
diff -r c6351a96bf80 -r a7e98deebb5c thys2/BasicIdentities.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/BasicIdentities.thy	Wed Mar 09 17:33:08 2022 +0000
@@ -0,0 +1,441 @@
+theory BasicIdentities imports 
+"Lexer"  "PDerivs"
+begin
+
+datatype rrexp = 
+  RZERO
+| RONE 
+| RCHAR char
+| RSEQ rrexp rrexp
+| RALTS "rrexp list"
+| RSTAR rrexp
+
+abbreviation
+  "RALT r1 r2 \<equiv> RALTS [r1, r2]"
+
+
+
+fun
+ rnullable :: "rrexp \<Rightarrow> bool"
+where
+  "rnullable (RZERO) = False"
+| "rnullable (RONE  ) = True"
+| "rnullable (RCHAR   c) = False"
+| "rnullable (RALTS   rs) = (\<exists>r \<in> set rs. rnullable r)"
+| "rnullable (RSEQ  r1 r2) = (rnullable r1 \<and> rnullable r2)"
+| "rnullable (RSTAR   r) = True"
+
+
+fun
+ rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
+where
+  "rder c (RZERO) = RZERO"
+| "rder c (RONE) = RZERO"
+| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
+| "rder c (RALTS rs) = RALTS (map (rder c) rs)"
+| "rder c (RSEQ r1 r2) = 
+     (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)"
+
+
+fun 
+  rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
+where
+  "rders r [] = r"
+| "rders r (c#s) = rders (rder c r) s"
+
+fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
+  where
+  "rdistinct [] acc = []"
+| "rdistinct (x#xs)  acc = 
+     (if x \<in> acc then rdistinct xs  acc 
+      else x # (rdistinct xs  ({x} \<union> acc)))"
+
+
+
+
+
+fun rflts :: "rrexp list \<Rightarrow> rrexp list"
+  where 
+  "rflts [] = []"
+| "rflts (RZERO # rs) = rflts rs"
+| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
+| "rflts (r1 # rs) = r1 # rflts rs"
+
+
+fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
+  where
+  "rsimp_ALTs  [] = RZERO"
+| "rsimp_ALTs [r] =  r"
+| "rsimp_ALTs rs = RALTS rs"
+
+fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
+  where
+  "rsimp_SEQ  RZERO _ = RZERO"
+| "rsimp_SEQ  _ RZERO = RZERO"
+| "rsimp_SEQ RONE r2 = r2"
+| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
+
+
+fun rsimp :: "rrexp \<Rightarrow> rrexp" 
+  where
+  "rsimp (RSEQ r1 r2) = rsimp_SEQ  (rsimp r1) (rsimp r2)"
+| "rsimp (RALTS rs) = rsimp_ALTs  (rdistinct (rflts (map rsimp rs))  {}) "
+| "rsimp r = r"
+
+
+fun 
+  rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
+where
+  "rders_simp r [] = r"
+| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
+
+fun rsize :: "rrexp \<Rightarrow> nat" where
+  "rsize RZERO = 1"
+| "rsize (RONE) = 1" 
+| "rsize (RCHAR  c) = 1"
+| "rsize (RALTS  rs) = Suc (sum_list (map rsize rs))"
+| "rsize (RSEQ  r1 r2) = Suc (rsize r1 + rsize r2)"
+| "rsize (RSTAR  r) = Suc (rsize r)"
+
+
+lemma rder_rsimp_ALTs_commute:
+  shows "  (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"
+  apply(induct rs)
+   apply simp
+  apply(case_tac rs)
+   apply simp
+  apply auto
+  done
+
+
+lemma rsimp_aalts_smaller:
+  shows "rsize (rsimp_ALTs  rs) \<le> rsize (RALTS rs)"
+  apply(induct rs)
+   apply simp
+  apply simp
+  apply(case_tac "rs = []")
+   apply simp
+  apply(subgoal_tac "\<exists>rsp ap. rs = ap # rsp")
+   apply(erule exE)+
+   apply simp
+  apply simp
+  by(meson neq_Nil_conv)
+  
+
+
+
+
+lemma rSEQ_mono:
+  shows "rsize (rsimp_SEQ r1 r2) \<le>rsize ( RSEQ r1 r2)"
+  apply auto
+  apply(induct r1)
+       apply auto
+      apply(case_tac "r2")
+       apply simp_all
+     apply(case_tac r2)
+          apply simp_all
+     apply(case_tac r2)
+         apply simp_all
+     apply(case_tac r2)
+        apply simp_all
+     apply(case_tac r2)
+  apply simp_all
+  done
+
+lemma ralts_cap_mono:
+  shows "rsize (RALTS rs) \<le> Suc ( sum_list (map rsize rs)) "
+  by simp
+
+lemma rflts_def_idiot:
+  shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk>
+       \<Longrightarrow> rflts (a # rs) = a # rflts rs"
+  apply(case_tac a)
+       apply simp_all
+  done
+
+
+lemma rflts_mono:
+  shows "sum_list (map rsize (rflts rs))\<le> sum_list (map rsize rs)"
+  apply(induct rs)
+  apply simp
+  apply(case_tac "a = RZERO")
+   apply simp
+  apply(case_tac "\<exists>rs1. a = RALTS rs1")
+  apply(erule exE)
+   apply simp
+  apply(subgoal_tac "rflts (a # rs) = a # (rflts rs)")
+  prefer 2
+  using rflts_def_idiot apply blast
+  apply simp
+  done
+
+lemma rdistinct_smaller: shows "sum_list (map rsize (rdistinct rs ss)) \<le>
+sum_list (map rsize rs )"
+  apply (induct rs arbitrary: ss)
+   apply simp
+  by (simp add: trans_le_add2)
+
+lemma rdistinct_phi_smaller: "sum_list (map rsize (rdistinct rs {})) \<le> sum_list (map rsize rs)"
+  by (simp add: rdistinct_smaller)
+
+
+lemma rsimp_alts_mono :
+  shows "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> rsize (rsimp xa) \<le> rsize xa)  \<Longrightarrow>
+rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {})) \<le> Suc (sum_list (map rsize x))"
+  apply(subgoal_tac "rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {} )) 
+                    \<le> rsize (RALTS (rdistinct (rflts (map rsimp x)) {} ))")
+  prefer 2
+  using rsimp_aalts_smaller apply auto[1]
+  apply(subgoal_tac "rsize (RALTS (rdistinct (rflts (map rsimp x)) {})) \<le>Suc( sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})))")
+  prefer 2
+  using ralts_cap_mono apply blast
+  apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})) \<le>
+                     sum_list (map rsize ( (rflts (map rsimp x))))")
+  prefer 2
+  using rdistinct_smaller apply presburger
+  apply(subgoal_tac "sum_list (map rsize (rflts (map rsimp x))) \<le> 
+                     sum_list (map rsize (map rsimp x))")
+  prefer 2
+  using rflts_mono apply blast
+  apply(subgoal_tac "sum_list (map rsize (map rsimp x)) \<le> sum_list (map rsize x)")
+  prefer 2
+  
+  apply (simp add: sum_list_mono)
+  by linarith
+
+
+
+
+
+lemma rsimp_mono:
+  shows "rsize (rsimp r) \<le> rsize r"
+  apply(induct r)
+  apply simp_all
+  apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")
+    apply force
+  using rSEQ_mono
+   apply presburger
+  using rsimp_alts_mono by auto
+
+lemma idiot:
+  shows "rsimp_SEQ RONE r = r"
+  apply(case_tac r)
+       apply simp_all
+  done
+
+lemma no_alt_short_list_after_simp:
+  shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
+  sorry
+
+lemma no_further_dB_after_simp:
+  shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
+  sorry
+
+
+lemma idiot2:
+  shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
+    \<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
+  apply(case_tac r1)
+       apply(case_tac r2)
+  apply simp_all
+     apply(case_tac r2)
+  apply simp_all
+     apply(case_tac r2)
+  apply simp_all
+   apply(case_tac r2)
+  apply simp_all
+  apply(case_tac r2)
+       apply simp_all
+  done
+
+lemma rders__onechar:
+  shows " (rders_simp r [c]) =  (rsimp (rders r [c]))"
+  by simp
+
+lemma rders_append:
+  "rders c (s1 @ s2) = rders (rders c s1) s2"
+  apply(induct s1 arbitrary: c s2)
+  apply(simp_all)
+  done
+
+lemma rders_simp_append:
+  "rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
+  apply(induct s1 arbitrary: c s2)
+  apply(simp_all)
+  done
+
+lemma inside_simp_removal:
+  shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
+  sorry
+
+lemma set_related_list:
+  shows "distinct rs  \<Longrightarrow> length rs = card (set rs)"
+  by (simp add: distinct_card)
+(*this section deals with the property of distinctBy: creates a list without duplicates*)
+lemma rdistinct_never_added_twice:
+  shows "rdistinct (a # rs) {a} = rdistinct rs {a}"
+  by force
+
+
+lemma rdistinct_does_the_job:
+  shows "distinct (rdistinct rs s)"
+  apply(induct rs arbitrary: s)
+   apply simp
+  apply simp
+  sorry
+
+lemma rders_simp_same_simpders:
+  shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
+  apply(induct s rule: rev_induct)
+   apply simp
+  apply(case_tac "xs = []")
+   apply simp
+  apply(simp add: rders_append rders_simp_append)
+  using inside_simp_removal by blast
+
+lemma simp_helps_der_pierce:
+  shows " rsimp
+            (rder x
+              (rsimp_ALTs rs)) = 
+          rsimp 
+            (rsimp_ALTs 
+              (map (rder x )
+                rs
+              )
+            )"
+  sorry
+
+
+lemma rders_simp_one_char:
+  shows "rders_simp r [c] = rsimp (rder c r)"
+  apply auto
+  done
+
+lemma rsimp_idem:
+  shows "rsimp (rsimp r) = rsimp r"
+  sorry
+
+corollary rsimp_inner_idem1:
+  shows "rsimp r = RSEQ r1 r2 \<Longrightarrow> rsimp r1 = r1 \<and> rsimp r2 = r2"
+  
+  sorry
+
+corollary rsimp_inner_idem2:
+  shows "rsimp r = RALTS rs \<Longrightarrow> \<forall>r' \<in> (set rs). rsimp r' = r'"
+  sorry
+
+corollary rsimp_inner_idem3:
+  shows "rsimp r = RALTS rs \<Longrightarrow> map rsimp rs = rs"
+  by (meson map_idI rsimp_inner_idem2)
+
+corollary rsimp_inner_idem4:
+  shows "rsimp r = RALTS rs \<Longrightarrow> flts rs = rs"
+  sorry
+
+
+lemma head_one_more_simp:
+  shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
+  by (simp add: rsimp_idem)
+
+lemma head_one_more_dersimp:
+  shows "map rsimp ((rder x (rders_simp r s) # rs)) = map rsimp ((rders_simp r (s@[x]) ) # rs)"
+  using head_one_more_simp rders_simp_append rders_simp_one_char by presburger
+
+
+
+
+lemma ders_simp_nullability:
+  shows "rnullable (rders r s) = rnullable (rders_simp r s)"
+  sorry
+
+lemma  first_elem_seqder:
+  shows "\<not>rnullable r1p \<Longrightarrow> map rsimp (rder x (RSEQ r1p r2)
+                   # rs) = map rsimp ((RSEQ (rder x r1p) r2) # rs) "
+  by auto
+
+lemma first_elem_seqder1:
+  shows  "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) = 
+                                          map rsimp ( (RSEQ (rsimp (rder x (rders_simp r xs))) r2) # rs)"
+  by (simp add: rsimp_idem)
+
+lemma first_elem_seqdersimps:
+  shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) = 
+                                          map rsimp ( (RSEQ (rders_simp r (xs @ [x])) r2) # rs)"
+  using first_elem_seqder1 rders_simp_append by auto
+
+
+
+
+
+
+lemma seq_ders_closed_form1:
+  shows "\<exists>Ss. rders_simp (RSEQ r1 r2) [c] = rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # 
+(map ( rders_simp r2 ) Ss)))"
+  apply(case_tac "rnullable r1")
+   apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] = 
+rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [[c]])))")
+    prefer 2
+    apply (simp add: rsimp_idem)
+   apply(rule_tac x = "[[c]]" in exI)
+   apply simp
+  apply(subgoal_tac  " rders_simp (RSEQ r1 r2) [c] = 
+rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [])))")
+   apply blast
+  apply(simp add: rsimp_idem)
+  sorry
+
+
+
+
+
+
+
+
+lemma simp_flatten2:
+  shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
+  sorry
+
+
+lemma simp_flatten:
+  shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
+
+  sorry
+
+
+
+fun vsuf :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list" where
+"vsuf [] _ = []"
+|"vsuf (c#cs) r1 = (if (rnullable r1) then  (vsuf cs (rder c r1)) @ [c # cs]
+                                      else  (vsuf cs (rder c r1))
+                   ) "
+
+
+
+
+
+
+fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list" where
+"star_update c r [] = []"
+|"star_update c r (s # Ss) = (if (rnullable (rders_simp r s)) 
+                                then (s@[c]) # [c] # (star_update c r Ss) 
+                               else   (s@[c]) # (star_update c r Ss) )"
+
+fun star_updates :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"
+  where
+"star_updates [] r Ss = Ss"
+| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"
+
+
+
+
+
+
+(*some basic facts about rsimp*)
+
+
+
+
+end
\ No newline at end of file




