author | blanchet |
Tue, 09 Sep 2014 20:51:36 +0200 | |
changeset 58259 | 52c35a59bbf5 |
parent 58249 | 180f1b3508ed |
child 58310 | 91ea607a34d8 |
permissions | -rw-r--r-- |
30439 | 1 |
(* Title: HOL/Decision_Procs/Cooper.thy |
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Author: Amine Chaieb |
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*) |
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theory Cooper |
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imports |
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Complex_Main |
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"~~/src/HOL/Library/Code_Target_Numeral" |
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"~~/src/HOL/Library/Old_Recdef" |
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begin |
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|
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(* Periodicity of dvd *) |
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(*********************************************************************************) |
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(**** SHADOW SYNTAX AND SEMANTICS ****) |
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(*********************************************************************************) |
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|
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use 'datatype_new' (soon to be renamed 'datatype') in Isabelle's libraries
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datatype_new num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num |
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| Mul int num |
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||
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primrec num_size :: "num \<Rightarrow> nat" -- {* A size for num to make inductive proofs simpler *} |
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where |
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"num_size (C c) = 1" |
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| "num_size (Bound n) = 1" |
25 |
| "num_size (Neg a) = 1 + num_size a" |
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26 |
| "num_size (Add a b) = 1 + num_size a + num_size b" |
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| "num_size (Sub a b) = 3 + num_size a + num_size b" |
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| "num_size (CN n c a) = 4 + num_size a" |
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| "num_size (Mul c a) = 1 + num_size a" |
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primrec Inum :: "int list \<Rightarrow> num \<Rightarrow> int" |
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where |
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"Inum bs (C c) = c" |
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| "Inum bs (Bound n) = bs!n" |
35 |
| "Inum bs (CN n c a) = c * (bs!n) + (Inum bs a)" |
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| "Inum bs (Neg a) = -(Inum bs a)" |
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| "Inum bs (Add a b) = Inum bs a + Inum bs b" |
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| "Inum bs (Sub a b) = Inum bs a - Inum bs b" |
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| "Inum bs (Mul c a) = c* Inum bs a" |
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datatype_new fm = |
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T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num| |
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NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm |
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| Closed nat | NClosed nat |
45 |
||
46 |
||
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fun fmsize :: "fm \<Rightarrow> nat" -- {* A size for fm *} |
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where |
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"fmsize (NOT p) = 1 + fmsize p" |
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| "fmsize (And p q) = 1 + fmsize p + fmsize q" |
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| "fmsize (Or p q) = 1 + fmsize p + fmsize q" |
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| "fmsize (Imp p q) = 3 + fmsize p + fmsize q" |
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| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" |
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| "fmsize (E p) = 1 + fmsize p" |
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| "fmsize (A p) = 4+ fmsize p" |
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| "fmsize (Dvd i t) = 2" |
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| "fmsize (NDvd i t) = 2" |
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| "fmsize p = 1" |
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lemma fmsize_pos: "fmsize p > 0" |
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by (induct p rule: fmsize.induct) simp_all |
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primrec Ifm :: "bool list \<Rightarrow> int list \<Rightarrow> fm \<Rightarrow> bool" -- {* Semantics of formulae (fm) *} |
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where |
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"Ifm bbs bs T \<longleftrightarrow> True" |
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| "Ifm bbs bs F \<longleftrightarrow> False" |
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| "Ifm bbs bs (Lt a) \<longleftrightarrow> Inum bs a < 0" |
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| "Ifm bbs bs (Gt a) \<longleftrightarrow> Inum bs a > 0" |
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| "Ifm bbs bs (Le a) \<longleftrightarrow> Inum bs a \<le> 0" |
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| "Ifm bbs bs (Ge a) \<longleftrightarrow> Inum bs a \<ge> 0" |
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| "Ifm bbs bs (Eq a) \<longleftrightarrow> Inum bs a = 0" |
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| "Ifm bbs bs (NEq a) \<longleftrightarrow> Inum bs a \<noteq> 0" |
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| "Ifm bbs bs (Dvd i b) \<longleftrightarrow> i dvd Inum bs b" |
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| "Ifm bbs bs (NDvd i b) \<longleftrightarrow> \<not> i dvd Inum bs b" |
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| "Ifm bbs bs (NOT p) \<longleftrightarrow> \<not> Ifm bbs bs p" |
|
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| "Ifm bbs bs (And p q) \<longleftrightarrow> Ifm bbs bs p \<and> Ifm bbs bs q" |
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| "Ifm bbs bs (Or p q) \<longleftrightarrow> Ifm bbs bs p \<or> Ifm bbs bs q" |
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| "Ifm bbs bs (Imp p q) \<longleftrightarrow> (Ifm bbs bs p \<longrightarrow> Ifm bbs bs q)" |
|
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| "Ifm bbs bs (Iff p q) \<longleftrightarrow> Ifm bbs bs p = Ifm bbs bs q" |
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| "Ifm bbs bs (E p) \<longleftrightarrow> (\<exists>x. Ifm bbs (x # bs) p)" |
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| "Ifm bbs bs (A p) \<longleftrightarrow> (\<forall>x. Ifm bbs (x # bs) p)" |
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| "Ifm bbs bs (Closed n) \<longleftrightarrow> bbs!n" |
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| "Ifm bbs bs (NClosed n) \<longleftrightarrow> \<not> bbs!n" |
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consts prep :: "fm \<Rightarrow> fm" |
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recdef prep "measure fmsize" |
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"prep (E T) = T" |
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"prep (E F) = F" |
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"prep (E (Or p q)) = Or (prep (E p)) (prep (E q))" |
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"prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))" |
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"prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" |
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"prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))" |
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"prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" |
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"prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" |
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"prep (E p) = E (prep p)" |
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"prep (A (And p q)) = And (prep (A p)) (prep (A q))" |
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"prep (A p) = prep (NOT (E (NOT p)))" |
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"prep (NOT (NOT p)) = prep p" |
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"prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))" |
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"prep (NOT (A p)) = prep (E (NOT p))" |
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"prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))" |
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"prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))" |
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"prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))" |
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"prep (NOT p) = NOT (prep p)" |
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"prep (Or p q) = Or (prep p) (prep q)" |
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"prep (And p q) = And (prep p) (prep q)" |
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"prep (Imp p q) = prep (Or (NOT p) q)" |
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"prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))" |
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"prep p = p" |
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(hints simp add: fmsize_pos) |
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lemma prep: "Ifm bbs bs (prep p) = Ifm bbs bs p" |
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by (induct p arbitrary: bs rule: prep.induct) auto |
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fun qfree :: "fm \<Rightarrow> bool" -- {* Quantifier freeness *} |
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where |
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"qfree (E p) \<longleftrightarrow> False" |
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| "qfree (A p) \<longleftrightarrow> False" |
|
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| "qfree (NOT p) \<longleftrightarrow> qfree p" |
|
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| "qfree (And p q) \<longleftrightarrow> qfree p \<and> qfree q" |
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| "qfree (Or p q) \<longleftrightarrow> qfree p \<and> qfree q" |
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| "qfree (Imp p q) \<longleftrightarrow> qfree p \<and> qfree q" |
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| "qfree (Iff p q) \<longleftrightarrow> qfree p \<and> qfree q" |
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| "qfree p \<longleftrightarrow> True" |
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text {* Boundedness and substitution *} |
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||
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primrec numbound0 :: "num \<Rightarrow> bool" -- {* a num is INDEPENDENT of Bound 0 *} |
|
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where |
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"numbound0 (C c) \<longleftrightarrow> True" |
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| "numbound0 (Bound n) \<longleftrightarrow> n > 0" |
|
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| "numbound0 (CN n i a) \<longleftrightarrow> n > 0 \<and> numbound0 a" |
|
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| "numbound0 (Neg a) \<longleftrightarrow> numbound0 a" |
|
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| "numbound0 (Add a b) \<longleftrightarrow> numbound0 a \<and> numbound0 b" |
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| "numbound0 (Sub a b) \<longleftrightarrow> numbound0 a \<and> numbound0 b" |
|
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| "numbound0 (Mul i a) \<longleftrightarrow> numbound0 a" |
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|
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lemma numbound0_I: |
|
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assumes nb: "numbound0 a" |
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shows "Inum (b # bs) a = Inum (b' # bs) a" |
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using nb by (induct a rule: num.induct) (auto simp add: gr0_conv_Suc) |
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primrec bound0 :: "fm \<Rightarrow> bool" -- {* A Formula is independent of Bound 0 *} |
146 |
where |
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"bound0 T \<longleftrightarrow> True" |
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| "bound0 F \<longleftrightarrow> True" |
|
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| "bound0 (Lt a) \<longleftrightarrow> numbound0 a" |
|
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| "bound0 (Le a) \<longleftrightarrow> numbound0 a" |
|
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| "bound0 (Gt a) \<longleftrightarrow> numbound0 a" |
|
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| "bound0 (Ge a) \<longleftrightarrow> numbound0 a" |
|
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| "bound0 (Eq a) \<longleftrightarrow> numbound0 a" |
|
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| "bound0 (NEq a) \<longleftrightarrow> numbound0 a" |
|
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| "bound0 (Dvd i a) \<longleftrightarrow> numbound0 a" |
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| "bound0 (NDvd i a) \<longleftrightarrow> numbound0 a" |
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| "bound0 (NOT p) \<longleftrightarrow> bound0 p" |
|
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| "bound0 (And p q) \<longleftrightarrow> bound0 p \<and> bound0 q" |
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| "bound0 (Or p q) \<longleftrightarrow> bound0 p \<and> bound0 q" |
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| "bound0 (Imp p q) \<longleftrightarrow> bound0 p \<and> bound0 q" |
|
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| "bound0 (Iff p q) \<longleftrightarrow> bound0 p \<and> bound0 q" |
|
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| "bound0 (E p) \<longleftrightarrow> False" |
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| "bound0 (A p) \<longleftrightarrow> False" |
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| "bound0 (Closed P) \<longleftrightarrow> True" |
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| "bound0 (NClosed P) \<longleftrightarrow> True" |
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|
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lemma bound0_I: |
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assumes bp: "bound0 p" |
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shows "Ifm bbs (b # bs) p = Ifm bbs (b' # bs) p" |
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using bp numbound0_I[where b="b" and bs="bs" and b'="b'"] |
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by (induct p rule: fm.induct) (auto simp add: gr0_conv_Suc) |
|
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|
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fun numsubst0 :: "num \<Rightarrow> num \<Rightarrow> num" |
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where |
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"numsubst0 t (C c) = (C c)" |
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| "numsubst0 t (Bound n) = (if n=0 then t else Bound n)" |
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| "numsubst0 t (CN 0 i a) = Add (Mul i t) (numsubst0 t a)" |
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| "numsubst0 t (CN n i a) = CN n i (numsubst0 t a)" |
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| "numsubst0 t (Neg a) = Neg (numsubst0 t a)" |
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| "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)" |
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| "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" |
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| "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)" |
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|
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lemma numsubst0_I: "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t" |
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by (induct t rule: numsubst0.induct) (auto simp: nth_Cons') |
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|
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lemma numsubst0_I': |
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"numbound0 a \<Longrightarrow> Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t" |
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by (induct t rule: numsubst0.induct) (auto simp: nth_Cons' numbound0_I[where b="b" and b'="b'"]) |
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|
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primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" -- {* substitue a num into a formula for Bound 0 *} |
192 |
where |
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"subst0 t T = T" |
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| "subst0 t F = F" |
195 |
| "subst0 t (Lt a) = Lt (numsubst0 t a)" |
|
196 |
| "subst0 t (Le a) = Le (numsubst0 t a)" |
|
197 |
| "subst0 t (Gt a) = Gt (numsubst0 t a)" |
|
198 |
| "subst0 t (Ge a) = Ge (numsubst0 t a)" |
|
199 |
| "subst0 t (Eq a) = Eq (numsubst0 t a)" |
|
200 |
| "subst0 t (NEq a) = NEq (numsubst0 t a)" |
|
201 |
| "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)" |
|
202 |
| "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)" |
|
203 |
| "subst0 t (NOT p) = NOT (subst0 t p)" |
|
204 |
| "subst0 t (And p q) = And (subst0 t p) (subst0 t q)" |
|
205 |
| "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)" |
|
206 |
| "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)" |
|
207 |
| "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)" |
|
208 |
| "subst0 t (Closed P) = (Closed P)" |
|
209 |
| "subst0 t (NClosed P) = (NClosed P)" |
|
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|
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lemma subst0_I: |
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assumes "qfree p" |
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shows "Ifm bbs (b # bs) (subst0 a p) = Ifm bbs (Inum (b # bs) a # bs) p" |
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using assms numsubst0_I[where b="b" and bs="bs" and a="a"] |
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215 |
by (induct p) (simp_all add: gr0_conv_Suc) |
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|
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fun decrnum:: "num \<Rightarrow> num" |
218 |
where |
|
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"decrnum (Bound n) = Bound (n - 1)" |
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| "decrnum (Neg a) = Neg (decrnum a)" |
221 |
| "decrnum (Add a b) = Add (decrnum a) (decrnum b)" |
|
222 |
| "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)" |
|
223 |
| "decrnum (Mul c a) = Mul c (decrnum a)" |
|
224 |
| "decrnum (CN n i a) = (CN (n - 1) i (decrnum a))" |
|
225 |
| "decrnum a = a" |
|
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|
226 |
|
50313 | 227 |
fun decr :: "fm \<Rightarrow> fm" |
228 |
where |
|
23274 | 229 |
"decr (Lt a) = Lt (decrnum a)" |
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| "decr (Le a) = Le (decrnum a)" |
231 |
| "decr (Gt a) = Gt (decrnum a)" |
|
232 |
| "decr (Ge a) = Ge (decrnum a)" |
|
233 |
| "decr (Eq a) = Eq (decrnum a)" |
|
234 |
| "decr (NEq a) = NEq (decrnum a)" |
|
235 |
| "decr (Dvd i a) = Dvd i (decrnum a)" |
|
236 |
| "decr (NDvd i a) = NDvd i (decrnum a)" |
|
50313 | 237 |
| "decr (NOT p) = NOT (decr p)" |
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| "decr (And p q) = And (decr p) (decr q)" |
239 |
| "decr (Or p q) = Or (decr p) (decr q)" |
|
240 |
| "decr (Imp p q) = Imp (decr p) (decr q)" |
|
241 |
| "decr (Iff p q) = Iff (decr p) (decr q)" |
|
242 |
| "decr p = p" |
|
23274 | 243 |
|
50313 | 244 |
lemma decrnum: |
245 |
assumes nb: "numbound0 t" |
|
55885 | 246 |
shows "Inum (x # bs) t = Inum bs (decrnum t)" |
50313 | 247 |
using nb by (induct t rule: decrnum.induct) (auto simp add: gr0_conv_Suc) |
23274 | 248 |
|
50313 | 249 |
lemma decr: |
250 |
assumes nb: "bound0 p" |
|
55885 | 251 |
shows "Ifm bbs (x # bs) p = Ifm bbs bs (decr p)" |
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using nb by (induct p rule: decr.induct) (simp_all add: gr0_conv_Suc decrnum) |
23274 | 253 |
|
254 |
lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)" |
|
50313 | 255 |
by (induct p) simp_all |
23274 | 256 |
|
50313 | 257 |
fun isatom :: "fm \<Rightarrow> bool" -- {* test for atomicity *} |
258 |
where |
|
55981 | 259 |
"isatom T \<longleftrightarrow> True" |
260 |
| "isatom F \<longleftrightarrow> True" |
|
261 |
| "isatom (Lt a) \<longleftrightarrow> True" |
|
262 |
| "isatom (Le a) \<longleftrightarrow> True" |
|
263 |
| "isatom (Gt a) \<longleftrightarrow> True" |
|
264 |
| "isatom (Ge a) \<longleftrightarrow> True" |
|
265 |
| "isatom (Eq a) \<longleftrightarrow> True" |
|
266 |
| "isatom (NEq a) \<longleftrightarrow> True" |
|
267 |
| "isatom (Dvd i b) \<longleftrightarrow> True" |
|
268 |
| "isatom (NDvd i b) \<longleftrightarrow> True" |
|
269 |
| "isatom (Closed P) \<longleftrightarrow> True" |
|
270 |
| "isatom (NClosed P) \<longleftrightarrow> True" |
|
271 |
| "isatom p \<longleftrightarrow> False" |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
272 |
|
50313 | 273 |
lemma numsubst0_numbound0: |
55844 | 274 |
assumes "numbound0 t" |
23274 | 275 |
shows "numbound0 (numsubst0 t a)" |
55844 | 276 |
using assms |
277 |
apply (induct a) |
|
50313 | 278 |
apply simp_all |
58259 | 279 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 280 |
apply simp_all |
50313 | 281 |
done |
23274 | 282 |
|
50313 | 283 |
lemma subst0_bound0: |
55844 | 284 |
assumes qf: "qfree p" |
285 |
and nb: "numbound0 t" |
|
23274 | 286 |
shows "bound0 (subst0 t p)" |
50313 | 287 |
using qf numsubst0_numbound0[OF nb] by (induct p) auto |
23274 | 288 |
|
289 |
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p" |
|
50313 | 290 |
by (induct p) simp_all |
23274 | 291 |
|
292 |
||
50313 | 293 |
definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" |
294 |
where |
|
295 |
"djf f p q = |
|
55885 | 296 |
(if q = T then T |
297 |
else if q = F then f p |
|
298 |
else |
|
299 |
let fp = f p |
|
300 |
in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)" |
|
50313 | 301 |
|
302 |
definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" |
|
303 |
where "evaldjf f ps = foldr (djf f) ps F" |
|
23274 | 304 |
|
305 |
lemma djf_Or: "Ifm bbs bs (djf f p q) = Ifm bbs bs (Or (f p) q)" |
|
55885 | 306 |
by (cases "q=T", simp add: djf_def, cases "q = F", simp add: djf_def) |
50313 | 307 |
(cases "f p", simp_all add: Let_def djf_def) |
23274 | 308 |
|
55885 | 309 |
lemma evaldjf_ex: "Ifm bbs bs (evaldjf f ps) \<longleftrightarrow> (\<exists>p \<in> set ps. Ifm bbs bs (f p))" |
50313 | 310 |
by (induct ps) (simp_all add: evaldjf_def djf_Or) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
311 |
|
50313 | 312 |
lemma evaldjf_bound0: |
313 |
assumes nb: "\<forall>x\<in> set xs. bound0 (f x)" |
|
23274 | 314 |
shows "bound0 (evaldjf f xs)" |
55422 | 315 |
using nb by (induct xs) (auto simp add: evaldjf_def djf_def Let_def, case_tac "f a", auto) |
23274 | 316 |
|
50313 | 317 |
lemma evaldjf_qf: |
318 |
assumes nb: "\<forall>x\<in> set xs. qfree (f x)" |
|
23274 | 319 |
shows "qfree (evaldjf f xs)" |
55422 | 320 |
using nb by (induct xs) (auto simp add: evaldjf_def djf_def Let_def, case_tac "f a", auto) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
321 |
|
50313 | 322 |
fun disjuncts :: "fm \<Rightarrow> fm list" |
323 |
where |
|
324 |
"disjuncts (Or p q) = disjuncts p @ disjuncts q" |
|
41837 | 325 |
| "disjuncts F = []" |
326 |
| "disjuncts p = [p]" |
|
23274 | 327 |
|
55885 | 328 |
lemma disjuncts: "(\<exists>q \<in> set (disjuncts p). Ifm bbs bs q) \<longleftrightarrow> Ifm bbs bs p" |
329 |
by (induct p rule: disjuncts.induct) auto |
|
23274 | 330 |
|
50313 | 331 |
lemma disjuncts_nb: |
55999 | 332 |
assumes "bound0 p" |
50313 | 333 |
shows "\<forall>q \<in> set (disjuncts p). bound0 q" |
334 |
proof - |
|
55999 | 335 |
from assms have "list_all bound0 (disjuncts p)" |
50313 | 336 |
by (induct p rule: disjuncts.induct) auto |
55999 | 337 |
then show ?thesis |
338 |
by (simp only: list_all_iff) |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
339 |
qed |
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
340 |
|
50313 | 341 |
lemma disjuncts_qf: |
55999 | 342 |
assumes "qfree p" |
50313 | 343 |
shows "\<forall>q \<in> set (disjuncts p). qfree q" |
344 |
proof - |
|
55999 | 345 |
from assms have "list_all qfree (disjuncts p)" |
50313 | 346 |
by (induct p rule: disjuncts.induct) auto |
55885 | 347 |
then show ?thesis by (simp only: list_all_iff) |
23274 | 348 |
qed |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
349 |
|
50313 | 350 |
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" |
351 |
where "DJ f p = evaldjf f (disjuncts p)" |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
352 |
|
50313 | 353 |
lemma DJ: |
55999 | 354 |
assumes "\<forall>p q. f (Or p q) = Or (f p) (f q)" |
355 |
and "f F = F" |
|
23274 | 356 |
shows "Ifm bbs bs (DJ f p) = Ifm bbs bs (f p)" |
50313 | 357 |
proof - |
55999 | 358 |
have "Ifm bbs bs (DJ f p) \<longleftrightarrow> (\<exists>q \<in> set (disjuncts p). Ifm bbs bs (f q))" |
50313 | 359 |
by (simp add: DJ_def evaldjf_ex) |
55999 | 360 |
also from assms have "\<dots> = Ifm bbs bs (f p)" |
361 |
by (induct p rule: disjuncts.induct) auto |
|
23274 | 362 |
finally show ?thesis . |
363 |
qed |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
364 |
|
50313 | 365 |
lemma DJ_qf: |
55999 | 366 |
assumes "\<forall>p. qfree p \<longrightarrow> qfree (f p)" |
23274 | 367 |
shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) " |
50313 | 368 |
proof clarify |
55844 | 369 |
fix p |
370 |
assume qf: "qfree p" |
|
371 |
have th: "DJ f p = evaldjf f (disjuncts p)" |
|
372 |
by (simp add: DJ_def) |
|
55925 | 373 |
from disjuncts_qf[OF qf] have "\<forall>q \<in> set (disjuncts p). qfree q" . |
55999 | 374 |
with assms have th': "\<forall>q \<in> set (disjuncts p). qfree (f q)" |
55844 | 375 |
by blast |
376 |
from evaldjf_qf[OF th'] th show "qfree (DJ f p)" |
|
377 |
by simp |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
378 |
qed |
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
379 |
|
50313 | 380 |
lemma DJ_qe: |
55885 | 381 |
assumes qe: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bbs bs (qe p) = Ifm bbs bs (E p)" |
382 |
shows "\<forall>bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> Ifm bbs bs ((DJ qe p)) = Ifm bbs bs (E p)" |
|
50313 | 383 |
proof clarify |
55844 | 384 |
fix p :: fm |
385 |
fix bs |
|
23274 | 386 |
assume qf: "qfree p" |
55844 | 387 |
from qe have qth: "\<forall>p. qfree p \<longrightarrow> qfree (qe p)" |
388 |
by blast |
|
55925 | 389 |
from DJ_qf[OF qth] qf have qfth: "qfree (DJ qe p)" |
55844 | 390 |
by auto |
50313 | 391 |
have "Ifm bbs bs (DJ qe p) = (\<exists>q\<in> set (disjuncts p). Ifm bbs bs (qe q))" |
23274 | 392 |
by (simp add: DJ_def evaldjf_ex) |
55999 | 393 |
also have "\<dots> \<longleftrightarrow> (\<exists>q \<in> set (disjuncts p). Ifm bbs bs (E q))" |
50313 | 394 |
using qe disjuncts_qf[OF qf] by auto |
55925 | 395 |
also have "\<dots> \<longleftrightarrow> Ifm bbs bs (E p)" |
50313 | 396 |
by (induct p rule: disjuncts.induct) auto |
397 |
finally show "qfree (DJ qe p) \<and> Ifm bbs bs (DJ qe p) = Ifm bbs bs (E p)" |
|
398 |
using qfth by blast |
|
23274 | 399 |
qed |
50313 | 400 |
|
401 |
||
402 |
text {* Simplification *} |
|
23274 | 403 |
|
50313 | 404 |
text {* Algebraic simplifications for nums *} |
41837 | 405 |
|
50313 | 406 |
fun bnds :: "num \<Rightarrow> nat list" |
407 |
where |
|
23274 | 408 |
"bnds (Bound n) = [n]" |
55921 | 409 |
| "bnds (CN n c a) = n # bnds a" |
41837 | 410 |
| "bnds (Neg a) = bnds a" |
55921 | 411 |
| "bnds (Add a b) = bnds a @ bnds b" |
55999 | 412 |
| "bnds (Sub a b) = bnds a @ bnds b" |
41837 | 413 |
| "bnds (Mul i a) = bnds a" |
414 |
| "bnds a = []" |
|
415 |
||
50313 | 416 |
fun lex_ns:: "nat list \<Rightarrow> nat list \<Rightarrow> bool" |
417 |
where |
|
55844 | 418 |
"lex_ns [] ms \<longleftrightarrow> True" |
419 |
| "lex_ns ns [] \<longleftrightarrow> False" |
|
420 |
| "lex_ns (n # ns) (m # ms) \<longleftrightarrow> n < m \<or> (n = m \<and> lex_ns ns ms)" |
|
23274 | 421 |
|
50313 | 422 |
definition lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool" |
423 |
where "lex_bnd t s = lex_ns (bnds t) (bnds s)" |
|
424 |
||
425 |
consts numadd:: "num \<times> num \<Rightarrow> num" |
|
55921 | 426 |
recdef numadd "measure (\<lambda>(t, s). num_size t + num_size s)" |
55981 | 427 |
"numadd (CN n1 c1 r1, CN n2 c2 r2) = |
50313 | 428 |
(if n1 = n2 then |
55981 | 429 |
let c = c1 + c2 |
430 |
in if c = 0 then numadd (r1, r2) else CN n1 c (numadd (r1, r2)) |
|
55921 | 431 |
else if n1 \<le> n2 then CN n1 c1 (numadd (r1, Add (Mul c2 (Bound n2)) r2)) |
50313 | 432 |
else CN n2 c2 (numadd (Add (Mul c1 (Bound n1)) r1, r2)))" |
433 |
"numadd (CN n1 c1 r1, t) = CN n1 c1 (numadd (r1, t))" |
|
434 |
"numadd (t, CN n2 c2 r2) = CN n2 c2 (numadd (t, r2))" |
|
435 |
"numadd (C b1, C b2) = C (b1 + b2)" |
|
436 |
"numadd (a, b) = Add a b" |
|
23274 | 437 |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
438 |
(*function (sequential) |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
439 |
numadd :: "num \<Rightarrow> num \<Rightarrow> num" |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
440 |
where |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
441 |
"numadd (Add (Mul c1 (Bound n1)) r1) (Add (Mul c2 (Bound n2)) r2) = |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
442 |
(if n1 = n2 then (let c = c1 + c2 |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
443 |
in (if c = 0 then numadd r1 r2 else |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
444 |
Add (Mul c (Bound n1)) (numadd r1 r2))) |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
445 |
else if n1 \<le> n2 then |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
446 |
Add (Mul c1 (Bound n1)) (numadd r1 (Add (Mul c2 (Bound n2)) r2)) |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
447 |
else |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
448 |
Add (Mul c2 (Bound n2)) (numadd (Add (Mul c1 (Bound n1)) r1) r2))" |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
449 |
| "numadd (Add (Mul c1 (Bound n1)) r1) t = |
50313 | 450 |
Add (Mul c1 (Bound n1)) (numadd r1 t)" |
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
451 |
| "numadd t (Add (Mul c2 (Bound n2)) r2) = |
50313 | 452 |
Add (Mul c2 (Bound n2)) (numadd t r2)" |
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
453 |
| "numadd (C b1) (C b2) = C (b1 + b2)" |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
454 |
| "numadd a b = Add a b" |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
455 |
apply pat_completeness apply auto*) |
50313 | 456 |
|
23274 | 457 |
lemma numadd: "Inum bs (numadd (t,s)) = Inum bs (Add t s)" |
55885 | 458 |
apply (induct t s rule: numadd.induct) |
459 |
apply (simp_all add: Let_def) |
|
55844 | 460 |
apply (case_tac "c1 + c2 = 0") |
461 |
apply (case_tac "n1 \<le> n2") |
|
462 |
apply simp_all |
|
50313 | 463 |
apply (case_tac "n1 = n2") |
55844 | 464 |
apply (simp_all add: algebra_simps) |
465 |
apply (simp add: distrib_right[symmetric]) |
|
50313 | 466 |
done |
23274 | 467 |
|
50313 | 468 |
lemma numadd_nb: "numbound0 t \<Longrightarrow> numbound0 s \<Longrightarrow> numbound0 (numadd (t, s))" |
469 |
by (induct t s rule: numadd.induct) (auto simp add: Let_def) |
|
23274 | 470 |
|
50313 | 471 |
fun nummul :: "int \<Rightarrow> num \<Rightarrow> num" |
472 |
where |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
473 |
"nummul i (C j) = C (i * j)" |
55844 | 474 |
| "nummul i (CN n c t) = CN n (c * i) (nummul i t)" |
41837 | 475 |
| "nummul i t = Mul i t" |
23274 | 476 |
|
50313 | 477 |
lemma nummul: "Inum bs (nummul i t) = Inum bs (Mul i t)" |
478 |
by (induct t arbitrary: i rule: nummul.induct) (auto simp add: algebra_simps numadd) |
|
23274 | 479 |
|
50313 | 480 |
lemma nummul_nb: "numbound0 t \<Longrightarrow> numbound0 (nummul i t)" |
481 |
by (induct t arbitrary: i rule: nummul.induct) (auto simp add: numadd_nb) |
|
23274 | 482 |
|
50313 | 483 |
definition numneg :: "num \<Rightarrow> num" |
484 |
where "numneg t = nummul (- 1) t" |
|
23274 | 485 |
|
50313 | 486 |
definition numsub :: "num \<Rightarrow> num \<Rightarrow> num" |
487 |
where "numsub s t = (if s = t then C 0 else numadd (s, numneg t))" |
|
23274 | 488 |
|
489 |
lemma numneg: "Inum bs (numneg t) = Inum bs (Neg t)" |
|
50313 | 490 |
using numneg_def nummul by simp |
23274 | 491 |
|
492 |
lemma numneg_nb: "numbound0 t \<Longrightarrow> numbound0 (numneg t)" |
|
50313 | 493 |
using numneg_def nummul_nb by simp |
23274 | 494 |
|
495 |
lemma numsub: "Inum bs (numsub a b) = Inum bs (Sub a b)" |
|
50313 | 496 |
using numneg numadd numsub_def by simp |
23274 | 497 |
|
50313 | 498 |
lemma numsub_nb: "numbound0 t \<Longrightarrow> numbound0 s \<Longrightarrow> numbound0 (numsub t s)" |
499 |
using numsub_def numadd_nb numneg_nb by simp |
|
23274 | 500 |
|
50313 | 501 |
fun simpnum :: "num \<Rightarrow> num" |
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
502 |
where |
23274 | 503 |
"simpnum (C j) = C j" |
50313 | 504 |
| "simpnum (Bound n) = CN n 1 (C 0)" |
505 |
| "simpnum (Neg t) = numneg (simpnum t)" |
|
506 |
| "simpnum (Add t s) = numadd (simpnum t, simpnum s)" |
|
507 |
| "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)" |
|
508 |
| "simpnum (Mul i t) = (if i = 0 then C 0 else nummul i (simpnum t))" |
|
509 |
| "simpnum t = t" |
|
23274 | 510 |
|
511 |
lemma simpnum_ci: "Inum bs (simpnum t) = Inum bs t" |
|
50313 | 512 |
by (induct t rule: simpnum.induct) (auto simp add: numneg numadd numsub nummul) |
23274 | 513 |
|
50313 | 514 |
lemma simpnum_numbound0: "numbound0 t \<Longrightarrow> numbound0 (simpnum t)" |
515 |
by (induct t rule: simpnum.induct) (auto simp add: numadd_nb numsub_nb nummul_nb numneg_nb) |
|
23274 | 516 |
|
50313 | 517 |
fun not :: "fm \<Rightarrow> fm" |
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
518 |
where |
23274 | 519 |
"not (NOT p) = p" |
50313 | 520 |
| "not T = F" |
521 |
| "not F = T" |
|
522 |
| "not p = NOT p" |
|
523 |
||
23274 | 524 |
lemma not: "Ifm bbs bs (not p) = Ifm bbs bs (NOT p)" |
41807 | 525 |
by (cases p) auto |
50313 | 526 |
|
23274 | 527 |
lemma not_qf: "qfree p \<Longrightarrow> qfree (not p)" |
41807 | 528 |
by (cases p) auto |
50313 | 529 |
|
23274 | 530 |
lemma not_bn: "bound0 p \<Longrightarrow> bound0 (not p)" |
41807 | 531 |
by (cases p) auto |
23274 | 532 |
|
50313 | 533 |
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" |
534 |
where |
|
55844 | 535 |
"conj p q = |
536 |
(if p = F \<or> q = F then F |
|
537 |
else if p = T then q |
|
538 |
else if q = T then p |
|
539 |
else And p q)" |
|
50313 | 540 |
|
23274 | 541 |
lemma conj: "Ifm bbs bs (conj p q) = Ifm bbs bs (And p q)" |
55844 | 542 |
by (cases "p = F \<or> q = F", simp_all add: conj_def) (cases p, simp_all) |
23274 | 543 |
|
50313 | 544 |
lemma conj_qf: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (conj p q)" |
545 |
using conj_def by auto |
|
23274 | 546 |
|
50313 | 547 |
lemma conj_nb: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (conj p q)" |
548 |
using conj_def by auto |
|
549 |
||
550 |
definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" |
|
551 |
where |
|
552 |
"disj p q = |
|
55844 | 553 |
(if p = T \<or> q = T then T |
554 |
else if p = F then q |
|
555 |
else if q = F then p |
|
556 |
else Or p q)" |
|
23274 | 557 |
|
558 |
lemma disj: "Ifm bbs bs (disj p q) = Ifm bbs bs (Or p q)" |
|
55885 | 559 |
by (cases "p = T \<or> q = T", simp_all add: disj_def) (cases p, simp_all) |
50313 | 560 |
|
55844 | 561 |
lemma disj_qf: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (disj p q)" |
50313 | 562 |
using disj_def by auto |
563 |
||
55844 | 564 |
lemma disj_nb: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (disj p q)" |
50313 | 565 |
using disj_def by auto |
23274 | 566 |
|
50313 | 567 |
definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" |
568 |
where |
|
55844 | 569 |
"imp p q = |
570 |
(if p = F \<or> q = T then T |
|
571 |
else if p = T then q |
|
572 |
else if q = F then not p |
|
573 |
else Imp p q)" |
|
50313 | 574 |
|
23274 | 575 |
lemma imp: "Ifm bbs bs (imp p q) = Ifm bbs bs (Imp p q)" |
55844 | 576 |
by (cases "p = F \<or> q = T", simp_all add: imp_def, cases p) (simp_all add: not) |
50313 | 577 |
|
578 |
lemma imp_qf: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (imp p q)" |
|
55844 | 579 |
using imp_def by (cases "p = F \<or> q = T", simp_all add: imp_def, cases p) (simp_all add: not_qf) |
50313 | 580 |
|
581 |
lemma imp_nb: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (imp p q)" |
|
55844 | 582 |
using imp_def by (cases "p = F \<or> q = T", simp_all add: imp_def, cases p) simp_all |
23274 | 583 |
|
50313 | 584 |
definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" |
55885 | 585 |
where |
586 |
"iff p q = |
|
587 |
(if p = q then T |
|
588 |
else if p = not q \<or> not p = q then F |
|
589 |
else if p = F then not q |
|
590 |
else if q = F then not p |
|
591 |
else if p = T then q |
|
592 |
else if q = T then p |
|
593 |
else Iff p q)" |
|
50313 | 594 |
|
23274 | 595 |
lemma iff: "Ifm bbs bs (iff p q) = Ifm bbs bs (Iff p q)" |
55885 | 596 |
by (unfold iff_def, cases "p = q", simp, cases "p = not q", simp add: not) |
597 |
(cases "not p = q", auto simp add: not) |
|
50313 | 598 |
|
55885 | 599 |
lemma iff_qf: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (iff p q)" |
600 |
by (unfold iff_def, cases "p = q", auto simp add: not_qf) |
|
23274 | 601 |
|
55885 | 602 |
lemma iff_nb: "bound0 p \<Longrightarrow> bound0 q \<Longrightarrow> bound0 (iff p q)" |
603 |
using iff_def by (unfold iff_def, cases "p = q", auto simp add: not_bn) |
|
50313 | 604 |
|
605 |
function (sequential) simpfm :: "fm \<Rightarrow> fm" |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
606 |
where |
23274 | 607 |
"simpfm (And p q) = conj (simpfm p) (simpfm q)" |
50313 | 608 |
| "simpfm (Or p q) = disj (simpfm p) (simpfm q)" |
609 |
| "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" |
|
610 |
| "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" |
|
611 |
| "simpfm (NOT p) = not (simpfm p)" |
|
55921 | 612 |
| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if v < 0 then T else F | _ \<Rightarrow> Lt a')" |
613 |
| "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')" |
|
614 |
| "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if v > 0 then T else F | _ \<Rightarrow> Gt a')" |
|
615 |
| "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')" |
|
616 |
| "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if v = 0 then T else F | _ \<Rightarrow> Eq a')" |
|
617 |
| "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')" |
|
50313 | 618 |
| "simpfm (Dvd i a) = |
55885 | 619 |
(if i = 0 then simpfm (Eq a) |
620 |
else if abs i = 1 then T |
|
55921 | 621 |
else let a' = simpnum a in case a' of C v \<Rightarrow> if i dvd v then T else F | _ \<Rightarrow> Dvd i a')" |
50313 | 622 |
| "simpfm (NDvd i a) = |
55885 | 623 |
(if i = 0 then simpfm (NEq a) |
624 |
else if abs i = 1 then F |
|
55921 | 625 |
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')" |
50313 | 626 |
| "simpfm p = p" |
627 |
by pat_completeness auto |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
628 |
termination by (relation "measure fmsize") auto |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
629 |
|
23274 | 630 |
lemma simpfm: "Ifm bbs bs (simpfm p) = Ifm bbs bs p" |
55844 | 631 |
proof (induct p rule: simpfm.induct) |
50313 | 632 |
case (6 a) |
633 |
let ?sa = "simpnum a" |
|
55925 | 634 |
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" |
635 |
by simp |
|
636 |
{ |
|
637 |
fix v |
|
638 |
assume "?sa = C v" |
|
639 |
then have ?case using sa |
|
640 |
by simp |
|
641 |
} |
|
50313 | 642 |
moreover { |
643 |
assume "\<not> (\<exists>v. ?sa = C v)" |
|
55925 | 644 |
then have ?case |
645 |
using sa by (cases ?sa) (simp_all add: Let_def) |
|
50313 | 646 |
} |
23274 | 647 |
ultimately show ?case by blast |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
648 |
next |
50313 | 649 |
case (7 a) |
650 |
let ?sa = "simpnum a" |
|
55925 | 651 |
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" |
652 |
by simp |
|
653 |
{ |
|
654 |
fix v |
|
655 |
assume "?sa = C v" |
|
656 |
then have ?case using sa |
|
657 |
by simp |
|
658 |
} |
|
50313 | 659 |
moreover { |
660 |
assume "\<not> (\<exists>v. ?sa = C v)" |
|
55925 | 661 |
then have ?case |
662 |
using sa by (cases ?sa) (simp_all add: Let_def) |
|
50313 | 663 |
} |
23274 | 664 |
ultimately show ?case by blast |
665 |
next |
|
50313 | 666 |
case (8 a) |
667 |
let ?sa = "simpnum a" |
|
55925 | 668 |
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" |
669 |
by simp |
|
670 |
{ |
|
671 |
fix v |
|
672 |
assume "?sa = C v" |
|
673 |
then have ?case using sa |
|
674 |
by simp |
|
675 |
} |
|
50313 | 676 |
moreover { |
677 |
assume "\<not> (\<exists>v. ?sa = C v)" |
|
55925 | 678 |
then have ?case |
679 |
using sa by (cases ?sa) (simp_all add: Let_def) |
|
50313 | 680 |
} |
23274 | 681 |
ultimately show ?case by blast |
682 |
next |
|
50313 | 683 |
case (9 a) |
684 |
let ?sa = "simpnum a" |
|
55925 | 685 |
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" |
686 |
by simp |
|
687 |
{ |
|
688 |
fix v |
|
689 |
assume "?sa = C v" |
|
690 |
then have ?case using sa |
|
691 |
by simp |
|
692 |
} |
|
50313 | 693 |
moreover { |
694 |
assume "\<not> (\<exists>v. ?sa = C v)" |
|
55925 | 695 |
then have ?case using sa |
696 |
by (cases ?sa) (simp_all add: Let_def) |
|
50313 | 697 |
} |
23274 | 698 |
ultimately show ?case by blast |
699 |
next |
|
50313 | 700 |
case (10 a) |
701 |
let ?sa = "simpnum a" |
|
55925 | 702 |
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" |
703 |
by simp |
|
704 |
{ |
|
705 |
fix v |
|
706 |
assume "?sa = C v" |
|
707 |
then have ?case |
|
708 |
using sa by simp |
|
709 |
} |
|
50313 | 710 |
moreover { |
711 |
assume "\<not> (\<exists>v. ?sa = C v)" |
|
55925 | 712 |
then have ?case |
713 |
using sa by (cases ?sa) (simp_all add: Let_def) |
|
50313 | 714 |
} |
23274 | 715 |
ultimately show ?case by blast |
716 |
next |
|
50313 | 717 |
case (11 a) |
718 |
let ?sa = "simpnum a" |
|
55925 | 719 |
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" |
720 |
by simp |
|
721 |
{ |
|
722 |
fix v |
|
723 |
assume "?sa = C v" |
|
724 |
then have ?case using sa |
|
725 |
by simp |
|
726 |
} |
|
50313 | 727 |
moreover { |
728 |
assume "\<not> (\<exists>v. ?sa = C v)" |
|
55925 | 729 |
then have ?case |
730 |
using sa by (cases ?sa) (simp_all add: Let_def) |
|
50313 | 731 |
} |
23274 | 732 |
ultimately show ?case by blast |
733 |
next |
|
50313 | 734 |
case (12 i a) |
735 |
let ?sa = "simpnum a" |
|
55925 | 736 |
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" |
737 |
by simp |
|
738 |
{ |
|
739 |
assume "i = 0" |
|
740 |
then have ?case using "12.hyps" |
|
741 |
by (simp add: dvd_def Let_def) |
|
742 |
} |
|
50313 | 743 |
moreover |
55925 | 744 |
{ |
745 |
assume i1: "abs i = 1" |
|
50313 | 746 |
from one_dvd[of "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"] |
55925 | 747 |
have ?case |
748 |
using i1 |
|
749 |
apply (cases "i = 0") |
|
750 |
apply (simp_all add: Let_def) |
|
751 |
apply (cases "i > 0") |
|
752 |
apply simp_all |
|
50313 | 753 |
done |
754 |
} |
|
755 |
moreover |
|
55925 | 756 |
{ |
757 |
assume inz: "i \<noteq> 0" and cond: "abs i \<noteq> 1" |
|
758 |
{ |
|
759 |
fix v |
|
760 |
assume "?sa = C v" |
|
761 |
then have ?case |
|
762 |
using sa[symmetric] inz cond |
|
763 |
by (cases "abs i = 1") auto |
|
764 |
} |
|
765 |
moreover |
|
766 |
{ |
|
50313 | 767 |
assume "\<not> (\<exists>v. ?sa = C v)" |
55925 | 768 |
then have "simpfm (Dvd i a) = Dvd i ?sa" |
769 |
using inz cond by (cases ?sa) (auto simp add: Let_def) |
|
770 |
then have ?case |
|
771 |
using sa by simp |
|
772 |
} |
|
773 |
ultimately have ?case by blast |
|
774 |
} |
|
50313 | 775 |
ultimately show ?case by blast |
776 |
next |
|
777 |
case (13 i a) |
|
55925 | 778 |
let ?sa = "simpnum a" |
779 |
from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" |
|
780 |
by simp |
|
781 |
{ |
|
782 |
assume "i = 0" |
|
783 |
then have ?case using "13.hyps" |
|
784 |
by (simp add: dvd_def Let_def) |
|
785 |
} |
|
50313 | 786 |
moreover |
55925 | 787 |
{ |
788 |
assume i1: "abs i = 1" |
|
50313 | 789 |
from one_dvd[of "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"] |
55925 | 790 |
have ?case |
791 |
using i1 |
|
792 |
apply (cases "i = 0") |
|
793 |
apply (simp_all add: Let_def) |
|
794 |
apply (cases "i > 0") |
|
795 |
apply simp_all |
|
50313 | 796 |
done |
797 |
} |
|
798 |
moreover |
|
55925 | 799 |
{ |
800 |
assume inz: "i \<noteq> 0" and cond: "abs i \<noteq> 1" |
|
801 |
{ |
|
802 |
fix v |
|
803 |
assume "?sa = C v" |
|
804 |
then have ?case |
|
805 |
using sa[symmetric] inz cond by (cases "abs i = 1") auto |
|
806 |
} |
|
807 |
moreover |
|
808 |
{ |
|
50313 | 809 |
assume "\<not> (\<exists>v. ?sa = C v)" |
55925 | 810 |
then have "simpfm (NDvd i a) = NDvd i ?sa" |
811 |
using inz cond by (cases ?sa) (auto simp add: Let_def) |
|
812 |
then have ?case using sa |
|
813 |
by simp |
|
814 |
} |
|
815 |
ultimately have ?case by blast |
|
816 |
} |
|
23274 | 817 |
ultimately show ?case by blast |
50313 | 818 |
qed (simp_all add: conj disj imp iff not) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
819 |
|
23274 | 820 |
lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)" |
50313 | 821 |
proof (induct p rule: simpfm.induct) |
55925 | 822 |
case (6 a) |
823 |
then have nb: "numbound0 a" by simp |
|
55885 | 824 |
then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
825 |
then show ?case by (cases "simpnum a") (auto simp add: Let_def) |
|
23274 | 826 |
next |
55925 | 827 |
case (7 a) |
828 |
then have nb: "numbound0 a" by simp |
|
55885 | 829 |
then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
830 |
then show ?case by (cases "simpnum a") (auto simp add: Let_def) |
|
23274 | 831 |
next |
55925 | 832 |
case (8 a) |
833 |
then have nb: "numbound0 a" by simp |
|
55885 | 834 |
then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
835 |
then show ?case by (cases "simpnum a") (auto simp add: Let_def) |
|
23274 | 836 |
next |
55925 | 837 |
case (9 a) |
838 |
then have nb: "numbound0 a" by simp |
|
55885 | 839 |
then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
840 |
then show ?case by (cases "simpnum a") (auto simp add: Let_def) |
|
23274 | 841 |
next |
55925 | 842 |
case (10 a) |
843 |
then have nb: "numbound0 a" by simp |
|
55885 | 844 |
then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
845 |
then show ?case by (cases "simpnum a") (auto simp add: Let_def) |
|
23274 | 846 |
next |
55925 | 847 |
case (11 a) |
848 |
then have nb: "numbound0 a" by simp |
|
55885 | 849 |
then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
850 |
then show ?case by (cases "simpnum a") (auto simp add: Let_def) |
|
23274 | 851 |
next |
55925 | 852 |
case (12 i a) |
853 |
then have nb: "numbound0 a" by simp |
|
55885 | 854 |
then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
855 |
then show ?case by (cases "simpnum a") (auto simp add: Let_def) |
|
23274 | 856 |
next |
55925 | 857 |
case (13 i a) |
858 |
then have nb: "numbound0 a" by simp |
|
55885 | 859 |
then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
860 |
then show ?case by (cases "simpnum a") (auto simp add: Let_def) |
|
50313 | 861 |
qed (auto simp add: disj_def imp_def iff_def conj_def not_bn) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
862 |
|
23274 | 863 |
lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)" |
50313 | 864 |
by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def) |
865 |
(case_tac "simpnum a", auto)+ |
|
23274 | 866 |
|
50313 | 867 |
text {* Generic quantifier elimination *} |
868 |
function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" |
|
869 |
where |
|
870 |
"qelim (E p) = (\<lambda>qe. DJ qe (qelim p qe))" |
|
871 |
| "qelim (A p) = (\<lambda>qe. not (qe ((qelim (NOT p) qe))))" |
|
872 |
| "qelim (NOT p) = (\<lambda>qe. not (qelim p qe))" |
|
873 |
| "qelim (And p q) = (\<lambda>qe. conj (qelim p qe) (qelim q qe))" |
|
874 |
| "qelim (Or p q) = (\<lambda>qe. disj (qelim p qe) (qelim q qe))" |
|
875 |
| "qelim (Imp p q) = (\<lambda>qe. imp (qelim p qe) (qelim q qe))" |
|
876 |
| "qelim (Iff p q) = (\<lambda>qe. iff (qelim p qe) (qelim q qe))" |
|
877 |
| "qelim p = (\<lambda>y. simpfm p)" |
|
878 |
by pat_completeness auto |
|
41837 | 879 |
termination by (relation "measure fmsize") auto |
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
880 |
|
23274 | 881 |
lemma qelim_ci: |
55885 | 882 |
assumes qe_inv: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bbs bs (qe p) = Ifm bbs bs (E p)" |
883 |
shows "\<And>bs. qfree (qelim p qe) \<and> Ifm bbs bs (qelim p qe) = Ifm bbs bs p" |
|
50313 | 884 |
using qe_inv DJ_qe[OF qe_inv] |
55964 | 885 |
by (induct p rule: qelim.induct) |
886 |
(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf |
|
887 |
simpfm simpfm_qf simp del: simpfm.simps) |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
888 |
|
50313 | 889 |
text {* Linearity for fm where Bound 0 ranges over @{text "\<int>"} *} |
890 |
||
891 |
fun zsplit0 :: "num \<Rightarrow> int \<times> num" -- {* splits the bounded from the unbounded part *} |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
892 |
where |
55885 | 893 |
"zsplit0 (C c) = (0, C c)" |
894 |
| "zsplit0 (Bound n) = (if n = 0 then (1, C 0) else (0, Bound n))" |
|
50313 | 895 |
| "zsplit0 (CN n i a) = |
55885 | 896 |
(let (i', a') = zsplit0 a |
897 |
in if n = 0 then (i + i', a') else (i', CN n i a'))" |
|
898 |
| "zsplit0 (Neg a) = (let (i', a') = zsplit0 a in (-i', Neg a'))" |
|
899 |
| "zsplit0 (Add a b) = |
|
900 |
(let |
|
901 |
(ia, a') = zsplit0 a; |
|
902 |
(ib, b') = zsplit0 b |
|
903 |
in (ia + ib, Add a' b'))" |
|
904 |
| "zsplit0 (Sub a b) = |
|
905 |
(let |
|
906 |
(ia, a') = zsplit0 a; |
|
907 |
(ib, b') = zsplit0 b |
|
908 |
in (ia - ib, Sub a' b'))" |
|
909 |
| "zsplit0 (Mul i a) = (let (i', a') = zsplit0 a in (i*i', Mul i a'))" |
|
23274 | 910 |
|
911 |
lemma zsplit0_I: |
|
55964 | 912 |
"\<And>n a. zsplit0 t = (n, a) \<Longrightarrow> |
55921 | 913 |
(Inum ((x::int) # bs) (CN 0 n a) = Inum (x # bs) t) \<and> numbound0 a" |
50313 | 914 |
(is "\<And>n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a") |
915 |
proof (induct t rule: zsplit0.induct) |
|
55844 | 916 |
case (1 c n a) |
917 |
then show ?case by auto |
|
23274 | 918 |
next |
55844 | 919 |
case (2 m n a) |
920 |
then show ?case by (cases "m = 0") auto |
|
23274 | 921 |
next |
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
922 |
case (3 m i a n a') |
23274 | 923 |
let ?j = "fst (zsplit0 a)" |
924 |
let ?b = "snd (zsplit0 a)" |
|
55844 | 925 |
have abj: "zsplit0 a = (?j, ?b)" by simp |
926 |
{ |
|
927 |
assume "m \<noteq> 0" |
|
928 |
with 3(1)[OF abj] 3(2) have ?case |
|
929 |
by (auto simp add: Let_def split_def) |
|
930 |
} |
|
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
931 |
moreover |
55844 | 932 |
{ |
933 |
assume m0: "m = 0" |
|
55964 | 934 |
with abj have th: "a' = ?b \<and> n = i + ?j" |
935 |
using 3 by (simp add: Let_def split_def) |
|
55844 | 936 |
from abj 3 m0 have th2: "(?I x (CN 0 ?j ?b) = ?I x a) \<and> ?N ?b" |
937 |
by blast |
|
55964 | 938 |
from th have "?I x (CN 0 n a') = ?I x (CN 0 (i + ?j) ?b)" |
55844 | 939 |
by simp |
940 |
also from th2 have "\<dots> = ?I x (CN 0 i (CN 0 ?j ?b))" |
|
941 |
by (simp add: distrib_right) |
|
942 |
finally have "?I x (CN 0 n a') = ?I x (CN 0 i a)" |
|
943 |
using th2 by simp |
|
944 |
with th2 th have ?case using m0 |
|
945 |
by blast |
|
946 |
} |
|
947 |
ultimately show ?case by blast |
|
23274 | 948 |
next |
949 |
case (4 t n a) |
|
950 |
let ?nt = "fst (zsplit0 t)" |
|
951 |
let ?at = "snd (zsplit0 t)" |
|
55964 | 952 |
have abj: "zsplit0 t = (?nt, ?at)" |
953 |
by simp |
|
954 |
then have th: "a = Neg ?at \<and> n = - ?nt" |
|
55844 | 955 |
using 4 by (simp add: Let_def split_def) |
956 |
from abj 4 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" |
|
957 |
by blast |
|
958 |
from th2[simplified] th[simplified] show ?case |
|
959 |
by simp |
|
23274 | 960 |
next |
961 |
case (5 s t n a) |
|
962 |
let ?ns = "fst (zsplit0 s)" |
|
963 |
let ?as = "snd (zsplit0 s)" |
|
964 |
let ?nt = "fst (zsplit0 t)" |
|
965 |
let ?at = "snd (zsplit0 t)" |
|
55844 | 966 |
have abjs: "zsplit0 s = (?ns, ?as)" |
967 |
by simp |
|
968 |
moreover have abjt: "zsplit0 t = (?nt, ?at)" |
|
969 |
by simp |
|
55964 | 970 |
ultimately have th: "a = Add ?as ?at \<and> n = ?ns + ?nt" |
55844 | 971 |
using 5 by (simp add: Let_def split_def) |
55964 | 972 |
from abjs[symmetric] have bluddy: "\<exists>x y. (x, y) = zsplit0 s" |
55844 | 973 |
by blast |
974 |
from 5 have "(\<exists>x y. (x, y) = zsplit0 s) \<longrightarrow> |
|
975 |
(\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \<and> numbound0 xb)" |
|
976 |
by auto |
|
977 |
with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" |
|
978 |
by blast |
|
979 |
from abjs 5 have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" |
|
980 |
by blast |
|
50313 | 981 |
from th3[simplified] th2[simplified] th[simplified] show ?case |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
982 |
by (simp add: distrib_right) |
23274 | 983 |
next |
984 |
case (6 s t n a) |
|
985 |
let ?ns = "fst (zsplit0 s)" |
|
986 |
let ?as = "snd (zsplit0 s)" |
|
987 |
let ?nt = "fst (zsplit0 t)" |
|
988 |
let ?at = "snd (zsplit0 t)" |
|
55844 | 989 |
have abjs: "zsplit0 s = (?ns, ?as)" |
990 |
by simp |
|
991 |
moreover have abjt: "zsplit0 t = (?nt, ?at)" |
|
992 |
by simp |
|
55964 | 993 |
ultimately have th: "a = Sub ?as ?at \<and> n = ?ns - ?nt" |
55844 | 994 |
using 6 by (simp add: Let_def split_def) |
55964 | 995 |
from abjs[symmetric] have bluddy: "\<exists>x y. (x, y) = zsplit0 s" |
55844 | 996 |
by blast |
50313 | 997 |
from 6 have "(\<exists>x y. (x,y) = zsplit0 s) \<longrightarrow> |
998 |
(\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \<and> numbound0 xb)" |
|
999 |
by auto |
|
55844 | 1000 |
with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" |
1001 |
by blast |
|
1002 |
from abjs 6 have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" |
|
1003 |
by blast |
|
50313 | 1004 |
from th3[simplified] th2[simplified] th[simplified] show ?case |
23274 | 1005 |
by (simp add: left_diff_distrib) |
1006 |
next |
|
1007 |
case (7 i t n a) |
|
1008 |
let ?nt = "fst (zsplit0 t)" |
|
1009 |
let ?at = "snd (zsplit0 t)" |
|
55844 | 1010 |
have abj: "zsplit0 t = (?nt,?at)" |
1011 |
by simp |
|
55964 | 1012 |
then have th: "a = Mul i ?at \<and> n = i * ?nt" |
55844 | 1013 |
using 7 by (simp add: Let_def split_def) |
1014 |
from abj 7 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" |
|
1015 |
by blast |
|
1016 |
then have "?I x (Mul i t) = i * ?I x (CN 0 ?nt ?at)" |
|
1017 |
by simp |
|
1018 |
also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" |
|
1019 |
by (simp add: distrib_left) |
|
1020 |
finally show ?case using th th2 |
|
1021 |
by simp |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1022 |
qed |
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1023 |
|
50313 | 1024 |
consts iszlfm :: "fm \<Rightarrow> bool" -- {* Linearity test for fm *} |
23274 | 1025 |
recdef iszlfm "measure size" |
55885 | 1026 |
"iszlfm (And p q) \<longleftrightarrow> iszlfm p \<and> iszlfm q" |
1027 |
"iszlfm (Or p q) \<longleftrightarrow> iszlfm p \<and> iszlfm q" |
|
1028 |
"iszlfm (Eq (CN 0 c e)) \<longleftrightarrow> c > 0 \<and> numbound0 e" |
|
1029 |
"iszlfm (NEq (CN 0 c e)) \<longleftrightarrow> c > 0 \<and> numbound0 e" |
|
1030 |
"iszlfm (Lt (CN 0 c e)) \<longleftrightarrow> c > 0 \<and> numbound0 e" |
|
1031 |
"iszlfm (Le (CN 0 c e)) \<longleftrightarrow> c > 0 \<and> numbound0 e" |
|
1032 |
"iszlfm (Gt (CN 0 c e)) \<longleftrightarrow> c > 0 \<and> numbound0 e" |
|
1033 |
"iszlfm (Ge (CN 0 c e)) \<longleftrightarrow> c > 0 \<and> numbound0 e" |
|
1034 |
"iszlfm (Dvd i (CN 0 c e)) \<longleftrightarrow> c > 0 \<and> i > 0 \<and> numbound0 e" |
|
1035 |
"iszlfm (NDvd i (CN 0 c e)) \<longleftrightarrow> c > 0 \<and> i > 0 \<and> numbound0 e" |
|
1036 |
"iszlfm p \<longleftrightarrow> isatom p \<and> bound0 p" |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1037 |
|
23274 | 1038 |
lemma zlin_qfree: "iszlfm p \<Longrightarrow> qfree p" |
1039 |
by (induct p rule: iszlfm.induct) auto |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1040 |
|
50313 | 1041 |
consts zlfm :: "fm \<Rightarrow> fm" -- {* Linearity transformation for fm *} |
23274 | 1042 |
recdef zlfm "measure fmsize" |
1043 |
"zlfm (And p q) = And (zlfm p) (zlfm q)" |
|
1044 |
"zlfm (Or p q) = Or (zlfm p) (zlfm q)" |
|
1045 |
"zlfm (Imp p q) = Or (zlfm (NOT p)) (zlfm q)" |
|
1046 |
"zlfm (Iff p q) = Or (And (zlfm p) (zlfm q)) (And (zlfm (NOT p)) (zlfm (NOT q)))" |
|
55964 | 1047 |
"zlfm (Lt a) = |
1048 |
(let (c, r) = zsplit0 a in |
|
1049 |
if c = 0 then Lt r else |
|
1050 |
if c > 0 then (Lt (CN 0 c r)) |
|
1051 |
else Gt (CN 0 (- c) (Neg r)))" |
|
1052 |
"zlfm (Le a) = |
|
1053 |
(let (c, r) = zsplit0 a in |
|
1054 |
if c = 0 then Le r |
|
1055 |
else if c > 0 then Le (CN 0 c r) |
|
1056 |
else Ge (CN 0 (- c) (Neg r)))" |
|
1057 |
"zlfm (Gt a) = |
|
1058 |
(let (c, r) = zsplit0 a in |
|
1059 |
if c = 0 then Gt r else |
|
1060 |
if c > 0 then Gt (CN 0 c r) |
|
1061 |
else Lt (CN 0 (- c) (Neg r)))" |
|
1062 |
"zlfm (Ge a) = |
|
1063 |
(let (c, r) = zsplit0 a in |
|
1064 |
if c = 0 then Ge r |
|
1065 |
else if c > 0 then Ge (CN 0 c r) |
|
1066 |
else Le (CN 0 (- c) (Neg r)))" |
|
1067 |
"zlfm (Eq a) = |
|
1068 |
(let (c, r) = zsplit0 a in |
|
1069 |
if c = 0 then Eq r |
|
1070 |
else if c > 0 then Eq (CN 0 c r) |
|
1071 |
else Eq (CN 0 (- c) (Neg r)))" |
|
1072 |
"zlfm (NEq a) = |
|
1073 |
(let (c, r) = zsplit0 a in |
|
1074 |
if c = 0 then NEq r |
|
1075 |
else if c > 0 then NEq (CN 0 c r) |
|
1076 |
else NEq (CN 0 (- c) (Neg r)))" |
|
1077 |
"zlfm (Dvd i a) = |
|
1078 |
(if i = 0 then zlfm (Eq a) |
|
1079 |
else |
|
1080 |
let (c, r) = zsplit0 a in |
|
1081 |
if c = 0 then Dvd (abs i) r |
|
1082 |
else if c > 0 then Dvd (abs i) (CN 0 c r) |
|
1083 |
else Dvd (abs i) (CN 0 (- c) (Neg r)))" |
|
1084 |
"zlfm (NDvd i a) = |
|
1085 |
(if i = 0 then zlfm (NEq a) |
|
1086 |
else |
|
1087 |
let (c, r) = zsplit0 a in |
|
1088 |
if c = 0 then NDvd (abs i) r |
|
1089 |
else if c > 0 then NDvd (abs i) (CN 0 c r) |
|
1090 |
else NDvd (abs i) (CN 0 (- c) (Neg r)))" |
|
23274 | 1091 |
"zlfm (NOT (And p q)) = Or (zlfm (NOT p)) (zlfm (NOT q))" |
1092 |
"zlfm (NOT (Or p q)) = And (zlfm (NOT p)) (zlfm (NOT q))" |
|
1093 |
"zlfm (NOT (Imp p q)) = And (zlfm p) (zlfm (NOT q))" |
|
1094 |
"zlfm (NOT (Iff p q)) = Or (And(zlfm p) (zlfm(NOT q))) (And (zlfm(NOT p)) (zlfm q))" |
|
1095 |
"zlfm (NOT (NOT p)) = zlfm p" |
|
1096 |
"zlfm (NOT T) = F" |
|
1097 |
"zlfm (NOT F) = T" |
|
1098 |
"zlfm (NOT (Lt a)) = zlfm (Ge a)" |
|
1099 |
"zlfm (NOT (Le a)) = zlfm (Gt a)" |
|
1100 |
"zlfm (NOT (Gt a)) = zlfm (Le a)" |
|
1101 |
"zlfm (NOT (Ge a)) = zlfm (Lt a)" |
|
1102 |
"zlfm (NOT (Eq a)) = zlfm (NEq a)" |
|
1103 |
"zlfm (NOT (NEq a)) = zlfm (Eq a)" |
|
1104 |
"zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)" |
|
1105 |
"zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)" |
|
1106 |
"zlfm (NOT (Closed P)) = NClosed P" |
|
1107 |
"zlfm (NOT (NClosed P)) = Closed P" |
|
1108 |
"zlfm p = p" (hints simp add: fmsize_pos) |
|
1109 |
||
1110 |
lemma zlfm_I: |
|
1111 |
assumes qfp: "qfree p" |
|
55981 | 1112 |
shows "Ifm bbs (i # bs) (zlfm p) = Ifm bbs (i # bs) p \<and> iszlfm (zlfm p)" |
1113 |
(is "?I (?l p) = ?I p \<and> ?L (?l p)") |
|
50313 | 1114 |
using qfp |
1115 |
proof (induct p rule: zlfm.induct) |
|
1116 |
case (5 a) |
|
23274 | 1117 |
let ?c = "fst (zsplit0 a)" |
1118 |
let ?r = "snd (zsplit0 a)" |
|
55844 | 1119 |
have spl: "zsplit0 a = (?c, ?r)" |
1120 |
by simp |
|
50313 | 1121 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
55964 | 1122 |
have Ia: "Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" |
55844 | 1123 |
by auto |
55964 | 1124 |
let ?N = "\<lambda>t. Inum (i # bs) t" |
1125 |
from 5 Ia nb show ?case |
|
50313 | 1126 |
apply (auto simp add: Let_def split_def algebra_simps) |
55844 | 1127 |
apply (cases "?r") |
1128 |
apply auto |
|
58259 | 1129 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 1130 |
apply auto |
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1131 |
done |
23274 | 1132 |
next |
50313 | 1133 |
case (6 a) |
23274 | 1134 |
let ?c = "fst (zsplit0 a)" |
1135 |
let ?r = "snd (zsplit0 a)" |
|
55844 | 1136 |
have spl: "zsplit0 a = (?c, ?r)" |
1137 |
by simp |
|
50313 | 1138 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
55964 | 1139 |
have Ia: "Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" |
55844 | 1140 |
by auto |
55964 | 1141 |
let ?N = "\<lambda>t. Inum (i # bs) t" |
50313 | 1142 |
from 6 Ia nb show ?case |
1143 |
apply (auto simp add: Let_def split_def algebra_simps) |
|
55844 | 1144 |
apply (cases "?r") |
1145 |
apply auto |
|
58259 | 1146 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 1147 |
apply auto |
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1148 |
done |
23274 | 1149 |
next |
50313 | 1150 |
case (7 a) |
23274 | 1151 |
let ?c = "fst (zsplit0 a)" |
1152 |
let ?r = "snd (zsplit0 a)" |
|
55844 | 1153 |
have spl: "zsplit0 a = (?c, ?r)" |
1154 |
by simp |
|
50313 | 1155 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
55844 | 1156 |
have Ia: "Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" |
1157 |
by auto |
|
55964 | 1158 |
let ?N = "\<lambda>t. Inum (i # bs) t" |
50313 | 1159 |
from 7 Ia nb show ?case |
1160 |
apply (auto simp add: Let_def split_def algebra_simps) |
|
55844 | 1161 |
apply (cases "?r") |
1162 |
apply auto |
|
58259 | 1163 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 1164 |
apply auto |
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1165 |
done |
23274 | 1166 |
next |
50313 | 1167 |
case (8 a) |
23274 | 1168 |
let ?c = "fst (zsplit0 a)" |
1169 |
let ?r = "snd (zsplit0 a)" |
|
55844 | 1170 |
have spl: "zsplit0 a = (?c, ?r)" |
1171 |
by simp |
|
50313 | 1172 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
55964 | 1173 |
have Ia: "Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" |
55844 | 1174 |
by auto |
55964 | 1175 |
let ?N = "\<lambda>t. Inum (i # bs) t" |
55844 | 1176 |
from 8 Ia nb show ?case |
50313 | 1177 |
apply (auto simp add: Let_def split_def algebra_simps) |
55844 | 1178 |
apply (cases "?r") |
1179 |
apply auto |
|
58259 | 1180 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 1181 |
apply auto |
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1182 |
done |
23274 | 1183 |
next |
50313 | 1184 |
case (9 a) |
23274 | 1185 |
let ?c = "fst (zsplit0 a)" |
1186 |
let ?r = "snd (zsplit0 a)" |
|
55844 | 1187 |
have spl: "zsplit0 a = (?c, ?r)" |
1188 |
by simp |
|
50313 | 1189 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
55844 | 1190 |
have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" |
1191 |
by auto |
|
55964 | 1192 |
let ?N = "\<lambda>t. Inum (i # bs) t" |
55844 | 1193 |
from 9 Ia nb show ?case |
50313 | 1194 |
apply (auto simp add: Let_def split_def algebra_simps) |
55844 | 1195 |
apply (cases "?r") |
1196 |
apply auto |
|
58259 | 1197 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 1198 |
apply auto |
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1199 |
done |
23274 | 1200 |
next |
50313 | 1201 |
case (10 a) |
23274 | 1202 |
let ?c = "fst (zsplit0 a)" |
1203 |
let ?r = "snd (zsplit0 a)" |
|
55844 | 1204 |
have spl: "zsplit0 a = (?c, ?r)" |
1205 |
by simp |
|
50313 | 1206 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
55844 | 1207 |
have Ia: "Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" |
1208 |
by auto |
|
55964 | 1209 |
let ?N = "\<lambda>t. Inum (i # bs) t" |
55844 | 1210 |
from 10 Ia nb show ?case |
50313 | 1211 |
apply (auto simp add: Let_def split_def algebra_simps) |
55844 | 1212 |
apply (cases "?r") |
1213 |
apply auto |
|
58259 | 1214 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 1215 |
apply auto |
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1216 |
done |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1217 |
next |
50313 | 1218 |
case (11 j a) |
23274 | 1219 |
let ?c = "fst (zsplit0 a)" |
1220 |
let ?r = "snd (zsplit0 a)" |
|
55844 | 1221 |
have spl: "zsplit0 a = (?c,?r)" |
1222 |
by simp |
|
50313 | 1223 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
55844 | 1224 |
have Ia: "Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" |
1225 |
by auto |
|
50313 | 1226 |
let ?N = "\<lambda>t. Inum (i#bs) t" |
55844 | 1227 |
have "j = 0 \<or> (j \<noteq> 0 \<and> ?c = 0) \<or> (j \<noteq> 0 \<and> ?c > 0) \<or> (j \<noteq> 0 \<and> ?c < 0)" |
1228 |
by arith |
|
23274 | 1229 |
moreover |
55844 | 1230 |
{ |
1231 |
assume "j = 0" |
|
1232 |
then have z: "zlfm (Dvd j a) = (zlfm (Eq a))" |
|
1233 |
by (simp add: Let_def) |
|
1234 |
then have ?case using 11 `j = 0` |
|
1235 |
by (simp del: zlfm.simps) |
|
1236 |
} |
|
23274 | 1237 |
moreover |
55844 | 1238 |
{ |
1239 |
assume "?c = 0" and "j \<noteq> 0" |
|
1240 |
then have ?case |
|
29700 | 1241 |
using zsplit0_I[OF spl, where x="i" and bs="bs"] |
50313 | 1242 |
apply (auto simp add: Let_def split_def algebra_simps) |
55844 | 1243 |
apply (cases "?r") |
1244 |
apply auto |
|
58259 | 1245 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 1246 |
apply auto |
1247 |
done |
|
1248 |
} |
|
23274 | 1249 |
moreover |
55844 | 1250 |
{ |
1251 |
assume cp: "?c > 0" and jnz: "j \<noteq> 0" |
|
1252 |
then have l: "?L (?l (Dvd j a))" |
|
23274 | 1253 |
by (simp add: nb Let_def split_def) |
55844 | 1254 |
then have ?case |
1255 |
using Ia cp jnz by (simp add: Let_def split_def) |
|
1256 |
} |
|
23274 | 1257 |
moreover |
55844 | 1258 |
{ |
1259 |
assume cn: "?c < 0" and jnz: "j \<noteq> 0" |
|
1260 |
then have l: "?L (?l (Dvd j a))" |
|
23274 | 1261 |
by (simp add: nb Let_def split_def) |
55844 | 1262 |
then have ?case |
1263 |
using Ia cn jnz dvd_minus_iff[of "abs j" "?c*i + ?N ?r"] |
|
1264 |
by (simp add: Let_def split_def) |
|
1265 |
} |
|
23274 | 1266 |
ultimately show ?case by blast |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1267 |
next |
50313 | 1268 |
case (12 j a) |
23274 | 1269 |
let ?c = "fst (zsplit0 a)" |
1270 |
let ?r = "snd (zsplit0 a)" |
|
55844 | 1271 |
have spl: "zsplit0 a = (?c, ?r)" |
1272 |
by simp |
|
50313 | 1273 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
55844 | 1274 |
have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" |
1275 |
by auto |
|
55964 | 1276 |
let ?N = "\<lambda>t. Inum (i # bs) t" |
55844 | 1277 |
have "j = 0 \<or> (j \<noteq> 0 \<and> ?c = 0) \<or> (j \<noteq> 0 \<and> ?c > 0) \<or> (j \<noteq> 0 \<and> ?c < 0)" |
1278 |
by arith |
|
23274 | 1279 |
moreover |
55844 | 1280 |
{ |
1281 |
assume "j = 0" |
|
55964 | 1282 |
then have z: "zlfm (NDvd j a) = zlfm (NEq a)" |
55844 | 1283 |
by (simp add: Let_def) |
1284 |
then have ?case |
|
1285 |
using assms 12 `j = 0` by (simp del: zlfm.simps) |
|
1286 |
} |
|
23274 | 1287 |
moreover |
55844 | 1288 |
{ |
1289 |
assume "?c = 0" and "j \<noteq> 0" |
|
1290 |
then have ?case |
|
29700 | 1291 |
using zsplit0_I[OF spl, where x="i" and bs="bs"] |
50313 | 1292 |
apply (auto simp add: Let_def split_def algebra_simps) |
55844 | 1293 |
apply (cases "?r") |
1294 |
apply auto |
|
58259 | 1295 |
apply (rename_tac nat a b, case_tac nat) |
55844 | 1296 |
apply auto |
1297 |
done |
|
1298 |
} |
|
23274 | 1299 |
moreover |
55844 | 1300 |
{ |
1301 |
assume cp: "?c > 0" and jnz: "j \<noteq> 0" |
|
1302 |
then have l: "?L (?l (Dvd j a))" |
|
23274 | 1303 |
by (simp add: nb Let_def split_def) |
55844 | 1304 |
then have ?case using Ia cp jnz |
1305 |
by (simp add: Let_def split_def) |
|
1306 |
} |
|
23274 | 1307 |
moreover |
55844 | 1308 |
{ |
1309 |
assume cn: "?c < 0" and jnz: "j \<noteq> 0" |
|
1310 |
then have l: "?L (?l (Dvd j a))" |
|
23274 | 1311 |
by (simp add: nb Let_def split_def) |
55844 | 1312 |
then have ?case |
1313 |
using Ia cn jnz dvd_minus_iff[of "abs j" "?c*i + ?N ?r"] |
|
1314 |
by (simp add: Let_def split_def) |
|
1315 |
} |
|
23274 | 1316 |
ultimately show ?case by blast |
1317 |
qed auto |
|
1318 |
||
50313 | 1319 |
consts minusinf :: "fm \<Rightarrow> fm" -- {* Virtual substitution of @{text "-\<infinity>"} *} |
23274 | 1320 |
recdef minusinf "measure size" |
50313 | 1321 |
"minusinf (And p q) = And (minusinf p) (minusinf q)" |
1322 |
"minusinf (Or p q) = Or (minusinf p) (minusinf q)" |
|
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1323 |
"minusinf (Eq (CN 0 c e)) = F" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1324 |
"minusinf (NEq (CN 0 c e)) = T" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1325 |
"minusinf (Lt (CN 0 c e)) = T" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1326 |
"minusinf (Le (CN 0 c e)) = T" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1327 |
"minusinf (Gt (CN 0 c e)) = F" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1328 |
"minusinf (Ge (CN 0 c e)) = F" |
23274 | 1329 |
"minusinf p = p" |
1330 |
||
1331 |
lemma minusinf_qfree: "qfree p \<Longrightarrow> qfree (minusinf p)" |
|
50313 | 1332 |
by (induct p rule: minusinf.induct) auto |
23274 | 1333 |
|
50313 | 1334 |
consts plusinf :: "fm \<Rightarrow> fm" -- {* Virtual substitution of @{text "+\<infinity>"} *} |
23274 | 1335 |
recdef plusinf "measure size" |
50313 | 1336 |
"plusinf (And p q) = And (plusinf p) (plusinf q)" |
1337 |
"plusinf (Or p q) = Or (plusinf p) (plusinf q)" |
|
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1338 |
"plusinf (Eq (CN 0 c e)) = F" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1339 |
"plusinf (NEq (CN 0 c e)) = T" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1340 |
"plusinf (Lt (CN 0 c e)) = F" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1341 |
"plusinf (Le (CN 0 c e)) = F" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1342 |
"plusinf (Gt (CN 0 c e)) = T" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1343 |
"plusinf (Ge (CN 0 c e)) = T" |
23274 | 1344 |
"plusinf p = p" |
1345 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51272
diff
changeset
|
1346 |
consts \<delta> :: "fm \<Rightarrow> int" -- {* Compute @{text "lcm {d| N\<^sup>? Dvd c*x+t \<in> p}"} *} |
23274 | 1347 |
recdef \<delta> "measure size" |
50313 | 1348 |
"\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)" |
1349 |
"\<delta> (Or p q) = lcm (\<delta> p) (\<delta> q)" |
|
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1350 |
"\<delta> (Dvd i (CN 0 c e)) = i" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1351 |
"\<delta> (NDvd i (CN 0 c e)) = i" |
23274 | 1352 |
"\<delta> p = 1" |
1353 |
||
50313 | 1354 |
consts d_\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool" -- {* check if a given l divides all the ds above *} |
50252 | 1355 |
recdef d_\<delta> "measure size" |
50313 | 1356 |
"d_\<delta> (And p q) = (\<lambda>d. d_\<delta> p d \<and> d_\<delta> q d)" |
1357 |
"d_\<delta> (Or p q) = (\<lambda>d. d_\<delta> p d \<and> d_\<delta> q d)" |
|
1358 |
"d_\<delta> (Dvd i (CN 0 c e)) = (\<lambda>d. i dvd d)" |
|
1359 |
"d_\<delta> (NDvd i (CN 0 c e)) = (\<lambda>d. i dvd d)" |
|
1360 |
"d_\<delta> p = (\<lambda>d. True)" |
|
23274 | 1361 |
|
50313 | 1362 |
lemma delta_mono: |
23274 | 1363 |
assumes lin: "iszlfm p" |
50313 | 1364 |
and d: "d dvd d'" |
1365 |
and ad: "d_\<delta> p d" |
|
50252 | 1366 |
shows "d_\<delta> p d'" |
55999 | 1367 |
using lin ad |
50313 | 1368 |
proof (induct p rule: iszlfm.induct) |
55844 | 1369 |
case (9 i c e) |
1370 |
then show ?case using d |
|
30042 | 1371 |
by (simp add: dvd_trans[of "i" "d" "d'"]) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1372 |
next |
55844 | 1373 |
case (10 i c e) |
1374 |
then show ?case using d |
|
30042 | 1375 |
by (simp add: dvd_trans[of "i" "d" "d'"]) |
23274 | 1376 |
qed simp_all |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1377 |
|
50313 | 1378 |
lemma \<delta>: |
55885 | 1379 |
assumes lin: "iszlfm p" |
50252 | 1380 |
shows "d_\<delta> p (\<delta> p) \<and> \<delta> p >0" |
50313 | 1381 |
using lin |
23274 | 1382 |
proof (induct p rule: iszlfm.induct) |
50313 | 1383 |
case (1 p q) |
23274 | 1384 |
let ?d = "\<delta> (And p q)" |
55844 | 1385 |
from 1 lcm_pos_int have dp: "?d > 0" |
1386 |
by simp |
|
1387 |
have d1: "\<delta> p dvd \<delta> (And p q)" |
|
1388 |
using 1 by simp |
|
1389 |
then have th: "d_\<delta> p ?d" |
|
1390 |
using delta_mono 1(2,3) by (simp only: iszlfm.simps) |
|
1391 |
have "\<delta> q dvd \<delta> (And p q)" |
|
1392 |
using 1 by simp |
|
1393 |
then have th': "d_\<delta> q ?d" |
|
1394 |
using delta_mono 1 by (simp only: iszlfm.simps) |
|
1395 |
from th th' dp show ?case |
|
1396 |
by simp |
|
23274 | 1397 |
next |
50313 | 1398 |
case (2 p q) |
23274 | 1399 |
let ?d = "\<delta> (And p q)" |
55844 | 1400 |
from 2 lcm_pos_int have dp: "?d > 0" |
1401 |
by simp |
|
1402 |
have "\<delta> p dvd \<delta> (And p q)" |
|
1403 |
using 2 by simp |
|
1404 |
then have th: "d_\<delta> p ?d" |
|
1405 |
using delta_mono 2 by (simp only: iszlfm.simps) |
|
1406 |
have "\<delta> q dvd \<delta> (And p q)" |
|
1407 |
using 2 by simp |
|
1408 |
then have th': "d_\<delta> q ?d" |
|
1409 |
using delta_mono 2 by (simp only: iszlfm.simps) |
|
1410 |
from th th' dp show ?case |
|
1411 |
by simp |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1412 |
qed simp_all |
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1413 |
|
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1414 |
|
55921 | 1415 |
consts a_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" -- {* adjust the coefficients of a formula *} |
50252 | 1416 |
recdef a_\<beta> "measure size" |
50313 | 1417 |
"a_\<beta> (And p q) = (\<lambda>k. And (a_\<beta> p k) (a_\<beta> q k))" |
1418 |
"a_\<beta> (Or p q) = (\<lambda>k. Or (a_\<beta> p k) (a_\<beta> q k))" |
|
1419 |
"a_\<beta> (Eq (CN 0 c e)) = (\<lambda>k. Eq (CN 0 1 (Mul (k div c) e)))" |
|
1420 |
"a_\<beta> (NEq (CN 0 c e)) = (\<lambda>k. NEq (CN 0 1 (Mul (k div c) e)))" |
|
1421 |
"a_\<beta> (Lt (CN 0 c e)) = (\<lambda>k. Lt (CN 0 1 (Mul (k div c) e)))" |
|
1422 |
"a_\<beta> (Le (CN 0 c e)) = (\<lambda>k. Le (CN 0 1 (Mul (k div c) e)))" |
|
1423 |
"a_\<beta> (Gt (CN 0 c e)) = (\<lambda>k. Gt (CN 0 1 (Mul (k div c) e)))" |
|
1424 |
"a_\<beta> (Ge (CN 0 c e)) = (\<lambda>k. Ge (CN 0 1 (Mul (k div c) e)))" |
|
1425 |
"a_\<beta> (Dvd i (CN 0 c e)) =(\<lambda>k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" |
|
1426 |
"a_\<beta> (NDvd i (CN 0 c e))=(\<lambda>k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" |
|
1427 |
"a_\<beta> p = (\<lambda>k. p)" |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1428 |
|
50313 | 1429 |
consts d_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" -- {* test if all coeffs c of c divide a given l *} |
50252 | 1430 |
recdef d_\<beta> "measure size" |
50313 | 1431 |
"d_\<beta> (And p q) = (\<lambda>k. (d_\<beta> p k) \<and> (d_\<beta> q k))" |
1432 |
"d_\<beta> (Or p q) = (\<lambda>k. (d_\<beta> p k) \<and> (d_\<beta> q k))" |
|
1433 |
"d_\<beta> (Eq (CN 0 c e)) = (\<lambda>k. c dvd k)" |
|
1434 |
"d_\<beta> (NEq (CN 0 c e)) = (\<lambda>k. c dvd k)" |
|
1435 |
"d_\<beta> (Lt (CN 0 c e)) = (\<lambda>k. c dvd k)" |
|
1436 |
"d_\<beta> (Le (CN 0 c e)) = (\<lambda>k. c dvd k)" |
|
1437 |
"d_\<beta> (Gt (CN 0 c e)) = (\<lambda>k. c dvd k)" |
|
1438 |
"d_\<beta> (Ge (CN 0 c e)) = (\<lambda>k. c dvd k)" |
|
1439 |
"d_\<beta> (Dvd i (CN 0 c e)) =(\<lambda>k. c dvd k)" |
|
1440 |
"d_\<beta> (NDvd i (CN 0 c e))=(\<lambda>k. c dvd k)" |
|
1441 |
"d_\<beta> p = (\<lambda>k. True)" |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1442 |
|
50313 | 1443 |
consts \<zeta> :: "fm \<Rightarrow> int" -- {* computes the lcm of all coefficients of x *} |
23274 | 1444 |
recdef \<zeta> "measure size" |
50313 | 1445 |
"\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)" |
1446 |
"\<zeta> (Or p q) = lcm (\<zeta> p) (\<zeta> q)" |
|
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1447 |
"\<zeta> (Eq (CN 0 c e)) = c" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1448 |
"\<zeta> (NEq (CN 0 c e)) = c" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1449 |
"\<zeta> (Lt (CN 0 c e)) = c" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1450 |
"\<zeta> (Le (CN 0 c e)) = c" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1451 |
"\<zeta> (Gt (CN 0 c e)) = c" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1452 |
"\<zeta> (Ge (CN 0 c e)) = c" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1453 |
"\<zeta> (Dvd i (CN 0 c e)) = c" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1454 |
"\<zeta> (NDvd i (CN 0 c e))= c" |
23274 | 1455 |
"\<zeta> p = 1" |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1456 |
|
50313 | 1457 |
consts \<beta> :: "fm \<Rightarrow> num list" |
23274 | 1458 |
recdef \<beta> "measure size" |
50313 | 1459 |
"\<beta> (And p q) = (\<beta> p @ \<beta> q)" |
1460 |
"\<beta> (Or p q) = (\<beta> p @ \<beta> q)" |
|
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1461 |
"\<beta> (Eq (CN 0 c e)) = [Sub (C -1) e]" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1462 |
"\<beta> (NEq (CN 0 c e)) = [Neg e]" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1463 |
"\<beta> (Lt (CN 0 c e)) = []" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1464 |
"\<beta> (Le (CN 0 c e)) = []" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1465 |
"\<beta> (Gt (CN 0 c e)) = [Neg e]" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1466 |
"\<beta> (Ge (CN 0 c e)) = [Sub (C -1) e]" |
23274 | 1467 |
"\<beta> p = []" |
19736 | 1468 |
|
50313 | 1469 |
consts \<alpha> :: "fm \<Rightarrow> num list" |
23274 | 1470 |
recdef \<alpha> "measure size" |
55999 | 1471 |
"\<alpha> (And p q) = \<alpha> p @ \<alpha> q" |
1472 |
"\<alpha> (Or p q) = \<alpha> p @ \<alpha> q" |
|
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1473 |
"\<alpha> (Eq (CN 0 c e)) = [Add (C -1) e]" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1474 |
"\<alpha> (NEq (CN 0 c e)) = [e]" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1475 |
"\<alpha> (Lt (CN 0 c e)) = [e]" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1476 |
"\<alpha> (Le (CN 0 c e)) = [Add (C -1) e]" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1477 |
"\<alpha> (Gt (CN 0 c e)) = []" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1478 |
"\<alpha> (Ge (CN 0 c e)) = []" |
23274 | 1479 |
"\<alpha> p = []" |
50313 | 1480 |
|
23274 | 1481 |
consts mirror :: "fm \<Rightarrow> fm" |
1482 |
recdef mirror "measure size" |
|
50313 | 1483 |
"mirror (And p q) = And (mirror p) (mirror q)" |
1484 |
"mirror (Or p q) = Or (mirror p) (mirror q)" |
|
23995
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1485 |
"mirror (Eq (CN 0 c e)) = Eq (CN 0 c (Neg e))" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1486 |
"mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1487 |
"mirror (Lt (CN 0 c e)) = Gt (CN 0 c (Neg e))" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1488 |
"mirror (Le (CN 0 c e)) = Ge (CN 0 c (Neg e))" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1489 |
"mirror (Gt (CN 0 c e)) = Lt (CN 0 c (Neg e))" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1490 |
"mirror (Ge (CN 0 c e)) = Le (CN 0 c (Neg e))" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1491 |
"mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))" |
c34490f1e0ff
Updated proofs; changed shadow syntax to improve (processing) time
chaieb
parents:
23984
diff
changeset
|
1492 |
"mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))" |
23274 | 1493 |
"mirror p = p" |
50313 | 1494 |
|
1495 |
text {* Lemmas for the correctness of @{text "\<sigma>_\<rho>"} *} |
|
1496 |
||
55844 | 1497 |
lemma dvd1_eq1: |
1498 |
fixes x :: int |
|
1499 |
shows "x > 0 \<Longrightarrow> x dvd 1 \<longleftrightarrow> x = 1" |
|
41807 | 1500 |
by simp |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1501 |
|
23274 | 1502 |
lemma minusinf_inf: |
1503 |
assumes linp: "iszlfm p" |
|
50313 | 1504 |
and u: "d_\<beta> p 1" |
55964 | 1505 |
shows "\<exists>z::int. \<forall>x < z. Ifm bbs (x # bs) (minusinf p) = Ifm bbs (x # bs) p" |
50313 | 1506 |
(is "?P p" is "\<exists>(z::int). \<forall>x < z. ?I x (?M p) = ?I x p") |
1507 |
using linp u |
|
23274 | 1508 |
proof (induct p rule: minusinf.induct) |
55844 | 1509 |
case (1 p q) |
1510 |
then show ?case |
|
55964 | 1511 |
by auto (rule_tac x = "min z za" in exI, simp) |
23274 | 1512 |
next |
55844 | 1513 |
case (2 p q) |
1514 |
then show ?case |
|
55964 | 1515 |
by auto (rule_tac x = "min z za" in exI, simp) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1516 |
next |
55844 | 1517 |
case (3 c e) |
1518 |
then have c1: "c = 1" and nb: "numbound0 e" |
|
1519 |
by simp_all |
|
26934 | 1520 |
fix a |
55999 | 1521 |
from 3 have "\<forall>x<(- Inum (a # bs) e). c * x + Inum (x # bs) e \<noteq> 0" |
55844 | 1522 |
proof clarsimp |
1523 |
fix x |
|
55999 | 1524 |
assume "x < (- Inum (a # bs) e)" and "x + Inum (x # bs) e = 0" |
23274 | 1525 |
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] |
55844 | 1526 |
show False by simp |
23274 | 1527 |
qed |
55844 | 1528 |
then show ?case by auto |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1529 |
next |
55844 | 1530 |
case (4 c e) |
1531 |
then have c1: "c = 1" and nb: "numbound0 e" |
|
1532 |
by simp_all |
|
26934 | 1533 |
fix a |
55964 | 1534 |
from 4 have "\<forall>x < (- Inum (a # bs) e). c * x + Inum (x # bs) e \<noteq> 0" |
55921 | 1535 |
proof clarsimp |
1536 |
fix x |
|
55964 | 1537 |
assume "x < (- Inum (a # bs) e)" and "x + Inum (x # bs) e = 0" |
23274 | 1538 |
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] |
1539 |
show "False" by simp |
|
1540 |
qed |
|
55885 | 1541 |
then show ?case by auto |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1542 |
next |
55921 | 1543 |
case (5 c e) |
1544 |
then have c1: "c = 1" and nb: "numbound0 e" |
|
1545 |
by simp_all |
|
26934 | 1546 |
fix a |
55999 | 1547 |
from 5 have "\<forall>x<(- Inum (a # bs) e). c * x + Inum (x # bs) e < 0" |
55921 | 1548 |
proof clarsimp |
1549 |
fix x |
|
55964 | 1550 |
assume "x < (- Inum (a # bs) e)" |
23274 | 1551 |
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] |
55921 | 1552 |
show "x + Inum (x # bs) e < 0" |
1553 |
by simp |
|
23274 | 1554 |
qed |
55885 | 1555 |
then show ?case by auto |
23274 | 1556 |
next |
55921 | 1557 |
case (6 c e) |
1558 |
then have c1: "c = 1" and nb: "numbound0 e" |
|
1559 |
by simp_all |
|
26934 | 1560 |
fix a |
55964 | 1561 |
from 6 have "\<forall>x<(- Inum (a # bs) e). c * x + Inum (x # bs) e \<le> 0" |
55921 | 1562 |
proof clarsimp |
1563 |
fix x |
|
55964 | 1564 |
assume "x < (- Inum (a # bs) e)" |
23274 | 1565 |
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] |
55964 | 1566 |
show "x + Inum (x # bs) e \<le> 0" by simp |
23274 | 1567 |
qed |
55885 | 1568 |
then show ?case by auto |
23274 | 1569 |
next |
55921 | 1570 |
case (7 c e) |
1571 |
then have c1: "c = 1" and nb: "numbound0 e" |
|
1572 |
by simp_all |
|
26934 | 1573 |
fix a |
55964 | 1574 |
from 7 have "\<forall>x<(- Inum (a # bs) e). \<not> (c * x + Inum (x # bs) e > 0)" |
55921 | 1575 |
proof clarsimp |
1576 |
fix x |
|
55964 | 1577 |
assume "x < - Inum (a # bs) e" and "x + Inum (x # bs) e > 0" |
23274 | 1578 |
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] |
55921 | 1579 |
show False by simp |
23274 | 1580 |
qed |
55885 | 1581 |
then show ?case by auto |
23274 | 1582 |
next |
55921 | 1583 |
case (8 c e) |
1584 |
then have c1: "c = 1" and nb: "numbound0 e" |
|
1585 |
by simp_all |
|
26934 | 1586 |
fix a |
55999 | 1587 |
from 8 have "\<forall>x<(- Inum (a # bs) e). \<not> c * x + Inum (x # bs) e \<ge> 0" |
55921 | 1588 |
proof clarsimp |
1589 |
fix x |
|
55999 | 1590 |
assume "x < (- Inum (a # bs) e)" and "x + Inum (x # bs) e \<ge> 0" |
23274 | 1591 |
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] |
55921 | 1592 |
show False by simp |
23274 | 1593 |
qed |
55885 | 1594 |
then show ?case by auto |
23274 | 1595 |
qed auto |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1596 |
|
23274 | 1597 |
lemma minusinf_repeats: |
55921 | 1598 |
assumes d: "d_\<delta> p d" |
1599 |
and linp: "iszlfm p" |
|
1600 |
shows "Ifm bbs ((x - k * d) # bs) (minusinf p) = Ifm bbs (x # bs) (minusinf p)" |
|
50313 | 1601 |
using linp d |
1602 |
proof (induct p rule: iszlfm.induct) |
|
1603 |
case (9 i c e) |
|
55921 | 1604 |
then have nbe: "numbound0 e" and id: "i dvd d" |
1605 |
by simp_all |
|
1606 |
then have "\<exists>k. d = i * k" |
|
1607 |
by (simp add: dvd_def) |
|
1608 |
then obtain "di" where di_def: "d = i * di" |
|
1609 |
by blast |
|
50313 | 1610 |
show ?case |
1611 |
proof (simp add: numbound0_I[OF nbe,where bs="bs" and b="x - k * d" and b'="x"] right_diff_distrib, |
|
1612 |
rule iffI) |
|
55921 | 1613 |
assume "i dvd c * x - c * (k * d) + Inum (x # bs) e" |
55999 | 1614 |
(is "?ri dvd ?rc * ?rx - ?rc * (?rk * ?rd) + ?I x e" is "?ri dvd ?rt") |
55921 | 1615 |
then have "\<exists>l::int. ?rt = i * l" |
1616 |
by (simp add: dvd_def) |
|
55964 | 1617 |
then have "\<exists>l::int. c * x + ?I x e = i * l + c * (k * i * di)" |
50313 | 1618 |
by (simp add: algebra_simps di_def) |
55964 | 1619 |
then have "\<exists>l::int. c * x + ?I x e = i* (l + c * k * di)" |
50313 | 1620 |
by (simp add: algebra_simps) |
55921 | 1621 |
then have "\<exists>l::int. c * x + ?I x e = i * l" |
1622 |
by blast |
|
1623 |
then show "i dvd c * x + Inum (x # bs) e" |
|
1624 |
by (simp add: dvd_def) |
|
50313 | 1625 |
next |
55964 | 1626 |
assume "i dvd c * x + Inum (x # bs) e" (is "?ri dvd ?rc * ?rx + ?e") |
55921 | 1627 |
then have "\<exists>l::int. c * x + ?e = i * l" |
1628 |
by (simp add: dvd_def) |
|
1629 |
then have "\<exists>l::int. c * x - c * (k * d) + ?e = i * l - c * (k * d)" |
|
1630 |
by simp |
|
1631 |
then have "\<exists>l::int. c * x - c * (k * d) + ?e = i * l - c * (k * i * di)" |
|
1632 |
by (simp add: di_def) |
|
55964 | 1633 |
then have "\<exists>l::int. c * x - c * (k * d) + ?e = i * (l - c * k * di)" |
55921 | 1634 |
by (simp add: algebra_simps) |
1635 |
then have "\<exists>l::int. c * x - c * (k * d) + ?e = i * l" |
|
1636 |
by blast |
|
1637 |
then show "i dvd c * x - c * (k * d) + Inum (x # bs) e" |
|
1638 |
by (simp add: dvd_def) |
|
50313 | 1639 |
qed |
23274 | 1640 |
next |
50313 | 1641 |
case (10 i c e) |
55921 | 1642 |
then have nbe: "numbound0 e" and id: "i dvd d" |
1643 |
by simp_all |
|
1644 |
then have "\<exists>k. d = i * k" |
|
1645 |
by (simp add: dvd_def) |
|
1646 |
then obtain di where di_def: "d = i * di" |
|
1647 |
by blast |
|
50313 | 1648 |
show ?case |
55999 | 1649 |
proof (simp add: numbound0_I[OF nbe,where bs="bs" and b="x - k * d" and b'="x"] right_diff_distrib, |
1650 |
rule iffI) |
|
55921 | 1651 |
assume "i dvd c * x - c * (k * d) + Inum (x # bs) e" |
55999 | 1652 |
(is "?ri dvd ?rc * ?rx - ?rc * (?rk * ?rd) + ?I x e" is "?ri dvd ?rt") |
55921 | 1653 |
then have "\<exists>l::int. ?rt = i * l" |
1654 |
by (simp add: dvd_def) |
|
1655 |
then have "\<exists>l::int. c * x + ?I x e = i * l + c * (k * i * di)" |
|
50313 | 1656 |
by (simp add: algebra_simps di_def) |
55921 | 1657 |
then have "\<exists>l::int. c * x+ ?I x e = i * (l + c * k * di)" |
50313 | 1658 |
by (simp add: algebra_simps) |
55921 | 1659 |
then have "\<exists>l::int. c * x + ?I x e = i * l" |
1660 |
by blast |
|
1661 |
then show "i dvd c * x + Inum (x # bs) e" |
|
1662 |
by (simp add: dvd_def) |
|
50313 | 1663 |
next |
55921 | 1664 |
assume "i dvd c * x + Inum (x # bs) e" (is "?ri dvd ?rc * ?rx + ?e") |
1665 |
then have "\<exists>l::int. c * x + ?e = i * l" |
|
1666 |
by (simp add: dvd_def) |
|
1667 |
then have "\<exists>l::int. c * x - c * (k * d) + ?e = i * l - c * (k * d)" |
|
1668 |
by simp |
|
1669 |
then have "\<exists>l::int. c * x - c * (k * d) + ?e = i * l - c * (k * i * di)" |
|
1670 |
by (simp add: di_def) |
|
55999 | 1671 |
then have "\<exists>l::int. c * x - c * (k * d) + ?e = i * (l - c * k * di)" |
55921 | 1672 |
by (simp add: algebra_simps) |
1673 |
then have "\<exists>l::int. c * x - c * (k * d) + ?e = i * l" |
|
50313 | 1674 |
by blast |
55921 | 1675 |
then show "i dvd c * x - c * (k * d) + Inum (x # bs) e" |
1676 |
by (simp add: dvd_def) |
|
50313 | 1677 |
qed |
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
1678 |
qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="x - k*d" and b'="x"]) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1679 |
|
50252 | 1680 |
lemma mirror_\<alpha>_\<beta>: |
23274 | 1681 |
assumes lp: "iszlfm p" |
55964 | 1682 |
shows "Inum (i # bs) ` set (\<alpha> p) = Inum (i # bs) ` set (\<beta> (mirror p))" |
50313 | 1683 |
using lp by (induct p rule: mirror.induct) auto |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1684 |
|
50313 | 1685 |
lemma mirror: |
23274 | 1686 |
assumes lp: "iszlfm p" |
55921 | 1687 |
shows "Ifm bbs (x # bs) (mirror p) = Ifm bbs ((- x) # bs) p" |
50313 | 1688 |
using lp |
1689 |
proof (induct p rule: iszlfm.induct) |
|
1690 |
case (9 j c e) |
|
55964 | 1691 |
then have nb: "numbound0 e" |
1692 |
by simp |
|
1693 |
have "Ifm bbs (x # bs) (mirror (Dvd j (CN 0 c e))) \<longleftrightarrow> j dvd c * x - Inum (x # bs) e" |
|
50313 | 1694 |
(is "_ = (j dvd c*x - ?e)") by simp |
55964 | 1695 |
also have "\<dots> \<longleftrightarrow> j dvd (- (c * x - ?e))" |
30042 | 1696 |
by (simp only: dvd_minus_iff) |
55964 | 1697 |
also have "\<dots> \<longleftrightarrow> j dvd (c * (- x)) + ?e" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
55999
diff
changeset
|
1698 |
by (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] ac_simps minus_add_distrib) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
1699 |
(simp add: algebra_simps) |
55964 | 1700 |
also have "\<dots> = Ifm bbs ((- x) # bs) (Dvd j (CN 0 c e))" |
50313 | 1701 |
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"] by simp |
23274 | 1702 |
finally show ?case . |
1703 |
next |
|
55964 | 1704 |
case (10 j c e) |
1705 |
then have nb: "numbound0 e" |
|
1706 |
by simp |
|
1707 |
have "Ifm bbs (x # bs) (mirror (Dvd j (CN 0 c e))) \<longleftrightarrow> j dvd c * x - Inum (x # bs) e" |
|
1708 |
(is "_ = (j dvd c * x - ?e)") by simp |
|
1709 |
also have "\<dots> \<longleftrightarrow> j dvd (- (c * x - ?e))" |
|
30042 | 1710 |
by (simp only: dvd_minus_iff) |
55964 | 1711 |
also have "\<dots> \<longleftrightarrow> j dvd (c * (- x)) + ?e" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
55999
diff
changeset
|
1712 |
by (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] ac_simps minus_add_distrib) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
1713 |
(simp add: algebra_simps) |
55964 | 1714 |
also have "\<dots> \<longleftrightarrow> Ifm bbs ((- x) # bs) (Dvd j (CN 0 c e))" |
50313 | 1715 |
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"] by simp |
23274 | 1716 |
finally show ?case by simp |
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
1717 |
qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] gr0_conv_Suc) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1718 |
|
50313 | 1719 |
lemma mirror_l: "iszlfm p \<and> d_\<beta> p 1 \<Longrightarrow> iszlfm (mirror p) \<and> d_\<beta> (mirror p) 1" |
41807 | 1720 |
by (induct p rule: mirror.induct) auto |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1721 |
|
23274 | 1722 |
lemma mirror_\<delta>: "iszlfm p \<Longrightarrow> \<delta> (mirror p) = \<delta> p" |
41807 | 1723 |
by (induct p rule: mirror.induct) auto |
23274 | 1724 |
|
50313 | 1725 |
lemma \<beta>_numbound0: |
1726 |
assumes lp: "iszlfm p" |
|
55964 | 1727 |
shows "\<forall>b \<in> set (\<beta> p). numbound0 b" |
41807 | 1728 |
using lp by (induct p rule: \<beta>.induct) auto |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1729 |
|
50313 | 1730 |
lemma d_\<beta>_mono: |
23274 | 1731 |
assumes linp: "iszlfm p" |
50313 | 1732 |
and dr: "d_\<beta> p l" |
1733 |
and d: "l dvd l'" |
|
50252 | 1734 |
shows "d_\<beta> p l'" |
50313 | 1735 |
using dr linp dvd_trans[of _ "l" "l'", simplified d] |
41807 | 1736 |
by (induct p rule: iszlfm.induct) simp_all |
23274 | 1737 |
|
50313 | 1738 |
lemma \<alpha>_l: |
55999 | 1739 |
assumes "iszlfm p" |
50313 | 1740 |
shows "\<forall>b \<in> set (\<alpha> p). numbound0 b" |
55999 | 1741 |
using assms by (induct p rule: \<alpha>.induct) auto |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1742 |
|
50313 | 1743 |
lemma \<zeta>: |
55999 | 1744 |
assumes "iszlfm p" |
50252 | 1745 |
shows "\<zeta> p > 0 \<and> d_\<beta> p (\<zeta> p)" |
55999 | 1746 |
using assms |
50313 | 1747 |
proof (induct p rule: iszlfm.induct) |
23274 | 1748 |
case (1 p q) |
55964 | 1749 |
from 1 have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" |
1750 |
by simp |
|
1751 |
from 1 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" |
|
1752 |
by simp |
|
50313 | 1753 |
from 1 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"] |
55964 | 1754 |
d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"] |
1755 |
dl1 dl2 |
|
1756 |
show ?case |
|
1757 |
by (auto simp add: lcm_pos_int) |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1758 |
next |
23274 | 1759 |
case (2 p q) |
55964 | 1760 |
from 2 have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" |
1761 |
by simp |
|
1762 |
from 2 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" |
|
1763 |
by simp |
|
50313 | 1764 |
from 2 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"] |
55964 | 1765 |
d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"] |
1766 |
dl1 dl2 |
|
1767 |
show ?case |
|
1768 |
by (auto simp add: lcm_pos_int) |
|
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31730
diff
changeset
|
1769 |
qed (auto simp add: lcm_pos_int) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1770 |
|
50313 | 1771 |
lemma a_\<beta>: |
55921 | 1772 |
assumes linp: "iszlfm p" |
1773 |
and d: "d_\<beta> p l" |
|
1774 |
and lp: "l > 0" |
|
55964 | 1775 |
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" |
50313 | 1776 |
using linp d |
23274 | 1777 |
proof (induct p rule: iszlfm.induct) |
50313 | 1778 |
case (5 c e) |
55964 | 1779 |
then have cp: "c > 0" and be: "numbound0 e" and d': "c dvd l" |
55921 | 1780 |
by simp_all |
1781 |
from lp cp have clel: "c \<le> l" |
|
1782 |
by (simp add: zdvd_imp_le [OF d' lp]) |
|
1783 |
from cp have cnz: "c \<noteq> 0" |
|
1784 |
by simp |
|
1785 |
have "c div c \<le> l div c" |
|
50313 | 1786 |
by (simp add: zdiv_mono1[OF clel cp]) |
55999 | 1787 |
then have ldcp: "0 < l div c" |
50313 | 1788 |
by (simp add: div_self[OF cnz]) |
55921 | 1789 |
have "c * (l div c) = c * (l div c) + l mod c" |
1790 |
using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
|
1791 |
then have cl: "c * (l div c) =l" |
|
1792 |
using zmod_zdiv_equality[where a="l" and b="c", symmetric] by simp |
|
55964 | 1793 |
then have "(l * x + (l div c) * Inum (x # bs) e < 0) \<longleftrightarrow> |
50313 | 1794 |
((c * (l div c)) * x + (l div c) * Inum (x # bs) e < 0)" |
1795 |
by simp |
|
55999 | 1796 |
also have "\<dots> \<longleftrightarrow> (l div c) * (c * x + Inum (x # bs) e) < (l div c) * 0" |
50313 | 1797 |
by (simp add: algebra_simps) |
55964 | 1798 |
also have "\<dots> \<longleftrightarrow> c * x + Inum (x # bs) e < 0" |
1799 |
using mult_less_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp |
|
1800 |
by simp |
|
50313 | 1801 |
finally show ?case |
55964 | 1802 |
using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be |
1803 |
by simp |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1804 |
next |
50313 | 1805 |
case (6 c e) |
55921 | 1806 |
then have cp: "c > 0" and be: "numbound0 e" and d': "c dvd l" |
1807 |
by simp_all |
|
1808 |
from lp cp have clel: "c \<le> l" |
|
1809 |
by (simp add: zdvd_imp_le [OF d' lp]) |
|
1810 |
from cp have cnz: "c \<noteq> 0" |
|
1811 |
by simp |
|
55964 | 1812 |
have "c div c \<le> l div c" |
50313 | 1813 |
by (simp add: zdiv_mono1[OF clel cp]) |
1814 |
then have ldcp:"0 < l div c" |
|
1815 |
by (simp add: div_self[OF cnz]) |
|
55964 | 1816 |
have "c * (l div c) = c * (l div c) + l mod c" |
55921 | 1817 |
using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
1818 |
then have cl: "c * (l div c) = l" |
|
1819 |
using zmod_zdiv_equality[where a="l" and b="c", symmetric] by simp |
|
55964 | 1820 |
then have "l * x + (l div c) * Inum (x # bs) e \<le> 0 \<longleftrightarrow> |
1821 |
(c * (l div c)) * x + (l div c) * Inum (x # bs) e \<le> 0" |
|
50313 | 1822 |
by simp |
55964 | 1823 |
also have "\<dots> \<longleftrightarrow> (l div c) * (c * x + Inum (x # bs) e) \<le> (l div c) * 0" |
50313 | 1824 |
by (simp add: algebra_simps) |
55964 | 1825 |
also have "\<dots> \<longleftrightarrow> c * x + Inum (x # bs) e \<le> 0" |
23274 | 1826 |
using mult_le_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp |
50313 | 1827 |
finally show ?case |
1828 |
using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be by simp |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1829 |
next |
50313 | 1830 |
case (7 c e) |
55921 | 1831 |
then have cp: "c > 0" and be: "numbound0 e" and d': "c dvd l" |
1832 |
by simp_all |
|
1833 |
from lp cp have clel: "c \<le> l" |
|
1834 |
by (simp add: zdvd_imp_le [OF d' lp]) |
|
1835 |
from cp have cnz: "c \<noteq> 0" |
|
1836 |
by simp |
|
1837 |
have "c div c \<le> l div c" |
|
1838 |
by (simp add: zdiv_mono1[OF clel cp]) |
|
55964 | 1839 |
then have ldcp: "0 < l div c" |
55921 | 1840 |
by (simp add: div_self[OF cnz]) |
55964 | 1841 |
have "c * (l div c) = c * (l div c) + l mod c" |
55921 | 1842 |
using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
55964 | 1843 |
then have cl: "c * (l div c) = l" |
55921 | 1844 |
using zmod_zdiv_equality[where a="l" and b="c", symmetric] by simp |
55964 | 1845 |
then have "l * x + (l div c) * Inum (x # bs) e > 0 \<longleftrightarrow> |
1846 |
(c * (l div c)) * x + (l div c) * Inum (x # bs) e > 0" |
|
55921 | 1847 |
by simp |
55964 | 1848 |
also have "\<dots> \<longleftrightarrow> (l div c) * (c * x + Inum (x # bs) e) > (l div c) * 0" |
55921 | 1849 |
by (simp add: algebra_simps) |
55964 | 1850 |
also have "\<dots> \<longleftrightarrow> c * x + Inum (x # bs) e > 0" |
55921 | 1851 |
using zero_less_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp |
1852 |
by simp |
|
1853 |
finally show ?case |
|
1854 |
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be |
|
1855 |
by simp |
|
1856 |
next |
|
1857 |
case (8 c e) |
|
1858 |
then have cp: "c > 0" and be: "numbound0 e" and d': "c dvd l" |
|
1859 |
by simp_all |
|
1860 |
from lp cp have clel: "c \<le> l" |
|
1861 |
by (simp add: zdvd_imp_le [OF d' lp]) |
|
1862 |
from cp have cnz: "c \<noteq> 0" |
|
1863 |
by simp |
|
1864 |
have "c div c \<le> l div c" |
|
1865 |
by (simp add: zdiv_mono1[OF clel cp]) |
|
1866 |
then have ldcp: "0 < l div c" |
|
1867 |
by (simp add: div_self[OF cnz]) |
|
55964 | 1868 |
have "c * (l div c) = c * (l div c) + l mod c" |
55921 | 1869 |
using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
1870 |
then have cl: "c * (l div c) =l" |
|
1871 |
using zmod_zdiv_equality[where a="l" and b="c", symmetric] |
|
1872 |
by simp |
|
55964 | 1873 |
then have "l * x + (l div c) * Inum (x # bs) e \<ge> 0 \<longleftrightarrow> |
1874 |
(c * (l div c)) * x + (l div c) * Inum (x # bs) e \<ge> 0" |
|
55921 | 1875 |
by simp |
55964 | 1876 |
also have "\<dots> \<longleftrightarrow> (l div c) * (c * x + Inum (x # bs) e) \<ge> (l div c) * 0" |
55921 | 1877 |
by (simp add: algebra_simps) |
55964 | 1878 |
also have "\<dots> \<longleftrightarrow> c * x + Inum (x # bs) e \<ge> 0" |
55921 | 1879 |
using ldcp zero_le_mult_iff [where a="l div c" and b="c*x + Inum (x # bs) e"] |
1880 |
by simp |
|
1881 |
finally show ?case |
|
1882 |
using be numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] |
|
1883 |
by simp |
|
1884 |
next |
|
1885 |
case (3 c e) |
|
1886 |
then have cp: "c > 0" and be: "numbound0 e" and d': "c dvd l" |
|
1887 |
by simp_all |
|
1888 |
from lp cp have clel: "c \<le> l" |
|
1889 |
by (simp add: zdvd_imp_le [OF d' lp]) |
|
1890 |
from cp have cnz: "c \<noteq> 0" |
|
1891 |
by simp |
|
1892 |
have "c div c \<le> l div c" |
|
50313 | 1893 |
by (simp add: zdiv_mono1[OF clel cp]) |
1894 |
then have ldcp:"0 < l div c" |
|
1895 |
by (simp add: div_self[OF cnz]) |
|
55964 | 1896 |
have "c * (l div c) = c * (l div c) + l mod c" |
50313 | 1897 |
using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
55885 | 1898 |
then have cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] |
50313 | 1899 |
by simp |
55964 | 1900 |
then have "l * x + (l div c) * Inum (x # bs) e = 0 \<longleftrightarrow> |
1901 |
(c * (l div c)) * x + (l div c) * Inum (x # bs) e = 0" |
|
23274 | 1902 |
by simp |
55964 | 1903 |
also have "\<dots> \<longleftrightarrow> (l div c) * (c * x + Inum (x # bs) e) = ((l div c)) * 0" |
50313 | 1904 |
by (simp add: algebra_simps) |
55964 | 1905 |
also have "\<dots> \<longleftrightarrow> c * x + Inum (x # bs) e = 0" |
55921 | 1906 |
using mult_eq_0_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp |
1907 |
by simp |
|
50313 | 1908 |
finally show ?case |
55921 | 1909 |
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be |
1910 |
by simp |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1911 |
next |
50313 | 1912 |
case (4 c e) |
55921 | 1913 |
then have cp: "c > 0" and be: "numbound0 e" and d': "c dvd l" |
1914 |
by simp_all |
|
1915 |
from lp cp have clel: "c \<le> l" |
|
1916 |
by (simp add: zdvd_imp_le [OF d' lp]) |
|
1917 |
from cp have cnz: "c \<noteq> 0" |
|
1918 |
by simp |
|
1919 |
have "c div c \<le> l div c" |
|
50313 | 1920 |
by (simp add: zdiv_mono1[OF clel cp]) |
1921 |
then have ldcp:"0 < l div c" |
|
1922 |
by (simp add: div_self[OF cnz]) |
|
55964 | 1923 |
have "c * (l div c) = c * (l div c) + l mod c" |
55921 | 1924 |
using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
1925 |
then have cl: "c * (l div c) = l" |
|
1926 |
using zmod_zdiv_equality[where a="l" and b="c", symmetric] by simp |
|
55964 | 1927 |
then have "l * x + (l div c) * Inum (x # bs) e \<noteq> 0 \<longleftrightarrow> |
55921 | 1928 |
(c * (l div c)) * x + (l div c) * Inum (x # bs) e \<noteq> 0" |
50313 | 1929 |
by simp |
55921 | 1930 |
also have "\<dots> \<longleftrightarrow> (l div c) * (c * x + Inum (x # bs) e) \<noteq> (l div c) * 0" |
50313 | 1931 |
by (simp add: algebra_simps) |
55921 | 1932 |
also have "\<dots> \<longleftrightarrow> c * x + Inum (x # bs) e \<noteq> 0" |
1933 |
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp |
|
1934 |
by simp |
|
50313 | 1935 |
finally show ?case |
55921 | 1936 |
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be |
1937 |
by simp |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
1938 |
next |
50313 | 1939 |
case (9 j c e) |
55921 | 1940 |
then have cp: "c > 0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" |
1941 |
by simp_all |
|
1942 |
from lp cp have clel: "c \<le> l" |
|
1943 |
by (simp add: zdvd_imp_le [OF d' lp]) |
|
50313 | 1944 |
from cp have cnz: "c \<noteq> 0" by simp |
1945 |
have "c div c\<le> l div c" |
|
1946 |
by (simp add: zdiv_mono1[OF clel cp]) |
|
1947 |
then have ldcp:"0 < l div c" |
|
1948 |
by (simp add: div_self[OF cnz]) |
|
55964 | 1949 |
have "c * (l div c) = c * (l div c) + l mod c" |
55885 | 1950 |
using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
55921 | 1951 |
then have cl: "c * (l div c) = l" |
55885 | 1952 |
using zmod_zdiv_equality[where a="l" and b="c", symmetric] by simp |
55921 | 1953 |
then have "(\<exists>k::int. l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) \<longleftrightarrow> |
55964 | 1954 |
(\<exists>k::int. (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)" |
55921 | 1955 |
by simp |
55964 | 1956 |
also have "\<dots> \<longleftrightarrow> (\<exists>k::int. (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c) * 0)" |
55921 | 1957 |
by (simp add: algebra_simps) |
1958 |
also |
|
1959 |
fix k |
|
1960 |
have "\<dots> \<longleftrightarrow> (\<exists>k::int. c * x + Inum (x # bs) e - j * k = 0)" |
|
1961 |
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp |
|
1962 |
by simp |
|
1963 |
also have "\<dots> \<longleftrightarrow> (\<exists>k::int. c * x + Inum (x # bs) e = j * k)" |
|
1964 |
by simp |
|
1965 |
finally show ?case |
|
1966 |
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] |
|
1967 |
be mult_strict_mono[OF ldcp jp ldcp ] |
|
1968 |
by (simp add: dvd_def) |
|
1969 |
next |
|
1970 |
case (10 j c e) |
|
1971 |
then have cp: "c > 0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" |
|
1972 |
by simp_all |
|
1973 |
from lp cp have clel: "c \<le> l" |
|
1974 |
by (simp add: zdvd_imp_le [OF d' lp]) |
|
1975 |
from cp have cnz: "c \<noteq> 0" |
|
50313 | 1976 |
by simp |
55921 | 1977 |
have "c div c \<le> l div c" |
1978 |
by (simp add: zdiv_mono1[OF clel cp]) |
|
1979 |
then have ldcp: "0 < l div c" |
|
1980 |
by (simp add: div_self[OF cnz]) |
|
1981 |
have "c * (l div c) = c* (l div c) + l mod c" |
|
1982 |
using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
|
1983 |
then have cl:"c * (l div c) =l" |
|
1984 |
using zmod_zdiv_equality[where a="l" and b="c", symmetric] |
|
1985 |
by simp |
|
1986 |
then have "(\<exists>k::int. l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) \<longleftrightarrow> |
|
1987 |
(\<exists>k::int. (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)" |
|
1988 |
by simp |
|
1989 |
also have "\<dots> \<longleftrightarrow> (\<exists>k::int. (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c) * 0)" |
|
1990 |
by (simp add: algebra_simps) |
|
1991 |
also |
|
1992 |
fix k |
|
55964 | 1993 |
have "\<dots> \<longleftrightarrow> (\<exists>k::int. c * x + Inum (x # bs) e - j * k = 0)" |
55921 | 1994 |
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp |
1995 |
by simp |
|
55964 | 1996 |
also have "\<dots> \<longleftrightarrow> (\<exists>k::int. c * x + Inum (x # bs) e = j * k)" |
55921 | 1997 |
by simp |
1998 |
finally show ?case |
|
1999 |
using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be |
|
2000 |
mult_strict_mono[OF ldcp jp ldcp ] |
|
2001 |
by (simp add: dvd_def) |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
2002 |
qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="(l * x)" and b'="x"]) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2003 |
|
55921 | 2004 |
lemma a_\<beta>_ex: |
2005 |
assumes linp: "iszlfm p" |
|
2006 |
and d: "d_\<beta> p l" |
|
2007 |
and lp: "l > 0" |
|
2008 |
shows "(\<exists>x. l dvd x \<and> Ifm bbs (x #bs) (a_\<beta> p l)) \<longleftrightarrow> (\<exists>x::int. Ifm bbs (x#bs) p)" |
|
2009 |
(is "(\<exists>x. l dvd x \<and> ?P x) \<longleftrightarrow> (\<exists>x. ?P' x)") |
|
23274 | 2010 |
proof- |
55999 | 2011 |
have "(\<exists>x. l dvd x \<and> ?P x) \<longleftrightarrow> (\<exists>x::int. ?P (l * x))" |
23274 | 2012 |
using unity_coeff_ex[where l="l" and P="?P", simplified] by simp |
55921 | 2013 |
also have "\<dots> = (\<exists>x::int. ?P' x)" |
2014 |
using a_\<beta>[OF linp d lp] by simp |
|
50313 | 2015 |
finally show ?thesis . |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2016 |
qed |
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2017 |
|
23274 | 2018 |
lemma \<beta>: |
55999 | 2019 |
assumes "iszlfm p" |
2020 |
and "d_\<beta> p 1" |
|
2021 |
and "d_\<delta> p d" |
|
55885 | 2022 |
and dp: "d > 0" |
55999 | 2023 |
and "\<not> (\<exists>j::int \<in> {1 .. d}. \<exists>b \<in> Inum (a # bs) ` set (\<beta> p). x = b + j)" |
55964 | 2024 |
and p: "Ifm bbs (x # bs) p" (is "?P x") |
23274 | 2025 |
shows "?P (x - d)" |
55999 | 2026 |
using assms |
55885 | 2027 |
proof (induct p rule: iszlfm.induct) |
2028 |
case (5 c e) |
|
2029 |
then have c1: "c = 1" and bn: "numbound0 e" |
|
2030 |
by simp_all |
|
55964 | 2031 |
with dp p c1 numbound0_I[OF bn,where b = "(x - d)" and b' = "x" and bs = "bs"] 5 |
41807 | 2032 |
show ?case by simp |
23274 | 2033 |
next |
55885 | 2034 |
case (6 c e) |
2035 |
then have c1: "c = 1" and bn: "numbound0 e" |
|
2036 |
by simp_all |
|
41807 | 2037 |
with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] 6 |
2038 |
show ?case by simp |
|
23274 | 2039 |
next |
55885 | 2040 |
case (7 c e) |
55964 | 2041 |
then have p: "Ifm bbs (x # bs) (Gt (CN 0 c e))" and c1: "c=1" and bn: "numbound0 e" |
55885 | 2042 |
by simp_all |
41807 | 2043 |
let ?e = "Inum (x # bs) e" |
55885 | 2044 |
{ |
55964 | 2045 |
assume "(x - d) + ?e > 0" |
55885 | 2046 |
then have ?case |
2047 |
using c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] by simp |
|
2048 |
} |
|
41807 | 2049 |
moreover |
55885 | 2050 |
{ |
55964 | 2051 |
assume H: "\<not> (x - d) + ?e > 0" |
2052 |
let ?v = "Neg e" |
|
2053 |
have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" |
|
2054 |
by simp |
|
57816
d8bbb97689d3
no need for 'set_simps' now that 'datatype_new' generates the desired 'set' property
blanchet
parents:
57514
diff
changeset
|
2055 |
from 7(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]] |
55964 | 2056 |
have nob: "\<not> (\<exists>j\<in> {1 ..d}. x = - ?e + j)" |
55885 | 2057 |
by auto |
2058 |
from H p have "x + ?e > 0 \<and> x + ?e \<le> d" |
|
2059 |
by (simp add: c1) |
|
2060 |
then have "x + ?e \<ge> 1 \<and> x + ?e \<le> d" |
|
2061 |
by simp |
|
55964 | 2062 |
then have "\<exists>j::int \<in> {1 .. d}. j = x + ?e" |
55885 | 2063 |
by simp |
55964 | 2064 |
then have "\<exists>j::int \<in> {1 .. d}. x = (- ?e + j)" |
41807 | 2065 |
by (simp add: algebra_simps) |
55885 | 2066 |
with nob have ?case |
2067 |
by auto |
|
2068 |
} |
|
2069 |
ultimately show ?case |
|
2070 |
by blast |
|
23274 | 2071 |
next |
55885 | 2072 |
case (8 c e) |
2073 |
then have p: "Ifm bbs (x # bs) (Ge (CN 0 c e))" and c1: "c = 1" and bn: "numbound0 e" |
|
50313 | 2074 |
by simp_all |
55885 | 2075 |
let ?e = "Inum (x # bs) e" |
2076 |
{ |
|
2077 |
assume "(x - d) + ?e \<ge> 0" |
|
2078 |
then have ?case |
|
2079 |
using c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] |
|
2080 |
by simp |
|
2081 |
} |
|
2082 |
moreover |
|
2083 |
{ |
|
2084 |
assume H: "\<not> (x - d) + ?e \<ge> 0" |
|
2085 |
let ?v = "Sub (C -1) e" |
|
2086 |
have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" |
|
2087 |
by simp |
|
57816
d8bbb97689d3
no need for 'set_simps' now that 'datatype_new' generates the desired 'set' property
blanchet
parents:
57514
diff
changeset
|
2088 |
from 8(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]] |
55885 | 2089 |
have nob: "\<not> (\<exists>j\<in> {1 ..d}. x = - ?e - 1 + j)" |
2090 |
by auto |
|
2091 |
from H p have "x + ?e \<ge> 0 \<and> x + ?e < d" |
|
2092 |
by (simp add: c1) |
|
2093 |
then have "x + ?e +1 \<ge> 1 \<and> x + ?e + 1 \<le> d" |
|
2094 |
by simp |
|
55964 | 2095 |
then have "\<exists>j::int \<in> {1 .. d}. j = x + ?e + 1" |
55885 | 2096 |
by simp |
55964 | 2097 |
then have "\<exists>j::int \<in> {1 .. d}. x= - ?e - 1 + j" |
55885 | 2098 |
by (simp add: algebra_simps) |
2099 |
with nob have ?case |
|
2100 |
by simp |
|
2101 |
} |
|
2102 |
ultimately show ?case |
|
2103 |
by blast |
|
23274 | 2104 |
next |
55885 | 2105 |
case (3 c e) |
2106 |
then |
|
2107 |
have p: "Ifm bbs (x #bs) (Eq (CN 0 c e))" (is "?p x") |
|
55964 | 2108 |
and c1: "c = 1" |
55885 | 2109 |
and bn: "numbound0 e" |
2110 |
by simp_all |
|
2111 |
let ?e = "Inum (x # bs) e" |
|
2112 |
let ?v="(Sub (C -1) e)" |
|
2113 |
have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" |
|
2114 |
by simp |
|
55964 | 2115 |
from p have "x= - ?e" |
2116 |
by (simp add: c1) with 3(5) |
|
2117 |
show ?case |
|
55885 | 2118 |
using dp |
2119 |
by simp (erule ballE[where x="1"], |
|
2120 |
simp_all add:algebra_simps numbound0_I[OF bn,where b="x"and b'="a"and bs="bs"]) |
|
23274 | 2121 |
next |
55885 | 2122 |
case (4 c e) |
2123 |
then |
|
55964 | 2124 |
have p: "Ifm bbs (x # bs) (NEq (CN 0 c e))" (is "?p x") |
55885 | 2125 |
and c1: "c = 1" |
2126 |
and bn: "numbound0 e" |
|
2127 |
by simp_all |
|
2128 |
let ?e = "Inum (x # bs) e" |
|
2129 |
let ?v="Neg e" |
|
55964 | 2130 |
have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" |
2131 |
by simp |
|
55885 | 2132 |
{ |
2133 |
assume "x - d + Inum (((x -d)) # bs) e \<noteq> 0" |
|
2134 |
then have ?case by (simp add: c1) |
|
2135 |
} |
|
2136 |
moreover |
|
2137 |
{ |
|
55964 | 2138 |
assume H: "x - d + Inum ((x - d) # bs) e = 0" |
2139 |
then have "x = - Inum ((x - d) # bs) e + d" |
|
55885 | 2140 |
by simp |
2141 |
then have "x = - Inum (a # bs) e + d" |
|
2142 |
by (simp add: numbound0_I[OF bn,where b="x - d"and b'="a"and bs="bs"]) |
|
2143 |
with 4(5) have ?case |
|
2144 |
using dp by simp |
|
2145 |
} |
|
2146 |
ultimately show ?case |
|
2147 |
by blast |
|
50313 | 2148 |
next |
55885 | 2149 |
case (9 j c e) |
2150 |
then |
|
2151 |
have p: "Ifm bbs (x # bs) (Dvd j (CN 0 c e))" (is "?p x") |
|
2152 |
and c1: "c = 1" |
|
2153 |
and bn: "numbound0 e" |
|
2154 |
by simp_all |
|
2155 |
let ?e = "Inum (x # bs) e" |
|
2156 |
from 9 have id: "j dvd d" |
|
2157 |
by simp |
|
55964 | 2158 |
from c1 have "?p x \<longleftrightarrow> j dvd (x + ?e)" |
55885 | 2159 |
by simp |
55964 | 2160 |
also have "\<dots> \<longleftrightarrow> j dvd x - d + ?e" |
55885 | 2161 |
using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] |
2162 |
by simp |
|
2163 |
finally show ?case |
|
2164 |
using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p |
|
2165 |
by simp |
|
23274 | 2166 |
next |
55885 | 2167 |
case (10 j c e) |
2168 |
then |
|
55964 | 2169 |
have p: "Ifm bbs (x # bs) (NDvd j (CN 0 c e))" (is "?p x") |
55885 | 2170 |
and c1: "c = 1" |
2171 |
and bn: "numbound0 e" |
|
2172 |
by simp_all |
|
2173 |
let ?e = "Inum (x # bs) e" |
|
2174 |
from 10 have id: "j dvd d" |
|
2175 |
by simp |
|
55964 | 2176 |
from c1 have "?p x \<longleftrightarrow> \<not> j dvd (x + ?e)" |
55885 | 2177 |
by simp |
55964 | 2178 |
also have "\<dots> \<longleftrightarrow> \<not> j dvd x - d + ?e" |
55885 | 2179 |
using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] |
2180 |
by simp |
|
2181 |
finally show ?case |
|
2182 |
using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p |
|
2183 |
by simp |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
2184 |
qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] gr0_conv_Suc) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2185 |
|
50313 | 2186 |
lemma \<beta>': |
23274 | 2187 |
assumes lp: "iszlfm p" |
50252 | 2188 |
and u: "d_\<beta> p 1" |
2189 |
and d: "d_\<delta> p d" |
|
23274 | 2190 |
and dp: "d > 0" |
55964 | 2191 |
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> |
2192 |
Ifm bbs (x # bs) p \<longrightarrow> Ifm bbs ((x - d) # bs) p" (is "\<forall>x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)") |
|
55885 | 2193 |
proof clarify |
50313 | 2194 |
fix x |
55885 | 2195 |
assume nb: "?b" |
2196 |
and px: "?P x" |
|
55964 | 2197 |
then have nb2: "\<not> (\<exists>j::int \<in> {1 .. d}. \<exists>b \<in> Inum (a # bs) ` set (\<beta> p). x = b + j)" |
23274 | 2198 |
by auto |
2199 |
from \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" . |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2200 |
qed |
55885 | 2201 |
|
2202 |
lemma cpmi_eq: |
|
55999 | 2203 |
fixes P P1 :: "int \<Rightarrow> bool" |
2204 |
assumes "0 < D" |
|
2205 |
and "\<exists>z. \<forall>x. x < z \<longrightarrow> P x = P1 x" |
|
2206 |
and "\<forall>x. \<not> (\<exists>j \<in> {1..D}. \<exists>b \<in> B. P (b + j)) \<longrightarrow> P x \<longrightarrow> P (x - D)" |
|
2207 |
and "\<forall>x k. P1 x = P1 (x - k * D)" |
|
2208 |
shows "(\<exists>x. P x) \<longleftrightarrow> (\<exists>j \<in> {1..D}. P1 j) \<or> (\<exists>j \<in> {1..D}. \<exists>b \<in> B. P (b + j))" |
|
2209 |
apply (insert assms) |
|
2210 |
apply (rule iffI) |
|
55885 | 2211 |
prefer 2 |
55981 | 2212 |
apply (drule minusinfinity) |
55885 | 2213 |
apply assumption+ |
55981 | 2214 |
apply fastforce |
55885 | 2215 |
apply clarsimp |
55981 | 2216 |
apply (subgoal_tac "\<And>k. 0 \<le> k \<Longrightarrow> \<forall>x. P x \<longrightarrow> P (x - k * D)") |
2217 |
apply (frule_tac x = x and z=z in decr_lemma) |
|
2218 |
apply (subgoal_tac "P1 (x - (\<bar>x - z\<bar> + 1) * D)") |
|
55885 | 2219 |
prefer 2 |
55981 | 2220 |
apply (subgoal_tac "0 \<le> \<bar>x - z\<bar> + 1") |
55885 | 2221 |
prefer 2 apply arith |
2222 |
apply fastforce |
|
55981 | 2223 |
apply (drule (1) periodic_finite_ex) |
55885 | 2224 |
apply blast |
55981 | 2225 |
apply (blast dest: decr_mult_lemma) |
55885 | 2226 |
done |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2227 |
|
23274 | 2228 |
theorem cp_thm: |
2229 |
assumes lp: "iszlfm p" |
|
55885 | 2230 |
and u: "d_\<beta> p 1" |
2231 |
and d: "d_\<delta> p d" |
|
2232 |
and dp: "d > 0" |
|
55999 | 2233 |
shows "(\<exists>x. Ifm bbs (x # bs) p) \<longleftrightarrow> |
55964 | 2234 |
(\<exists>j \<in> {1.. d}. Ifm bbs (j # bs) (minusinf p) \<or> |
2235 |
(\<exists>b \<in> set (\<beta> p). Ifm bbs ((Inum (i # bs) b + j) # bs) p))" |
|
55999 | 2236 |
(is "(\<exists>x. ?P x) \<longleftrightarrow> (\<exists>j \<in> ?D. ?M j \<or> (\<exists>b \<in> ?B. ?P (?I b + j)))") |
55885 | 2237 |
proof - |
50313 | 2238 |
from minusinf_inf[OF lp u] |
55999 | 2239 |
have th: "\<exists>z. \<forall>x<z. ?P x = ?M x" |
55885 | 2240 |
by blast |
55964 | 2241 |
let ?B' = "{?I b | b. b \<in> ?B}" |
2242 |
have BB': "(\<exists>j\<in>?D. \<exists>b \<in> ?B. ?P (?I b + j)) \<longleftrightarrow> (\<exists>j \<in> ?D. \<exists>b \<in> ?B'. ?P (b + j))" |
|
55885 | 2243 |
by auto |
55964 | 2244 |
then have th2: "\<forall>x. \<not> (\<exists>j \<in> ?D. \<exists>b \<in> ?B'. ?P (b + j)) \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)" |
23274 | 2245 |
using \<beta>'[OF lp u d dp, where a="i" and bbs = "bbs"] by blast |
2246 |
from minusinf_repeats[OF d lp] |
|
55885 | 2247 |
have th3: "\<forall>x k. ?M x = ?M (x-k*d)" |
2248 |
by simp |
|
2249 |
from cpmi_eq[OF dp th th2 th3] BB' show ?thesis |
|
2250 |
by blast |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2251 |
qed |
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2252 |
|
55885 | 2253 |
(* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *) |
50313 | 2254 |
lemma mirror_ex: |
55999 | 2255 |
assumes "iszlfm p" |
2256 |
shows "(\<exists>x. Ifm bbs (x#bs) (mirror p)) \<longleftrightarrow> (\<exists>x. Ifm bbs (x#bs) p)" |
|
50313 | 2257 |
(is "(\<exists>x. ?I x ?mp) = (\<exists>x. ?I x p)") |
55964 | 2258 |
proof auto |
2259 |
fix x |
|
2260 |
assume "?I x ?mp" |
|
2261 |
then have "?I (- x) p" |
|
55999 | 2262 |
using mirror[OF assms] by blast |
55964 | 2263 |
then show "\<exists>x. ?I x p" |
2264 |
by blast |
|
23274 | 2265 |
next |
55964 | 2266 |
fix x |
2267 |
assume "?I x p" |
|
2268 |
then have "?I (- x) ?mp" |
|
55999 | 2269 |
using mirror[OF assms, where x="- x", symmetric] by auto |
55964 | 2270 |
then show "\<exists>x. ?I x ?mp" |
2271 |
by blast |
|
23274 | 2272 |
qed |
24349 | 2273 |
|
50313 | 2274 |
lemma cp_thm': |
55999 | 2275 |
assumes "iszlfm p" |
2276 |
and "d_\<beta> p 1" |
|
2277 |
and "d_\<delta> p d" |
|
2278 |
and "d > 0" |
|
55964 | 2279 |
shows "(\<exists>x. Ifm bbs (x # bs) p) \<longleftrightarrow> |
2280 |
((\<exists>j\<in> {1 .. d}. Ifm bbs (j#bs) (minusinf p)) \<or> |
|
2281 |
(\<exists>j\<in> {1.. d}. \<exists>b\<in> (Inum (i#bs)) ` set (\<beta> p). Ifm bbs ((b + j) # bs) p))" |
|
55999 | 2282 |
using cp_thm[OF assms,where i="i"] by auto |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2283 |
|
50313 | 2284 |
definition unit :: "fm \<Rightarrow> fm \<times> num list \<times> int" |
2285 |
where |
|
55964 | 2286 |
"unit p = |
2287 |
(let |
|
2288 |
p' = zlfm p; |
|
2289 |
l = \<zeta> p'; |
|
2290 |
q = And (Dvd l (CN 0 1 (C 0))) (a_\<beta> p' l); |
|
2291 |
d = \<delta> q; |
|
2292 |
B = remdups (map simpnum (\<beta> q)); |
|
2293 |
a = remdups (map simpnum (\<alpha> q)) |
|
2294 |
in if length B \<le> length a then (q, B, d) else (mirror q, a, d))" |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2295 |
|
50313 | 2296 |
lemma unit: |
2297 |
assumes qf: "qfree p" |
|
55964 | 2298 |
shows "\<And>q B d. unit p = (q, B, d) \<Longrightarrow> |
2299 |
((\<exists>x. Ifm bbs (x # bs) p) \<longleftrightarrow> (\<exists>x. Ifm bbs (x # bs) q)) \<and> |
|
2300 |
(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> |
|
2301 |
iszlfm q \<and> (\<forall>b\<in> set B. numbound0 b)" |
|
50313 | 2302 |
proof - |
2303 |
fix q B d |
|
23274 | 2304 |
assume qBd: "unit p = (q,B,d)" |
55981 | 2305 |
let ?thes = "((\<exists>x. Ifm bbs (x#bs) p) \<longleftrightarrow> (\<exists>x. Ifm bbs (x#bs) q)) \<and> |
23274 | 2306 |
Inum (i#bs) ` set B = Inum (i#bs) ` set (\<beta> q) \<and> |
50313 | 2307 |
d_\<beta> q 1 \<and> d_\<delta> q d \<and> 0 < d \<and> iszlfm q \<and> (\<forall>b\<in> set B. numbound0 b)" |
2308 |
let ?I = "\<lambda>x p. Ifm bbs (x#bs) p" |
|
23274 | 2309 |
let ?p' = "zlfm p" |
2310 |
let ?l = "\<zeta> ?p'" |
|
50252 | 2311 |
let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a_\<beta> ?p' ?l)" |
23274 | 2312 |
let ?d = "\<delta> ?q" |
2313 |
let ?B = "set (\<beta> ?q)" |
|
2314 |
let ?B'= "remdups (map simpnum (\<beta> ?q))" |
|
2315 |
let ?A = "set (\<alpha> ?q)" |
|
2316 |
let ?A'= "remdups (map simpnum (\<alpha> ?q))" |
|
50313 | 2317 |
from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] |
2318 |
have pp': "\<forall>i. ?I i ?p' = ?I i p" by auto |
|
23274 | 2319 |
from conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]] |
50313 | 2320 |
have lp': "iszlfm ?p'" . |
50252 | 2321 |
from lp' \<zeta>[where p="?p'"] have lp: "?l >0" and dl: "d_\<beta> ?p' ?l" by auto |
2322 |
from a_\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp' dl lp] pp' |
|
50313 | 2323 |
have pq_ex:"(\<exists>(x::int). ?I x p) = (\<exists>x. ?I x ?q)" by simp |
50252 | 2324 |
from lp' lp a_\<beta>[OF lp' dl lp] have lq:"iszlfm ?q" and uq: "d_\<beta> ?q 1" by auto |
2325 |
from \<delta>[OF lq] have dp:"?d >0" and dd: "d_\<delta> ?q ?d" by blast+ |
|
50313 | 2326 |
let ?N = "\<lambda>t. Inum (i#bs) t" |
55981 | 2327 |
have "?N ` set ?B' = ((?N \<circ> simpnum) ` ?B)" |
2328 |
by auto |
|
2329 |
also have "\<dots> = ?N ` ?B" |
|
2330 |
using simpnum_ci[where bs="i#bs"] by auto |
|
23274 | 2331 |
finally have BB': "?N ` set ?B' = ?N ` ?B" . |
55981 | 2332 |
have "?N ` set ?A' = ((?N \<circ> simpnum) ` ?A)" |
2333 |
by auto |
|
2334 |
also have "\<dots> = ?N ` ?A" |
|
2335 |
using simpnum_ci[where bs="i#bs"] by auto |
|
23274 | 2336 |
finally have AA': "?N ` set ?A' = ?N ` ?A" . |
50313 | 2337 |
from \<beta>_numbound0[OF lq] have B_nb:"\<forall>b\<in> set ?B'. numbound0 b" |
23274 | 2338 |
by (simp add: simpnum_numbound0) |
50313 | 2339 |
from \<alpha>_l[OF lq] have A_nb: "\<forall>b\<in> set ?A'. numbound0 b" |
23274 | 2340 |
by (simp add: simpnum_numbound0) |
55981 | 2341 |
{ |
2342 |
assume "length ?B' \<le> length ?A'" |
|
2343 |
then have q: "q = ?q" and "B = ?B'" and d: "d = ?d" |
|
23274 | 2344 |
using qBd by (auto simp add: Let_def unit_def) |
55981 | 2345 |
with BB' B_nb |
2346 |
have b: "?N ` (set B) = ?N ` set (\<beta> q)" and bn: "\<forall>b\<in> set B. numbound0 b" |
|
2347 |
by simp_all |
|
2348 |
with pq_ex dp uq dd lq q d have ?thes |
|
2349 |
by simp |
|
2350 |
} |
|
50313 | 2351 |
moreover |
55981 | 2352 |
{ |
2353 |
assume "\<not> (length ?B' \<le> length ?A')" |
|
55885 | 2354 |
then have q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d" |
23274 | 2355 |
using qBd by (auto simp add: Let_def unit_def) |
50313 | 2356 |
with AA' mirror_\<alpha>_\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)" |
2357 |
and bn: "\<forall>b\<in> set B. numbound0 b" by simp_all |
|
2358 |
from mirror_ex[OF lq] pq_ex q |
|
55981 | 2359 |
have pqm_eq:"(\<exists>(x::int). ?I x p) = (\<exists>(x::int). ?I x q)" |
2360 |
by simp |
|
23274 | 2361 |
from lq uq q mirror_l[where p="?q"] |
55981 | 2362 |
have lq': "iszlfm q" and uq: "d_\<beta> q 1" |
2363 |
by auto |
|
2364 |
from \<delta>[OF lq'] mirror_\<delta>[OF lq] q d have dq: "d_\<delta> q d" |
|
2365 |
by auto |
|
2366 |
from pqm_eq b bn uq lq' dp dq q dp d have ?thes |
|
2367 |
by simp |
|
23274 | 2368 |
} |
2369 |
ultimately show ?thes by blast |
|
2370 |
qed |
|
50313 | 2371 |
|
2372 |
||
2373 |
text {* Cooper's Algorithm *} |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2374 |
|
55981 | 2375 |
definition cooper :: "fm \<Rightarrow> fm" |
2376 |
where |
|
50313 | 2377 |
"cooper p = |
2378 |
(let |
|
2379 |
(q, B, d) = unit p; |
|
2380 |
js = [1..d]; |
|
2381 |
mq = simpfm (minusinf q); |
|
2382 |
md = evaldjf (\<lambda>j. simpfm (subst0 (C j) mq)) js |
|
2383 |
in |
|
2384 |
if md = T then T |
|
2385 |
else |
|
2386 |
(let |
|
2387 |
qd = evaldjf (\<lambda>(b, j). simpfm (subst0 (Add b (C j)) q)) [(b, j). b \<leftarrow> B, j \<leftarrow> js] |
|
2388 |
in decr (disj md qd)))" |
|
2389 |
||
2390 |
lemma cooper: |
|
2391 |
assumes qf: "qfree p" |
|
2392 |
shows "((\<exists>x. Ifm bbs (x#bs) p) = (Ifm bbs bs (cooper p))) \<and> qfree (cooper p)" |
|
23274 | 2393 |
(is "(?lhs = ?rhs) \<and> _") |
50313 | 2394 |
proof - |
2395 |
let ?I = "\<lambda>x p. Ifm bbs (x#bs) p" |
|
23274 | 2396 |
let ?q = "fst (unit p)" |
2397 |
let ?B = "fst (snd(unit p))" |
|
2398 |
let ?d = "snd (snd (unit p))" |
|
41836 | 2399 |
let ?js = "[1..?d]" |
23274 | 2400 |
let ?mq = "minusinf ?q" |
2401 |
let ?smq = "simpfm ?mq" |
|
50313 | 2402 |
let ?md = "evaldjf (\<lambda>j. simpfm (subst0 (C j) ?smq)) ?js" |
26934 | 2403 |
fix i |
50313 | 2404 |
let ?N = "\<lambda>t. Inum (i#bs) t" |
24336 | 2405 |
let ?Bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]" |
50313 | 2406 |
let ?qd = "evaldjf (\<lambda>(b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs" |
23274 | 2407 |
have qbf:"unit p = (?q,?B,?d)" by simp |
55981 | 2408 |
from unit[OF qf qbf] |
2409 |
have pq_ex: "(\<exists>(x::int). ?I x p) \<longleftrightarrow> (\<exists>(x::int). ?I x ?q)" |
|
2410 |
and B: "?N ` set ?B = ?N ` set (\<beta> ?q)" |
|
2411 |
and uq: "d_\<beta> ?q 1" |
|
2412 |
and dd: "d_\<delta> ?q ?d" |
|
2413 |
and dp: "?d > 0" |
|
2414 |
and lq: "iszlfm ?q" |
|
2415 |
and Bn: "\<forall>b\<in> set ?B. numbound0 b" |
|
2416 |
by auto |
|
23274 | 2417 |
from zlin_qfree[OF lq] have qfq: "qfree ?q" . |
55921 | 2418 |
from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq" . |
55981 | 2419 |
have jsnb: "\<forall>j \<in> set ?js. numbound0 (C j)" |
2420 |
by simp |
|
55885 | 2421 |
then have "\<forall>j\<in> set ?js. bound0 (subst0 (C j) ?smq)" |
23274 | 2422 |
by (auto simp only: subst0_bound0[OF qfmq]) |
55885 | 2423 |
then have th: "\<forall>j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))" |
23274 | 2424 |
by (auto simp add: simpfm_bound0) |
55981 | 2425 |
from evaldjf_bound0[OF th] have mdb: "bound0 ?md" |
2426 |
by simp |
|
50313 | 2427 |
from Bn jsnb have "\<forall>(b,j) \<in> set ?Bjs. numbound0 (Add b (C j))" |
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23515
diff
changeset
|
2428 |
by simp |
55885 | 2429 |
then have "\<forall>(b,j) \<in> set ?Bjs. bound0 (subst0 (Add b (C j)) ?q)" |
23274 | 2430 |
using subst0_bound0[OF qfq] by blast |
55885 | 2431 |
then have "\<forall>(b,j) \<in> set ?Bjs. bound0 (simpfm (subst0 (Add b (C j)) ?q))" |
55981 | 2432 |
using simpfm_bound0 by blast |
55885 | 2433 |
then have th': "\<forall>x \<in> set ?Bjs. bound0 ((\<lambda>(b,j). simpfm (subst0 (Add b (C j)) ?q)) x)" |
50313 | 2434 |
by auto |
55981 | 2435 |
from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" |
2436 |
by simp |
|
2437 |
from mdb qdb have mdqdb: "bound0 (disj ?md ?qd)" |
|
2438 |
unfolding disj_def by (cases "?md = T \<or> ?qd = T") simp_all |
|
23274 | 2439 |
from trans [OF pq_ex cp_thm'[OF lq uq dd dp,where i="i"]] B |
55981 | 2440 |
have "?lhs \<longleftrightarrow> (\<exists>j \<in> {1.. ?d}. ?I j ?mq \<or> (\<exists>b \<in> ?N ` set ?B. Ifm bbs ((b + j) # bs) ?q))" |
55921 | 2441 |
by auto |
55981 | 2442 |
also have "\<dots> \<longleftrightarrow> (\<exists>j \<in> {1.. ?d}. ?I j ?mq \<or> (\<exists>b \<in> set ?B. Ifm bbs ((?N b + j) # bs) ?q))" |
55921 | 2443 |
by simp |
55981 | 2444 |
also have "\<dots> \<longleftrightarrow> (\<exists>j \<in> {1.. ?d}. ?I j ?mq ) \<or> |
2445 |
(\<exists>j\<in> {1.. ?d}. \<exists>b \<in> set ?B. Ifm bbs ((?N (Add b (C j))) # bs) ?q)" |
|
50313 | 2446 |
by (simp only: Inum.simps) blast |
55981 | 2447 |
also have "\<dots> \<longleftrightarrow> (\<exists>j \<in> {1.. ?d}. ?I j ?smq) \<or> |
2448 |
(\<exists>j \<in> {1.. ?d}. \<exists>b \<in> set ?B. Ifm bbs ((?N (Add b (C j))) # bs) ?q)" |
|
50313 | 2449 |
by (simp add: simpfm) |
55981 | 2450 |
also have "\<dots> \<longleftrightarrow> (\<exists>j\<in> set ?js. (\<lambda>j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or> |
2451 |
(\<exists>j\<in> set ?js. \<exists>b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q)" |
|
41836 | 2452 |
by (simp only: simpfm subst0_I[OF qfmq] set_upto) auto |
55981 | 2453 |
also have "\<dots> \<longleftrightarrow> ?I i (evaldjf (\<lambda>j. simpfm (subst0 (C j) ?smq)) ?js) \<or> |
2454 |
(\<exists>j\<in> set ?js. \<exists>b\<in> set ?B. ?I i (subst0 (Add b (C j)) ?q))" |
|
50313 | 2455 |
by (simp only: evaldjf_ex subst0_I[OF qfq]) |
55981 | 2456 |
also have "\<dots> \<longleftrightarrow> ?I i ?md \<or> |
2457 |
(\<exists>(b,j) \<in> set ?Bjs. (\<lambda>(b,j). ?I i (simpfm (subst0 (Add b (C j)) ?q))) (b,j))" |
|
50313 | 2458 |
by (simp only: simpfm set_concat set_map concat_map_singleton UN_simps) blast |
55981 | 2459 |
also have "\<dots> \<longleftrightarrow> ?I i ?md \<or> ?I i (evaldjf (\<lambda>(b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs)" |
50313 | 2460 |
by (simp only: evaldjf_ex[where bs="i#bs" and f="\<lambda>(b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="?Bjs"]) |
2461 |
(auto simp add: split_def) |
|
55981 | 2462 |
finally have mdqd: "?lhs \<longleftrightarrow> ?I i ?md \<or> ?I i ?qd" |
55921 | 2463 |
by simp |
55981 | 2464 |
also have "\<dots> \<longleftrightarrow> ?I i (disj ?md ?qd)" |
55921 | 2465 |
by (simp add: disj) |
55981 | 2466 |
also have "\<dots> \<longleftrightarrow> Ifm bbs bs (decr (disj ?md ?qd))" |
55921 | 2467 |
by (simp only: decr [OF mdqdb]) |
55981 | 2468 |
finally have mdqd2: "?lhs \<longleftrightarrow> Ifm bbs bs (decr (disj ?md ?qd))" . |
55921 | 2469 |
{ |
2470 |
assume mdT: "?md = T" |
|
2471 |
then have cT: "cooper p = T" |
|
23274 | 2472 |
by (simp only: cooper_def unit_def split_def Let_def if_True) simp |
55921 | 2473 |
from mdT have lhs: "?lhs" |
2474 |
using mdqd by simp |
|
2475 |
from mdT have "?rhs" |
|
2476 |
by (simp add: cooper_def unit_def split_def) |
|
55981 | 2477 |
with lhs cT have ?thesis |
2478 |
by simp |
|
55921 | 2479 |
} |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2480 |
moreover |
55921 | 2481 |
{ |
2482 |
assume mdT: "?md \<noteq> T" |
|
2483 |
then have "cooper p = decr (disj ?md ?qd)" |
|
50313 | 2484 |
by (simp only: cooper_def unit_def split_def Let_def if_False) |
55921 | 2485 |
with mdqd2 decr_qf[OF mdqdb] have ?thesis |
2486 |
by simp |
|
2487 |
} |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2488 |
ultimately show ?thesis by blast |
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2489 |
qed |
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2490 |
|
55921 | 2491 |
definition pa :: "fm \<Rightarrow> fm" |
2492 |
where "pa p = qelim (prep p) cooper" |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2493 |
|
55921 | 2494 |
theorem mirqe: "Ifm bbs bs (pa p) = Ifm bbs bs p \<and> qfree (pa p)" |
23274 | 2495 |
using qelim_ci cooper prep by (auto simp add: pa_def) |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2496 |
|
50313 | 2497 |
definition cooper_test :: "unit \<Rightarrow> fm" |
2498 |
where |
|
2499 |
"cooper_test u = |
|
2500 |
pa (E (A (Imp (Ge (Sub (Bound 0) (Bound 1))) |
|
2501 |
(E (E (Eq (Sub (Add (Mul 3 (Bound 1)) (Mul 5 (Bound 0))) (Bound 2))))))))" |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2502 |
|
51272 | 2503 |
ML_val {* @{code cooper_test} () *} |
27456 | 2504 |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2505 |
(*code_reflect Cooper_Procedure |
55685 | 2506 |
functions pa T Bound nat_of_integer integer_of_nat int_of_integer integer_of_int |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2507 |
file "~~/src/HOL/Tools/Qelim/cooper_procedure.ML"*) |
27456 | 2508 |
|
28290 | 2509 |
oracle linzqe_oracle = {* |
27456 | 2510 |
let |
2511 |
||
55814 | 2512 |
fun num_of_term vs (t as Free (xn, xT)) = |
2513 |
(case AList.lookup (op =) vs t of |
|
2514 |
NONE => error "Variable not found in the list!" |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2515 |
| SOME n => @{code Bound} (@{code nat_of_integer} n)) |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2516 |
| num_of_term vs @{term "0::int"} = @{code C} (@{code int_of_integer} 0) |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2517 |
| num_of_term vs @{term "1::int"} = @{code C} (@{code int_of_integer} 1) |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2518 |
| num_of_term vs @{term "- 1::int"} = @{code C} (@{code int_of_integer} (~ 1)) |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2519 |
| num_of_term vs (@{term "numeral :: _ \<Rightarrow> int"} $ t) = |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2520 |
@{code C} (@{code int_of_integer} (HOLogic.dest_num t)) |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2521 |
| num_of_term vs (@{term "- numeral :: _ \<Rightarrow> int"} $ t) = |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2522 |
@{code C} (@{code int_of_integer} (~(HOLogic.dest_num t))) |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2523 |
| num_of_term vs (Bound i) = @{code Bound} (@{code nat_of_integer} i) |
27456 | 2524 |
| num_of_term vs (@{term "uminus :: int \<Rightarrow> int"} $ t') = @{code Neg} (num_of_term vs t') |
2525 |
| num_of_term vs (@{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) = |
|
2526 |
@{code Add} (num_of_term vs t1, num_of_term vs t2) |
|
2527 |
| num_of_term vs (@{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) = |
|
2528 |
@{code Sub} (num_of_term vs t1, num_of_term vs t2) |
|
2529 |
| num_of_term vs (@{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} $ t1 $ t2) = |
|
55814 | 2530 |
(case try HOLogic.dest_number t1 of |
2531 |
SOME (_, i) => @{code Mul} (@{code int_of_integer} i, num_of_term vs t2) |
|
2532 |
| NONE => |
|
2533 |
(case try HOLogic.dest_number t2 of |
|
2534 |
SOME (_, i) => @{code Mul} (@{code int_of_integer} i, num_of_term vs t1) |
|
2535 |
| NONE => error "num_of_term: unsupported multiplication")) |
|
28264 | 2536 |
| num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t); |
27456 | 2537 |
|
2538 |
fun fm_of_term ps vs @{term True} = @{code T} |
|
2539 |
| fm_of_term ps vs @{term False} = @{code F} |
|
2540 |
| fm_of_term ps vs (@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) = |
|
2541 |
@{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) |
|
2542 |
| fm_of_term ps vs (@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) = |
|
2543 |
@{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) |
|
2544 |
| fm_of_term ps vs (@{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) = |
|
50313 | 2545 |
@{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) |
27456 | 2546 |
| fm_of_term ps vs (@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} $ t1 $ t2) = |
55814 | 2547 |
(case try HOLogic.dest_number t1 of |
2548 |
SOME (_, i) => @{code Dvd} (@{code int_of_integer} i, num_of_term vs t2) |
|
2549 |
| NONE => error "num_of_term: unsupported dvd") |
|
27456 | 2550 |
| fm_of_term ps vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) = |
2551 |
@{code Iff} (fm_of_term ps vs t1, fm_of_term ps vs t2) |
|
38795
848be46708dc
formerly unnamed infix conjunction and disjunction now named HOL.conj and HOL.disj
haftmann
parents:
38786
diff
changeset
|
2552 |
| fm_of_term ps vs (@{term HOL.conj} $ t1 $ t2) = |
27456 | 2553 |
@{code And} (fm_of_term ps vs t1, fm_of_term ps vs t2) |
38795
848be46708dc
formerly unnamed infix conjunction and disjunction now named HOL.conj and HOL.disj
haftmann
parents:
38786
diff
changeset
|
2554 |
| fm_of_term ps vs (@{term HOL.disj} $ t1 $ t2) = |
27456 | 2555 |
@{code Or} (fm_of_term ps vs t1, fm_of_term ps vs t2) |
38786
e46e7a9cb622
formerly unnamed infix impliciation now named HOL.implies
haftmann
parents:
38558
diff
changeset
|
2556 |
| fm_of_term ps vs (@{term HOL.implies} $ t1 $ t2) = |
27456 | 2557 |
@{code Imp} (fm_of_term ps vs t1, fm_of_term ps vs t2) |
2558 |
| fm_of_term ps vs (@{term "Not"} $ t') = |
|
2559 |
@{code NOT} (fm_of_term ps vs t') |
|
38558 | 2560 |
| fm_of_term ps vs (Const (@{const_name Ex}, _) $ Abs (xn, xT, p)) = |
27456 | 2561 |
let |
42284 | 2562 |
val (xn', p') = Syntax_Trans.variant_abs (xn, xT, p); (* FIXME !? *) |
27456 | 2563 |
val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs; |
2564 |
in @{code E} (fm_of_term ps vs' p) end |
|
38558 | 2565 |
| fm_of_term ps vs (Const (@{const_name All}, _) $ Abs (xn, xT, p)) = |
27456 | 2566 |
let |
42284 | 2567 |
val (xn', p') = Syntax_Trans.variant_abs (xn, xT, p); (* FIXME !? *) |
27456 | 2568 |
val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs; |
2569 |
in @{code A} (fm_of_term ps vs' p) end |
|
28264 | 2570 |
| fm_of_term ps vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t); |
23515 | 2571 |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2572 |
fun term_of_num vs (@{code C} i) = HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i) |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2573 |
| term_of_num vs (@{code Bound} n) = |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2574 |
let |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2575 |
val q = @{code integer_of_nat} n |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2576 |
in fst (the (find_first (fn (_, m) => q = m) vs)) end |
27456 | 2577 |
| term_of_num vs (@{code Neg} t') = @{term "uminus :: int \<Rightarrow> int"} $ term_of_num vs t' |
2578 |
| term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} $ |
|
2579 |
term_of_num vs t1 $ term_of_num vs t2 |
|
2580 |
| term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} $ |
|
2581 |
term_of_num vs t1 $ term_of_num vs t2 |
|
2582 |
| term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} $ |
|
2583 |
term_of_num vs (@{code C} i) $ term_of_num vs t2 |
|
55814 | 2584 |
| term_of_num vs (@{code CN} (n, i, t)) = |
2585 |
term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t)); |
|
27456 | 2586 |
|
50313 | 2587 |
fun term_of_fm ps vs @{code T} = @{term True} |
45740 | 2588 |
| term_of_fm ps vs @{code F} = @{term False} |
27456 | 2589 |
| term_of_fm ps vs (@{code Lt} t) = |
2590 |
@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"} |
|
2591 |
| term_of_fm ps vs (@{code Le} t) = |
|
2592 |
@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"} |
|
2593 |
| term_of_fm ps vs (@{code Gt} t) = |
|
2594 |
@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t |
|
2595 |
| term_of_fm ps vs (@{code Ge} t) = |
|
2596 |
@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} $ @{term "0::int"} $ term_of_num vs t |
|
2597 |
| term_of_fm ps vs (@{code Eq} t) = |
|
2598 |
@{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::int"} |
|
2599 |
| term_of_fm ps vs (@{code NEq} t) = |
|
2600 |
term_of_fm ps vs (@{code NOT} (@{code Eq} t)) |
|
2601 |
| term_of_fm ps vs (@{code Dvd} (i, t)) = |
|
2602 |
@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} $ term_of_num vs (@{code C} i) $ term_of_num vs t |
|
2603 |
| term_of_fm ps vs (@{code NDvd} (i, t)) = |
|
2604 |
term_of_fm ps vs (@{code NOT} (@{code Dvd} (i, t))) |
|
2605 |
| term_of_fm ps vs (@{code NOT} t') = |
|
2606 |
HOLogic.Not $ term_of_fm ps vs t' |
|
2607 |
| term_of_fm ps vs (@{code And} (t1, t2)) = |
|
2608 |
HOLogic.conj $ term_of_fm ps vs t1 $ term_of_fm ps vs t2 |
|
2609 |
| term_of_fm ps vs (@{code Or} (t1, t2)) = |
|
2610 |
HOLogic.disj $ term_of_fm ps vs t1 $ term_of_fm ps vs t2 |
|
2611 |
| term_of_fm ps vs (@{code Imp} (t1, t2)) = |
|
2612 |
HOLogic.imp $ term_of_fm ps vs t1 $ term_of_fm ps vs t2 |
|
2613 |
| term_of_fm ps vs (@{code Iff} (t1, t2)) = |
|
2614 |
@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm ps vs t1 $ term_of_fm ps vs t2 |
|
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2615 |
| term_of_fm ps vs (@{code Closed} n) = |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2616 |
let |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2617 |
val q = @{code integer_of_nat} n |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50313
diff
changeset
|
2618 |
in (fst o the) (find_first (fn (_, m) => m = q) ps) end |
29788 | 2619 |
| term_of_fm ps vs (@{code NClosed} n) = term_of_fm ps vs (@{code NOT} (@{code Closed} n)); |
27456 | 2620 |
|
2621 |
fun term_bools acc t = |
|
2622 |
let |
|
55814 | 2623 |
val is_op = |
2624 |
member (op =) [@{term HOL.conj}, @{term HOL.disj}, @{term HOL.implies}, |
|
2625 |
@{term "op = :: bool => _"}, |
|
27456 | 2626 |
@{term "op = :: int => _"}, @{term "op < :: int => _"}, |
2627 |
@{term "op <= :: int => _"}, @{term "Not"}, @{term "All :: (int => _) => _"}, |
|
2628 |
@{term "Ex :: (int => _) => _"}, @{term "True"}, @{term "False"}] |
|
50313 | 2629 |
fun is_ty t = not (fastype_of t = HOLogic.boolT) |
55814 | 2630 |
in |
2631 |
(case t of |
|
2632 |
(l as f $ a) $ b => |
|
2633 |
if is_ty t orelse is_op t then term_bools (term_bools acc l) b |
|
27456 | 2634 |
else insert (op aconv) t acc |
55814 | 2635 |
| f $ a => |
2636 |
if is_ty t orelse is_op t then term_bools (term_bools acc f) a |
|
27456 | 2637 |
else insert (op aconv) t acc |
42284 | 2638 |
| Abs p => term_bools acc (snd (Syntax_Trans.variant_abs p)) (* FIXME !? *) |
55814 | 2639 |
| _ => if is_ty t orelse is_op t then acc else insert (op aconv) t acc) |
27456 | 2640 |
end; |
2641 |
||
55814 | 2642 |
in |
2643 |
fn ct => |
|
2644 |
let |
|
2645 |
val thy = Thm.theory_of_cterm ct; |
|
2646 |
val t = Thm.term_of ct; |
|
2647 |
val fs = Misc_Legacy.term_frees t; |
|
2648 |
val bs = term_bools [] t; |
|
2649 |
val vs = map_index swap fs; |
|
2650 |
val ps = map_index swap bs; |
|
2651 |
val t' = (term_of_fm ps vs o @{code pa} o fm_of_term ps vs) t; |
|
2652 |
in Thm.cterm_of thy (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t'))) end |
|
27456 | 2653 |
end; |
2654 |
*} |
|
2655 |
||
48891 | 2656 |
ML_file "cooper_tac.ML" |
47432 | 2657 |
|
2658 |
method_setup cooper = {* |
|
53168 | 2659 |
Scan.lift (Args.mode "no_quantify") >> |
47432 | 2660 |
(fn q => fn ctxt => SIMPLE_METHOD' (Cooper_Tac.linz_tac ctxt (not q))) |
2661 |
*} "decision procedure for linear integer arithmetic" |
|
2662 |
||
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2663 |
|
27456 | 2664 |
text {* Tests *} |
2665 |
||
55814 | 2666 |
lemma "\<exists>(j::int). \<forall>x\<ge>j. \<exists>a b. x = 3*a+5*b" |
27456 | 2667 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2668 |
|
55814 | 2669 |
lemma "\<forall>(x::int) \<ge> 8. \<exists>i j. 5*i + 3*j = x" |
27456 | 2670 |
by cooper |
2671 |
||
55814 | 2672 |
theorem "(\<forall>(y::int). 3 dvd y) \<Longrightarrow> \<forall>(x::int). b < x \<longrightarrow> a \<le> x" |
23274 | 2673 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2674 |
|
55814 | 2675 |
theorem "\<And>(y::int) (z::int) (n::int). 3 dvd z \<Longrightarrow> 2 dvd (y::int) \<Longrightarrow> |
2676 |
(\<exists>(x::int). 2*x = y) \<and> (\<exists>(k::int). 3*k = z)" |
|
23274 | 2677 |
by cooper |
2678 |
||
55814 | 2679 |
theorem "\<And>(y::int) (z::int) n. Suc n < 6 \<Longrightarrow> 3 dvd z \<Longrightarrow> |
2680 |
2 dvd (y::int) \<Longrightarrow> (\<exists>(x::int). 2*x = y) \<and> (\<exists>(k::int). 3*k = z)" |
|
23274 | 2681 |
by cooper |
2682 |
||
55814 | 2683 |
theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 \<longrightarrow> y = 5 + x" |
23274 | 2684 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2685 |
|
55814 | 2686 |
lemma "\<forall>(x::int) \<ge> 8. \<exists>i j. 5*i + 3*j = x" |
50313 | 2687 |
by cooper |
27456 | 2688 |
|
55814 | 2689 |
lemma "\<forall>(y::int) (z::int) (n::int). |
2690 |
3 dvd z \<longrightarrow> 2 dvd (y::int) \<longrightarrow> (\<exists>(x::int). 2*x = y) \<and> (\<exists>(k::int). 3*k = z)" |
|
27456 | 2691 |
by cooper |
2692 |
||
55814 | 2693 |
lemma "\<forall>(x::int) y. x < y \<longrightarrow> 2 * x + 1 < 2 * y" |
27456 | 2694 |
by cooper |
2695 |
||
55814 | 2696 |
lemma "\<forall>(x::int) y. 2 * x + 1 \<noteq> 2 * y" |
27456 | 2697 |
by cooper |
2698 |
||
55814 | 2699 |
lemma "\<exists>(x::int) y. 0 < x \<and> 0 \<le> y \<and> 3 * x - 5 * y = 1" |
27456 | 2700 |
by cooper |
2701 |
||
55814 | 2702 |
lemma "\<not> (\<exists>(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)" |
27456 | 2703 |
by cooper |
2704 |
||
55814 | 2705 |
lemma "\<forall>(x::int). 2 dvd x \<longrightarrow> (\<exists>(y::int). x = 2*y)" |
27456 | 2706 |
by cooper |
2707 |
||
55814 | 2708 |
lemma "\<forall>(x::int). 2 dvd x \<longleftrightarrow> (\<exists>(y::int). x = 2*y)" |
27456 | 2709 |
by cooper |
2710 |
||
55814 | 2711 |
lemma "\<forall>(x::int). 2 dvd x \<longleftrightarrow> (\<forall>(y::int). x \<noteq> 2*y + 1)" |
27456 | 2712 |
by cooper |
2713 |
||
55814 | 2714 |
lemma "\<not> (\<forall>(x::int). |
55921 | 2715 |
(2 dvd x \<longleftrightarrow> (\<forall>(y::int). x \<noteq> 2*y+1) \<or> |
55814 | 2716 |
(\<exists>(q::int) (u::int) i. 3*i + 2*q - u < 17) \<longrightarrow> 0 < x \<or> (\<not> 3 dvd x \<and> x + 8 = 0)))" |
23274 | 2717 |
by cooper |
27456 | 2718 |
|
55814 | 2719 |
lemma "\<not> (\<forall>(i::int). 4 \<le> i \<longrightarrow> (\<exists>x y. 0 \<le> x \<and> 0 \<le> y \<and> 3 * x + 5 * y = i))" |
27456 | 2720 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2721 |
|
55814 | 2722 |
lemma "\<exists>j. \<forall>(x::int) \<ge> j. \<exists>i j. 5*i + 3*j = x" |
2723 |
by cooper |
|
2724 |
||
2725 |
theorem "(\<forall>(y::int). 3 dvd y) \<Longrightarrow> \<forall>(x::int). b < x \<longrightarrow> a \<le> x" |
|
23274 | 2726 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2727 |
|
55814 | 2728 |
theorem "\<And>(y::int) (z::int) (n::int). 3 dvd z \<Longrightarrow> 2 dvd (y::int) \<Longrightarrow> |
2729 |
(\<exists>(x::int). 2*x = y) \<and> (\<exists>(k::int). 3*k = z)" |
|
23274 | 2730 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2731 |
|
55814 | 2732 |
theorem "\<And>(y::int) (z::int) n. Suc n < 6 \<Longrightarrow> 3 dvd z \<Longrightarrow> |
2733 |
2 dvd (y::int) \<Longrightarrow> (\<exists>(x::int). 2*x = y) \<and> (\<exists>(k::int). 3*k = z)" |
|
23274 | 2734 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2735 |
|
55814 | 2736 |
theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 \<longrightarrow> y = 5 + x" |
23274 | 2737 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2738 |
|
55814 | 2739 |
theorem "\<forall>(x::nat). \<exists>(y::nat). y = 5 + x \<or> x div 6 + 1 = 2" |
23274 | 2740 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2741 |
|
23274 | 2742 |
theorem "\<exists>(x::int). 0 < x" |
2743 |
by cooper |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2744 |
|
55814 | 2745 |
theorem "\<forall>(x::int) y. x < y \<longrightarrow> 2 * x + 1 < 2 * y" |
23274 | 2746 |
by cooper |
50313 | 2747 |
|
23274 | 2748 |
theorem "\<forall>(x::int) y. 2 * x + 1 \<noteq> 2 * y" |
2749 |
by cooper |
|
50313 | 2750 |
|
23274 | 2751 |
theorem "\<exists>(x::int) y. 0 < x & 0 \<le> y & 3 * x - 5 * y = 1" |
2752 |
by cooper |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2753 |
|
55814 | 2754 |
theorem "\<not> (\<exists>(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)" |
23274 | 2755 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2756 |
|
55814 | 2757 |
theorem "\<not> (\<exists>(x::int). False)" |
23274 | 2758 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2759 |
|
55814 | 2760 |
theorem "\<forall>(x::int). 2 dvd x \<longrightarrow> (\<exists>(y::int). x = 2*y)" |
50313 | 2761 |
by cooper |
23274 | 2762 |
|
55814 | 2763 |
theorem "\<forall>(x::int). 2 dvd x \<longrightarrow> (\<exists>(y::int). x = 2*y)" |
50313 | 2764 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2765 |
|
55814 | 2766 |
theorem "\<forall>(x::int). 2 dvd x \<longleftrightarrow> (\<exists>(y::int). x = 2*y)" |
50313 | 2767 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2768 |
|
55814 | 2769 |
theorem "\<forall>(x::int). 2 dvd x \<longleftrightarrow> (\<forall>(y::int). x \<noteq> 2*y + 1)" |
50313 | 2770 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2771 |
|
55814 | 2772 |
theorem |
2773 |
"\<not> (\<forall>(x::int). |
|
2774 |
(2 dvd x \<longleftrightarrow> |
|
2775 |
(\<forall>(y::int). x \<noteq> 2*y+1) \<or> |
|
2776 |
(\<exists>(q::int) (u::int) i. 3*i + 2*q - u < 17) |
|
2777 |
\<longrightarrow> 0 < x \<or> (\<not> 3 dvd x \<and> x + 8 = 0)))" |
|
23274 | 2778 |
by cooper |
50313 | 2779 |
|
55814 | 2780 |
theorem "\<not> (\<forall>(i::int). 4 \<le> i \<longrightarrow> (\<exists>x y. 0 \<le> x \<and> 0 \<le> y \<and> 3 * x + 5 * y = i))" |
23274 | 2781 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2782 |
|
55814 | 2783 |
theorem "\<forall>(i::int). 8 \<le> i \<longrightarrow> (\<exists>x y. 0 \<le> x \<and> 0 \<le> y \<and> 3 * x + 5 * y = i)" |
23274 | 2784 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2785 |
|
55814 | 2786 |
theorem "\<exists>(j::int). \<forall>i. j \<le> i \<longrightarrow> (\<exists>x y. 0 \<le> x \<and> 0 \<le> y \<and> 3 * x + 5 * y = i)" |
23274 | 2787 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2788 |
|
55814 | 2789 |
theorem "\<not> (\<forall>j (i::int). j \<le> i \<longrightarrow> (\<exists>x y. 0 \<le> x \<and> 0 \<le> y \<and> 3 * x + 5 * y = i))" |
23274 | 2790 |
by cooper |
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2791 |
|
55814 | 2792 |
theorem "(\<exists>m::nat. n = 2 * m) \<longrightarrow> (n + 1) div 2 = n div 2" |
23274 | 2793 |
by cooper |
17388 | 2794 |
|
17378
105519771c67
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
chaieb
parents:
diff
changeset
|
2795 |
end |