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r", auto)
apply (case_tac nat, auto)
done
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 (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda>t. Inum (i#bs) t"
from 7 Ia nb show ?case
apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r", auto)
apply (case_tac nat, auto)
done
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 (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda>t. Inum (i#bs) t"
from 8 Ia nb show ?case
apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r", auto)
apply (case_tac nat, auto)
done
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 (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda>t. Inum (i#bs) t"
from 9 Ia nb show ?case
apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r", auto)
apply (case_tac nat, auto)
done
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 (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda>t. Inum (i#bs) t"
from 10 Ia nb show ?case
apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done
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 (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda>t. Inum (i#bs) t"
have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0) \<or> (j\<noteq> 0 \<and> ?c<0)" by arith
moreover
{assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def)
hence ?case using 11 `j = 0` by (simp del: zlfm.simps) }
moreover
{assume "?c=0" and "j\<noteq>0" hence ?case
using zsplit0_I[OF spl, where x="i" and bs="bs"]
apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done}
moreover
{assume cp: "?c > 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))"
by (simp add: nb Let_def split_def)
hence ?case using Ia cp jnz by (simp add: Let_def split_def)}
moreover
{assume cn: "?c < 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))"
by (simp add: nb Let_def split_def)
hence ?case using Ia cn jnz dvd_minus_iff[of "abs j" "?c*i + ?N ?r" ]
by (simp add: Let_def split_def) }
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 (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda>t. Inum (i#bs) t"
have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0) \<or> (j\<noteq> 0 \<and> ?c<0)" by arith
moreover
{assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def)
hence ?case using assms 12 `j = 0` by (simp del: zlfm.simps)}
moreover
{assume "?c=0" and "j\<noteq>0" hence ?case
using zsplit0_I[OF spl, where x="i" and bs="bs"]
apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done}
moreover
{assume cp: "?c > 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))"
by (simp add: nb Let_def split_def)
hence ?case using Ia cp jnz by (simp add: Let_def split_def) }
moreover
{assume cn: "?c < 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))"
by (simp add: nb Let_def split_def)
hence ?case using Ia cn jnz dvd_minus_iff[of "abs j" "?c*i + ?N ?r"]
by (simp add: Let_def split_def)}
ultimately show ?case by blast
qed auto
consts minusinf :: "fm \<Rightarrow> fm" -- {* Virtual substitution of @{text "-\<infinity>"} *}
recdef minusinf "measure size"
"minusinf (And p q) = And (minusinf p) (minusinf q)"
"minusinf (Or p q) = Or (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
consts plusinf :: "fm \<Rightarrow> fm" -- {* Virtual substitution of @{text "+\<infinity>"} *}
recdef plusinf "measure size"
"plusinf (And p q) = And (plusinf p) (plusinf q)"
"plusinf (Or p q) = Or (plusinf p) (plusinf q)"
"plusinf (Eq (CN 0 c e)) = F"
"plusinf (NEq (CN 0 c e)) = T"
"plusinf (Lt (CN 0 c e)) = F"
"plusinf (Le (CN 0 c e)) = F"
"plusinf (Gt (CN 0 c e)) = T"
"plusinf (Ge (CN 0 c e)) = T"
"plusinf p = p"
consts \<delta> :: "fm \<Rightarrow> int" -- {* Compute @{text "lcm {d| N\<^isup>? Dvd c*x+t \<in> p}"} *}
recdef \<delta> "measure size"
"\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)"
"\<delta> (Or p q) = lcm (\<delta> p) (\<delta> q)"
"\<delta> (Dvd i (CN 0 c e)) = i"
"\<delta> (NDvd i (CN 0 c e)) = i"
"\<delta> p = 1"
consts d_\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool" -- {* check if a given l divides all the ds above *}
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"
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"
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 1 lcm_pos_int have dp: "?d >0" by simp
have d1: "\<delta> p dvd \<delta> (And p q)" using 1 by simp
hence th: "d_\<delta> p ?d" using delta_mono 1(2,3) by(simp only: iszlfm.simps)
have "\<delta> q dvd \<delta> (And p q)" using 1 by simp
hence th': "d_\<delta> q ?d" using delta_mono 1 by(simp only: iszlfm.simps)
from th th' dp show ?case by simp
next
case (2 p q)
let ?d = "\<delta> (And p q)"
from 2 lcm_pos_int have dp: "?d >0" by simp
have "\<delta> p dvd \<delta> (And p q)" using 2 by simp
hence th: "d_\<delta> p ?d" using delta_mono 2 by(simp only: iszlfm.simps)
have "\<delta> q dvd \<delta> (And p q)" using 2 by simp
hence th': "d_\<delta> q ?d" using delta_mono 2 by(simp only: iszlfm.simps)
from th th' dp show ?case by simp
qed simp_all
consts a_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" -- {* adjust the coeffitients of a formula *}
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)"
consts d_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" -- {* test if all coeffs c of c divide a given l *}
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)"
consts \<zeta> :: "fm \<Rightarrow> int" -- {* computes the lcm of all coefficients of x *}
recdef \<zeta> "measure size"
"\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)"
"\<zeta> (Or p q) = lcm (\<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"
consts \<beta> :: "fm \<Rightarrow> num list"
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 = []"
consts \<alpha> :: "fm \<Rightarrow> num list"
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"
text {* Lemmas for the correctness of @{text "\<sigma>_\<rho>"} *}
lemma dvd1_eq1: "x >0 \<Longrightarrow> (x::int) dvd 1 = (x = 1)"
by simp
lemma minusinf_inf:
assumes linp: "iszlfm p"
and u: "d_\<beta> p 1"
shows "\<exists>(z::int). \<forall>x < z. Ifm bbs (x#bs) (minusinf p) = Ifm bbs (x#bs) p"
(is "?P p" is "\<exists>(z::int). \<forall>x < z. ?I x (?M p) = ?I x p")
using linp u
proof (induct p rule: minusinf.induct)
case (1 p q) thus ?case
by auto (rule_tac x="min z za" in exI,simp)
next
case (2 p q) thus ?case
by auto (rule_tac x="min z za" in exI,simp)
next
case (3 c e) hence c1: "c=1" and nb: "numbound0 e" by simp_all
fix a
from 3 have "\<forall>x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \<noteq> 0"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e = 0"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "False" by simp
qed
thus ?case by auto
next
case (4 c e) hence c1: "c=1" and nb: "numbound0 e" by simp_all
fix a
from 4 have "\<forall>x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \<noteq> 0"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e = 0"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "False" by simp
qed
thus ?case by auto
next
case (5 c e) hence c1: "c=1" and nb: "numbound0 e" by simp_all
fix a
from 5 have "\<forall>x<(- Inum (a#bs) e). c*x + Inum (x#bs) e < 0"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "x + Inum (x#bs) e < 0" by simp
qed
thus ?case by auto
next
case (6 c e) hence c1: "c=1" and nb: "numbound0 e" by simp_all
fix a
from 6 have "\<forall>x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \<le> 0"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "x + Inum (x#bs) e \<le> 0" by simp
qed
thus ?case by auto
next
case (7 c e) hence c1: "c=1" and nb: "numbound0 e" by simp_all
fix a
from 7 have "\<forall>x<(- Inum (a#bs) e). \<not> (c*x + Inum (x#bs) e > 0)"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e > 0"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "False" by simp
qed
thus ?case by auto
next
case (8 c e) hence c1: "c=1" and nb: "numbound0 e" by simp_all
fix a
from 8 have "\<forall>x<(- Inum (a#bs) e). \<not> (c*x + Inum (x#bs) e \<ge> 0)"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e \<ge> 0"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "False" by simp
qed
thus ?case by auto
qed auto
lemma minusinf_repeats:
assumes d: "d_\<delta> p d" and linp: "iszlfm p"
shows "Ifm bbs ((x - k*d)#bs) (minusinf p) = Ifm bbs (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_all
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="x - k * d" and b'="x"] right_diff_distrib,
rule iffI)
assume "i dvd c * x - c*(k*d) + Inum (x # bs) e"
(is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt")
hence "\<exists>(l::int). ?rt = i * l" by (simp add: dvd_def)
hence "\<exists>(l::int). c*x+ ?I x e = i*l+c*(k * i*di)"
by (simp add: algebra_simps di_def)
hence "\<exists>(l::int). c*x+ ?I x e = i*(l + c*k*di)"
by (simp add: algebra_simps)
hence "\<exists>(l::int). c*x+ ?I x e = i*l" by blast
thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def)
next
assume "i dvd c*x + Inum (x # bs) e" (is "?ri dvd ?rc*?rx+?e")
hence "\<exists>(l::int). c*x+?e = i*l" by (simp add: dvd_def)
hence "\<exists>(l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp
hence "\<exists>(l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def)
hence "\<exists>(l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists>(l::int). c*x - c * (k*d) +?e = i*l" by blast
thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
qed
next
case (10 i c e)
hence nbe: "numbound0 e" and id: "i dvd d" by simp_all
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="x - k * d" and b'="x"] right_diff_distrib, rule iffI)
assume "i dvd c * x - c*(k*d) + Inum (x # bs) e"
(is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt")
hence "\<exists>(l::int). ?rt = i * l" by (simp add: dvd_def)
hence "\<exists>(l::int). c*x+ ?I x e = i*l+c*(k * i*di)"
by (simp add: algebra_simps di_def)
hence "\<exists>(l::int). c*x+ ?I x e = i*(l + c*k*di)"
by (simp add: algebra_simps)
hence "\<exists>(l::int). c*x+ ?I x e = i*l" by blast
thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def)
next
assume
"i dvd c*x + Inum (x # bs) e" (is "?ri dvd ?rc*?rx+?e")
hence "\<exists>(l::int). c*x+?e = i*l" by (simp add: dvd_def)
hence "\<exists>(l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp
hence "\<exists>(l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def)
hence "\<exists>(l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists>(l::int). c*x - c * (k*d) +?e = i*l"
by blast
thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
qed
qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="x - k*d" and b'="x"])
lemma mirror_\<alpha>_\<beta>:
assumes lp: "iszlfm p"
shows "(Inum (i#bs)) ` set (\<alpha> p) = (Inum (i#bs)) ` set (\<beta> (mirror p))"
using lp by (induct p rule: mirror.induct) auto
lemma mirror:
assumes lp: "iszlfm p"
shows "Ifm bbs (x#bs) (mirror p) = Ifm bbs ((- x)#bs) p"
using lp
proof (induct p rule: iszlfm.induct)
case (9 j c e)
hence nb: "numbound0 e" by simp
have "Ifm bbs (x#bs) (mirror (Dvd j (CN 0 c e))) = (j dvd c*x - Inum (x#bs) e)"
(is "_ = (j dvd c*x - ?e)") by simp
also have "\<dots> = (j dvd (- (c*x - ?e)))"
by (simp only: dvd_minus_iff)
also have "\<dots> = (j dvd (c* (- x)) + ?e)"
apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_minus add_ac minus_add_distrib)
apply (simp add: algebra_simps)
done
also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"] by simp
finally show ?case .
next
case (10 j c e) hence nb: "numbound0 e" by simp
have "Ifm bbs (x#bs) (mirror (Dvd j (CN 0 c e))) = (j dvd c*x - Inum (x#bs) e)"
(is "_ = (j dvd c*x - ?e)") by simp
also have "\<dots> = (j dvd (- (c*x - ?e)))"
by (simp only: dvd_minus_iff)
also have "\<dots> = (j dvd (c* (- x)) + ?e)"
apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_minus add_ac minus_add_distrib)
apply (simp add: algebra_simps)
done
also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"] by simp
finally show ?case by simp
qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] gr0_conv_Suc)
lemma mirror_l: "iszlfm p \<and> d_\<beta> p 1 \<Longrightarrow> iszlfm (mirror p) \<and> d_\<beta> (mirror p) 1"
by (induct p rule: mirror.induct) auto
lemma mirror_\<delta>: "iszlfm p \<Longrightarrow> \<delta> (mirror p) = \<delta> p"
by (induct p rule: mirror.induct) auto
lemma \<beta>_numbound0:
assumes lp: "iszlfm p"
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"
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"
shows "\<forall>b \<in> set (\<alpha> p). numbound0 b"
using lp by (induct p rule: \<alpha>.induct) auto
lemma \<zeta>:
assumes linp: "iszlfm p"
shows "\<zeta> p > 0 \<and> d_\<beta> p (\<zeta> p)"
using linp
proof (induct p rule: iszlfm.induct)
case (1 p q)
from 1 have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
from 1 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
from 1 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"]
dl1 dl2 show ?case by (auto simp add: lcm_pos_int)
next
case (2 p q)
from 2 have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
from 2 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
from 2 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"]
dl1 dl2 show ?case by (auto simp add: lcm_pos_int)
qed (auto simp add: lcm_pos_int)
lemma a_\<beta>:
assumes linp: "iszlfm p" and d: "d_\<beta> p l" and lp: "l > 0"
shows "iszlfm (a_\<beta> p l) \<and> d_\<beta> (a_\<beta> p l) 1 \<and> (Ifm bbs (l*x #bs) (a_\<beta> p l) = Ifm bbs (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 d': "c dvd l" by simp_all
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: div_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 "(l*x + (l div c) * Inum (x # bs) e < 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e < 0)"
by simp
also have "\<dots> = ((l div c) * (c*x + Inum (x # bs) e) < (l div c) * 0)"
by (simp add: algebra_simps)
also have "\<dots> = (c*x + Inum (x # bs) e < 0)"
using mult_less_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp
finally show ?case
using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be by simp
next
case (6 c e)
hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp_all
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: div_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 "(l*x + (l div c) * Inum (x# bs) e \<le> 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e \<le> 0)" by simp
also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<le> ((l div c)) * 0)"
by (simp add: algebra_simps)
also have "\<dots> = (c*x + Inum (x # bs) e \<le> 0)"
using mult_le_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp
finally show ?case
using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be by simp
next
case (7 c e)
hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp_all
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: div_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 "(l*x + (l div c)* Inum (x # bs) e > 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e > 0)" by simp
also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) > ((l div c)) * 0)"
by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e > 0)"
using zero_less_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp
next
case (8 c e)
hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp_all
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: div_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 "(l*x + (l div c)* Inum (x # bs) e \<ge> 0) =
((c*(l div c))*x + (l div c)* Inum (x # bs) e \<ge> 0)" by simp
also have "\<dots> = ((l div c)*(c*x + Inum (x # bs) e) \<ge> ((l div c)) * 0)"
by (simp add: algebra_simps)
also have "\<dots> = (c*x + Inum (x # bs) e \<ge> 0)"
using ldcp zero_le_mult_iff [where a="l div c" and b="c*x + Inum (x # bs) e"] by simp
finally show ?case
using be numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] by simp
next
case (3 c e)
hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp_all
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: div_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 "(l * x + (l div c) * Inum (x # bs) e = 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e = 0)" by simp
also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) = ((l div c)) * 0)"
by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e = 0)"
using mult_eq_0_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp
next
case (4 c e)
hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp_all
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: div_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 "(l * x + (l div c) * Inum (x # bs) e \<noteq> 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e \<noteq> 0)" by simp
also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<noteq> ((l div c)) * 0)"
by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e \<noteq> 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp
next
case (9 j c e)
hence cp: "c>0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" by simp_all
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: div_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). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) =
(\<exists>(k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)" by simp
also have "\<dots> = (\<exists>(k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)"
by (simp add: algebra_simps)
also have "\<dots> = (\<exists>(k::int). c * x + Inum (x # bs) e - j * k = 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp
by simp
also have "\<dots> = (\<exists>(k::int). c * x + Inum (x # bs) e = j * k)" by simp
finally show ?case
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be mult_strict_mono[OF ldcp jp ldcp ]
by (simp add: dvd_def)
next
case (10 j c e)
hence cp: "c>0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" by simp_all
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: div_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). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\<exists>(k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)" by simp
also have "\<dots> = (\<exists>(k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists>(k::int). c * x + Inum (x # bs) e - j * k = 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp
also have "\<dots> = (\<exists>(k::int). c * x + Inum (x # bs) e = j * k)" by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def)
qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="(l * x)" and b'="x"])
lemma a_\<beta>_ex: assumes linp: "iszlfm p" and d: "d_\<beta> p l" and lp: "l>0"
shows "(\<exists>x. l dvd x \<and> Ifm bbs (x #bs) (a_\<beta> p l)) = (\<exists>(x::int). Ifm bbs (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"
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). x = b + j)"
and p: "Ifm bbs (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" by simp_all
with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] 5
show ?case by simp
next
case (6 c e) hence c1: "c=1" and bn:"numbound0 e" by simp_all
with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] 6
show ?case by simp
next
case (7 c e) hence p: "Ifm bbs (x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" by simp_all
let ?e = "Inum (x # bs) e"
{assume "(x-d) +?e > 0" hence ?case using c1
numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] by simp}
moreover
{assume H: "\<not> (x-d) + ?e > 0"
let ?v="Neg e"
have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp
from 7(5)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]]
have nob: "\<not> (\<exists>j\<in> {1 ..d}. x = - ?e + j)" by auto
from H p have "x + ?e > 0 \<and> x + ?e \<le> d" by (simp add: c1)
hence "x + ?e \<ge> 1 \<and> x + ?e \<le> d" by simp
hence "\<exists>(j::int) \<in> {1 .. d}. j = x + ?e" by simp
hence "\<exists>(j::int) \<in> {1 .. d}. x = (- ?e + j)"
by (simp add: algebra_simps)
with nob have ?case by auto}
ultimately show ?case by blast
next
case (8 c e) hence p: "Ifm bbs (x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e"
by simp_all
let ?e = "Inum (x # bs) e"
{assume "(x-d) +?e \<ge> 0" hence ?case using c1
numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"]
by simp}
moreover
{assume H: "\<not> (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 8(5)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]]
have nob: "\<not> (\<exists>j\<in> {1 ..d}. x = - ?e - 1 + j)" by auto
from H p have "x + ?e \<ge> 0 \<and> x + ?e < d" by (simp add: c1)
hence "x + ?e +1 \<ge> 1 \<and> x + ?e + 1 \<le> d" by simp
hence "\<exists>(j::int) \<in> {1 .. d}. j = x + ?e + 1" by simp
hence "\<exists>(j::int) \<in> {1 .. d}. x= - ?e - 1 + j" by (simp add: algebra_simps)
with nob have ?case by simp }
ultimately show ?case by blast
next
case (3 c e) hence p: "Ifm bbs (x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp_all
let ?e = "Inum (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 "x= - ?e" by (simp add: c1) with 3(5) show ?case using dp
by simp (erule ballE[where x="1"],
simp_all add:algebra_simps numbound0_I[OF bn,where b="x"and b'="a"and bs="bs"])
next
case (4 c e)hence p: "Ifm bbs (x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp_all
let ?e = "Inum (x # bs) e"
let ?v="Neg e"
have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp
{assume "x - d + Inum (((x -d)) # bs) e \<noteq> 0"
hence ?case by (simp add: c1)}
moreover
{assume H: "x - d + Inum (((x -d)) # bs) e = 0"
hence "x = - Inum (((x -d)) # bs) e + d" by simp
hence "x = - Inum (a # bs) e + d"
by (simp add: numbound0_I[OF bn,where b="x - d"and b'="a"and bs="bs"])
with 4(5) have ?case using dp by simp}
ultimately show ?case by blast
next
case (9 j c e) hence p: "Ifm bbs (x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp_all
let ?e = "Inum (x # bs) e"
from 9 have id: "j dvd d" by simp
from c1 have "?p x = (j dvd (x+ ?e))" by simp
also have "\<dots> = (j dvd x - d + ?e)"
using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
finally show ?case
using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp
next
case (10 j c e) hence p: "Ifm bbs (x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp_all
let ?e = "Inum (x # bs) e"
from 10 have id: "j dvd d" by simp
from c1 have "?p x = (\<not> j dvd (x+ ?e))" by simp
also have "\<dots> = (\<not> j dvd x - d + ?e)"
using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
finally show ?case using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp
qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] gr0_conv_Suc)
lemma \<beta>':
assumes lp: "iszlfm p"
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 bbs ((Inum (a#bs) b + j) #bs) p) \<longrightarrow> Ifm bbs (x#bs) p \<longrightarrow> Ifm bbs ((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). x = b + j)"
by auto
from \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" .
qed
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(fastforce)
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 fastforce
apply(drule (1) periodic_finite_ex)
apply blast
apply(blast dest:decr_mult_lemma)
done
theorem cp_thm:
assumes lp: "iszlfm p"
and u: "d_\<beta> p 1"
and d: "d_\<delta> p d"
and dp: "d > 0"
shows "(\<exists>(x::int). Ifm bbs (x #bs) p) = (\<exists>j\<in> {1.. d}. Ifm bbs (j #bs) (minusinf p) \<or> (\<exists>b \<in> set (\<beta> p). Ifm bbs ((Inum (i#bs) b + j) #bs) p))"
(is "(\<exists>(x::int). ?P (x)) = (\<exists>j\<in> ?D. ?M j \<or> (\<exists>b\<in> ?B. ?P (?I b + j)))")
proof-
from minusinf_inf[OF lp u]
have th: "\<exists>(z::int). \<forall>x<z. ?P (x) = ?M x" by blast
let ?B' = "{?I b | b. b\<in> ?B}"
have BB': "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b +j)) = (\<exists>j \<in> ?D. \<exists>b \<in> ?B'. ?P (b + j))" by auto
hence th2: "\<forall>x. \<not> (\<exists>j \<in> ?D. \<exists>b \<in> ?B'. ?P ((b + j))) \<longrightarrow> ?P (x) \<longrightarrow> ?P ((x - d))"
using \<beta>'[OF lp u d dp, where a="i" and bbs = "bbs"] 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
(* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *)
lemma mirror_ex:
assumes lp: "iszlfm p"
shows "(\<exists>x. Ifm bbs (x#bs) (mirror p)) = (\<exists>x. Ifm bbs (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 cp_thm':
assumes lp: "iszlfm p"
and up: "d_\<beta> p 1" and dd: "d_\<delta> p d" and dp: "d > 0"
shows "(\<exists>x. Ifm bbs (x#bs) p) = ((\<exists>j\<in> {1 .. d}. Ifm bbs (j#bs) (minusinf p)) \<or> (\<exists>j\<in> {1.. d}. \<exists>b\<in> (Inum (i#bs)) ` set (\<beta> p). Ifm bbs ((b+j)#bs) p))"
using cp_thm[OF lp up dd dp,where i="i"] by auto
definition unit :: "fm \<Rightarrow> fm \<times> num list \<times> int"
where
"unit p = (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. Ifm bbs (x#bs) p) = (\<exists>x. Ifm bbs (x#bs) q)) \<and> (Inum (i#bs)) ` set B = (Inum (i#bs)) ` set (\<beta> q) \<and> d_\<beta> q 1 \<and> d_\<delta> q d \<and> d >0 \<and> iszlfm q \<and> (\<forall>b\<in> set B. numbound0 b)"
proof -
fix q B d
assume qBd: "unit p = (q,B,d)"
let ?thes = "((\<exists>x. Ifm bbs (x#bs) p) = (\<exists>x. Ifm bbs (x#bs) q)) \<and>
Inum (i#bs) ` set B = Inum (i#bs) ` set (\<beta> q) \<and>
d_\<beta> q 1 \<and> d_\<delta> q d \<and> 0 < d \<and> iszlfm q \<and> (\<forall>b\<in> set B. numbound0 b)"
let ?I = "\<lambda>x p. Ifm bbs (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 conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]
have lp': "iszlfm ?p'" .
from lp' \<zeta>[where p="?p'"] 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
from lp' lp a_\<beta>[OF lp' dl lp] have lq:"iszlfm ?q" and uq: "d_\<beta> ?q 1" by auto
from \<delta>[OF lq] have dp:"?d >0" and dd: "d_\<delta> ?q ?d" by blast+
let ?N = "\<lambda>t. Inum (i#bs) t"
have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by auto
also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="i#bs"] by auto
finally have BB': "?N ` set ?B' = ?N ` ?B" .
have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by auto
also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="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_all
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_all
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_l[where p="?q"]
have lq': "iszlfm q" 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
text {* Cooper's Algorithm *}
definition cooper :: "fm \<Rightarrow> fm" where
"cooper p =
(let
(q, B, d) = unit p;
js = [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>(b, j). simpfm (subst0 (Add b (C j)) q)) [(b, j). b \<leftarrow> B, j \<leftarrow> js]
in decr (disj md qd)))"
lemma cooper:
assumes qf: "qfree p"
shows "((\<exists>x. Ifm bbs (x#bs) p) = (Ifm bbs bs (cooper p))) \<and> qfree (cooper p)"
(is "(?lhs = ?rhs) \<and> _")
proof -
let ?I = "\<lambda>x p. Ifm bbs (x#bs) p"
let ?q = "fst (unit p)"
let ?B = "fst (snd(unit p))"
let ?d = "snd (snd (unit p))"
let ?js = "[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 (i#bs) t"
let ?Bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]"
let ?qd = "evaldjf (\<lambda>(b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs"
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" 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. bound0 (subst0 (Add b (C j)) ?q)"
using subst0_bound0[OF qfq] by blast
hence "\<forall>(b,j) \<in> set ?Bjs. bound0 (simpfm (subst0 (Add b (C j)) ?q))"
using simpfm_bound0 by blast
hence th': "\<forall>x \<in> set ?Bjs. bound0 ((\<lambda>(b,j). simpfm (subst0 (Add b (C j)) ?q)) x)"
by auto
from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp
from mdb qdb
have mdqdb: "bound0 (disj ?md ?qd)" unfolding disj_def by (cases "?md=T \<or> ?qd=T") simp_all
from trans [OF pq_ex cp_thm'[OF lq uq dd dp,where i="i"]] B
have "?lhs = (\<exists>j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists>b\<in> ?N ` set ?B. Ifm bbs ((b+ j)#bs) ?q))" by auto
also have "\<dots> = (\<exists>j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists>b\<in> set ?B. Ifm bbs ((?N b+ j)#bs) ?q))" by simp
also have "\<dots> = ((\<exists>j\<in> {1.. ?d}. ?I j ?mq ) \<or>
(\<exists>j\<in> {1.. ?d}. \<exists>b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))"
by (simp only: Inum.simps) blast
also have "\<dots> = ((\<exists>j\<in> {1.. ?d}. ?I j ?smq ) \<or>
(\<exists>j\<in> {1.. ?d}. \<exists>b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))"
by (simp add: simpfm)
also have "\<dots> = ((\<exists>j\<in> set ?js. (\<lambda>j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or>
(\<exists>j\<in> set ?js. \<exists>b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))"
by (simp only: simpfm subst0_I[OF qfmq] set_upto) auto
also have "\<dots> = (?I i (evaldjf (\<lambda>j. simpfm (subst0 (C j) ?smq)) ?js) \<or>
(\<exists>j\<in> set ?js. \<exists>b\<in> set ?B. ?I i (subst0 (Add b (C j)) ?q)))"
by (simp only: evaldjf_ex subst0_I[OF qfq])
also have "\<dots>= (?I i ?md \<or> (\<exists>(b,j) \<in> set ?Bjs. (\<lambda>(b,j). ?I i (simpfm (subst0 (Add b (C j)) ?q))) (b,j)))"
by (simp only: simpfm set_concat set_map concat_map_singleton UN_simps) blast
also have "\<dots> = (?I i ?md \<or> (?I i (evaldjf (\<lambda>(b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs)))"
by (simp only: evaldjf_ex[where bs="i#bs" and f="\<lambda>(b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="?Bjs"])
(auto simp add: split_def)
finally have mdqd: "?lhs = (?I i ?md \<or> ?I i ?qd)" by simp
also have "\<dots> = (?I i (disj ?md ?qd))" by (simp add: disj)
also have "\<dots> = (Ifm bbs bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb])
finally have mdqd2: "?lhs = (Ifm bbs bs (decr (disj ?md ?qd)))" .
{ 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 have lhs:"?lhs" using mdqd by simp
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
definition pa :: "fm \<Rightarrow> fm" where
"pa p = qelim (prep p) cooper"
theorem mirqe: "(Ifm bbs bs (pa p) = Ifm bbs bs p) \<and> qfree (pa p)"
using qelim_ci cooper prep by (auto simp add: pa_def)
definition cooper_test :: "unit \<Rightarrow> fm"
where
"cooper_test u =
pa (E (A (Imp (Ge (Sub (Bound 0) (Bound 1)))
(E (E (Eq (Sub (Add (Mul 3 (Bound 1)) (Mul 5 (Bound 0))) (Bound 2))))))))"
ML {* @{code cooper_test} () *}
(*code_reflect Cooper_Procedure
functions pa
file "~~/src/HOL/Tools/Qelim/cooper_procedure.ML"*)
oracle linzqe_oracle = {*
let
fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
of NONE => error "Variable not found in the list!"
| SOME n => @{code Bound} (@{code nat_of_integer} n))
| num_of_term vs @{term "0::int"} = @{code C} (@{code int_of_integer} 0)
| num_of_term vs @{term "1::int"} = @{code C} (@{code int_of_integer} 1)
| num_of_term vs (@{term "numeral :: _ \<Rightarrow> int"} $ t) =
@{code C} (@{code int_of_integer} (HOLogic.dest_num t))
| num_of_term vs (@{term "neg_numeral :: _ \<Rightarrow> int"} $ t) =
@{code C} (@{code int_of_integer} (~(HOLogic.dest_num t)))
| num_of_term vs (Bound i) = @{code Bound} (@{code nat_of_integer} i)
| num_of_term vs (@{term "uminus :: int \<Rightarrow> int"} $ t') = @{code Neg} (num_of_term vs t')
| num_of_term vs (@{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
@{code Add} (num_of_term vs t1, num_of_term vs t2)
| num_of_term vs (@{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
@{code Sub} (num_of_term vs t1, num_of_term vs t2)
| num_of_term vs (@{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) =
(case try HOLogic.dest_number t1
of SOME (_, i) => @{code Mul} (@{code int_of_integer} i, num_of_term vs t2)
| NONE => (case try HOLogic.dest_number t2
of SOME (_, i) => @{code Mul} (@{code int_of_integer} i, num_of_term vs t1)
| NONE => error "num_of_term: unsupported multiplication"))
| num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);
fun fm_of_term ps vs @{term True} = @{code T}
| fm_of_term ps vs @{term False} = @{code F}
| fm_of_term ps vs (@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
@{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term ps vs (@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
@{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term ps vs (@{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
@{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term ps vs (@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) =
(case try HOLogic.dest_number t1
of SOME (_, i) => @{code Dvd} (@{code int_of_integer} i, num_of_term vs t2)
| NONE => error "num_of_term: unsupported dvd")
| fm_of_term ps vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
@{code Iff} (fm_of_term ps vs t1, fm_of_term ps vs t2)
| fm_of_term ps vs (@{term HOL.conj} $ t1 $ t2) =
@{code And} (fm_of_term ps vs t1, fm_of_term ps vs t2)
| fm_of_term ps vs (@{term HOL.disj} $ t1 $ t2) =
@{code Or} (fm_of_term ps vs t1, fm_of_term ps vs t2)
| fm_of_term ps vs (@{term HOL.implies} $ t1 $ t2) =
@{code Imp} (fm_of_term ps vs t1, fm_of_term ps vs t2)
| fm_of_term ps vs (@{term "Not"} $ t') =
@{code NOT} (fm_of_term ps vs t')
| fm_of_term ps vs (Const (@{const_name Ex}, _) $ Abs (xn, xT, p)) =
let
val (xn', p') = Syntax_Trans.variant_abs (xn, xT, p); (* FIXME !? *)
val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs;
in @{code E} (fm_of_term ps vs' p) end
| fm_of_term ps vs (Const (@{const_name All}, _) $ Abs (xn, xT, p)) =
let
val (xn', p') = Syntax_Trans.variant_abs (xn, xT, p); (* FIXME !? *)
val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs;
in @{code A} (fm_of_term ps vs' p) end
| fm_of_term ps vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
fun term_of_num vs (@{code C} i) = HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i)
| term_of_num vs (@{code Bound} n) =
let
val q = @{code integer_of_nat} n
in fst (the (find_first (fn (_, m) => q = m) vs)) end
| term_of_num vs (@{code Neg} t') = @{term "uminus :: int \<Rightarrow> int"} $ term_of_num vs t'
| term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} $
term_of_num vs t1 $ term_of_num vs t2
| term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} $
term_of_num vs t1 $ term_of_num vs t2
| term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} $
term_of_num vs (@{code C} i) $ term_of_num vs t2
| term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
fun term_of_fm ps vs @{code T} = @{term True}
| term_of_fm ps vs @{code F} = @{term False}
| term_of_fm ps vs (@{code Lt} t) =
@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
| term_of_fm ps vs (@{code Le} t) =
@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
| term_of_fm ps vs (@{code Gt} t) =
@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t
| term_of_fm ps vs (@{code Ge} t) =
@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t
| term_of_fm ps vs (@{code Eq} t) =
@{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"}
| term_of_fm ps vs (@{code NEq} t) =
term_of_fm ps vs (@{code NOT} (@{code Eq} t))
| term_of_fm ps vs (@{code Dvd} (i, t)) =
@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs (@{code C} i) $ term_of_num vs t
| term_of_fm ps vs (@{code NDvd} (i, t)) =
term_of_fm ps vs (@{code NOT} (@{code Dvd} (i, t)))
| term_of_fm ps vs (@{code NOT} t') =
HOLogic.Not $ term_of_fm ps vs t'
| term_of_fm ps vs (@{code And} (t1, t2)) =
HOLogic.conj $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
| term_of_fm ps vs (@{code Or} (t1, t2)) =
HOLogic.disj $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
| term_of_fm ps vs (@{code Imp} (t1, t2)) =
HOLogic.imp $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
| term_of_fm ps vs (@{code Iff} (t1, t2)) =
@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm ps vs t1 $ term_of_fm ps vs t2
| term_of_fm ps vs (@{code Closed} n) =
let
val q = @{code integer_of_nat} n
in (fst o the) (find_first (fn (_, m) => m = q) ps) end
| term_of_fm ps vs (@{code NClosed} n) = term_of_fm ps vs (@{code NOT} (@{code Closed} n));
fun term_bools acc t =
let
val is_op = member (op =) [@{term HOL.conj}, @{term HOL.disj}, @{term HOL.implies}, @{term "op = :: bool => _"},
@{term "op = :: int => _"}, @{term "op < :: int => _"},
@{term "op <= :: int => _"}, @{term "Not"}, @{term "All :: (int => _) => _"},
@{term "Ex :: (int => _) => _"}, @{term "True"}, @{term "False"}]
fun is_ty t = not (fastype_of t = HOLogic.boolT)
in case t
of (l as f $ a) $ b => if is_ty t orelse is_op t then term_bools (term_bools acc l)b
else insert (op aconv) t acc
| f $ a => if is_ty t orelse is_op t then term_bools (term_bools acc f) a
else insert (op aconv) t acc
| Abs p => term_bools acc (snd (Syntax_Trans.variant_abs p)) (* FIXME !? *)
| _ => if is_ty t orelse is_op t then acc else insert (op aconv) t acc
end;
in fn ct =>
let
val thy = Thm.theory_of_cterm ct;
val t = Thm.term_of ct;
val fs = Misc_Legacy.term_frees t;
val bs = term_bools [] t;
val vs = map_index swap fs;
val ps = map_index swap bs;
val t' = (term_of_fm ps vs o @{code pa} o fm_of_term ps vs) t;
in (Thm.cterm_of thy o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end
end;
*}
ML_file "cooper_tac.ML"
method_setup cooper = {*
Args.mode "no_quantify" >>
(fn q => fn ctxt => SIMPLE_METHOD' (Cooper_Tac.linz_tac ctxt (not q)))
*} "decision procedure for linear integer arithmetic"
text {* Tests *}
lemma "\<exists>(j::int). \<forall>x\<ge>j. (\<exists>a b. x = 3*a+5*b)"
by cooper
lemma "ALL (x::int) >=8. EX i j. 5*i + 3*j = x"
by cooper
theorem "(\<forall>(y::int). 3 dvd y) ==> \<forall>(x::int). b < x --> a \<le> x"
by cooper
theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==>
(\<exists>(x::int). 2*x = y) & (\<exists>(k::int). 3*k = z)"
by cooper
theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==> 3 dvd z ==>
2 dvd (y::int) ==> (\<exists>(x::int). 2*x = y) & (\<exists>(k::int). 3*k = z)"
by cooper
theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 --> y = 5 + x "
by cooper
lemma "ALL (x::int) >=8. EX i j. 5*i + 3*j = x"
by cooper
lemma "ALL (y::int) (z::int) (n::int). 3 dvd z --> 2 dvd (y::int) --> (EX (x::int). 2*x = y) & (EX (k::int). 3*k = z)"
by cooper
lemma "ALL(x::int) y. x < y --> 2 * x + 1 < 2 * y"
by cooper
lemma "ALL(x::int) y. 2 * x + 1 ~= 2 * y"
by cooper
lemma "EX(x::int) y. 0 < x & 0 <= y & 3 * x - 5 * y = 1"
by cooper
lemma "~ (EX(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)"
by cooper
lemma "ALL(x::int). (2 dvd x) --> (EX(y::int). x = 2*y)"
by cooper
lemma "ALL(x::int). (2 dvd x) = (EX(y::int). x = 2*y)"
by cooper
lemma "ALL(x::int). ((2 dvd x) = (ALL(y::int). x ~= 2*y + 1))"
by cooper
lemma "~ (ALL(x::int). ((2 dvd x) = (ALL(y::int). x ~= 2*y+1) | (EX(q::int) (u::int) i. 3*i + 2*q - u < 17) --> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))"
by cooper
lemma "~ (ALL(i::int). 4 <= i --> (EX x y. 0 <= x & 0 <= y & 3 * x + 5 * y = i))"
by cooper
lemma "EX j. ALL (x::int) >= j. EX i j. 5*i + 3*j = x"
by cooper
theorem "(\<forall>(y::int). 3 dvd y) ==> \<forall>(x::int). b < x --> a \<le> x"
by cooper
theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==>
(\<exists>(x::int). 2*x = y) & (\<exists>(k::int). 3*k = z)"
by cooper
theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==> 3 dvd z ==>
2 dvd (y::int) ==> (\<exists>(x::int). 2*x = y) & (\<exists>(k::int). 3*k = z)"
by cooper
theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 --> y = 5 + x "
by cooper
theorem "\<forall>(x::nat). \<exists>(y::nat). y = 5 + x | x div 6 + 1= 2"
by cooper
theorem "\<exists>(x::int). 0 < x"
by cooper
theorem "\<forall>(x::int) y. x < y --> 2 * x + 1 < 2 * y"
by cooper
theorem "\<forall>(x::int) y. 2 * x + 1 \<noteq> 2 * y"
by cooper
theorem "\<exists>(x::int) y. 0 < x & 0 \<le> y & 3 * x - 5 * y = 1"
by cooper
theorem "~ (\<exists>(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)"
by cooper
theorem "~ (\<exists>(x::int). False)"
by cooper
theorem "\<forall>(x::int). (2 dvd x) --> (\<exists>(y::int). x = 2*y)"
by cooper
theorem "\<forall>(x::int). (2 dvd x) --> (\<exists>(y::int). x = 2*y)"
by cooper
theorem "\<forall>(x::int). (2 dvd x) = (\<exists>(y::int). x = 2*y)"
by cooper
theorem "\<forall>(x::int). ((2 dvd x) = (\<forall>(y::int). x \<noteq> 2*y + 1))"
by cooper
theorem "~ (\<forall>(x::int).
((2 dvd x) = (\<forall>(y::int). x \<noteq> 2*y+1) |
(\<exists>(q::int) (u::int) i. 3*i + 2*q - u < 17)
--> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))"
by cooper
theorem "~ (\<forall>(i::int). 4 \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i))"
by cooper
theorem "\<forall>(i::int). 8 \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i)"
by cooper
theorem "\<exists>(j::int). \<forall>i. j \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i)"
by cooper
theorem "~ (\<forall>j (i::int). j \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i))"
by cooper
theorem "(\<exists>m::nat. n = 2 * m) --> (n + 1) div 2 = n div 2"
by cooper
end