--- a/thys/BitCoded.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys/BitCoded.thy Wed Jul 13 08:35:45 2022 +0100
@@ -29,7 +29,7 @@
where
"decode' ds ZERO = (Void, [])"
| "decode' ds ONE = (Void, ds)"
-| "decode' ds (CHAR d) = (Char d, ds)"
+| "decode' ds (CH d) = (Char d, ds)"
| "decode' [] (ALT r1 r2) = (Void, [])"
| "decode' (Z # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r1 in (Left v, ds'))"
| "decode' (S # ds) (ALT r1 r2) = (let (v, ds') = decode' ds r2 in (Right v, ds'))"
@@ -111,7 +111,7 @@
where
"erase AZERO = ZERO"
| "erase (AONE _) = ONE"
-| "erase (ACHAR _ c) = CHAR c"
+| "erase (ACHAR _ c) = CH c"
| "erase (AALTs _ []) = ZERO"
| "erase (AALTs _ [r]) = (erase r)"
| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"
@@ -165,7 +165,7 @@
fun intern :: "rexp \<Rightarrow> arexp" where
"intern ZERO = AZERO"
| "intern ONE = AONE []"
-| "intern (CHAR c) = ACHAR [] c"
+| "intern (CH c) = ACHAR [] c"
| "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1))
(fuse [S] (intern r2))"
| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"
--- a/thys3/Paper.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/Paper.thy Wed Jul 13 08:35:45 2022 +0100
@@ -32,6 +32,7 @@
ALT ("_ + _" [77,77] 78) and
SEQ ("_ \<cdot> _" [77,77] 78) and
STAR ("_\<^sup>*" [79] 78) and
+ NTIMES ("_\<^sup>{\<^sup>_\<^sup>}" [79] 78) and
val.Void ("Empty" 78) and
val.Char ("Char _" [1000] 78) and
@@ -94,7 +95,13 @@
Brzozowski's derivatives \cite{Brzozowski1964} is that they are neatly
expressible in any functional language, and easily definable and
reasoned about in theorem provers---the definitions just consist of
-inductive datatypes and simple recursive functions. Derivatives of a
+inductive datatypes and simple recursive functions. Another neat
+feature of derivatives is that they can be easily extended to bounded
+regular expressions, such as @{term "NTIMES r n"}, where numbers or
+intervals specify how many times a regular expression should be used
+during matching.
+
+Derivatives of a
regular expression, written @{term "der c r"}, give a simple solution
to the problem of matching a string @{term s} with a regular
expression @{term r}: if the derivative of @{term r} w.r.t.\ (in
@@ -231,7 +238,8 @@
@{term "CH c"} $\mid$
@{term "ALT r\<^sub>1 r\<^sub>2"} $\mid$
@{term "SEQ r\<^sub>1 r\<^sub>2"} $\mid$
- @{term "STAR r"}
+ @{term "STAR r"} $\mid$
+ @{term "NTIMES r n"}
\end{center}
\noindent where @{const ZERO} stands for the regular expression that does
@@ -239,7 +247,13 @@
only the empty string and @{term c} for matching a character literal.
The constructors $+$ and $\cdot$ represent alternatives and sequences, respectively.
We sometimes omit the $\cdot$ in a sequence regular expression for brevity.
- The
+ In our work here we also add to the usual ``basic'' regular expressions
+ the bounded regular expression @{term "NTIMES r n"} where the @{term n}
+ specifies that @{term r} should match exactly @{term n}-times. For
+ brevity we omit the other bounded regular expressions
+ @{text "r"}$^{\{..n\}}$, @{text "r"}$^{\{n..\}}$ and @{text "r"}$^{\{n..m\}}$
+ which specify an interval for how many times @{text r} should match. Our
+ results extend straightforwardly also to them. The
\emph{language} of a regular expression, written $L(r)$, is defined as usual
and we omit giving the definition here (see for example \cite{AusafDyckhoffUrban2016}).
--- a/thys3/ROOT Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/ROOT Wed Jul 13 08:35:45 2022 +0100
@@ -3,8 +3,8 @@
"HOL-Library.Sublist"
"RegLangs"
"PosixSpec"
- "Positions"
- "PDerivs"
+ (*"Positions"*)
+ (*"PDerivs"*)
"Lexer"
"LexerSimp"
"Blexer"
--- a/thys3/src/BasicIdentities.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/BasicIdentities.thy Wed Jul 13 08:35:45 2022 +0100
@@ -9,6 +9,7 @@
| RSEQ rrexp rrexp
| RALTS "rrexp list"
| RSTAR rrexp
+| RNTIMES rrexp nat
abbreviation
"RALT r1 r2 \<equiv> RALTS [r1, r2]"
@@ -23,7 +24,7 @@
| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"
| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"
| "rnullable (RSTAR r) = True"
-
+| "rnullable (RNTIMES r n) = (if n = 0 then True else rnullable r)"
fun
rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
@@ -36,8 +37,8 @@
(if rnullable r1
then RALT (RSEQ (rder c r1) r2) (rder c r2)
else RSEQ (rder c r1) r2)"
-| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
-
+| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
+| "rder c (RNTIMES r n) = (if n = 0 then RZERO else RSEQ (rder c r) (RNTIMES r (n - 1)))"
fun
rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
@@ -191,6 +192,7 @@
| "rsize (RALTS rs) = Suc (sum_list (map rsize rs))"
| "rsize (RSEQ r1 r2) = Suc (rsize r1 + rsize r2)"
| "rsize (RSTAR r) = Suc (rsize r)"
+| "rsize (RNTIMES r n) = Suc (rsize r) + n"
abbreviation rsizes where
"rsizes rs \<equiv> sum_list (map rsize rs)"
@@ -237,7 +239,9 @@
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
- apply simp_all
+ apply simp_all
+ apply(case_tac r2)
+ apply simp_all
done
lemma ralts_cap_mono:
@@ -329,7 +333,9 @@
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
- apply simp_all
+ apply simp_all
+apply(case_tac r2)
+ apply simp_all
done
lemma rders__onechar:
@@ -374,7 +380,7 @@
| "good (RSEQ _ RZERO) = False"
| "good (RSEQ r1 r2) = (good r1 \<and> good r2)"
| "good (RSTAR r) = True"
-
+| "good (RNTIMES r n) = True"
lemma k0a:
shows "rflts [RALTS rs] = rs"
@@ -424,9 +430,8 @@
apply (metis flts1 good.simps(5) good.simps(6) k0a neq_Nil_conv)
apply (metis flts1 good.simps(5) good.simps(6) k0a neq_Nil_conv)
apply fastforce
- apply(simp)
- done
-
+ apply(simp)
+ by simp
lemma flts3:
@@ -458,7 +463,9 @@
apply(case_tac r2)
apply(simp_all)
apply(case_tac r2)
- apply(simp_all)
+ apply(simp_all)
+apply(case_tac r2)
+ apply(simp_all)
done
lemma rsize0:
@@ -509,7 +516,7 @@
apply(induct rs )
apply(auto)
apply (metis list.set_intros(1) nn1qq nonalt.elims(3))
- apply (metis nonalt.elims(2) nonnested.simps(3) nonnested.simps(4) nonnested.simps(5) nonnested.simps(6) nonnested.simps(7))
+ apply (metis nonalt.elims(2) nonnested.simps(3) nonnested.simps(4) nonnested.simps(5) nonnested.simps(6) nonnested.simps(7) nonnested.simps(8))
using bbbbs1 apply fastforce
by (metis bbbbs1 list.set_intros(2) nn1qq)
@@ -552,8 +559,8 @@
apply(case_tac "\<exists>bs. rsimp r1 = RONE")
apply(auto)[1]
using idiot apply fastforce
- using idiot2 nonnested.simps(11) apply presburger
- by (metis (mono_tags, lifting) Diff_empty image_iff list.set_map nn1bb nn1c rdistinct_set_equality1)
+ apply (simp add: idiot2)
+ by (metis (mono_tags, lifting) image_iff list.set_map nn1bb nn1c rdistinct_set_equality)
lemma nonalt_flts_rd:
shows "\<lbrakk>xa \<in> set (rdistinct (rflts (map rsimp rs)) {})\<rbrakk>
@@ -603,6 +610,7 @@
apply (simp add: elem_smaller_than_set)
by (metis Diff_empty flts3 rdistinct_set_equality1)
+thm Diff_empty flts3 rdistinct_set_equality1
lemma good1:
shows "good (rsimp a) \<or> rsimp a = RZERO"
@@ -632,8 +640,9 @@
apply blast
apply fastforce
using less_add_Suc2 apply blast
- using less_iff_Suc_add by blast
-
+ using less_iff_Suc_add apply blast
+ using good.simps(45) rsimp.simps(7) by presburger
+
fun
@@ -645,28 +654,51 @@
| "RL (RSEQ r1 r2) = (RL r1) ;; (RL r2)"
| "RL (RALTS rs) = (\<Union> (set (map RL rs)))"
| "RL (RSTAR r) = (RL r)\<star>"
+| "RL (RNTIMES r n) = (RL r) ^^ n"
+lemma pow_rempty_iff:
+ shows "[] \<in> (RL r) ^^ n \<longleftrightarrow> (if n = 0 then True else [] \<in> (RL r))"
+ by (induct n) (auto simp add: Sequ_def)
lemma RL_rnullable:
shows "rnullable r = ([] \<in> RL r)"
apply(induct r)
- apply(auto simp add: Sequ_def)
+ apply(auto simp add: Sequ_def pow_rempty_iff)
done
+lemma concI_if_Nil1: "[] \<in> A \<Longrightarrow> xs : B \<Longrightarrow> xs \<in> A ;; B"
+by (metis append_Nil concI)
+
+
+lemma empty_pow_add:
+ fixes A::"string set"
+ assumes "[] \<in> A" "s \<in> A ^^ n"
+ shows "s \<in> A ^^ (n + m)"
+ using assms
+ apply(induct m arbitrary: n)
+ apply(auto simp add: Sequ_def)
+ done
+
+(*
+lemma der_pow:
+ shows "Der c (A ^^ n) = (if n = 0 then {} else (Der c A) ;; (A ^^ (n - 1)))"
+ apply(induct n arbitrary: A)
+ apply(auto)
+ by (smt (verit, best) Suc_pred concE concI concI_if_Nil2 conc_pow_comm lang_pow.simps(2))
+*)
+
lemma RL_rder:
shows "RL (rder c r) = Der c (RL r)"
apply(induct r)
- apply(auto simp add: Sequ_def Der_def)
+ apply(auto simp add: Sequ_def Der_def)[5]
apply (metis append_Cons)
using RL_rnullable apply blast
apply (metis append_eq_Cons_conv)
apply (metis append_Cons)
- apply (metis RL_rnullable append_eq_Cons_conv)
- apply (metis Star.step append_Cons)
- using Star_decomp by auto
-
-
-
+ apply (metis RL_rnullable append_eq_Cons_conv)
+ apply simp
+ apply(simp)
+ done
lemma RL_rsimp_RSEQ:
shows "RL (rsimp_SEQ r1 r2) = (RL r1 ;; RL r2)"
@@ -843,12 +875,14 @@
apply(case_tac "rsimp aa")
apply simp+
apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
- by simp
+ apply(simp)
+ apply(simp)
+ done
lemma identity_wwo0:
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
- by (metis idem_after_simp1 list.exhaust list.simps(8) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
-
+ apply (metis idem_after_simp1 list.exhaust list.simps(8) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
+ done
lemma distinct_removes_last:
shows "\<lbrakk>a \<in> set as\<rbrakk>
@@ -994,12 +1028,15 @@
apply(subgoal_tac "\<forall>r \<in> set( rflts (map rsimp rsa)). good r")
apply(case_tac "rdistinct (rflts (map rsimp rsa)) {}")
- apply simp
- apply(case_tac "listb")
- apply simp+
- apply (metis Cons_eq_appendI good1_flatten rflts.simps(3) rsimp.simps(2) rsimp_ALTs.simps(3))
- by (metis (mono_tags, lifting) flts3 good1 image_iff list.set_map)
-
+ apply simp
+ apply auto[1]
+ apply simp
+ apply(simp)
+ apply(case_tac "lista")
+ apply simp_all
+
+ apply (metis append_Cons append_Nil good1_flatten rflts.simps(2) rsimp.simps(2) rsimp_ALTs.elims)
+ by (metis (no_types, opaque_lifting) append_Cons append_Nil good1_flatten rflts.simps(2) rsimp.simps(2) rsimp_ALTs.elims)
lemma last_elem_out:
shows "\<lbrakk>x \<notin> set xs; x \<notin> rset \<rbrakk> \<Longrightarrow> rdistinct (xs @ [x]) rset = rdistinct xs rset @ [x]"
@@ -1127,8 +1164,9 @@
apply fastforce
apply fastforce
apply fastforce
- by fastforce
-
+ apply fastforce
+ by simp
+
lemma distinct_removes_duplicate_flts:
shows " a \<in> set rsa
@@ -1143,29 +1181,19 @@
apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
rdistinct (rflts (map rsimp rsa @ [RONE])) {}")
apply (simp only:)
- apply(subst flts_keeps1)
- apply (metis distinct_removes_last(1) rflts_def_idiot2 rrexp.simps(20) rrexp.simps(6))
+ apply(subst flts_keeps1)
+ apply (metis distinct_removes_last(1) flts_append in_set_conv_decomp rflts.simps(4))
apply presburger
apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
rdistinct ((rflts (map rsimp rsa)) @ [RCHAR x]) {}")
apply (simp only:)
- prefer 2
- apply (metis flts_keeps_others rrexp.distinct(21) rrexp.distinct(3))
- apply (metis distinct_removes_last(1) rflts_def_idiot2 rrexp.distinct(21) rrexp.distinct(3))
-
- apply (metis distinct_removes_last(1) flts_keeps_others rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
- prefer 2
- apply (metis distinct_removes_last(1) flts_keeps_others flts_removes0 rflts_def_idiot2 rrexp.distinct(29))
- apply(subgoal_tac "rflts (map rsimp rsa @ [rsimp a]) = rflts (map rsimp rsa) @ x")
- prefer 2
- apply (simp add: rflts_spills_last)
- apply(subgoal_tac "\<forall> r \<in> set x. r \<in> set (rflts (map rsimp rsa))")
- prefer 2
- apply (metis (mono_tags, lifting) image_iff image_set spilled_alts_contained)
- apply (metis rflts_spills_last)
- by (metis distinct_removes_list spilled_alts_contained)
-
-
+ prefer 2
+ apply (metis flts_append rflts.simps(1) rflts.simps(5))
+ apply (metis distinct_removes_last(1) rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(3))
+ apply (metis distinct_removes_last(1) flts_append rflts.simps(1) rflts.simps(6) rflts_def_idiot2 rrexp.distinct(31) rrexp.distinct(5))
+ apply (metis distinct_removes_list rflts_spills_last spilled_alts_contained)
+ apply (metis distinct_removes_last(1) flts_append good.simps(1) good.simps(44) rflts.simps(1) rflts.simps(7) rflts_def_idiot2 rrexp.distinct(37))
+ by (metis distinct_removes_last(1) flts_append rflts.simps(1) rflts.simps(8) rflts_def_idiot2 rrexp.distinct(11) rrexp.distinct(39))
(*some basic facts about rsimp*)
--- a/thys3/src/Blexer.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/Blexer.thy Wed Jul 13 08:35:45 2022 +0100
@@ -1,6 +1,6 @@
theory Blexer
- imports "Lexer" "PDerivs"
+ imports "Lexer"
begin
section \<open>Bit-Encodings\<close>
@@ -17,13 +17,22 @@
| "code (Stars []) = [S]"
| "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)"
+fun sz where
+ "sz ZERO = 0"
+| "sz ONE = 0"
+| "sz (CH _) = 0"
+| "sz (SEQ r1 r2) = 1 + sz r1 + sz r2"
+| "sz (ALT r1 r2) = 1 + sz r1 + sz r2"
+| "sz (STAR r) = 1 + sz r"
+| "sz (NTIMES r n) = 1 + n + sz r"
+
fun
Stars_add :: "val \<Rightarrow> val \<Rightarrow> val"
where
"Stars_add v (Stars vs) = Stars (v # vs)"
-function
+function (sequential)
decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)"
where
"decode' bs ZERO = (undefined, bs)"
@@ -39,6 +48,12 @@
| "decode' (Z # bs) (STAR r) = (let (v, bs') = decode' bs r in
let (vs, bs'') = decode' bs' (STAR r)
in (Stars_add v vs, bs''))"
+| "decode' [] (NTIMES r n) = (Void, [])"
+| "decode' (S # bs) (NTIMES r n) = (Stars [], bs)"
+(*| "decode' (Z # bs) (NTIMES r 0) = (undefined, bs)"*)
+| "decode' (Z # bs) (NTIMES r n) = (let (v, bs') = decode' bs r in
+ let (vs, bs'') = decode' bs' (NTIMES r (n - 1))
+ in (Stars_add v vs, bs''))"
by pat_completeness auto
lemma decode'_smaller:
@@ -48,12 +63,15 @@
apply(induct bs r)
apply(auto simp add: decode'.psimps split: prod.split)
using dual_order.trans apply blast
-by (meson dual_order.trans le_SucI)
+apply (meson dual_order.trans le_SucI)
+ apply (meson le_SucI le_trans)
+ done
termination "decode'"
-apply(relation "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))")
+apply(relation "inv_image (measure(%cs. sz cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))")
apply(auto dest!: decode'_smaller)
-by (metis less_Suc_eq_le snd_conv)
+ apply (metis less_Suc_eq_le snd_conv)
+ by (metis less_Suc_eq_le snd_conv)
definition
decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option"
@@ -69,13 +87,23 @@
apply(auto)
done
+lemma decode'_code_NTIMES:
+ assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x))"
+ shows "decode' (code (Stars vs) @ ds) (NTIMES r n) = (Stars vs, ds)"
+ using assms
+ apply(induct vs arbitrary: n r ds)
+ apply(auto)
+ done
+
+
lemma decode'_code:
assumes "\<Turnstile> v : r"
shows "decode' ((code v) @ ds) r = (v, ds)"
using assms
apply(induct v r arbitrary: ds)
apply(auto)
- using decode'_code_Stars by blast
+ using decode'_code_Stars apply blast
+ by (metis Un_iff decode'_code_NTIMES set_append)
lemma decode_code:
assumes "\<Turnstile> v : r"
@@ -93,6 +121,7 @@
| ASEQ "bit list" arexp arexp
| AALTs "bit list" "arexp list"
| ASTAR "bit list" arexp
+| ANTIMES "bit list" arexp nat
abbreviation
"AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]"
@@ -104,6 +133,7 @@
| "asize (AALTs cs rs) = Suc (sum_list (map asize rs))"
| "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)"
| "asize (ASTAR cs r) = Suc (asize r)"
+| "asize (ANTIMES cs r n) = Suc (asize r) + n"
fun
erase :: "arexp \<Rightarrow> rexp"
@@ -116,6 +146,7 @@
| "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))"
| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
| "erase (ASTAR _ r) = STAR (erase r)"
+| "erase (ANTIMES _ r n) = NTIMES (erase r) n"
fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where
@@ -125,6 +156,7 @@
| "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs"
| "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2"
| "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r"
+| "fuse bs (ANTIMES cs r n) = ANTIMES (bs @ cs) r n"
lemma fuse_append:
shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)"
@@ -141,6 +173,7 @@
(fuse [S] (intern r2))"
| "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)"
| "intern (STAR r) = ASTAR [] (intern r)"
+| "intern (NTIMES r n) = ANTIMES [] (intern r) n"
fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where
@@ -153,7 +186,9 @@
| "retrieve (ASTAR bs r) (Stars []) = bs @ [S]"
| "retrieve (ASTAR bs r) (Stars (v#vs)) =
bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)"
-
+| "retrieve (ANTIMES bs r 0) (Stars []) = bs @ [S]"
+| "retrieve (ANTIMES bs r (Suc n)) (Stars (v#vs)) =
+ bs @ [Z] @ retrieve r v @ retrieve (ANTIMES [] r n) (Stars vs)"
fun
@@ -165,27 +200,44 @@
| "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)"
| "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)"
| "bnullable (ASTAR bs r) = True"
+| "bnullable (ANTIMES bs r n) = (if n = 0 then True else bnullable r)"
abbreviation
bnullables :: "arexp list \<Rightarrow> bool"
where
"bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)"
-fun
- bmkeps :: "arexp \<Rightarrow> bit list" and
- bmkepss :: "arexp list \<Rightarrow> bit list"
+function (sequential)
+ bmkeps :: "arexp \<Rightarrow> bit list"
where
"bmkeps(AONE bs) = bs"
| "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)"
-| "bmkeps(AALTs bs rs) = bs @ (bmkepss rs)"
+| "bmkeps(AALTs bs (r#rs)) =
+ (if bnullable(r) then (bs @ bmkeps r) else (bmkeps (AALTs bs rs)))"
| "bmkeps(ASTAR bs r) = bs @ [S]"
-| "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))"
+| "bmkeps(ANTIMES bs r 0) = bs @ [S]"
+| "bmkeps(ANTIMES bs r (Suc n)) = bs @ [Z] @ (bmkeps r) @ bmkeps(ANTIMES [] r n)"
+apply(pat_completeness)
+apply(auto)
+done
+
+termination "bmkeps"
+apply(relation "measure asize")
+ apply(auto)
+ using asize.elims by force
+
+fun
+ bmkepss :: "arexp list \<Rightarrow> bit list"
+where
+ "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))"
+
lemma bmkepss1:
assumes "\<not> bnullables rs1"
shows "bmkepss (rs1 @ rs2) = bmkepss rs2"
using assms
- by (induct rs1) (auto)
+ by(induct rs1) (auto)
+
lemma bmkepss2:
assumes "bnullables rs1"
@@ -206,7 +258,7 @@
then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2))
else ASEQ bs (bder c r1) r2)"
| "bder c (ASTAR bs r) = ASEQ (bs @ [Z]) (bder c r) (ASTAR [] r)"
-
+| "bder c (ANTIMES bs r n) = (if n = 0 then AZERO else ASEQ (bs @ [Z]) (bder c r) (ANTIMES [] r (n - 1)))"
fun
bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
@@ -264,6 +316,15 @@
apply(simp_all)
done
+lemma retrieve_encode_NTIMES:
+ assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v" "length vs = n"
+ shows "code (Stars vs) = retrieve (ANTIMES [] (intern r) n) (Stars vs)"
+ using assms
+ apply(induct vs arbitrary: n)
+ apply(simp_all)
+ by force
+
+
lemma retrieve_fuse2:
assumes "\<Turnstile> v : (erase r)"
shows "retrieve (fuse bs r) v = bs @ retrieve r v"
@@ -285,7 +346,13 @@
apply(auto elim!: Prf_elims)[1]
apply(case_tac vs)
apply(simp)
- apply(simp)
+ apply(simp)
+ (* NTIMES *)
+ apply(auto elim!: Prf_elims)[1]
+ apply(case_tac vs1)
+ apply(simp_all)
+ apply(case_tac vs2)
+ apply(simp_all)
done
lemma retrieve_fuse:
@@ -300,8 +367,11 @@
shows "code v = retrieve (intern r) v"
using assms
apply(induct v r )
- apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
- done
+ apply(simp_all add: retrieve_fuse retrieve_encode_STARS)
+ apply(subst retrieve_encode_NTIMES)
+ apply(auto)
+ done
+
lemma retrieve_AALTs_bnullable1:
@@ -340,14 +410,26 @@
apply (simp add: retrieve_AALTs_bnullable1)
by (metis retrieve_AALTs_bnullable2)
-
+lemma bmkeps_retrieve_ANTIMES:
+ assumes "if n = 0 then True else bmkeps r = retrieve r (mkeps (erase r))"
+ and "bnullable (ANTIMES bs r n)"
+ shows "bmkeps (ANTIMES bs r n) = retrieve (ANTIMES bs r n) (Stars (replicate n (mkeps (erase r))))"
+ using assms
+ apply(induct n arbitrary: r bs)
+ apply(auto)[1]
+ apply(simp)
+ done
+
lemma bmkeps_retrieve:
assumes "bnullable r"
shows "bmkeps r = retrieve r (mkeps (erase r))"
using assms
- apply(induct r)
- apply(auto)
- using bmkeps_retrieve_AALTs by auto
+ apply(induct r rule: bmkeps.induct)
+ apply(auto)
+ apply (simp add: retrieve_AALTs_bnullable1)
+ using retrieve_AALTs_bnullable1 apply force
+ by (metis retrieve_AALTs_bnullable2)
+
lemma bder_retrieve:
assumes "\<Turnstile> v : der c (erase r)"
@@ -388,7 +470,15 @@
apply(clarify)
apply(erule Prf_elims)
apply(clarify)
- by (simp add: retrieve_fuse2)
+ apply (simp add: retrieve_fuse2)
+ (* ANTIMES case *)
+ apply(auto)
+ apply(erule Prf_elims)
+ apply(erule Prf_elims)
+ apply(clarify)
+ apply(erule Prf_elims)
+ apply(clarify)
+ by (metis (full_types) Suc_pred append_assoc injval.simps(8) retrieve.simps(10) retrieve.simps(6))
lemma MAIN_decode:
--- a/thys3/src/BlexerSimp.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/BlexerSimp.thy Wed Jul 13 08:35:45 2022 +0100
@@ -18,6 +18,7 @@
| "(AALTs bs1 []) ~1 (AALTs bs2 []) = True"
| "(AALTs bs1 (r1 # rs1)) ~1 (AALTs bs2 (r2 # rs2)) = (r1 ~1 r2 \<and> (AALTs bs1 rs1) ~1 (AALTs bs2 rs2))"
| "(ASTAR bs1 r1) ~1 (ASTAR bs2 r2) = r1 ~1 r2"
+| "(ANTIMES bs1 r1 n1) ~1 (ANTIMES bs2 r2 n2) = (r1 ~1 r2 \<and> n1 = n2)"
| "_ ~1 _ = False"
@@ -102,7 +103,12 @@
assumes "bnullable r"
shows "bmkeps (fuse bs r) = bs @ bmkeps r"
using assms
- by (induct r rule: bnullable.induct) (auto)
+ apply(induct r rule: bnullable.induct)
+ apply(auto)
+ apply (metis append.assoc bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
+ apply(case_tac n)
+ apply(auto)
+ done
lemma bmkepss_fuse:
assumes "bnullables rs"
@@ -363,7 +369,9 @@
using rs1 srewrites7 apply presburger
using srewrites7 apply force
apply (simp add: srewrites7)
+ apply(simp add: srewrites7)
by (simp add: srewrites7)
+
lemma bnullable0:
shows "r1 \<leadsto> r2 \<Longrightarrow> bnullable r1 = bnullable r2"
@@ -398,11 +406,21 @@
next
case (ss5 bs1 rs1 rsb)
then show ?case
- by (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
+ apply (simp add: bmkepss1 bmkepss2 bmkepss_fuse bnullable_fuse)
+ apply(case_tac rs1)
+ apply(auto simp add: bnullable_fuse)
+ apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
+ apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
+ apply (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
+ by (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
next
case (ss6 a1 a2 rsa rsb rsc)
then show ?case
by (smt (verit, best) Nil_is_append_conv bmkepss1 bmkepss2 bnullable_correctness in_set_conv_decomp list.distinct(1) list.set_intros(1) nullable_correctness set_ConsD subsetD)
+next
+ case (bs10 rs1 rs2 bs)
+ then show ?case
+ by (metis bmkeps_retrieve bmkeps_retrieve_AALTs bnullable.simps(4))
qed (auto)
lemma rewrites_bmkeps:
--- a/thys3/src/ClosedForms.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/ClosedForms.thy Wed Jul 13 08:35:45 2022 +0100
@@ -539,8 +539,10 @@
apply (simp add: frewrites_alt)
apply (simp add: frewrites_cons)
apply (simp add: frewrites_append)
- by (simp add: frewrites_cons)
-
+ apply (simp add: frewrites_cons)
+ apply (auto simp add: frewrites_cons)
+ using frewrite.intros(1) many_steps_later by blast
+
lemma gstar0:
shows "rsa @ (rdistinct rs (set rsa)) \<leadsto>g* rsa @ (rdistinct rs (insert RZERO (set rsa)))"
@@ -642,7 +644,8 @@
apply(induct r)
apply simp+
apply (metis list.set_intros(1))
- by blast
+ apply blast
+ by simp
@@ -1047,7 +1050,7 @@
apply (meson flts_gstar greal_trans grewrites_ralts_rsimpalts gstar_rdistinct)
apply (simp add: grewrites_ralts hrewrites_list)
- by simp
+ by simp_all
lemma interleave_aux1:
shows " RALT (RSEQ RZERO r1) r h\<leadsto>* r"
@@ -1121,7 +1124,7 @@
apply(subgoal_tac "rder x (RALTS xa) h\<leadsto>* rder x (rsimp (RALTS xa))")
using hrewrites_simpeq apply presburger
using interleave_star1 simp_hrewrites apply presburger
- by simp
+ by simp_all
@@ -1248,11 +1251,15 @@
apply simp+
using created_by_seq.cases apply blast
-
- apply (meson created_by_seq.cases rrexp.distinct(19) rrexp.distinct(21))
- apply (metis created_by_seq.simps rder.simps(5))
- apply (smt (verit, ccfv_threshold) created_by_seq.simps list.set_intros(1) list.simps(8) list.simps(9) rder.simps(4) rrexp.distinct(25) rrexp.inject(3))
- using created_by_seq.intros(1) by force
+ apply(auto)
+ apply (meson created_by_seq.cases rrexp.distinct(23) rrexp.distinct(25))
+ using created_by_seq.simps apply blast
+ apply (meson created_by_seq.simps)
+ using created_by_seq.intros(1) apply blast
+ apply (metis (no_types, lifting) created_by_seq.simps k0a list.set_intros(1) list.simps(8) list.simps(9) rrexp.distinct(31))
+ apply (simp add: created_by_seq.intros(1))
+ using created_by_seq.simps apply blast
+ by (simp add: created_by_seq.intros(1))
lemma createdbyseq_left_creatable:
shows "created_by_seq (RALT r1 r2) \<Longrightarrow> created_by_seq r1"
@@ -1473,8 +1480,6 @@
| "hElem r = [r]"
-
-
lemma cbs_ders_cbs:
shows "created_by_star r \<Longrightarrow> created_by_star (rder c r)"
apply(induct r rule: created_by_star.induct)
@@ -1491,11 +1496,6 @@
apply simp
using cbs_ders_cbs by auto
-(*
-lemma created_by_star_cases:
- shows "created_by_star r \<Longrightarrow> \<exists>ra rb. (r = RALT ra rb \<and> created_by_star ra \<and> created_by_star rb) \<or> r = RSEQ ra rb "
- by (meson created_by_star.cases)
-*)
lemma hfau_pushin:
@@ -1549,6 +1549,8 @@
apply(subst stupdates_append[symmetric])
using stupdates_join_general by blast
+
+
lemma starders_hfau_also1:
shows "hflat_aux (rders (RSTAR r) (c # xs)) = map (\<lambda>s1. RSEQ (rders r s1) (RSTAR r)) (star_updates xs r [[c]])"
using star_hfau_induct by force
@@ -1566,7 +1568,7 @@
apply simp
apply (metis (no_types, lifting) append_Cons append_eq_append_conv2 gmany_steps_later greal_trans grewrite.intros(2) grewrites_append self_append_conv)
apply simp
- by simp
+ by simp_all
@@ -1600,7 +1602,9 @@
apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux a @ hflat_aux aa))")
using hflat_aux.simps(1) apply presburger
apply simp
- using cbs_hfau_rsimpeq1 by fastforce
+ using cbs_hfau_rsimpeq1 apply(fastforce)
+ by simp
+
lemma star_closed_form1:
shows "rsimp (rders (RSTAR r0) (c#s)) =
@@ -1644,8 +1648,11 @@
using hrewrites_simpeq srewritescf_alt1 apply fastforce
using star_closed_form6_hrewrites by blast
+
+
+
lemma stupdate_nonempty:
- shows "\<forall>s \<in> set Ss. s \<noteq> [] \<Longrightarrow> \<forall>s \<in> set (star_update c r Ss). s \<noteq> []"
+ shows "\<forall>s \<in> set Ss. s \<noteq> [] \<Longrightarrow> \<forall>s \<in> set (star_update c r Ss). s \<noteq> []"
apply(induct Ss)
apply simp
apply(case_tac "rnullable (rders r a)")
@@ -1671,12 +1678,518 @@
lemma star_closed_form:
shows "rders_simp (RSTAR r0) (c#s) =
rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
- apply(induct s)
+ apply(case_tac s)
apply simp
apply (metis idem_after_simp1 rsimp.simps(1) rsimp.simps(6) rsimp_idem)
using star_closed_form4 star_closed_form5 star_closed_form6 star_closed_form8 by presburger
-unused_thms
+
+
+fun nupdate :: "char \<Rightarrow> rrexp \<Rightarrow> (string * nat) option list \<Rightarrow> (string * nat) option list" where
+ "nupdate c r [] = []"
+| "nupdate c r (Some (s, Suc n) # Ss) = (if (rnullable (rders r s))
+ then Some (s@[c], Suc n) # Some ([c], n) # (nupdate c r Ss)
+ else Some ((s@[c]), Suc n) # (nupdate c r Ss)
+ )"
+| "nupdate c r (Some (s, 0) # Ss) = (if (rnullable (rders r s))
+ then Some (s@[c], 0) # None # (nupdate c r Ss)
+ else Some ((s@[c]), 0) # (nupdate c r Ss)
+ ) "
+| "nupdate c r (None # Ss) = (None # nupdate c r Ss)"
+
+
+fun nupdates :: "char list \<Rightarrow> rrexp \<Rightarrow> (string * nat) option list \<Rightarrow> (string * nat) option list"
+ where
+ "nupdates [] r Ss = Ss"
+| "nupdates (c # cs) r Ss = nupdates cs r (nupdate c r Ss)"
+
+fun ntset :: "rrexp \<Rightarrow> nat \<Rightarrow> string \<Rightarrow> (string * nat) option list" where
+ "ntset r (Suc n) (c # cs) = nupdates cs r [Some ([c], n)]"
+| "ntset r 0 _ = [None]"
+| "ntset r _ [] = []"
+
+inductive created_by_ntimes :: "rrexp \<Rightarrow> bool" where
+ "created_by_ntimes RZERO"
+| "created_by_ntimes (RSEQ ra (RNTIMES rb n))"
+| "\<lbrakk>created_by_ntimes r1; created_by_ntimes r2\<rbrakk> \<Longrightarrow> created_by_ntimes (RALT r1 r2)"
+| "\<lbrakk>created_by_ntimes r \<rbrakk> \<Longrightarrow> created_by_ntimes (RALT r RZERO)"
+
+fun highest_power_aux :: "(string * nat) option list \<Rightarrow> nat \<Rightarrow> nat" where
+ "highest_power_aux [] n = n"
+| "highest_power_aux (None # rs) n = highest_power_aux rs n"
+| "highest_power_aux (Some (s, n) # rs) m = highest_power_aux rs (max n m)"
+
+fun hpower :: "(string * nat) option list \<Rightarrow> nat" where
+ "hpower rs = highest_power_aux rs 0"
+
+
+lemma nupdate_mono:
+ shows " (highest_power_aux (nupdate c r optlist) m) \<le> (highest_power_aux optlist m)"
+ apply(induct optlist arbitrary: m)
+ apply simp
+ apply(case_tac a)
+ apply simp
+ apply(case_tac aa)
+ apply(case_tac b)
+ apply simp+
+ done
+
+lemma nupdate_mono1:
+ shows "hpower (nupdate c r optlist) \<le> hpower optlist"
+ by (simp add: nupdate_mono)
+
+
+
+lemma cbn_ders_cbn:
+ shows "created_by_ntimes r \<Longrightarrow> created_by_ntimes (rder c r)"
+ apply(induct r rule: created_by_ntimes.induct)
+ apply simp
+
+ using created_by_ntimes.intros(1) created_by_ntimes.intros(2) created_by_ntimes.intros(3) apply presburger
+
+ apply (metis created_by_ntimes.simps rder.simps(5) rder.simps(7))
+ using created_by_star.intros(1) created_by_star.intros(2) apply auto[1]
+ using created_by_ntimes.intros(1) created_by_ntimes.intros(3) apply auto[1]
+ by (metis (mono_tags, lifting) created_by_ntimes.simps list.simps(8) list.simps(9) rder.simps(1) rder.simps(4))
+
+lemma ntimes_ders_cbn:
+ shows "created_by_ntimes (rders (RSEQ r' (RNTIMES r n)) s)"
+ apply(induct s rule: rev_induct)
+ apply simp
+ apply (simp add: created_by_ntimes.intros(2))
+ apply(subst rders_append)
+ using cbn_ders_cbn by auto
+
+lemma always0:
+ shows "rders RZERO s = RZERO"
+ apply(induct s)
+ by simp+
+
+lemma ntimes_ders_cbn1:
+ shows "created_by_ntimes (rders (RNTIMES r n) (c#s))"
+ apply(case_tac n)
+ apply simp
+ using always0 created_by_ntimes.intros(1) apply auto[1]
+ by (simp add: ntimes_ders_cbn)
+
+
+lemma ntimes_hfau_pushin:
+ shows "created_by_ntimes r \<Longrightarrow> hflat_aux (rder c r) = concat (map hflat_aux (map (rder c) (hflat_aux r)))"
+ apply(induct r rule: created_by_ntimes.induct)
+ apply simp+
+ done
+
+
+abbreviation
+ "opterm r SN \<equiv> case SN of
+ Some (s, n) \<Rightarrow> RSEQ (rders r s) (RNTIMES r n)
+ | None \<Rightarrow> RZERO
+
+
+"
+
+fun nonempty_string :: "(string * nat) option \<Rightarrow> bool" where
+ "nonempty_string None = True"
+| "nonempty_string (Some ([], n)) = False"
+| "nonempty_string (Some (c#s, n)) = True"
+
+
+lemma nupdate_nonempty:
+ shows "\<lbrakk>\<forall>opt \<in> set Ss. nonempty_string opt \<rbrakk> \<Longrightarrow> \<forall>opt \<in> set (nupdate c r Ss). nonempty_string opt"
+ apply(induct c r Ss rule: nupdate.induct)
+ apply(auto)
+ apply (metis Nil_is_append_conv neq_Nil_conv nonempty_string.simps(3))
+ by (metis Nil_is_append_conv neq_Nil_conv nonempty_string.simps(3))
+
+
+
+lemma nupdates_nonempty:
+ shows "\<lbrakk>\<forall>opt \<in> set Ss. nonempty_string opt \<rbrakk> \<Longrightarrow> \<forall>opt \<in> set (nupdates s r Ss). nonempty_string opt"
+ apply(induct s arbitrary: Ss)
+ apply simp
+ apply simp
+ using nupdate_nonempty by presburger
+
+lemma nullability1: shows "rnullable (rders r s) = rnullable (rders_simp r s)"
+ by (metis der_simp_nullability rders.simps(1) rders_simp.simps(1) rders_simp_same_simpders)
+
+lemma nupdate_induct1:
+ shows
+ "concat (map (hflat_aux \<circ> (rder c \<circ> (opterm r))) sl ) =
+ map (opterm r) (nupdate c r sl)"
+ apply(induct sl)
+ apply simp
+ apply(simp add: rders_append)
+ apply(case_tac "a")
+ apply simp+
+ apply(case_tac "aa")
+ apply(case_tac "b")
+ apply(case_tac "rnullable (rders r ab)")
+ apply(subgoal_tac "rnullable (rders_simp r ab)")
+ apply simp
+ using rders.simps(1) rders.simps(2) rders_append apply presburger
+ using nullability1 apply blast
+ apply simp
+ using rders.simps(1) rders.simps(2) rders_append apply presburger
+ apply simp
+ using rders.simps(1) rders.simps(2) rders_append by presburger
+
+
+lemma nupdates_join_general:
+ shows "concat (map hflat_aux (map (rder x) (map (opterm r) (nupdates xs r Ss)) )) =
+ map (opterm r) (nupdates (xs @ [x]) r Ss)"
+ apply(induct xs arbitrary: Ss)
+ apply (simp)
+ prefer 2
+ apply auto[1]
+ using nupdate_induct1 by blast
+
+
+lemma nupdates_join_general1:
+ shows "concat (map (hflat_aux \<circ> (rder x) \<circ> (opterm r)) (nupdates xs r Ss)) =
+ map (opterm r) (nupdates (xs @ [x]) r Ss)"
+ by (metis list.map_comp nupdates_join_general)
+
+lemma nupdates_append: shows
+"nupdates (s @ [c]) r Ss = nupdate c r (nupdates s r Ss)"
+ apply(induct s arbitrary: Ss)
+ apply simp
+ apply simp
+ done
+
+lemma nupdates_mono:
+ shows "highest_power_aux (nupdates s r optlist) m \<le> highest_power_aux optlist m"
+ apply(induct s rule: rev_induct)
+ apply simp
+ apply(subst nupdates_append)
+ by (meson le_trans nupdate_mono)
+
+lemma nupdates_mono1:
+ shows "hpower (nupdates s r optlist) \<le> hpower optlist"
+ by (simp add: nupdates_mono)
+
+
+(*"\<forall>r \<in> set (nupdates s r [Some ([c], n)]). r = None \<or>( \<exists>s' m. r = Some (s', m) \<and> m \<le> n)"*)
+lemma nupdates_mono2:
+ shows "hpower (nupdates s r [Some ([c], n)]) \<le> n"
+ by (metis highest_power_aux.simps(1) highest_power_aux.simps(3) hpower.simps max_nat.right_neutral nupdates_mono1)
+
+lemma hpow_arg_mono:
+ shows "m \<ge> n \<Longrightarrow> highest_power_aux rs m \<ge> highest_power_aux rs n"
+ apply(induct rs arbitrary: m n)
+ apply simp
+ apply(case_tac a)
+ apply simp
+ apply(case_tac aa)
+ apply simp
+ done
+
+
+lemma hpow_increase:
+ shows "highest_power_aux (a # rs') m \<ge> highest_power_aux rs' m"
+ apply(case_tac a)
+ apply simp
+ apply simp
+ apply(case_tac aa)
+ apply(case_tac b)
+ apply simp+
+ apply(case_tac "Suc nat > m")
+ using hpow_arg_mono max.cobounded2 apply blast
+ using hpow_arg_mono max.cobounded2 by blast
+
+lemma hpow_append:
+ shows "highest_power_aux (rsa @ rsb) m = highest_power_aux rsb (highest_power_aux rsa m)"
+ apply (induct rsa arbitrary: rsb m)
+ apply simp
+ apply simp
+ apply(case_tac a)
+ apply simp
+ apply(case_tac aa)
+ apply simp
+ done
+
+lemma hpow_aux_mono:
+ shows "highest_power_aux (rsa @ rsb) m \<ge> highest_power_aux rsb m"
+ apply(induct rsa arbitrary: rsb rule: rev_induct)
+ apply simp
+ apply simp
+ using hpow_increase order.trans by blast
+
+
+
+
+lemma hpow_mono:
+ shows "hpower (rsa @ rsb) \<le> n \<Longrightarrow> hpower rsb \<le> n"
+ apply(induct rsb arbitrary: rsa)
+ apply simp
+ apply(subgoal_tac "hpower rsb \<le> n")
+ apply simp
+ apply (metis dual_order.trans hpow_aux_mono)
+ by (metis hpow_append hpow_increase hpower.simps nat_le_iff_add trans_le_add1)
+
+
+lemma hpower_rs_elems_aux:
+ shows "highest_power_aux rs k \<le> n \<Longrightarrow> \<forall>r\<in>set rs. r = None \<or> (\<exists>s' m. r = Some (s', m) \<and> m \<le> n)"
+apply(induct rs k arbitrary: n rule: highest_power_aux.induct)
+ apply(auto)
+ by (metis dual_order.trans highest_power_aux.simps(1) hpow_append hpow_aux_mono linorder_le_cases max.absorb1 max.absorb2)
+
+lemma hpower_rs_elems:
+ shows "hpower rs \<le> n \<Longrightarrow> \<forall>r \<in> set rs. r = None \<or>( \<exists>s' m. r = Some (s', m) \<and> m \<le> n)"
+ by (simp add: hpower_rs_elems_aux)
+
+lemma nupdates_elems_leqn:
+ shows "\<forall>r \<in> set (nupdates s r [Some ([c], n)]). r = None \<or>( \<exists>s' m. r = Some (s', m) \<and> m \<le> n)"
+ by (meson hpower_rs_elems nupdates_mono2)
+
+lemma ntimes_hfau_induct:
+ shows "hflat_aux (rders (RSEQ (rder c r) (RNTIMES r n)) s) =
+ map (opterm r) (nupdates s r [Some ([c], n)])"
+ apply(induct s rule: rev_induct)
+ apply simp
+ apply(subst rders_append)+
+ apply simp
+ apply(subst nupdates_append)
+ apply(subgoal_tac "created_by_ntimes (rders (RSEQ (rder c r) (RNTIMES r n)) xs)")
+ prefer 2
+ apply (simp add: ntimes_ders_cbn)
+ apply(subst ntimes_hfau_pushin)
+ apply simp
+ apply(subgoal_tac "concat (map hflat_aux (map (rder x) (hflat_aux (rders (RSEQ (rder c r) (RNTIMES r n)) xs)))) =
+ concat (map hflat_aux (map (rder x) ( map (opterm r) (nupdates xs r [Some ([c], n)])))) ")
+ apply(simp only:)
+ prefer 2
+ apply presburger
+ apply(subst nupdates_append[symmetric])
+ using nupdates_join_general by blast
+
+
+(*nupdates s r [Some ([c], n)]*)
+lemma ntimes_ders_hfau_also1:
+ shows "hflat_aux (rders (RNTIMES r (Suc n)) (c # xs)) = map (opterm r) (nupdates xs r [Some ([c], n)])"
+ using ntimes_hfau_induct by force
+
+
+
+lemma hfau_rsimpeq2_ntimes:
+ shows "created_by_ntimes r \<Longrightarrow> rsimp r = rsimp ( (RALTS (hflat_aux r)))"
+ apply(induct r)
+ apply simp+
+
+ apply (metis rsimp_seq_equal1)
+ prefer 2
+ apply simp
+ apply(case_tac x)
+ apply simp
+ apply(case_tac "list")
+ apply simp
+
+ apply (metis idem_after_simp1)
+ apply(case_tac "lista")
+ prefer 2
+ apply (metis hflat_aux.simps(8) idem_after_simp1 list.simps(8) list.simps(9) rsimp.simps(2))
+ apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux (RALT a aa)))")
+ apply simp
+ apply(subgoal_tac "rsimp (RALT a aa) = rsimp (RALTS (hflat_aux a @ hflat_aux aa))")
+ using hflat_aux.simps(1) apply presburger
+ apply simp
+ using cbs_hfau_rsimpeq1 apply(fastforce)
+ by simp
+
+
+lemma ntimes_closed_form1:
+ shows "rsimp (rders (RNTIMES r (Suc n)) (c#s)) =
+rsimp ( ( RALTS ( map (opterm r) (nupdates s r [Some ([c], n)]) )))"
+ apply(subgoal_tac "created_by_ntimes (rders (RNTIMES r (Suc n)) (c#s))")
+ apply(subst hfau_rsimpeq2_ntimes)
+ apply linarith
+ using ntimes_ders_hfau_also1 apply auto[1]
+ using ntimes_ders_cbn1 by blast
+
+
+lemma ntimes_closed_form2:
+ shows "rsimp (rders_simp (RNTIMES r (Suc n)) (c#s) ) =
+rsimp ( ( RALTS ( (map (opterm r ) (nupdates s r [Some ([c], n)]) ) )))"
+ by (metis list.distinct(1) ntimes_closed_form1 rders_simp_same_simpders rsimp_idem)
+
+
+lemma ntimes_closed_form3:
+ shows "rsimp (rders_simp (RNTIMES r n) (c#s)) = (rders_simp (RNTIMES r n) (c#s))"
+ by (metis list.distinct(1) rders_simp_same_simpders rsimp_idem)
+
+
+lemma ntimes_closed_form4:
+ shows " (rders_simp (RNTIMES r (Suc n)) (c#s)) =
+rsimp ( ( RALTS ( (map (opterm r ) (nupdates s r [Some ([c], n)]) ) )))"
+ using ntimes_closed_form2 ntimes_closed_form3
+ by metis
+
+
+
+
+lemma ntimes_closed_form5:
+ shows " rsimp ( RALTS (map (\<lambda>s1. RSEQ (rders r0 s1) (RNTIMES r n) ) Ss)) =
+ rsimp ( RALTS (map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RNTIMES r n)) ) Ss))"
+ by (smt (verit, ccfv_SIG) list.map_comp map_eq_conv o_apply simp_flatten_aux0)
+
+
+
+lemma ntimes_closed_form6_hrewrites:
+ shows "
+(map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RNTIMES r0 n)) ) Ss )
+ scf\<leadsto>*
+(map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RNTIMES r0 n)) ) Ss )"
+ apply(induct Ss)
+ apply simp
+ apply (simp add: ss1)
+ by (metis (no_types, lifting) list.simps(9) rsimp.simps(1) rsimp_idem simp_hrewrites ss2)
+
+
+
+lemma ntimes_closed_form6:
+ shows " rsimp ( ( RALTS ( (map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RNTIMES r0 n)) ) Ss )))) =
+ rsimp ( ( RALTS ( (map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RNTIMES r0 n)) ) Ss ))))"
+ apply(subgoal_tac " map (\<lambda>s1. (RSEQ (rsimp (rders r0 s1)) (RNTIMES r0 n)) ) Ss scf\<leadsto>*
+ map (\<lambda>s1. rsimp (RSEQ (rders r0 s1) (RNTIMES r0 n)) ) Ss ")
+ using hrewrites_simpeq srewritescf_alt1 apply fastforce
+ using ntimes_closed_form6_hrewrites by blast
+
+abbreviation
+ "optermsimp r SN \<equiv> case SN of
+ Some (s, n) \<Rightarrow> RSEQ (rders_simp r s) (RNTIMES r n)
+ | None \<Rightarrow> RZERO
+
+
+"
+
+abbreviation
+ "optermOsimp r SN \<equiv> case SN of
+ Some (s, n) \<Rightarrow> rsimp (RSEQ (rders r s) (RNTIMES r n))
+ | None \<Rightarrow> RZERO
+
+
+"
+
+abbreviation
+ "optermosimp r SN \<equiv> case SN of
+ Some (s, n) \<Rightarrow> RSEQ (rsimp (rders r s)) (RNTIMES r n)
+ | None \<Rightarrow> RZERO
+"
+
+lemma ntimes_closed_form51:
+ shows "rsimp (RALTS (map (opterm r) (nupdates s r [Some ([c], n)]))) =
+ rsimp (RALTS (map (rsimp \<circ> (opterm r)) (nupdates s r [Some ([c], n)])))"
+ by (metis map_map simp_flatten_aux0)
+
+
+
+lemma osimp_Osimp:
+ shows " nonempty_string sn \<Longrightarrow> optermosimp r sn = optermsimp r sn"
+ apply(induct rule: nonempty_string.induct)
+ apply force
+ apply auto[1]
+ apply simp
+ by (metis list.distinct(1) rders.simps(2) rders_simp.simps(2) rders_simp_same_simpders)
+
+
+
+lemma osimp_Osimp_list:
+ shows "\<forall>sn \<in> set snlist. nonempty_string sn \<Longrightarrow> map (optermosimp r) snlist = map (optermsimp r) snlist"
+ by (simp add: osimp_Osimp)
+
+
+lemma ntimes_closed_form8:
+ shows
+"rsimp (RALTS (map (optermosimp r) (nupdates s r [Some ([c], n)]))) =
+ rsimp (RALTS (map (optermsimp r) (nupdates s r [Some ([c], n)])))"
+ apply(subgoal_tac "\<forall>opt \<in> set (nupdates s r [Some ([c], n)]). nonempty_string opt")
+ using osimp_Osimp_list apply presburger
+ by (metis list.distinct(1) list.set_cases nonempty_string.simps(3) nupdates_nonempty set_ConsD)
+
+
+
+lemma ntimes_closed_form9aux:
+ shows "\<forall>snopt \<in> set (nupdates s r [Some ([c], n)]). nonempty_string snopt"
+ by (metis list.distinct(1) list.set_cases nonempty_string.simps(3) nupdates_nonempty set_ConsD)
+
+lemma ntimes_closed_form9aux1:
+ shows "\<forall>snopt \<in> set snlist. nonempty_string snopt \<Longrightarrow>
+rsimp (RALTS (map (optermosimp r) snlist)) =
+rsimp (RALTS (map (optermOsimp r) snlist))"
+ apply(induct snlist)
+ apply simp+
+ apply(case_tac "a")
+ apply simp+
+ by (smt (z3) case_prod_conv idem_after_simp1 map_eq_conv nonempty_string.elims(2) o_apply option.simps(4) option.simps(5) rsimp.simps(1) rsimp.simps(7) rsimp_idem)
+
+
+
+
+lemma ntimes_closed_form9:
+ shows
+"rsimp (RALTS (map (optermosimp r) (nupdates s r [Some ([c], n)]))) =
+ rsimp (RALTS (map (optermOsimp r) (nupdates s r [Some ([c], n)])))"
+ using ntimes_closed_form9aux ntimes_closed_form9aux1 by presburger
+
+
+lemma ntimes_closed_form10rewrites_aux:
+ shows " map (rsimp \<circ> (opterm r)) optlist scf\<leadsto>*
+ map (optermOsimp r) optlist"
+ apply(induct optlist)
+ apply simp
+ apply (simp add: ss1)
+ apply simp
+ apply(case_tac a)
+ using ss2 apply fastforce
+ using ss2 by force
+
+
+lemma ntimes_closed_form10rewrites:
+ shows " map (rsimp \<circ> (opterm r)) (nupdates s r [Some ([c], n)]) scf\<leadsto>*
+ map (optermOsimp r) (nupdates s r [Some ([c], n)])"
+ using ntimes_closed_form10rewrites_aux by blast
+
+lemma ntimes_closed_form10:
+ shows "rsimp (RALTS (map (rsimp \<circ> (opterm r)) (nupdates s r [Some ([c], n)]))) =
+ rsimp (RALTS (map (optermOsimp r) (nupdates s r [Some ([c], n)])))"
+ by (smt (verit, best) case_prod_conv hpower_rs_elems map_eq_conv nupdates_mono2 o_apply option.case(2) option.simps(4) rsimp.simps(3))
+
+
+lemma rders_simp_cons:
+ shows "rders_simp r (c # s) = rders_simp (rsimp (rder c r)) s"
+ by simp
+
+lemma rder_ntimes:
+ shows "rder c (RNTIMES r (Suc n)) = RSEQ (rder c r) (RNTIMES r n)"
+ by simp
+
+
+lemma ntimes_closed_form:
+ shows "rders_simp (RNTIMES r0 (Suc n)) (c#s) =
+rsimp ( RALTS ( (map (optermsimp r0 ) (nupdates s r0 [Some ([c], n)]) ) ))"
+ apply (subst rders_simp_cons)
+ apply(subst rder_ntimes)
+ using ntimes_closed_form10 ntimes_closed_form4 ntimes_closed_form51 ntimes_closed_form8 ntimes_closed_form9 by force
+
+
+
+
+
+
+(*
+lemma ntimes_closed_form:
+ assumes "s \<noteq> []"
+ shows "rders_simp (RNTIMES r (Suc n)) s =
+rsimp ( RALTS ( map
+ (\<lambda> optSN. case optSN of
+ Some (s, n) \<Rightarrow> RSEQ (rders_simp r s) (RNTIMES r n)
+ | None \<Rightarrow> RZERO
+ )
+ (ntset r n s)
+ )
+ )"
+
+*)
end
\ No newline at end of file
--- a/thys3/src/ClosedFormsBounds.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/ClosedFormsBounds.thy Wed Jul 13 08:35:45 2022 +0100
@@ -141,7 +141,7 @@
apply(simp only:)
apply(subgoal_tac "rsizes (rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))
\<le> rsizes (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))")
- apply (smt (verit, best) dual_order.trans insert_iff rrexp.distinct(15))
+ apply (smt (verit, ccfv_threshold) dual_order.trans insertE rrexp.distinct(17))
apply (metis (no_types, opaque_lifting) le_add_same_cancel2 list.simps(9) sum_list.Cons zero_le)
apply fastforce
apply fastforce
@@ -151,8 +151,8 @@
using rflts.simps(4) apply presburger
apply simp
apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
- apply(simp only:)
- apply (metis Un_insert_left insertE rrexp.distinct(15))
+ apply(simp only:)
+ apply (metis Un_insert_left insertE rrexp.distinct(17))
apply fastforce
apply(case_tac "a \<in> noalts_set")
apply simp
@@ -174,14 +174,14 @@
apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
apply(simp only:)
apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
- apply(simp only:)
- apply (metis insertE rrexp.distinct(21))
+ apply(simp only:)
+ apply (metis insertE nonalt.simps(1) nonalt.simps(4))
apply blast
apply fastforce
apply force
- apply simp
- apply (metis Un_insert_left insert_iff rrexp.distinct(21))
+ apply simp
+ apply (metis Un_insert_left insertE nonalt.simps(1) nonalt.simps(4))
apply(case_tac "a \<in> noalts_set")
apply simp
apply(subgoal_tac "a \<notin> alts_set")
@@ -204,14 +204,13 @@
apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
apply(simp only:)
-
- apply (metis insertE rrexp.distinct(25))
+ apply (metis insertE rrexp.distinct(31))
apply blast
apply fastforce
apply force
apply simp
- apply (metis Un_insert_left insertE rrexp.distinct(25))
+ apply (metis Un_insert_left insertE rrexp.distinct(31))
using Suc_le_mono flts_size_reduction_alts apply presburger
apply(case_tac "a \<in> noalts_set")
@@ -234,16 +233,42 @@
apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
apply(simp only:)
apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
- apply(simp only:)
- apply (metis insertE rrexp.distinct(29))
+ apply(simp only:)
+ apply (metis insertE rrexp.distinct(37))
apply blast
apply fastforce
apply force
apply simp
- apply (metis Un_insert_left insert_iff rrexp.distinct(29))
- done
+ apply (metis Un_insert_left insert_iff rrexp.distinct(37))
+ apply(case_tac "a \<in> noalts_set")
+ apply simp
+ apply(subgoal_tac "a \<notin> alts_set")
+ prefer 2
+ apply blast
+ apply(case_tac "a \<in> corr_set")
+ apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
+ prefer 2
+ apply fastforce
+ apply(simp only:)
+ apply(subgoal_tac "rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))) \<le>
+ rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))")
+
+ apply(subgoal_tac "rsizes (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)) \<le>
+ rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set)))")
+ apply fastforce
+ apply simp
+ apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
+ apply(simp only:)
+ apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
+ apply(simp only:)
+ apply (metis insertE nonalt.simps(1) nonalt.simps(7))
+ apply blast
+ apply blast
+ apply force
+ apply(auto)
+ by (metis Un_insert_left insert_iff rrexp.distinct(39))
lemma flts_vs_nflts:
@@ -379,6 +404,220 @@
qed
+thm ntimes_closed_form
+
+thm rsize.simps
+
+lemma nupdates_snoc:
+ shows " (nupdates (xs @ [x]) r optlist) = nupdate x r (nupdates xs r optlist)"
+ by (simp add: nupdates_append)
+
+lemma nupdate_elems:
+ shows "\<forall>opt \<in> set (nupdate c r optlist). opt = None \<or> (\<exists>s n. opt = Some (s, n))"
+ using nonempty_string.cases by auto
+
+lemma nupdates_elems:
+ shows "\<forall>opt \<in> set (nupdates s r optlist). opt = None \<or> (\<exists>s n. opt = Some (s, n))"
+ by (meson nonempty_string.cases)
+
+
+lemma opterm_optlist_result_shape:
+ shows "\<forall>r' \<in> set (map (optermsimp r) optlist). r' = RZERO \<or> (\<exists>s m. r' = RSEQ (rders_simp r s) (RNTIMES r m))"
+ apply(induct optlist)
+ apply simp
+ apply(case_tac a)
+ apply simp+
+ by fastforce
+
+
+lemma opterm_optlist_result_shape2:
+ shows "\<And>optlist. \<forall>r' \<in> set (map (optermsimp r) optlist). r' = RZERO \<or> (\<exists>s m. r' = RSEQ (rders_simp r s) (RNTIMES r m))"
+ using opterm_optlist_result_shape by presburger
+
+
+lemma nupdate_n_leq_n:
+ shows "\<forall>r \<in> set (nupdate c' r [Some ([c], n)]). r = None \<or>( \<exists>s' m. r = Some (s', m) \<and> m \<le> n)"
+ apply(case_tac n)
+ apply simp
+ apply simp
+ done
+(*
+lemma nupdate_induct_leqn:
+ shows "\<lbrakk>\<forall>opt \<in> set optlist. opt = None \<or> (\<exists>s' m. opt = Some(s', m) \<and> m \<le> n) \<rbrakk> \<Longrightarrow>
+ \<forall>opt \<in> set (nupdate c' r optlist). opt = None \<or> (\<exists>s' m. opt = Some (s', m) \<and> m \<le> n)"
+ apply (case_tac optlist)
+ apply simp
+ apply(case_tac a)
+ apply simp
+ sledgehammer
+*)
+
+
+lemma nupdates_n_leq_n:
+ shows "\<forall>r \<in> set (nupdates s r [Some ([c], n)]). r = None \<or>( \<exists>s' m. r = Some (s', m) \<and> m \<le> n)"
+ apply(induct s rule: rev_induct)
+ apply simp
+ apply(subst nupdates_append)
+ by (metis nupdates_elems_leqn nupdates_snoc)
+
+
+
+lemma ntimes_closed_form_list_elem_shape:
+ shows "\<forall>r' \<in> set (map (optermsimp r) (nupdates s r [Some ([c], n)])).
+r' = RZERO \<or> (\<exists>s' m. r' = RSEQ (rders_simp r s') (RNTIMES r m) \<and> m \<le> n)"
+ apply(insert opterm_optlist_result_shape2)
+ apply(case_tac s)
+ apply(auto)
+ apply (metis rders_simp_one_char)
+ by (metis case_prod_conv nupdates.simps(2) nupdates_n_leq_n option.simps(4) option.simps(5))
+
+
+lemma ntimes_trivial1:
+ shows "rsize RZERO \<le> N + rsize (RNTIMES r n)"
+ by simp
+
+
+lemma ntimes_trivial20:
+ shows "m \<le> n \<Longrightarrow> rsize (RNTIMES r m) \<le> rsize (RNTIMES r n)"
+ by simp
+
+
+lemma ntimes_trivial2:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
+ shows " r' = RSEQ (rders_simp r s1) (RNTIMES r m) \<and> m \<le> n
+ \<Longrightarrow> rsize r' \<le> Suc (N + rsize (RNTIMES r n))"
+ apply simp
+ by (simp add: add_mono_thms_linordered_semiring(1) assms)
+
+lemma ntimes_closed_form_list_elem_bounded:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
+ shows "\<forall>r' \<in> set (map (optermsimp r) (nupdates s r [Some ([c], n)])). rsize r' \<le> Suc (N + rsize (RNTIMES r n))"
+ apply(rule ballI)
+ apply(subgoal_tac "r' = RZERO \<or> (\<exists>s' m. r' = RSEQ (rders_simp r s') (RNTIMES r m) \<and> m \<le> n)")
+ prefer 2
+ using ntimes_closed_form_list_elem_shape apply blast
+ apply(case_tac "r' = RZERO")
+ using le_SucI ntimes_trivial1 apply presburger
+ apply(subgoal_tac "\<exists>s1 m. r' = RSEQ (rders_simp r s1) (RNTIMES r m) \<and> m \<le> n")
+ apply(erule exE)+
+ using assms ntimes_trivial2 apply presburger
+ by blast
+
+
+lemma P_holds_after_distinct:
+ assumes "\<forall>r \<in> set rs. P r"
+ shows "\<forall>r \<in> set (rdistinct rs rset). P r"
+ by (simp add: assms rdistinct_set_equality1)
+
+lemma ntimes_control_bounded:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
+ shows "rsizes (rdistinct (map (optermsimp r) (nupdates s r [Some ([c], n)])) {})
+ \<le> (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * (Suc (N + rsize (RNTIMES r n)))"
+ apply(subgoal_tac "\<forall>r' \<in> set (rdistinct (map (optermsimp r) (nupdates s r [Some ([c], n)])) {}).
+ rsize r' \<le> Suc (N + rsize (RNTIMES r n))")
+ apply (meson distinct_list_rexp_upto rdistinct_same_set)
+ apply(subgoal_tac "\<forall>r' \<in> set (map (optermsimp r) (nupdates s r [Some ([c], n)])). rsize r' \<le> Suc (N + rsize (RNTIMES r n))")
+ apply (simp add: rdistinct_set_equality)
+ by (metis assms nat_le_linear not_less_eq_eq ntimes_closed_form_list_elem_bounded)
+
+
+
+lemma ntimes_closed_form_bounded0:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
+ shows " (rders_simp (RNTIMES r 0) s) = RZERO \<or> (rders_simp (RNTIMES r 0) s) = RNTIMES r 0
+ "
+ apply(induct s)
+ apply simp
+ by (metis always0 list.simps(3) rder.simps(7) rders.simps(2) rders_simp_same_simpders rsimp.simps(3))
+
+lemma ntimes_closed_form_bounded1:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
+ shows " rsize (rders_simp (RNTIMES r 0) s) \<le> max (rsize RZERO) (rsize (RNTIMES r 0))"
+
+ by (metis assms max.cobounded1 max.cobounded2 ntimes_closed_form_bounded0)
+
+lemma self_smaller_than_bound:
+ shows "\<forall>s. rsize (rders_simp r s) \<le> N \<Longrightarrow> rsize r \<le> N"
+ apply(drule_tac x = "[]" in spec)
+ apply simp
+ done
+
+lemma ntimes_closed_form_bounded_nil_aux:
+ shows "max (rsize RZERO) (rsize (RNTIMES r 0)) = 1 + rsize r"
+ by auto
+
+lemma ntimes_closed_form_bounded_nil:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
+ shows " rsize (rders_simp (RNTIMES r 0) s) \<le> 1 + rsize r"
+ using assms ntimes_closed_form_bounded1 by auto
+
+lemma ntimes_ineq1:
+ shows "(rsize (RNTIMES r n)) \<ge> 1 + rsize r"
+ by simp
+
+lemma ntimes_ineq2:
+ shows "1 + rsize r \<le>
+max ((Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * (Suc (N + rsize (RNTIMES r n))))) (rsize (RNTIMES r n))"
+ by (meson le_max_iff_disj ntimes_ineq1)
+
+lemma ntimes_closed_form_bounded:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
+ shows "rsize (rders_simp (RNTIMES r (Suc n)) s) \<le>
+ max ((Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * (Suc (N + rsize (RNTIMES r n))))) (rsize (RNTIMES r n))"
+proof(cases s)
+ case Nil
+ then show "rsize (rders_simp (RNTIMES r (Suc n)) s)
+ \<le> max (Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * Suc (N + rsize (RNTIMES r n))) (rsize (RNTIMES r n))"
+ by simp
+next
+ case (Cons a list)
+
+ then have "rsize (rders_simp (RNTIMES r (Suc n)) s) =
+ rsize (rsimp (RALTS ((map (optermsimp r) (nupdates list r [Some ([a], n)])))))"
+ using ntimes_closed_form by fastforce
+ also have "... \<le> Suc (rsizes (rdistinct ((map (optermsimp r) (nupdates list r [Some ([a], n)]))) {}))"
+ using alts_simp_control by blast
+ also have "... \<le> Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * (Suc (N + rsize (RNTIMES r n)))"
+ using ntimes_control_bounded[OF assms]
+ by (metis add_mono le_add1 mult_Suc plus_1_eq_Suc)
+ also have "... \<le> max (Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * Suc (N + rsize (RNTIMES r n))) (rsize (RNTIMES r n))"
+ by simp
+ finally show ?thesis by simp
+qed
+
+
+lemma ntimes_closed_form_boundedA:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
+ shows "\<exists>N'. \<forall>s. rsize (rders_simp (RNTIMES r n) s) \<le> N'"
+ apply(case_tac n)
+ using assms ntimes_closed_form_bounded_nil apply blast
+ using assms ntimes_closed_form_bounded by blast
+
+
+lemma star_closed_form_nonempty_bounded:
+ assumes "\<forall>s. rsize (rders_simp r s) \<le> N" and "s \<noteq> []"
+ shows "rsize (rders_simp (RSTAR r) s) \<le>
+ ((Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r))))) "
+proof(cases s)
+ case Nil
+ then show ?thesis
+ using local.Nil by fastforce
+next
+ case (Cons a list)
+ then have "rsize (rders_simp (RSTAR r) s) =
+ rsize (rsimp (RALTS ((map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates list r [[a]])))))"
+ using star_closed_form by fastforce
+ also have "... \<le> Suc (rsizes (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates list r [[a]])) {}))"
+ using alts_simp_control by blast
+ also have "... \<le> Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r)))"
+ by (smt (z3) add_mono_thms_linordered_semiring(1) assms(1) le_add1 map_eq_conv mult_Suc plus_1_eq_Suc star_control_bounded)
+ also have "... \<le> max (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * Suc (N + rsize (RSTAR r))) (rsize (RSTAR r))"
+ by simp
+ finally show ?thesis by simp
+qed
+
+
+
lemma seq_estimate_bounded:
assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1"
and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
@@ -425,7 +664,6 @@
by auto
qed
-
lemma rders_simp_bounded:
shows "\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
apply(induct r)
@@ -440,9 +678,12 @@
apply(assumption)
apply(assumption)
apply (metis alts_closed_form_bounded size_list_estimation')
- using star_closed_form_bounded by blast
+ using star_closed_form_bounded apply blast
+ using ntimes_closed_form_boundedA by blast
+
+
+unused_thms
+export_code rders_simp rsimp rder in Scala module_name Example
-unused_thms
-
end
--- a/thys3/src/FBound.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/FBound.thy Wed Jul 13 08:35:45 2022 +0100
@@ -18,6 +18,7 @@
| "rerase (AALTs bs rs) = RALTS (map rerase rs)"
| "rerase (ASEQ _ r1 r2) = RSEQ (rerase r1) (rerase r2)"
| "rerase (ASTAR _ r) = RSTAR (rerase r)"
+| "rerase (ANTIMES _ r n) = RNTIMES (rerase r) n"
lemma eq1_rerase:
shows "x ~1 y \<longleftrightarrow> (rerase x) = (rerase y)"
--- a/thys3/src/GeneralRegexBound.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/GeneralRegexBound.thy Wed Jul 13 08:35:45 2022 +0100
@@ -18,6 +18,10 @@
definition RALTs_set where
"RALTs_set A n \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A \<and> rsizes rs \<le> n}"
+definition RNTIMES_set where
+ "RNTIMES_set A n \<equiv> {RNTIMES r m | m r. r \<in> A \<and> rsize r + m \<le> n}"
+
+
definition
"sizeNregex N \<equiv> {r. rsize r \<le> N}"
@@ -26,7 +30,8 @@
"sizeNregex (Suc n) = (({RZERO, RONE} \<union> {RCHAR c| c. True})
\<union> (RSTAR ` sizeNregex n)
\<union> (RSEQ_set (sizeNregex n) n)
- \<union> (RALTs_set (sizeNregex n) n))"
+ \<union> (RALTs_set (sizeNregex n) n))
+ \<union> (RNTIMES_set (sizeNregex n) n)"
apply(auto)
apply(case_tac x)
apply(auto simp add: RSEQ_set_def)
@@ -37,15 +42,21 @@
apply (simp add: RALTs_set_def)
apply (metis imageI list.set_map member_le_sum_list order_trans)
apply (simp add: sizeNregex_def)
- apply (simp add: sizeNregex_def)
+ apply (simp add: sizeNregex_def)
+ apply (simp add: RNTIMES_set_def)
apply (simp add: sizeNregex_def)
using sizeNregex_def apply force
apply (simp add: sizeNregex_def)
apply (simp add: sizeNregex_def)
- apply (simp add: RALTs_set_def)
+ apply (simp add: sizeNregex_def)
+ apply (simp add: RALTs_set_def)
apply(simp add: sizeNregex_def)
apply(auto)
- using ex_in_conv by fastforce
+ using ex_in_conv apply fastforce
+ apply (simp add: RNTIMES_set_def)
+ apply(simp add: sizeNregex_def)
+ by force
+
lemma s4:
"RSEQ_set A n \<subseteq> RSEQ_set_cartesian A"
@@ -155,6 +166,22 @@
apply simp
by (simp add: full_SetCompr_eq)
+thm RNTIMES_set_def
+
+lemma s9_aux0:
+ shows "RNTIMES_set (insert r A) n \<subseteq> RNTIMES_set A n \<union> (\<Union> i \<in> {..n}. {RNTIMES r i})"
+apply(auto simp add: RNTIMES_set_def)
+ done
+
+lemma s9_aux:
+ assumes "finite A"
+ shows "finite (RNTIMES_set A n)"
+ using assms
+ apply(induct A arbitrary: n)
+ apply(auto simp add: RNTIMES_set_def)[1]
+ apply(subgoal_tac "finite (RNTIMES_set F n \<union> (\<Union> i \<in> {..n}. {RNTIMES x i}))")
+ apply (metis finite_subset s9_aux0)
+ by blast
lemma finite_size_n:
shows "finite (sizeNregex n)"
@@ -175,8 +202,8 @@
apply(rule finite_subset)
apply(rule t2)
apply(rule s8_aux)
- apply(simp)
- done
+ apply(simp)
+ by (simp add: s9_aux)
lemma three_easy_cases0:
shows "rsize (rders_simp RZERO s) \<le> Suc 0"
--- a/thys3/src/Lexer.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/Lexer.thy Wed Jul 13 08:35:45 2022 +0100
@@ -3,7 +3,7 @@
imports PosixSpec
begin
-section {* The Lexer Functions by Sulzmann and Lu (without simplification) *}
+section \<open>The Lexer Functions by Sulzmann and Lu (without simplification)\<close>
fun
mkeps :: "rexp \<Rightarrow> val"
@@ -12,6 +12,7 @@
| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)"
| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
| "mkeps(STAR r) = Stars []"
+| "mkeps(NTIMES r n) = Stars (replicate n (mkeps r))"
fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val"
where
@@ -22,6 +23,7 @@
| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
| "injval (STAR r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
+| "injval (NTIMES r n) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)"
fun
lexer :: "rexp \<Rightarrow> string \<Rightarrow> val option"
@@ -33,20 +35,32 @@
-section {* Mkeps, Injval Properties *}
+section \<open>Mkeps, Injval Properties\<close>
+
+lemma mkeps_flat:
+ assumes "nullable(r)"
+ shows "flat (mkeps r) = []"
+using assms
+ by (induct rule: mkeps.induct) (auto)
+
+lemma Prf_NTimes_empty:
+ assumes "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v = []"
+ and "length vs = n"
+ shows "\<Turnstile> Stars vs : NTIMES r n"
+ using assms
+ by (metis Prf.intros(7) empty_iff eq_Nil_appendI list.set(1))
+
lemma mkeps_nullable:
assumes "nullable(r)"
shows "\<Turnstile> mkeps r : r"
using assms
-by (induct rule: nullable.induct)
- (auto intro: Prf.intros)
-
-lemma mkeps_flat:
- assumes "nullable(r)"
- shows "flat (mkeps r) = []"
-using assms
-by (induct rule: nullable.induct) (auto)
+ apply (induct rule: mkeps.induct)
+ apply(auto intro: Prf.intros split: if_splits)
+ apply (metis Prf.intros(7) append_is_Nil_conv empty_iff list.set(1) list.size(3))
+ apply(rule Prf_NTimes_empty)
+ apply(auto simp add: mkeps_flat)
+ done
lemma Prf_injval_flat:
assumes "\<Turnstile> v : der c r"
@@ -62,14 +76,20 @@
using assms
apply(induct r arbitrary: c v rule: rexp.induct)
apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)
+(* Star *)
apply(simp add: Prf_injval_flat)
-done
+(* NTimes *)
+ apply(case_tac x2)
+ apply(simp)
+ apply(simp)
+ apply(subst append.simps(2)[symmetric])
+ apply(rule Prf.intros)
+ apply(auto simp add: Prf_injval_flat)
+ done
+text \<open>Mkeps and injval produce, or preserve, Posix values.\<close>
-text {*
- Mkeps and injval produce, or preserve, Posix values.
-*}
lemma Posix_mkeps:
assumes "nullable r"
@@ -80,7 +100,7 @@
apply(subst append.simps(1)[symmetric])
apply(rule Posix.intros)
apply(auto)
-done
+by (simp add: Posix_NTIMES2 pow_empty_iff)
lemma Posix_injval:
assumes "s \<in> (der c r) \<rightarrow> v"
@@ -228,11 +248,60 @@
ultimately
have "((c # s1) @ s2) \<in> STAR r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros)
then show "(c # s) \<in> STAR r \<rightarrow> injval (STAR r) c v" using cons by(simp)
- qed
+ qed
+next
+ case (NTIMES r n)
+ have IH: "\<And>s v. s \<in> der c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
+ have "s \<in> der c (NTIMES r n) \<rightarrow> v" by fact
+ then consider
+ (cons) v1 vs s1 s2 where
+ "v = Seq v1 (Stars vs)" "s = s1 @ s2"
+ "s1 \<in> der c r \<rightarrow> v1" "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> (Stars vs)" "0 < n"
+ "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))"
+
+ apply(auto elim: Posix_elims simp add: der_correctness Der_def intro: Posix.intros split: if_splits)
+ apply(erule Posix_elims)
+ apply(simp)
+ apply(subgoal_tac "\<exists>vss. v2 = Stars vss")
+ apply(clarify)
+ apply(drule_tac x="vss" in meta_spec)
+ apply(drule_tac x="s1" in meta_spec)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp add: der_correctness Der_def)
+ apply(erule Posix_elims)
+ apply(auto)
+ done
+ then show "(c # s) \<in> (NTIMES r n) \<rightarrow> injval (NTIMES r n) c v"
+ proof (cases)
+ case cons
+ have "s1 \<in> der c r \<rightarrow> v1" by fact
+ then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
+ moreover
+ have "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> Stars vs" by fact
+ moreover
+ have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact
+ then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
+ then have "flat (injval r c v1) \<noteq> []" by simp
+ moreover
+ have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L (der c r) \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))" by fact
+ then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))"
+ by (simp add: der_correctness Der_def)
+ ultimately
+ have "((c # s1) @ s2) \<in> NTIMES r n \<rightarrow> Stars (injval r c v1 # vs)"
+ apply (rule_tac Posix.intros)
+ apply(simp_all)
+ apply(case_tac n)
+ apply(simp)
+ using Posix_elims(1) NTIMES.prems apply auto[1]
+ apply(simp)
+ done
+ then show "(c # s) \<in> NTIMES r n \<rightarrow> injval (NTIMES r n) c v" using cons by(simp)
+ qed
+
qed
-section {* Lexer Correctness *}
+section \<open>Lexer Correctness\<close>
lemma lexer_correct_None:
@@ -354,7 +423,8 @@
apply(erule Prf_elims)
apply(auto)
apply (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5))
- by (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5))
+ apply (smt Prf_elims(6) injval.simps(7) list.inject val.inject(5))
+ by (smt (verit, best) Prf_elims(1) Prf_elims(2) Prf_elims(7) injval.simps(8) list.inject val.simps(5))
@@ -375,7 +445,7 @@
apply (simp add: lexer_correctness(1))
apply(subgoal_tac "\<Turnstile> a : (der c r)")
prefer 2
- using Posix_Prf apply blast
+ using Posix1a apply blast
using injval_inj by blast
--- a/thys3/src/PosixSpec.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/PosixSpec.thy Wed Jul 13 08:35:45 2022 +0100
@@ -46,6 +46,9 @@
| "\<Turnstile> Void : ONE"
| "\<Turnstile> Char c : CH c"
| "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> flat v \<noteq> [] \<Longrightarrow> \<Turnstile> Stars vs : STAR r"
+| "\<lbrakk>\<forall>v \<in> set vs1. \<Turnstile> v : r \<and> flat v \<noteq> [];
+ \<forall>v \<in> set vs2. \<Turnstile> v : r \<and> flat v = [];
+ length (vs1 @ vs2) = n\<rbrakk> \<Longrightarrow> \<Turnstile> Stars (vs1 @ vs2) : NTIMES r n"
inductive_cases Prf_elims:
"\<Turnstile> v : ZERO"
@@ -54,6 +57,7 @@
"\<Turnstile> v : ONE"
"\<Turnstile> v : CH c"
"\<Turnstile> vs : STAR r"
+ "\<Turnstile> vs : NTIMES r n"
lemma Prf_Stars_appendE:
assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
@@ -61,6 +65,28 @@
using assms
by (auto intro: Prf.intros elim!: Prf_elims)
+lemma Pow_cstring:
+ fixes A::"string set"
+ assumes "s \<in> A ^^ n"
+ shows "\<exists>ss1 ss2. concat (ss1 @ ss2) = s \<and> length (ss1 @ ss2) = n \<and>
+ (\<forall>s \<in> set ss1. s \<in> A \<and> s \<noteq> []) \<and> (\<forall>s \<in> set ss2. s \<in> A \<and> s = [])"
+using assms
+apply(induct n arbitrary: s)
+ apply(auto)[1]
+ apply(auto simp add: Sequ_def)
+ apply(drule_tac x="s2" in meta_spec)
+ apply(simp)
+apply(erule exE)+
+ apply(clarify)
+apply(case_tac "s1 = []")
+apply(simp)
+apply(rule_tac x="ss1" in exI)
+apply(rule_tac x="s1 # ss2" in exI)
+apply(simp)
+apply(rule_tac x="s1 # ss1" in exI)
+apply(rule_tac x="ss2" in exI)
+ apply(simp)
+ done
lemma flats_Prf_value:
assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"
@@ -75,15 +101,48 @@
apply(simp)
apply(rule_tac x="v#vs" in exI)
apply(simp)
-done
+ done
+
+lemma Aux:
+ assumes "\<forall>s\<in>set ss. s = []"
+ shows "concat ss = []"
+using assms
+by (induct ss) (auto)
+lemma flats_cval:
+ assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<Turnstile> v : r"
+ shows "\<exists>vs1 vs2. flats (vs1 @ vs2) = concat ss \<and> length (vs1 @ vs2) = length ss \<and>
+ (\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []) \<and>
+ (\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = [])"
+using assms
+apply(induct ss rule: rev_induct)
+apply(rule_tac x="[]" in exI)+
+apply(simp)
+apply(simp)
+apply(clarify)
+apply(case_tac "flat v = []")
+apply(rule_tac x="vs1" in exI)
+apply(rule_tac x="v#vs2" in exI)
+apply(simp)
+apply(rule_tac x="vs1 @ [v]" in exI)
+apply(rule_tac x="vs2" in exI)
+apply(simp)
+by (simp add: Aux)
+
+lemma pow_Prf:
+ assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> flat v \<in> A"
+ shows "flats vs \<in> A ^^ (length vs)"
+ using assms
+ by (induct vs) (auto)
lemma L_flat_Prf1:
assumes "\<Turnstile> v : r"
shows "flat v \<in> L r"
-using assms
-by (induct) (auto simp add: Sequ_def Star_concat)
-
+ using assms
+ apply (induct v r rule: Prf.induct)
+ apply(auto simp add: Sequ_def Star_concat lang_pow_add)
+ by (metis pow_Prf)
+
lemma L_flat_Prf2:
assumes "s \<in> L r"
shows "\<exists>v. \<Turnstile> v : r \<and> flat v = s"
@@ -105,7 +164,30 @@
next
case (ALT r1 r2 s)
then show "\<exists>v. \<Turnstile> v : ALT r1 r2 \<and> flat v = s"
- unfolding L.simps by (fastforce intro: Prf.intros)
+ unfolding L.simps by (fastforce intro: Prf.intros)
+next
+ case (NTIMES r n)
+ have IH: "\<And>s. s \<in> L r \<Longrightarrow> \<exists>v. \<Turnstile> v : r \<and> flat v = s" by fact
+ have "s \<in> L (NTIMES r n)" by fact
+ then obtain ss1 ss2 where "concat (ss1 @ ss2) = s" "length (ss1 @ ss2) = n"
+ "\<forall>s \<in> set ss1. s \<in> L r \<and> s \<noteq> []" "\<forall>s \<in> set ss2. s \<in> L r \<and> s = []"
+ using Pow_cstring by force
+ then obtain vs1 vs2 where "flats (vs1 @ vs2) = s" "length (vs1 @ vs2) = n"
+ "\<forall>v\<in>set vs1. \<Turnstile> v : r \<and> flat v \<noteq> []" "\<forall>v\<in>set vs2. \<Turnstile> v : r \<and> flat v = []"
+ using IH flats_cval
+ apply -
+ apply(drule_tac x="ss1 @ ss2" in meta_spec)
+ apply(drule_tac x="r" in meta_spec)
+ apply(drule meta_mp)
+ apply(simp)
+ apply (metis Un_iff)
+ apply(clarify)
+ apply(drule_tac x="vs1" in meta_spec)
+ apply(drule_tac x="vs2" in meta_spec)
+ apply(simp)
+ done
+ then show "\<exists>v. \<Turnstile> v : NTIMES r n \<and> flat v = s"
+ using Prf.intros(7) flat_Stars by blast
qed (auto intro: Prf.intros)
@@ -130,9 +212,11 @@
and "LV ONE s = (if s = [] then {Void} else {})"
and "LV (CH c) s = (if s = [c] then {Char c} else {})"
and "LV (ALT r1 r2) s = Left ` LV r1 s \<union> Right ` LV r2 s"
+ and "LV (NTIMES r 0) s = (if s = [] then {Stars []} else {})"
unfolding LV_def
-by (auto intro: Prf.intros elim: Prf.cases)
-
+ apply (auto intro: Prf.intros elim: Prf.cases)
+ by (metis Prf.intros(7) append.right_neutral empty_iff list.set(1) list.size(3))
+
abbreviation
"Prefixes s \<equiv> {s'. prefix s' s}"
@@ -174,6 +258,64 @@
ultimately show "finite (Prefixes s)" by simp
qed
+definition
+ "Stars_Append Vs1 Vs2 \<equiv> {Stars (vs1 @ vs2) | vs1 vs2. Stars vs1 \<in> Vs1 \<and> Stars vs2 \<in> Vs2}"
+
+lemma finite_Stars_Append:
+ assumes "finite Vs1" "finite Vs2"
+ shows "finite (Stars_Append Vs1 Vs2)"
+ using assms
+proof -
+ define UVs1 where "UVs1 \<equiv> Stars -` Vs1"
+ define UVs2 where "UVs2 \<equiv> Stars -` Vs2"
+ from assms have "finite UVs1" "finite UVs2"
+ unfolding UVs1_def UVs2_def
+ by(simp_all add: finite_vimageI inj_on_def)
+ then have "finite ((\<lambda>(vs1, vs2). Stars (vs1 @ vs2)) ` (UVs1 \<times> UVs2))"
+ by simp
+ moreover
+ have "Stars_Append Vs1 Vs2 = (\<lambda>(vs1, vs2). Stars (vs1 @ vs2)) ` (UVs1 \<times> UVs2)"
+ unfolding Stars_Append_def UVs1_def UVs2_def by auto
+ ultimately show "finite (Stars_Append Vs1 Vs2)"
+ by simp
+qed
+
+lemma LV_NTIMES_subset:
+ "LV (NTIMES r n) s \<subseteq> Stars_Append (LV (STAR r) s) (\<Union>i\<le>n. LV (NTIMES r i) [])"
+apply(auto simp add: LV_def)
+apply(auto elim!: Prf_elims)
+ apply(auto simp add: Stars_Append_def)
+ apply(rule_tac x="vs1" in exI)
+ apply(rule_tac x="vs2" in exI)
+ apply(auto)
+ using Prf.intros(6) apply(auto)
+ apply(rule_tac x="length vs2" in bexI)
+ thm Prf.intros
+ apply(subst append.simps(1)[symmetric])
+ apply(rule Prf.intros)
+ apply(auto)[1]
+ apply(auto)[1]
+ apply(simp)
+ apply(simp)
+ done
+
+lemma LV_NTIMES_Suc_empty:
+ shows "LV (NTIMES r (Suc n)) [] =
+ (\<lambda>(v, vs). Stars (v#vs)) ` (LV r [] \<times> (Stars -` (LV (NTIMES r n) [])))"
+unfolding LV_def
+apply(auto elim!: Prf_elims simp add: image_def)
+apply(case_tac vs1)
+apply(auto)
+apply(case_tac vs2)
+apply(auto)
+apply(subst append.simps(1)[symmetric])
+apply(rule Prf.intros)
+apply(auto)
+apply(subst append.simps(1)[symmetric])
+apply(rule Prf.intros)
+apply(auto)
+ done
+
lemma LV_STAR_finite:
assumes "\<forall>s. finite (LV r s)"
shows "finite (LV (STAR r) s)"
@@ -215,7 +357,22 @@
ultimately
show "finite (LV (STAR r) s)" by (simp add: finite_subset)
qed
-
+
+lemma finite_NTimes_empty:
+ assumes "\<And>s. finite (LV r s)"
+ shows "finite (LV (NTIMES r n) [])"
+ using assms
+ apply(induct n)
+ apply(auto simp add: LV_simps)
+ apply(subst LV_NTIMES_Suc_empty)
+ apply(rule finite_imageI)
+ apply(rule finite_cartesian_product)
+ using assms apply simp
+ apply(rule finite_vimageI)
+ apply(simp)
+ apply(simp add: inj_on_def)
+ done
+
lemma LV_finite:
shows "finite (LV r s)"
@@ -251,6 +408,15 @@
next
case (STAR r s)
then show "finite (LV (STAR r) s)" by (simp add: LV_STAR_finite)
+next
+ case (NTIMES r n s)
+ have "\<And>s. finite (LV r s)" by fact
+ then have "finite (Stars_Append (LV (STAR r) s) (\<Union>i\<le>n. LV (NTIMES r i) []))"
+ apply(rule_tac finite_Stars_Append)
+ apply (simp add: LV_STAR_finite)
+ using finite_NTimes_empty by blast
+ then show "finite (LV (NTIMES r n) s)"
+ by (metis LV_NTIMES_subset finite_subset)
qed
@@ -271,6 +437,11 @@
\<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))\<rbrakk>
\<Longrightarrow> (s1 @ s2) \<in> STAR r \<rightarrow> Stars (v # vs)"
| Posix_STAR2: "[] \<in> STAR r \<rightarrow> Stars []"
+| Posix_NTIMES1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> NTIMES r (n - 1) \<rightarrow> Stars vs; flat v \<noteq> []; 0 < n;
+ \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1)))\<rbrakk>
+ \<Longrightarrow> (s1 @ s2) \<in> NTIMES r n \<rightarrow> Stars (v # vs)"
+| Posix_NTIMES2: "\<lbrakk>\<forall>v \<in> set vs. [] \<in> r \<rightarrow> v; length vs = n\<rbrakk>
+ \<Longrightarrow> [] \<in> NTIMES r n \<rightarrow> Stars vs"
inductive_cases Posix_elims:
"s \<in> ZERO \<rightarrow> v"
@@ -279,19 +450,47 @@
"s \<in> ALT r1 r2 \<rightarrow> v"
"s \<in> SEQ r1 r2 \<rightarrow> v"
"s \<in> STAR r \<rightarrow> v"
+ "s \<in> NTIMES r n \<rightarrow> v"
lemma Posix1:
assumes "s \<in> r \<rightarrow> v"
shows "s \<in> L r" "flat v = s"
using assms
- by(induct s r v rule: Posix.induct)
- (auto simp add: Sequ_def)
+ apply(induct s r v rule: Posix.induct)
+ apply(auto simp add: pow_empty_iff)
+ apply (metis Suc_pred concI lang_pow.simps(2))
+ by (meson ex_in_conv set_empty)
+
+lemma Posix1a:
+ assumes "s \<in> r \<rightarrow> v"
+ shows "\<Turnstile> v : r"
+using assms
+ apply(induct s r v rule: Posix.induct)
+ apply(auto intro: Prf.intros)
+ apply (metis Prf.intros(6) Prf_elims(6) set_ConsD val.inject(5))
+ prefer 2
+ apply (metis Posix1(2) Prf.intros(7) append_Nil empty_iff list.set(1))
+ apply(erule Prf_elims)
+ apply(auto)
+ apply(subst append.simps(2)[symmetric])
+ apply(rule Prf.intros)
+ apply(auto)
+ done
text \<open>
For a give value and string, our Posix definition
determines a unique value.
\<close>
+lemma List_eq_zipI:
+ assumes "list_all2 (\<lambda>v1 v2. v1 = v2) vs1 vs2"
+ and "length vs1 = length vs2"
+ shows "vs1 = vs2"
+ using assms
+ apply(induct vs1 vs2 rule: list_all2_induct)
+ apply(auto)
+ done
+
lemma Posix_determ:
assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
shows "v1 = v2"
@@ -356,6 +555,29 @@
case (Posix_STAR2 r v2)
have "[] \<in> STAR r \<rightarrow> v2" by fact
then show "Stars [] = v2" by cases (auto simp add: Posix1)
+next
+ case (Posix_NTIMES2 vs r n v2)
+ then show "Stars vs = v2"
+ apply(erule_tac Posix_elims)
+ apply(auto)
+ apply (simp add: Posix1(2))
+ apply(rule List_eq_zipI)
+ apply(auto simp add: list_all2_iff)
+ by (meson in_set_zipE)
+next
+ case (Posix_NTIMES1 s1 r v s2 n vs)
+ have "(s1 @ s2) \<in> NTIMES r n \<rightarrow> v2"
+ "s1 \<in> r \<rightarrow> v" "s2 \<in> NTIMES r (n - 1) \<rightarrow> Stars vs" "flat v \<noteq> []"
+ "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (NTIMES r (n - 1 )))" by fact+
+ then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (NTIMES r (n - 1)) \<rightarrow> (Stars vs')"
+ apply(cases) apply (auto simp add: append_eq_append_conv2)
+ using Posix1(1) apply fastforce
+ apply (metis One_nat_def Posix1(1) Posix_NTIMES1.hyps(7) append.right_neutral append_self_conv2)
+ using Posix1(2) by blast
+ moreover
+ have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
+ "\<And>v2. s2 \<in> NTIMES r (n - 1) \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
+ ultimately show "Stars (v # vs) = v2" by auto
qed
@@ -368,13 +590,9 @@
shows "v \<in> LV r s"
using assms unfolding LV_def
apply(induct rule: Posix.induct)
- apply(auto simp add: intro!: Prf.intros elim!: Prf_elims)
- done
+ apply(auto simp add: intro!: Prf.intros elim!: Prf_elims Posix1a)
+ apply (smt (verit, best) One_nat_def Posix1a Posix_NTIMES1 L.simps(7))
+ using Posix1a Posix_NTIMES2 by blast
-lemma Posix_Prf:
- assumes "s \<in> r \<rightarrow> v"
- shows "\<Turnstile> v : r"
- using assms Posix_LV LV_def
- by simp
end
--- a/thys3/src/RegLangs.thy Wed Jul 13 08:35:09 2022 +0100
+++ b/thys3/src/RegLangs.thy Wed Jul 13 08:35:45 2022 +0100
@@ -21,6 +21,42 @@
and "{} ;; A = {}"
by (simp_all add: Sequ_def)
+lemma concI[simp,intro]: "u : A \<Longrightarrow> v : B \<Longrightarrow> u@v : A ;; B"
+by (auto simp add: Sequ_def)
+
+lemma concE[elim]:
+assumes "w \<in> A ;; B"
+obtains u v where "u \<in> A" "v \<in> B" "w = u@v"
+using assms by (auto simp: Sequ_def)
+
+lemma concI_if_Nil2: "[] \<in> B \<Longrightarrow> xs : A \<Longrightarrow> xs \<in> A ;; B"
+by (metis append_Nil2 concI)
+
+lemma conc_assoc: "(A ;; B) ;; C = A ;; (B ;; C)"
+by (auto elim!: concE) (simp only: append_assoc[symmetric] concI)
+
+
+text \<open>Language power operations\<close>
+
+overloading lang_pow == "compow :: nat \<Rightarrow> string set \<Rightarrow> string set"
+begin
+ primrec lang_pow :: "nat \<Rightarrow> string set \<Rightarrow> string set" where
+ "lang_pow 0 A = {[]}" |
+ "lang_pow (Suc n) A = A ;; (lang_pow n A)"
+end
+
+
+lemma conc_pow_comm:
+ shows "A ;; (A ^^ n) = (A ^^ n) ;; A"
+by (induct n) (simp_all add: conc_assoc[symmetric])
+
+lemma lang_pow_add: "A ^^ (n + m) = (A ^^ n) ;; (A ^^ m)"
+ by (induct n) (auto simp: conc_assoc)
+
+lemma lang_empty:
+ fixes A::"string set"
+ shows "A ^^ 0 = {[]}"
+ by simp
section \<open>Semantic Derivative (Left Quotient) of Languages\<close>
@@ -88,6 +124,11 @@
unfolding Der_def Sequ_def
by(auto simp add: Star_decomp)
+lemma Der_inter[simp]: "Der a (A \<inter> B) = Der a A \<inter> Der a B"
+ and Der_compl[simp]: "Der a (-A) = - Der a A"
+ and Der_Union[simp]: "Der a (Union M) = Union(Der a ` M)"
+ and Der_UN[simp]: "Der a (UN x:I. S x) = (UN x:I. Der a (S x))"
+by (auto simp: Der_def)
lemma Der_star[simp]:
shows "Der c (A\<star>) = (Der c A) ;; A\<star>"
@@ -103,6 +144,13 @@
finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .
qed
+lemma Der_pow[simp]:
+ shows "Der c (A ^^ n) = (if n = 0 then {} else (Der c A) ;; (A ^^ (n - 1)))"
+ apply(induct n arbitrary: A)
+ apply(auto simp add: Cons_eq_append_conv)
+ by (metis Suc_pred concI_if_Nil2 conc_assoc conc_pow_comm lang_pow.simps(2))
+
+
lemma Star_concat:
assumes "\<forall>s \<in> set ss. s \<in> A"
shows "concat ss \<in> A\<star>"
@@ -119,6 +167,7 @@
+
section \<open>Regular Expressions\<close>
datatype rexp =
@@ -128,6 +177,7 @@
| SEQ rexp rexp
| ALT rexp rexp
| STAR rexp
+| NTIMES rexp nat
section \<open>Semantics of Regular Expressions\<close>
@@ -140,7 +190,7 @@
| "L (SEQ r1 r2) = (L r1) ;; (L r2)"
| "L (ALT r1 r2) = (L r1) \<union> (L r2)"
| "L (STAR r) = (L r)\<star>"
-
+| "L (NTIMES r n) = (L r) ^^ n"
section \<open>Nullable, Derivatives\<close>
@@ -153,7 +203,7 @@
| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
| "nullable (STAR r) = True"
-
+| "nullable (NTIMES r n) = (if n = 0 then True else nullable r)"
fun
der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
@@ -167,6 +217,8 @@
then ALT (SEQ (der c r1) r2) (der c r2)
else SEQ (der c r1) r2)"
| "der c (STAR r) = SEQ (der c r) (STAR r)"
+| "der c (NTIMES r n) = (if n = 0 then ZERO else SEQ (der c r) (NTIMES r (n - 1)))"
+
fun
ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
@@ -175,13 +227,23 @@
| "ders (c # s) r = ders s (der c r)"
+lemma pow_empty_iff:
+ shows "[] \<in> (L r) ^^ n \<longleftrightarrow> (if n = 0 then True else [] \<in> (L r))"
+ by (induct n) (auto simp add: Sequ_def)
+
lemma nullable_correctness:
shows "nullable r \<longleftrightarrow> [] \<in> (L r)"
-by (induct r) (auto simp add: Sequ_def)
+ by (induct r) (auto simp add: Sequ_def pow_empty_iff)
lemma der_correctness:
shows "L (der c r) = Der c (L r)"
-by (induct r) (simp_all add: nullable_correctness)
+ apply (induct r)
+ apply(auto simp add: nullable_correctness Sequ_def)
+ using Der_def apply force
+ using Der_def apply auto[1]
+ apply (smt (verit, ccfv_SIG) Der_def append_eq_Cons_conv mem_Collect_eq)
+ using Der_def apply force
+ using Der_Sequ Sequ_def by auto
lemma ders_correctness:
shows "L (ders s r) = Ders s (L r)"
@@ -197,40 +259,4 @@
by (simp add: ders_append)
-(*
-datatype ctxt =
- SeqC rexp bool
- | AltCL rexp
- | AltCH rexp
- | StarC rexp
-
-function
- down :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"
-and up :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"
-where
- "down c (SEQ r1 r2) ctxts =
- (if (nullable r1) then down c r1 (SeqC r2 True # ctxts)
- else down c r1 (SeqC r2 False # ctxts))"
-| "down c (CH d) ctxts =
- (if c = d then up c ONE ctxts else up c ZERO ctxts)"
-| "down c ONE ctxts = up c ZERO ctxts"
-| "down c ZERO ctxts = up c ZERO ctxts"
-| "down c (ALT r1 r2) ctxts = down c r1 (AltCH r2 # ctxts)"
-| "down c (STAR r1) ctxts = down c r1 (StarC r1 # ctxts)"
-| "up c r [] = (r, [])"
-| "up c r (SeqC r2 False # ctxts) = up c (SEQ r r2) ctxts"
-| "up c r (SeqC r2 True # ctxts) = down c r2 (AltCL (SEQ r r2) # ctxts)"
-| "up c r (AltCL r1 # ctxts) = up c (ALT r1 r) ctxts"
-| "up c r (AltCH r2 # ctxts) = down c r2 (AltCL r # ctxts)"
-| "up c r (StarC r1 # ctxts) = up c (SEQ r (STAR r1)) ctxts"
- apply(pat_completeness)
- apply(auto)
- done
-
-termination
- sorry
-
-*)
-
-
end
\ No newline at end of file