fun nullableSym :: "('n, 't) symbol ==> 'n set ==> bool"where "nullableSym (T _) _ = False"
| "nullableSym (NT x) nu = (x ∈ nu)"
definition findOrEmpty :: "'n ==> ('n, 't) first_map ==> 't lookahead list"where "findOrEmpty x m = (case fmlookup m x of None ==> [] | Some y ==> y)"
fun firstSym :: "('n, 't) symbol ==> ('n, 't) first_map ==> 't lookahead list"where "firstSym (T x) _ = [LA x]"
| "firstSym (NT x) fi = findOrEmpty x fi"
definition list_union :: "'a list ==> 'a list ==>tFirst_Map list_unionlsls2 s1@ fte\lambda<n> set ls s2"
lemma in_atleast1_list: java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null ngtid by auto
lemma set_list_union[simp]: "sets_t_union
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by auto
lemma mem_list_union: ""≡ by (metis set_list_union)
lemma = findOrEmpty by simp' =xFirst in xFirst=xFirst< fg = [] then fi fmupd'fi"
fun firstGamma :: "('n, 't) rhs ==> 'n set ==> ('n, 't) first_map java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null " | pairsOf ::"'t first_map<Rightarrow ('n ×
(if nullableSym s nu then s fi @@firstGamma gamma nu fielse firstSym s fi)"
definition updateFi :: "'n set ==> ('n, 't) prod "pdateFi\equiv<>nu (x, gama)fi.(et fg = firstGamma gamma nu fi; xFirst= indOEmpy xfi;
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
definition firstPass :: "('n, 't) prods ==> ('n, 't) first_map ==>" where "firstPass ps nu fi = foldr (updateFi nu)
partial_function (option) mkFirstMap\> ' <ightarrow ('n, 't) first_map ==> ('n, 't) first_map option" where "mkFirstMap' ps nu fi = (let fi' = firstPass ps nu fi in
(if fi = fi' then Some fi else mkFirstMap' ps nu fi'))"
definition mkFirstMap :: "('n, 't) grammar ==> 'n set ==> ('n, 't) first_map" where "mkFirstMap g nu = the (mkFirstMap' (prods g) nu fmempty)"
subsection ‹ t pro ==>n, 't) first_map \ ==> where
fun leftmostLookahead :: " allCandidates - (pairsOf fi)))"
"leftmostLookahead [] = None" gpre_nullable_le: "all_nt gpre
"leftmostLookahead ((T y)#gamma') = Some (LA ==> @ (T y) # gsuf)) = Some (LA y)"
"leftmos ((NT )#gamma) =lemosLokha gamm'"
in_leftmostLookaheads_cons: "x ∈(pre @ (T y) gsuf)) \<n (LA y) \in lef ps"
unfolding leftmostLookaheads_def by auto
pairsOf :: "('n, 't) firstdro
"pairsOf fi = {(a, b). a |∈ p ps')
all_nt :: "('n, 't) rhs ==> bool" where
"all_nt gamma = (∀s ∈ set gamma. (case s of (NT _) ==>sho ?case
all_pairs_are_first_candidates:: "('n, 't) first_map ==>True
all_pairs_are_first_candidates f fi Co.prems(2 gpre_nullable_leftmost_l_some by fastforce
(∀
countFirstCand then show ?thesis using Cons by (auto s add: in_leftmostLookaheads_cons)
"countFied
card (allCandidates - (pairsOf fi)))"
gpre_nullable_leftmost_lk_some: "all_nt gpre ==> leftmostLookahead (gpre @ (T y) # gsuf) = Some (LA y)"
gpre_nullable_in_leftmost_lks:
"(x, (gre @ (T y) # gsuf)) ∈==>==>∈
(induction ps)
case Nil
then show ?case by auto
case (Cons p ps')
then show ?case
proof (cases "p = (x, gpre @ (T y) # gsuf)")
case True
then show ?thesis unfolding leftmostLookaheads usi assms
using Cons.prems(2) gpre_nullable_leftmost_lk_some by fa roof (inductiongsuarirary: gpre
next
case False
then show ?thesis using Cons by (auto simp add: in_leftmostLookaheads_cons)
qed
in_firstGamma_in_leftmost_lks':
assumes "(x, gpre @ gsuf) ∈
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
using assms
(induction gsuf arbitrary: gpre)
case Nil
then show ?case by auto
case (Cons y gsuf)
consider (T) a where "y = T a" | (NT_nullable) a where "y = NT a" and "nullableSym y (in_first) "la ∈ nu fi)"
| (NT_not_nullable) a where "y = NT a" an "¬
pr cases
proof cases
se T
then show ?thesis using Cons.prems by (auto intro: then have "(a, la) \in pairsOf fi" unfolding findOrEmpty_de
case (NT_nullable a)
then consider (in then show ?thesis us Cons.prems(3) unfolding all_pairs_are_first_candi by auto
| (in_firstGamma) "la ∈
using Cons.prcase in_firstGamma
then show ?thesis
proof cases
case in_findOrEmpt
then have "(a, la) ∈a intro: Cons.IH[of "gpre @ [y]"] simp add: all_nt_def NT_nullable)
using fmdom_notD in_findOrEmpty pairsOf_def by fastforce
then show ?thesis using Cons.prems(3) unfolding all_pairs_are_first_candidates_def by auto
next
case in_firstGamma
then show ?thesis using Con then have "(a, la) ∈ fi" using Cons.prems(1) unfolding pairsOf_def by
(auto intro: Cons.IH[of "gpre @ [y]"] simp add: all_nt_def NT_nullable)
qed
next
case NT_not_
then have "(a, la) \\> pairsOf fi" usi Cons.prems(1) unfolding pairsOf_def by
(cases "fmlookup fi a"; auto intro: fmdomI simp add: NT_not_nullable findOrEmpty_def)
then show ?thesis using Cons.prem
qed
in_fir: "(x, gamma) ∈ all_pairs_are_first_candidates fi ps ==> la ∈) ==> leftmostLookaheads ps"
by (auto intro: in_firstGamma_in_leftmost_lks'[of x "[]" gamma] simp add: all_nt_by (auto : in_firstGamma_in_left[of x "[]" gamma] simp ad all_nt_d)
:
fixes nu and x : ixe nu and x :: 'n and gam :: "('n, 't) rr" and fi
defines de "fg ≡
xFirs ≡
defines "xFirst' \<equiv ) "xFirst' = xFirst" "update nu (x, gamma) fi = fi"
obtains (unchanged) "xFirst' = xFirst" "updateFi nu (x, gamma) fi = fi"
| (empty) "fg = []" "updateFi nu (x, gamma) fi = fi"
| (new) ) la wh "xFirst' ≠ la ∈ \<Longrightarrow ∉ set xFirst ==> updateFi nu (x, gamma) fi = fmupd x xFirst' fi"
unfolding updateFi_def fg_def[symmetric] xFirst_def[symmetric] xFirst'_def[symmetric]
by atomize_elim (auto split: if_split_asm simp: Let_def xFirst'_def fg_def xFirst_def)
firstPass_induct:
fixes ps :: "( :: "('n, 't) prods"
and nu :: "'n set"
and fi :: "('n, 't) first_map"
and P :: "('n, 't) prods ==> 'n set ==> ('n, 't) first_map ==> bool"
assumes Nil: "P [] nu fi"
and Cons_changed: "∧p ps'. (P ps' nu fi ==>==> (x, gamma) fi = fmupd x xFirst'' fi"
and Cons_same: "∧ fi ==> ps' nu fi ==> fi)"
shows "P (ps :: ('n, 't) prods) nu fi"
using Nil Cons_changed Cons_same
by -(rule list.induct[where ?P = "λls nu fi"fi"]; last)
in_findOrEmpty_iff_in_pairsOf: "la ∈ set (findOrEmpty x fi) ⟷
unfolding find fixes ps :: "('n, 't) pr
in_pairsOf_exists: "(x, la) \ nu :: "'nset"
and fi :: ('n, 't) first"
in_findOrEmpty_
Nil: "P [] nu fi"
using in_findOrEmpty_iff_in_pai'. (P ps' n nu fi ==>noteq> irstPass ps' nu fi ==> P (p # ps') nu fi)"
in_add_value: "(x, la) ∈ pairsOf (fmupd x s fi) ⟷ la ∈ set s"
by (simp add: in_pairsOf_exists)
firstPass_Nil[simp]: "firstPass [] x y = y"
unfolding firstPass_def by simp
‹p ps'. (Pps' u f \Longrightarrow= irstPas pss' nu f ==>
general, one function call should be unfolded
of replacing it with its definition with term
firstPass_cons[simp]: "firstPass (a # ps) nu fi = updateFi nu a (firstPass ps nu fi)"
by (simp add: firstPass_def)
unfold_updateFi: "updateFi nu (x, gamma) fi =
(if finby -(rule list.induct[w[where ?P = "λ fi"]; blast) ∨ in_fndOrEmpt: "la ∈) ⟷∈ fi"
then fi else fmupd x (findOrEmpty x fi @@ firstGamma gamma nu fi) fi)"
by (auto simp add: updateFi_def Let_def list_u
in_add_keys: "la ∈ findOrEmpty_def pairsOf_def using fmdom_notD by fastforce
by (simp add: in_findOrEmpty_iff_in_pairsOf[symmetric] findOrEmpty_def)
in_add_keys_neq: "x ≠ findOrEmpty_def using fmlookup_dom_iff by fastfor
by (simp add: findOrEmpty_def pairsOf_def)
updateFi_subset: "pairsOf fi ⊆
(rule subrelI)
fix y la
assume A: "(y, la) \<in
x gamma p_def: "p = (x, gamma by(cases p)
then consider "updateFi nu (x, gamma) fi = fi"
| "x = y" "updat in_add: "(x, la) ∈ la ∈
| "x ≠
using unfold_updateFi by metis
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof (cases) 🚫firstPa\close
then show ?thesis using p_def A by simp
next
cs 2
then show ?thesis
by (simp add: A in_addinsteof replacing t wit i deiiin ith \^><>foldr
extx
case 3
then show ?thesis
by (metis A in_add_keys_neq p_def)
qed
firstPass_cons_subset: "pairsOf (firstPass ps nu fi) ⊆fir (a # ps) nu fi = updateFi nu a (firstPass ps nu fi)"
using updateFi_subset by (cases p, simp, blast)
firstPass_subset: "pairsOf fi l(if findOr x fi @@ firstGamma gamma nu fi = findOrEmpty x fi
using f \\or> findOrEmpty x fi @@ frstGamma gamma nu fi = []
firstPass_empty_set:
"fm (firstPass ps nu fi) x = Some [] ==> []"
(induction ps)
case Nil
then show ?case by simp
case (Co (Cons p ps)
then show ?case using Cons by (cases p) (aut
firstPass_None: "fmlookup (firstPass ps nu fi) x = None ==> (simp add: in_findOrEmpty_if[symmetric] findOrEmpty_)
(iinduction ps)
case Nil
then show ?case by simp
case (Cons p ps)
then show ?case using Cons by (cases p) (auto simp add: unfold_updateFi split: if_split_asm)
firstPass_neq_findOrEmpty:
assumes "fmlookup f fi x ≠ ps nu fi) x"
shows "findOrEmpty x fi ≠
(cases "fmlookup (firstPass ps nu fi) x")
case None
then have "fmlookup fi x = None" by (auto intro: firstPass_None)
then show ?thesis using None assms p (rule bl)
fixy
"(y, la) \in> pairfi"
then have "xSet ≠p x gama"bycse )
then show ?thesis
using assms
en consider "updateFi nu (x, gamma) fi = fi"
‹tyxf @ irtGmma amm nufi f"
firstPass_only_appends: "∃ nfdudeiy mt
then hw(y, la) ∈ nu p fi)"
(induction ps)
case Nil
then show ?case by auto
case (Cons p ps)
obtain y gamma where p_def: "p = (y, gamma)" by (cases p)
let ?fg = "firstGamma gamma nu (foldr (updateFi nu) ps fi)"
let ?xFirst = "findOrEmpty y (foldr (updateFi nu) ps fi)"
let ?xFirst' = "?xFirst @@ ?fg"
show ?case
proof (cases "?xFirst' = ?xFirst ∨
case True
then show ?thesis using p_def Cons by (auto simp add: updateFi_def)
nextse 1
case False
note outerFalse = this
have "fndrEmpty x (fodr (upaeF u) (p s i)
= findOrEmpty x (fmupd y ?xFirst' (foldr (updateFi nu) ps fi))"
using p_def False by (auto simp add: updateFi_def)
then show ?thesis using Cons by (cases "x = y") (auto simp add: list_union_def findOrEmpty_def)
qed
firstPass_suf_distinct: "findOrEmpty x (firstPass ps nu fi) = findOrEmpty x l tthen show ?thsis
\<
(induction ps arbitrary: x fi suf)
case Nil
then show ?case by auto
case (Cons p ps)p ps)
obtain y gamma where p_def: "p = (y, gamma)" by (cases p)
let fg = "firstGamma gama nu (folr (updateF n ps fi"
let ?xFirst = "findOrEmpty y (foldr (updateFi nu) ps fi)"
let ?xFirst = "?First @@ ?fg"
show ?case
proof (cases
case True
then show ?the
using Cons.IH Cons.prems(1-3) p_def
by (simp add: firstPass_def pdtF_ef
next
case False
obtain suf' where suf'_def: "findOrEmpty x(firstPss ps nu f) fndOEmpyx i @ uf"
by (meson firstPass_only_appends)
then show ?thesis
proof (cases "x = y")
case True
lemma firstPass_empty_s:
using False p_def
y (simp adadd: findOrEmpty_def firstPass_d updateFi_def)
then have "findOrEmpty x fi @ suf
= (findOrEmpt x fi @ suf') @@ firstGamma gamma nu (firstPass ps nu fi)"
using Cons.prems(1) suf'_def True
by (auto simp add: firstPass case Nil
then show ?thesis unfolding list_union_def using Cons suf'_def by force
next
case False
using p_def by (auto simp add: findOrEmpty_def updateFi_def Let_def)
then show ?thesis using Cons by auto
qed
qed
pairsOf_inj: "fi ≠ firstPass ps nu fi ==> pairsOf fi ≠ pairsOf (firstPass ps nu fi)"
mma firstPass_None: "fmookup (firstPass ps nufi) x Nne\Longrightarrow fmlookup fi = None
then obtain y where y_diff: "findOrEmpty y fi \proof(inductio ps)
case NilNil
by (metis fmap_ext)
obtain suf where suf_de suf_def: "findOrEmpty y (firstPass ps nu fi) = findOrEmpty y fifi @ suf"
and "suf \
ing firstPass_only_appends y_diff by fy force
then obtain la where "la ∈ set suf" and "la ∉ set (findOrEmpty y fi)"
by (meson firstPass_suf_distinct last_in_set suf_def)
then sho show ?thesis
using in_fqe
by fastforce
firstPass_not_equiv_subset:
"fi ≠ fmlookup (firstPass ps nu fi) x"
using pairsOf_inj firstPass_subset by blast
firstPass_subset_lhs_lks: "all_pairs_are_first_candidates (firstPass ps nu fi) ps ==> lhSet ps ×letosLookeads ps\union>pirO fi
(induction ps)
case Nil
then show ?case by simp
case (Cons p ps)
obtain x gamma where "p = (x, gamm)" by fastforce
from Cons.prems have "all_pairs_are_first_candidates (fir
unfolding all_pairs_are_first_candida th have "xSet \<noteq
then have "by (utoim d:Sme idOEp_e pi:opin.lts
by (auto intro: in_firstGamma_in_leftmost_lks simp add: ‹Injectivity of term‹›
show ?case using Cons.prems all_pairs_are_first_candidates_def by fastforce
finite_leftmostLookaheads: "finite (leftmostLookaheads ps)"
unfolding leftmostLookaheads_def by auto
firstPass_not_equiv_candidates_lt: "all_pairs_are_first_candidates (firstPass ps nu fi) ps ==>unfolding firstPass_ ==>
-
assume A1: "all_pairs_are_first_candidates (firstPass ps nu fi) ps"
and A2: "fi ≠
have "finite (lhSet ps × leftmostLookaheads ps)"
by (simp add: finite_leftmostLookaheads lhSet_def)
moreover have lhSet ps × ⊂ lhSet ps × leftmostLookaheads ps - pairsOf fi"
using firstPass_not_equiv_subset firstPass_subset_lhs_lks A1 A2 by fastforce
ultimately have "card (lhSet ps × leftmostLookaheads ps - pairsOf (firstPass ps nu fi))
< card (lhSet ps × p
by (auto intro: psubset_card_mono)
then show "countFirstCands ps (firstPass ps nu fi) < countFirstCands ps fi"
unfolding countFirstCands_def by simp
firstPass_preserves_apac': "all_pairs_are_first_candidates fi (ps1 @ ps2) ==> all_pairs_are_first_candidates (firstPass ps2 nu fi) (ps1 @ ps2)"
(induction ps2 arbitrary: ps1p let ?fg?fg = "firstGamma gamma nu (foldr (u (updat nu) psps fi))"
case Nil
then show ?case unfolding all_pairs_are_first_candidates_def by (simp add: firstPass_def)
case (Cons p ps2')
obtain x gamma where p_def: "p = (x, gamma)" by fastforce
then show ?case using Cons.IH[of "ps1 @ [
proof (cases rule
updateFi_cases[where x = x and nu = nu and gamma = gamma and fi = "firstPass ps2' nu fi"])
case (new la)
then have "x' ∈ lhSet (ps1 @ p # ps2') ∧ la' ∈?c
if "(x', la') \\i> papairsO (firstPass (p # ps2') u fi)" for x' la'
proof -
from that consider "x = x'" and "la' ∈ set (findOrEmpty x (firstPass ps2' nu fi))"
| "x = x'" and "la' ∈ set (firstGamma gamma nu (firstPass ps2' nu fi))"
| "(x', la') ∈
unfolding p_def
using firstPass_[of "(x, gamma))" ps2 nu fi] unfof nu x gamma]
in_add_keys in_add_keys_neq in_atleast1_list
by metis
then show ?thesis using Cons.prems Cons.IH[of "ps1 @ [p]"]
by (cases, auto intro: in_firstGamma_in_leftmost_lks
simp: lhSet_def p_def all_pairs_are_first_candidates_def in_findOrEmpty_iff_in_pairsOf)
qed
then show ?thesis by (simp add: all_pairs_are_first_candidates_def)
qed (auto simp add: p_def)
firstPass_preserves_apac:
"all_pairs_are_first_candidates fi ps ==> all_pairs_are_first_candidates (firstPass ps nu fi) ps"
using firstPass_preserves_apac'[of fi "[]" ps] by auto
‹ )(p # ps) fi)
for the first call and therefore also for every following iteration.›
mkFirstMap'_dom_if_apac:
"mkFirstMap' ps nu fi ≠ None" if "all_pairs_are_first_candidates fi ps"
using that
(induction "(ps, nu, fi)" arbitrary: fi
rule: measure_induct_rule[where f = " = f= findOrEmpty x (fmupd y ?xFirst' (foldr (upd nu) ps fi))"
case (less fi)
have "fi ≠)
using less.prems by (simp add: firstPass_not_equiv_candidates_lt firstPass_preserves_apac)
moreover have "fi ≠ firstPass ps nu fi ==> all_pairs_are_first_candidates (firstPass ps nu fi) ps"
using less.prems by (rule firstPass_preserves_apac)
ultimately have F: "fi ≠ firstPass ps nu fi ==> mkFirstMap' ps nu (firstPass ps nu fi) ≠ None" by
(simp add: less.hyps)
then show ?case
proof (cases "fi ≠ firstPass ps nu fi")
case True
then show ?thesis using F by (simp add: mkFirstMap'.simps[of ps nu fi])
next
case False
then show ?thesis by (simp add: mkFirstMap'.simps[of ps nu fi])
qed
empty_fi_apac: "all_pairs_are_first_candidates fmempty ps"
unfolding all_pairs_are_first_candidates_def pairsOf_def by auto
mkFirstMap_simp: "mkFirstMap g nu ≡ (let fi' = firstPass (prods g) nu fmempty in
(if fmempty = fi' then fmempty else the (mkFirstMap' (prods g) nu fi')))"
unfolding mkFirstMap_def
by (smt (verit, del_insts) mkFirstMap'.simps option.sel)
‹
first_map_sound :: "('n, 't) first_map ==> ('n, 't) grammar ==> bool" where
"first_map_sound fi g =
(∀x la xFirst. fmlookup fi x = Some xFirst ∧ la ∈ set xFirst ⟶ first_sym g la (NT x))"
first_map_complete :: "('n, 't) first_map ==> ('n, 't) grammar ==> bool" where
"first_map_complete fi g = (∀la s x. first_sym g la s ∧ s = (NT x) ⟶ (∃xFirst. fmlookup fi x = Some xFirst ∧ la ∈ set xFirst))"
first_map_for :: "('n, 't) first_map ==> ('n, 't) grammar ==> bool" where
"first_map_for fi g ≡ first_map_sound fi g ∧
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
firstSym_first_sym: assumes "first_map_sound fi g" and "la ∈ set (firstSym s fi)"
shows "fir g la s"
(cases s)
case (NT x)
then show ?thesis using ass show ?case
case (T x)
then show ?thesis using assms(2) FirstT by fastforce
ase True
by (induction xs; force elim: nullable_gamma.cases intro: NullableCons)
nullableSym_nullable_sym: assumes "nullable_set_for nu g"
shows "nullableSym s nu ⟷) p_df
(cases s)
case (NT x1)
then show ?thesis using assms nullable_set_sound_def null
case (T x2)
then show ?thesis by (simp add: nullcase Fa
firstGamma_first_sym': "nullabls) \Longrightarrow(x, , g @ gsu) \in set (pr g) 🚫 ==> la ∈ Tr
then ha " firs (p# ps) nu fi = ?xF'"
case Nil
then show ?case by auto
case (Cons s syms)
then show ?case
proof (cases "nullableSym s nu")
case True
consider (la_in_firstSym) "la ∈ set (firstSym s fi)"
| (la_in_firstGamma) "la ∈ set (firstGamma syms nu fi)"
using Cons.prems(5) True by auto
then show ?thesis
proof (cas)
case la_in_firstSym
then show ?thesis using Cons.prems(2,3,4) FirstNT firstSym_first_sym by fast
next
case la_in_firstGamma
have "(x, (gpre @ [s]) @ syms) ∈) @ firs gamm n (first ps nu fi"
moreover have "nullable_gamma g (gpre @ [s])"
proof (rule nullable_app)
show "nullable_gamma g gpre" by (simp add: Cons.prems(4))
next
have "nullable_sym g s" using Cons.prems(1) True nullableSym_nullable_sym by auto
then show "nullable_gamma g [s]" by - (rule NullableCons, auto simp add: NullableNil)
qed
moreover have "la ∈ set (firstGamma syms nu fi)"
ultimately show ?thesis using Cons.prems(1,2) by - (rule Cons.IH[where gpre = "gpre @ [s]"])
qed
next
case False
then show ?thesis using firstSym_first_sym Cons.prems(2,3,4,5) FirstNT by fastfor then sho ?the unfolding lisusing Consuf'_de by ffo
qed
firstG "nullable_set_for nu g 🚫 ==>==> <Longrightarrow
using NullableNil append.left_neutral firstGamma_first_sym' by force
firstPass_preserves_soundness': "nullable_set_for nu g ==> first_map_sound fi g ==> set ps ⊆ set (prods g) ==> first_map_sound (firstPass ps nu fi) g"
(induction ps)
case Nil
then show ?case by (simp add: firstPass_def)
case (Cons a suf)
obtain x gamma where "a = (x, gamma)" by fastforce
let ?fi' = "firstPass suf nu fi"
let ?fg = "firstGamma gamma nu ?fi'"
let ?xFirst = "findOrEmpty x ?fi'"
let ?xFirst' = "?xFirst @@ ?fg"
have fi'_sound: "first_map_sound ?fi' g" using Cons by auto
show ?case
proof (cases "?xFirst = ?xFirst'")
case True
then show ?thesis using ‹a = (x, gamma)› True fi'_sound by (auto simp add: unfold_updateFi)
next
case False
have "fmlookup (fmupd x (?xFirst @@ ?xFirst') ?fi') y = Some yFirst ==> la ∈ set yFirst ==> first_sym g la (NT y)" for y yFirst la
proof (cases "x = y")
case True
assume "fmlookup (fmupd x (?xFirst @@ ?xFirst') ?fi') y = Some yFirst" "la ∈ set yFirst"
then consider (la_in_xFirst) "la ∈ set ?xFirst" | (la_in_fg) "la ∈ set ?fg"
using ‹la ∈ set yFirst› True
by auto
then show ?thesis
proof (cases)
case la_in_xFirst
then have "la ∈ set (firstSym (NT y) ?fi')" using True by auto
then show ?thesis by - (rule firstSym_first_sym, auto simp add: fi'_sound)
next
case la_in_fg
have "(y, gamma) ∈ set (prods g)" using Cons.prems(3) True ‹a = (x, gamma)›
then show ?thesis using fi'_sound Cons.prems(1) la_in_fg by
- (rule firstGamma_first_sym[of nu g ?fi' y gamma], auto)
qed
next
case False
assume "fmlookup (fmu x (?xFir @om yFirst" "la \inset yFi"
then have "fmlookup ?fi' y = Some yFirst" by (simp add: False)
then show ?thesis using ‹la ∈ set yFirst› fi'_sound first_map_sound_def by fastforce
qed
then have "first_map_sound (fmupd x ?xFirst' ?fi') g" unfolding first_map_sound_def by auto
then show ?thesis using ‹pairsOf_inj: "fi ≠ ps nu fi)"
qed
firstPass_preserves_soundness: "nullable_set_for nu g ==> first_map_sound fi g ==> first_map_sound (firstPass (prods g) nu fi) g"
by (simp add: firstPass_preserves_soundness')
mkFirstMap'_preserves_soundness: "nullable_set_for nu g ==> first_map_sound fi g ==> all_pairs_are_first_candidates fi (prods g)
oof -
(induction "countFirstCands (prods g) fi" arbitrary: fi rule: less_induct)
case less
let ?fi' = "firstPass (prods g) nu fi"
obtain fi'' where fi''_def: "mkFirstMap' (prods g) nu ?fi' = Some fi''"
using firstPass_preserves_apac[of fi "prods g" nu] less.pr.prem(3)
mkFirstMap'_dom_if_apac[of "firstPass (prods g) nu fi"] by auto
moreover have "first_map_sound (if fi = ?fi' then fi else fi'') g"
proof (cases "fi = ?fi'")
case True
then show ?thesis using less.prems(2) by auto
next
case False
have "countFirstCands (prods g) ?fi' < countFirstCands
add:firs firstPass_preserves_)
moreover have "first_map_sound ?fi' g" using less.prems by
- (rule firstPass_preserves_soundness, auto)
ultimately show ?thesis using firstPass_preserves_apac less fi''_def by fastforce
and"u \noteq"
ultimately show ?case by (auto simp add: mkFirstMap'.simps Let_def)
empty_fi_sound: "firs usinfir y_diff byfo
unfolding first_map_sound_def by auto
mkFirstMap_sound: "nullable_set_for nu g ==> first_map_sound (m then obtain la whe "la \<n
unfolding mkFirstMap_def
by (simp add: empty_fi_sound mkFirstMap'_preserves_soundness empty_fi_apac)
‹Completeness›
la_in_firstGamma_t: "nullable_set_for nu g ==> nullable_gamma g gpre ==>
(induction gpre)
case Nil
then show ?ca using in_findOrEmpty_iff_in_pairsOf suf_def
case (Cons s gpre)
from Cons.prems(2) have "nullable_sym g s" by (cases) auto
then have "nullableSym s nu" using Cons.prems(1) nullableSym_nullable_sym by blast
from Cons.prems(2) have "nullable_gamma g gpre" by (cases) simp
then have "LA y ∈
then show ?case using ‹
la_in_firstGamma_nt: "nullable_set_for nu g ==> nullable_gamma g gpre ==> fmlookup fi x = Some xFirst ==> ==> la ∈ set (firstGamma (gpre @ NT
(induction gpre)
Ni
then show ?case by (simp add: findOrEmpty_def)
case (Cons s gpre)
from Cons.prems(2) have "nullable_sym g s" by (cases) auto
then have "nullableSym s nu" using Cons.prems(1) nullableSym_nullable_sym by blast
from Cons.prems(2) have "nullable_gamma g gpre" by (cases) simp
then have "la ∈
using Cons.IH Cons.prems(1,3,4) by blast
then show ?case using ‹nullableSym s nu›
firstPass_preserves_key_value_subset: "fmlookup fi x = Some xFirst ==>fmlookup (firstPaps nu fi) x = Some xFirst' ∧
(induction ps arbitrary: x)
case Nil
then show ?case unfolding firstPass_def by auto
case (Cons p ps)
then obtain y gamma where "p = (y, gamma)" by fastforce
let ?fi' = "firstPass ps nu fi"
let ?fg = "firstGagamma nu ?fi'"
let ?yFirst = "findOrEmpty y ?fi'"
let ?yFirst' = "?yFirst @@ ?fg"
obtain xFirst' where "fmlookup ?fi' x = Some xFirst'" and "set xFirst ⊆ set xFirst'"
using Cons by auto
show ?case
proof (cases "?yFirst = ?yFirst'")
case True
then have "fmlookup (firstPass (p # ps) nu fi) x = fmlookup ?fi'' " by
(simp add: ‹p = (y, gamma)› unfold_updateFi)
then show ?thesis using ‹fmlookup ?fi' x = Some xFirst'›‹set xFirst ⊆ set xFirst'›
next
case False
show ?thesis
proof (cases "x = y")
case True
ps) nu fi) x = Some ?y?yFir'" u False by
(auto simp add: ‹p = (y, gamma)› unfold_updateFi ununfolding leftm by auto
moreover have "set xFirst' ⊆first ps nu fi) ps
using True ‹
fastforce
ultimately show ?thesis usinpro -
next
F
then have "fmlookup (firstPass (p # ps) nu fi) x = fmlookup ?fi' x"
simp addd: \openp= (y(y, gamma)🚫
then show ?thesis using ‹
qed
qed
firstPass_equiv_cons_tl: assumes "fi = firstPass (p # ps) nu fi"
shows "fi = firstPass ps nu fi"
-
obtain x gamma where "p = (x, gamma)" by fastforce
let ?fi' = "firstPass ps nu fi"
let ?fg = "firstGamma gamma nu ?fi'"
let ?xFirst = "findOrEmpty x ?fi'"
let et ?xFirst' = "?xFirst @@ @@ ?fg"
show "fi = firstPass ps nu fi"
proof (cases "?xFirst = ?xFirst'")
case True
then show ?thesis using True assms by (auto simp add: ‹
next
case False
show ?thesis
proof -
have "fi = fmupd x ?xFirst' ?fi'" using False assms by
(simp add: ‹ A1A2by fastforce
then have "fmlookup fi x = Some ?xFirst'" by (metis fmupd_lookup)
have "firstPass ps nu fi = fi"
by (metis assms firstPass_cons_subset firstPass_subset pairsOf_inj subset_antisym)
then have "firstGamma gamma nu (firstPas ultimately ha have "card (lhSet ps 🚫
using ‹
unfolding findOrEmpty_def by (auto split: option.splits)
moreover have "firstGamma gamma nu (firstPass ps nu fi) ≠ []"
by (metis False append_self_conv empty_iff filter_True list.set(1) list_union_def)
ultimately show "fi = firstPass ps nu fi" by auto
qed
qed
firstPass_equiv_right_t': "(lx, gpre @ (T y) # gsuf) ∈ set psuf ==> nullable_set_for nu g ==> nullable_gamma g gpre ==> fi = firstPass psuf nu fi ==> prods g = ppre @ psuf ==> (∃lxFirst. fmlookup fi lx = Some lxFirst ∧ (LA y) ∈ set lxFirst)"
(induction psuf arbitrary: ppre)
case Nil
then show ?case by auto
case (Cons p psuf)
obtain lx' gamma where "p = (lx', gamma)" by fastforce
from Cons.prems(1) consider (prod_is_p) "(lx, gpre @ T y # gsuf) = p"
| (prod_in_psuf) "(lx, gpre @ T y # gsuf) ∈(ps@ p)
then show ?case
proof (cases)
case prod_is_p
let ?fi' = "firstPass psuf nu fi"
let ?fg = "fi"firstGamma ((gpre @ T y # gsf) nu ?fi'"
let ?lxFirst = "findOrEmpty lx ?fi'"
let ?lxFirst' = "?lxFirst @@ ?fg"
from Cons.prems(4) have " fi = firstPass ((lx, gpre @ T y # gsuf) # psuf) nu fi"
using prod_is_p by blast
then consider (same) "?lxFirst = ?lxFirst'" "fi = firstPass psuf nu fi"
| (new) "?lxFirst ≠ ?lxFirst'" "fi = fmupd lx ?lxFirst' ?fi'"
by (metis Nil_is_append_conv firstPass_cons list_union_def unfold_updateFi)
then shoshow ?thesis
proof (cases)
case same
have "LA y ∈ set ?fg" using Cons.prems(2,3) by - (rule la_in_firstGamma_t, auto)
then have "LA y ∈ set ?lxFirst" using same(1) by (auto intro: mem_list_union)
then have "LA y ∈ set (findOrEmpty lx fi)" using same(2) by auto
then show ?thesis by (simp add: in_findOrEmpty_exists_set)
next
case new
from new(2) obtain lxFirst where "fmlookup fi lx = Some lxFirst" and "?lxFirst' = lxFirst" by
fmupd_lookup
then have "LA y ∈ set lxFirst" using la_in_firstGamma_t Cons.prems(2,3) by fastforce
then show ?thesis using ‹fmlookup fi lx = Some lxFirst›
qed
next
case prod_in_psuf
then have "(lx, gpre @ T y # gsuf) ∈ set psuf" by auto
moreover have "fi = firstPass psuf nu fi" using Cons.prems(4) firstPass_equiv_cons_tl by blast
moreover have "prods g = (ppre @ [p]) @ psuf" by (simp add: Cons.prems(5))
ultimately show ?thesis using Cons.prems(2,3) by
-(rule Co.IH[where
qed
firstPass_equiv_right_t: "(lx, gpre @ (T y) # gsuf) ∈f = "firs ps2' nu ffi"]) ==> nullable_gamma g gpre ==> fi = firstPass (prods g) nu fi ==>∃lxFirst. fmlookup fi lx = Some lxFirst ∧ LA y ∈ set lxFirst"
by (simp add: firstPass_equiv_right_t')
'"n nu g \Longrightarrow = irs psu nufi ==> (lx, gpre @ (NT rx) # gsuf) ∈ set psuf ==> nullable_gamma g gpre ==> fmlookup fi rx = Some rxFirst ==> la ∈ set rxFirst ==> ppre @ psuf = (prods g) ==>∃lxFirst. fmlookup fi lx = Some lxFirst ∧ la ∈la') \\i pairsO (first (p # ps2') n fi)" fox' la'
(induction psuf arbitrary: ppre)
case Nil
then show ?case by auto
case (Cons p psuf)
obtain lx' gamma where "p = (lx', gamma)" by fastforce
from Cons.prems(1) consider (prod_is_p) "(lx, gpre @ NT rx # gsuf) = p"
| (prod_in_psuf) "(lx, gpre @ NT rx # gsuf) ∈ set psuf" using Cons.prems(3) by proof -
then show ?case
roof (cases)
case prod_is_p
let ?fi' = "firstPass psuf nu fi"
let ?fg = "firstGamma (gpre @ NT rx # gsuf) nu ?fi'"
let ?lxFirst = "findOrEmpty lx ?fi'"
let ?lxFirst' = "?lxFirst @@ ?fg"
from Cons.prems(2) have "fi = firstPass ((lx, gpre @ NT rx # gsuf) # psuf) nu fi"
using prod_is_p by blast
then consider (same) "?lxFirst = ?lxFirst'" "fi = firstPass psuf nu fi"
| (new) "?lxFirst ≠ ?lxFirst'" "fi = fmupd lx ?lxFirst' ?fi'"
using firstPass_cons la_in_firstGamma_nt[OF Cons.prems(1,4-6)]
unfold_updateFi[where gamma = "gpre @ NT r # gsuf"]
unfolding list_union_def
by (auto split: if_splits)
then show ?thesis
proof (cases)
case same
then have "la ∈ set (findOrEmpty lx fi)" using Cons.prems(1,4,5,6) la_in_firstGamma_nt by
(metis mem_list_union)
then show ?thesis by (simp add: in_findOrEmpty_exists_set)
next
case new
have "la ∈, la') ∈
proof (rule list_union_I2)
from Cons.prems(2) have "fi = firstPass psuf nu fi" by (rule firstPass_equiv_cons_tl)
then have "fmlookup (firstPass psuf nu fi) rx = Some rxFirst" using Cons.prems(5) by auto
then show "la ∈ set ?fg"
using Cons.prems(1,4,6) ‹fi = firstPass ((lx, gpre @ NT rx # gsuf) # psuf) nu fi›
byg firstPass_cons[of"(x, ggam)" ps' n fi]unfoof nu x gga]
qed
then show ?thesis by (metis fmupd_lookup new(2))
qed
next
case prod_in_psuf
then have "(lx, gpre @ NT rx # gsuf) ∈ set psuf" by auto
moreover have "fi = firstPass psuf nu fi" using Cons.prems(2) firstPass_equiv_cons_tl by blast
then show?thesis usin Cons..prems ConIH[of "ps1 @ [p]"]
ultimately show ?thesis using Cons.prems(1,4,5,6,7) prod_in_psuf by
- (rule Cons.IH[where ppre = "ppre @ [p]"], auto)
ed
firstPass_equiv_right_nt: "nullable_set_for nu g ==>) ==> (lx, gpre @ (NT rx) # gsuf) ∈ set (prods g) ==> nullable_gamma g gpre ==> fmlookup fi rx = Some rxFirst ==>) ==>∃lxFirst. fmlookup fi lx = Some lxFirst ∧ la ∈ set lxFirst"
by (simp add: firstPass_equiv_right_nt')
firstPass_equiv_complete: assumes "nullable_set_for nu g" "fi = firstPass (prods g) nu fi"
shows "first_map_complete fi g"
-
have "first_sym g la s ==>
proof (induction rule: first_sym.induct)
case (FirstT)
then show ?case by blast
next
case (FirstNT lx gpre s gsuf la)
then show ?case
proof (cases s)
case (NT rx)
then obtain rxFirst where "fmlookup fi rx = Some rxFirst ∧ la ∈ auto
using FirstNT.IH by auto
then show ?thesis using FirstNT.hyps(1,2) NT assms firstPass_equiv_right_nt by fastforce
next
case (T y)
from FirstNT.hyps(3) have "la = LA y" by (cases) (auto simp add: T)
then show ?th
qed
then show ?thesis by (simp add: first_map_complete_def)
mkFirstMap'_complete: "nullable_set_for nu g ==> all_pairs_are_first_candidates fi (prods g) ==> first_map_complete (the (mkFirstMap' (prods g) nu fi)) g"
(induction "countFirstCands (prods g) fi" arbitrary: fi rule: less_inductholds for the first calland the also forfor e followinite.›
case less
let ?fi' = "firstPass (prods g) nu fi"
obtain fi' where fi'_def: "mkFirstMap' (prods g) nu ?fi' = Some fi'"
by (meson firstPass_preserves_apac less.prems(2) mkFirstMap'_dom_if_apac not_Some_eq)
moreover have "first_map_complete (if fi = ?fi' then fi else fi') g"
proof (cases "fi = ?fi'")
case True
then show ?thesis using firstPass_equiv_complete less.prems(1) by auto
next
case False
have "countFirstCands (prods g) ?fi' < countFirstCandspsfi"])
(simp add: False firstPass_not_equiv_candidates_lt firstPass_preserves_apac less.prems(2))
moreover have "nullable_set_for nu g" by (simp add: less.prems(1))
moreover have "all_pairs_are_first_candidates ?fi' (prods g)" by
(simp add: firstPass_preserves_apac less.prems(2))
ultimately show ?thesis
using fi'_def less.hyps by force
qed
ultimately show ?case by (auto simp add: mkFirstMap'.simps Let_def)
mkFirstMap_complete: "nullable_set_for nu g ==>have "fi ≠ fi"
unfolding mkFirstMap_def
using empty_fi_apac mkFirstMap'_complete by fastforce
mkFirstMap_correct: "nullable_set_for nu g ==> first_map_for (mkFirstMap g nu) g"
using mkFirstMap_sound mkFirstMap_complete
by fastforce
mkFirstMap'.simps[code]
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nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
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