src/HOL/MicroJava/BV/Kildall.thy

-(*  Title:      HOL/MicroJava/BV/Kildall.thy
-    ID:         $Id$
-    Author:     Tobias Nipkow, Gerwin Klein
-    Copyright   2000 TUM
-
-Kildall's algorithm
-*)
-
-header {* \isaheader{Kildall's Algorithm}\label{sec:Kildall} *}
-
-theory Kildall
-imports SemilatAlg While_Combinator
-begin
-
-
-consts
- iter :: "'s binop \<Rightarrow> 's step_type \<Rightarrow>
-          's list \<Rightarrow> nat set \<Rightarrow> 's list \<times> nat set"
- propa :: "'s binop \<Rightarrow> (nat \<times> 's) list \<Rightarrow> 's list \<Rightarrow> nat set \<Rightarrow> 's list * nat set"
-
-primrec
-"propa f []      ss w = (ss,w)"
-"propa f (q'#qs) ss w = (let (q,t) = q';
-                             u = t +_f ss!q;
-                             w' = (if u = ss!q then w else insert q w)
-                         in propa f qs (ss[q := u]) w')"
-
-defs iter_def:
-"iter f step ss w ==
- while (%(ss,w). w \<noteq> {})
-       (%(ss,w). let p = SOME p. p \<in> w
-                 in propa f (step p (ss!p)) ss (w-{p}))
-       (ss,w)"
-
-constdefs
- unstables :: "'s ord \<Rightarrow> 's step_type \<Rightarrow> 's list \<Rightarrow> nat set"
-"unstables r step ss == {p. p < size ss \<and> \<not>stable r step ss p}"
-
- kildall :: "'s ord \<Rightarrow> 's binop \<Rightarrow> 's step_type \<Rightarrow> 's list \<Rightarrow> 's list"
-"kildall r f step ss == fst(iter f step ss (unstables r step ss))"
-
-consts merges :: "'s binop \<Rightarrow> (nat \<times> 's) list \<Rightarrow> 's list \<Rightarrow> 's list"
-primrec
-"merges f []      ss = ss"
-"merges f (p'#ps) ss = (let (p,s) = p' in merges f ps (ss[p := s +_f ss!p]))"
-
-
-lemmas [simp] = Let_def Semilat.le_iff_plus_unchanged [OF Semilat.intro, symmetric]
-
-
-lemma (in Semilat) nth_merges:
- "\<And>ss. \<lbrakk>p < length ss; ss \<in> list n A; \<forall>(p,t)\<in>set ps. p<n \<and> t\<in>A \<rbrakk> \<Longrightarrow>
-  (merges f ps ss)!p = map snd [(p',t') \<leftarrow> ps. p'=p] ++_f ss!p"
-  (is "\<And>ss. \<lbrakk>_; _; ?steptype ps\<rbrakk> \<Longrightarrow> ?P ss ps")
-proof (induct ps)
-  show "\<And>ss. ?P ss []" by simp
-
-  fix ss p' ps'
-  assume ss: "ss \<in> list n A"
-  assume l:  "p < length ss"
-  assume "?steptype (p'#ps')"
-  then obtain a b where
-    p': "p'=(a,b)" and ab: "a<n" "b\<in>A" and ps': "?steptype ps'"
-    by (cases p') auto
-  assume "\<And>ss. p< length ss \<Longrightarrow> ss \<in> list n A \<Longrightarrow> ?steptype ps' \<Longrightarrow> ?P ss ps'"
-  from this [OF _ _ ps'] have IH: "\<And>ss. ss \<in> list n A \<Longrightarrow> p < length ss \<Longrightarrow> ?P ss ps'" .
-
-  from ss ab
-  have "ss[a := b +_f ss!a] \<in> list n A" by (simp add: closedD)
-  moreover
-  from calculation l
-  have "p < length (ss[a := b +_f ss!a])" by simp
-  ultimately
-  have "?P (ss[a := b +_f ss!a]) ps'" by (rule IH)
-  with p' l
-  show "?P ss (p'#ps')" by simp
-qed
-
-
-(** merges **)
-
-lemma length_merges [rule_format, simp]:
-  "\<forall>ss. size(merges f ps ss) = size ss"
-  by (induct_tac ps, auto)
-
-
-lemma (in Semilat) merges_preserves_type_lemma:
-shows "\<forall>xs. xs \<in> list n A \<longrightarrow> (\<forall>(p,x) \<in> set ps. p<n \<and> x\<in>A)
-          \<longrightarrow> merges f ps xs \<in> list n A"
-apply (insert closedI)
-apply (unfold closed_def)
-apply (induct_tac ps)
- apply simp
-apply clarsimp
-done
-
-lemma (in Semilat) merges_preserves_type [simp]:
- "\<lbrakk> xs \<in> list n A; \<forall>(p,x) \<in> set ps. p<n \<and> x\<in>A \<rbrakk>
-  \<Longrightarrow> merges f ps xs \<in> list n A"
-by (simp add: merges_preserves_type_lemma)
-
-lemma (in Semilat) merges_incr_lemma:
- "\<forall>xs. xs \<in> list n A \<longrightarrow> (\<forall>(p,x)\<in>set ps. p<size xs \<and> x \<in> A) \<longrightarrow> xs <=[r] merges f ps xs"
-apply (induct_tac ps)
- apply simp
-apply simp
-apply clarify
-apply (rule order_trans)
-  apply simp
- apply (erule list_update_incr)
-  apply simp
- apply simp
-apply (blast intro!: listE_set intro: closedD listE_length [THEN nth_in])
-done
-
-lemma (in Semilat) merges_incr:
- "\<lbrakk> xs \<in> list n A; \<forall>(p,x)\<in>set ps. p<size xs \<and> x \<in> A \<rbrakk> 
-  \<Longrightarrow> xs <=[r] merges f ps xs"
-  by (simp add: merges_incr_lemma)
-
-
-lemma (in Semilat) merges_same_conv [rule_format]:
- "(\<forall>xs. xs \<in> list n A \<longrightarrow> (\<forall>(p,x)\<in>set ps. p<size xs \<and> x\<in>A) \<longrightarrow> 
-     (merges f ps xs = xs) = (\<forall>(p,x)\<in>set ps. x <=_r xs!p))"
-  apply (induct_tac ps)
-   apply simp
-  apply clarsimp
-  apply (rename_tac p x ps xs)
-  apply (rule iffI)
-   apply (rule context_conjI)
-    apply (subgoal_tac "xs[p := x +_f xs!p] <=[r] xs")
-     apply (drule_tac p = p in le_listD)
-      apply simp
-     apply simp
-    apply (erule subst, rule merges_incr)
-       apply (blast intro!: listE_set intro: closedD listE_length [THEN nth_in])
-      apply clarify
-      apply (rule conjI)
-       apply simp
-       apply (blast dest: boundedD)
-      apply blast
-   apply clarify
-   apply (erule allE)
-   apply (erule impE)
-    apply assumption
-   apply (drule bspec)
-    apply assumption
-   apply (simp add: le_iff_plus_unchanged [THEN iffD1] list_update_same_conv [THEN iffD2])
-   apply blast
-  apply clarify 
-  apply (simp add: le_iff_plus_unchanged [THEN iffD1] list_update_same_conv [THEN iffD2])
-  done
-
-
-lemma (in Semilat) list_update_le_listI [rule_format]:
-  "set xs <= A \<longrightarrow> set ys <= A \<longrightarrow> xs <=[r] ys \<longrightarrow> p < size xs \<longrightarrow>  
-   x <=_r ys!p \<longrightarrow> x\<in>A \<longrightarrow> xs[p := x +_f xs!p] <=[r] ys"
-  apply(insert semilat)
-  apply (unfold Listn.le_def lesub_def semilat_def)
-  apply (simp add: list_all2_conv_all_nth nth_list_update)
-  done
-
-lemma (in Semilat) merges_pres_le_ub:
-  assumes "set ts <= A" and "set ss <= A"
-    and "\<forall>(p,t)\<in>set ps. t <=_r ts!p \<and> t \<in> A \<and> p < size ts" and "ss <=[r] ts"
-  shows "merges f ps ss <=[r] ts"
-proof -
-  { fix t ts ps
-    have
-    "\<And>qs. \<lbrakk>set ts <= A; \<forall>(p,t)\<in>set ps. t <=_r ts!p \<and> t \<in> A \<and> p< size ts \<rbrakk> \<Longrightarrow>
-    set qs <= set ps  \<longrightarrow> 
-    (\<forall>ss. set ss <= A \<longrightarrow> ss <=[r] ts \<longrightarrow> merges f qs ss <=[r] ts)"
-    apply (induct_tac qs)
-     apply simp
-    apply (simp (no_asm_simp))
-    apply clarify
-    apply (rotate_tac -2)
-    apply simp
-    apply (erule allE, erule impE, erule_tac [2] mp)
-     apply (drule bspec, assumption)
-     apply (simp add: closedD)
-    apply (drule bspec, assumption)
-    apply (simp add: list_update_le_listI)
-    done 
-  } note this [dest]
-  
-  from prems show ?thesis by blast
-qed
-
-
-(** propa **)
-
-
-lemma decomp_propa:
-  "\<And>ss w. (\<forall>(q,t)\<in>set qs. q < size ss) \<Longrightarrow> 
-   propa f qs ss w = 
-   (merges f qs ss, {q. \<exists>t. (q,t)\<in>set qs \<and> t +_f ss!q \<noteq> ss!q} Un w)"
-  apply (induct qs)
-   apply simp   
-  apply (simp (no_asm))
-  apply clarify  
-  apply simp
-  apply (rule conjI) 
-   apply blast
-  apply (simp add: nth_list_update)
-  apply blast
-  done 
-
-(** iter **)
-
-lemma (in Semilat) stable_pres_lemma:
-shows "\<lbrakk>pres_type step n A; bounded step n; 
-     ss \<in> list n A; p \<in> w; \<forall>q\<in>w. q < n; 
-     \<forall>q. q < n \<longrightarrow> q \<notin> w \<longrightarrow> stable r step ss q; q < n; 
-     \<forall>s'. (q,s') \<in> set (step p (ss ! p)) \<longrightarrow> s' +_f ss ! q = ss ! q; 
-     q \<notin> w \<or> q = p \<rbrakk> 
-  \<Longrightarrow> stable r step (merges f (step p (ss!p)) ss) q"
-  apply (unfold stable_def)
-  apply (subgoal_tac "\<forall>s'. (q,s') \<in> set (step p (ss!p)) \<longrightarrow> s' : A")
-   prefer 2
-   apply clarify
-   apply (erule pres_typeD)
-    prefer 3 apply assumption
-    apply (rule listE_nth_in)
-     apply assumption
-    apply simp
-   apply simp
-  apply simp
-  apply clarify
-  apply (subst nth_merges)
-       apply simp
-       apply (blast dest: boundedD)
-      apply assumption
-     apply clarify
-     apply (rule conjI)
-      apply (blast dest: boundedD)
-     apply (erule pres_typeD)
-       prefer 3 apply assumption
-      apply simp
-     apply simp
-apply(subgoal_tac "q < length ss")
-prefer 2 apply simp
-  apply (frule nth_merges [of q _ _ "step p (ss!p)"]) (* fixme: why does method subst not work?? *)
-apply assumption
-  apply clarify
-  apply (rule conjI)
-   apply (blast dest: boundedD)
-  apply (erule pres_typeD)
-     prefer 3 apply assumption
-    apply simp
-   apply simp
-  apply (drule_tac P = "\<lambda>x. (a, b) \<in> set (step q x)" in subst)
-   apply assumption
-
- apply (simp add: plusplus_empty)
- apply (cases "q \<in> w")
-  apply simp
-  apply (rule ub1')
-     apply (rule semilat)
-    apply clarify
-    apply (rule pres_typeD)
-       apply assumption
-      prefer 3 apply assumption
-     apply (blast intro: listE_nth_in dest: boundedD)
-    apply (blast intro: pres_typeD dest: boundedD)
-   apply (blast intro: listE_nth_in dest: boundedD)
-  apply assumption
-
- apply simp
- apply (erule allE, erule impE, assumption, erule impE, assumption)
- apply (rule order_trans)
-   apply simp
-  defer
- apply (rule pp_ub2)(*
-    apply assumption*)
-   apply simp
-   apply clarify
-   apply simp
-   apply (rule pres_typeD)
-      apply assumption
-     prefer 3 apply assumption
-    apply (blast intro: listE_nth_in dest: boundedD)
-   apply (blast intro: pres_typeD dest: boundedD)
-  apply (blast intro: listE_nth_in dest: boundedD)
- apply blast
- done
-
-
-lemma (in Semilat) merges_bounded_lemma:
- "\<lbrakk> mono r step n A; bounded step n; 
-    \<forall>(p',s') \<in> set (step p (ss!p)). s' \<in> A; ss \<in> list n A; ts \<in> list n A; p < n; 
-    ss <=[r] ts; \<forall>p. p < n \<longrightarrow> stable r step ts p \<rbrakk> 
-  \<Longrightarrow> merges f (step p (ss!p)) ss <=[r] ts" 
-  apply (unfold stable_def)
-  apply (rule merges_pres_le_ub)
-     apply simp
-    apply simp
-   prefer 2 apply assumption
-
-  apply clarsimp
-  apply (drule boundedD, assumption+)
-  apply (erule allE, erule impE, assumption)
-  apply (drule bspec, assumption)
-  apply simp
-
-  apply (drule monoD [of _ _ _ _ p "ss!p"  "ts!p"])
-     apply assumption
-    apply simp
-   apply (simp add: le_listD)
-  
-  apply (drule lesub_step_typeD, assumption) 
-  apply clarify
-  apply (drule bspec, assumption)
-  apply simp
-  apply (blast intro: order_trans)
-  done
-
-lemma termination_lemma:
-  assumes semilat: "semilat (A, r, f)"
-  shows "\<lbrakk> ss \<in> list n A; \<forall>(q,t)\<in>set qs. q<n \<and> t\<in>A; p\<in>w \<rbrakk> \<Longrightarrow> 
-  ss <[r] merges f qs ss \<or> 
-  merges f qs ss = ss \<and> {q. \<exists>t. (q,t)\<in>set qs \<and> t +_f ss!q \<noteq> ss!q} Un (w-{p}) < w" (is "PROP ?P")
-proof -
-  interpret Semilat A r f using assms by (rule Semilat.intro)
-  show "PROP ?P" apply(insert semilat)
-    apply (unfold lesssub_def)
-    apply (simp (no_asm_simp) add: merges_incr)
-    apply (rule impI)
-    apply (rule merges_same_conv [THEN iffD1, elim_format]) 
-    apply assumption+
-    defer
-    apply (rule sym, assumption)
-    defer apply simp
-    apply (subgoal_tac "\<forall>q t. \<not>((q, t) \<in> set qs \<and> t +_f ss ! q \<noteq> ss ! q)")
-    apply (blast intro!: psubsetI elim: equalityE)
-    apply clarsimp
-    apply (drule bspec, assumption) 
-    apply (drule bspec, assumption)
-    apply clarsimp
-    done
-qed
-
-lemma iter_properties[rule_format]:
-  assumes semilat: "semilat (A, r, f)"
-  shows "\<lbrakk> acc r ; pres_type step n A; mono r step n A;
-     bounded step n; \<forall>p\<in>w0. p < n; ss0 \<in> list n A;
-     \<forall>p<n. p \<notin> w0 \<longrightarrow> stable r step ss0 p \<rbrakk> \<Longrightarrow>
-   iter f step ss0 w0 = (ss',w')
-   \<longrightarrow>
-   ss' \<in> list n A \<and> stables r step ss' \<and> ss0 <=[r] ss' \<and>
-   (\<forall>ts\<in>list n A. ss0 <=[r] ts \<and> stables r step ts \<longrightarrow> ss' <=[r] ts)"
-  (is "PROP ?P")
-proof -
-  interpret Semilat A r f using assms by (rule Semilat.intro)
-  show "PROP ?P" apply(insert semilat)
-apply (unfold iter_def stables_def)
-apply (rule_tac P = "%(ss,w).
- ss \<in> list n A \<and> (\<forall>p<n. p \<notin> w \<longrightarrow> stable r step ss p) \<and> ss0 <=[r] ss \<and>
- (\<forall>ts\<in>list n A. ss0 <=[r] ts \<and> stables r step ts \<longrightarrow> ss <=[r] ts) \<and>
- (\<forall>p\<in>w. p < n)" and
- r = "{(ss',ss) . ss <[r] ss'} <*lex*> finite_psubset"
-       in while_rule)
-
--- "Invariant holds initially:"
-apply (simp add:stables_def)
-
--- "Invariant is preserved:"
-apply(simp add: stables_def split_paired_all)
-apply(rename_tac ss w)
-apply(subgoal_tac "(SOME p. p \<in> w) \<in> w")
- prefer 2; apply (fast intro: someI)
-apply(subgoal_tac "\<forall>(q,t) \<in> set (step (SOME p. p \<in> w) (ss ! (SOME p. p \<in> w))). q < length ss \<and> t \<in> A")
- prefer 2
- apply clarify
- apply (rule conjI)
-  apply(clarsimp, blast dest!: boundedD)
- apply (erule pres_typeD)
-  prefer 3
-  apply assumption
-  apply (erule listE_nth_in)
-  apply simp
- apply simp
-apply (subst decomp_propa)
- apply fast
-apply simp
-apply (rule conjI)
- apply (rule merges_preserves_type)
- apply blast
- apply clarify
- apply (rule conjI)
-  apply(clarsimp, fast dest!: boundedD)
- apply (erule pres_typeD)
-  prefer 3
-  apply assumption
-  apply (erule listE_nth_in)
-  apply blast
- apply blast
-apply (rule conjI)
- apply clarify
- apply (blast intro!: stable_pres_lemma)
-apply (rule conjI)
- apply (blast intro!: merges_incr intro: le_list_trans)
-apply (rule conjI)
- apply clarsimp
- apply (blast intro!: merges_bounded_lemma)
-apply (blast dest!: boundedD)
-
-
--- "Postcondition holds upon termination:"
-apply(clarsimp simp add: stables_def split_paired_all)
-
--- "Well-foundedness of the termination relation:"
-apply (rule wf_lex_prod)
- apply (insert orderI [THEN acc_le_listI])
- apply (simp add: acc_def lesssub_def wfP_wf_eq [symmetric])
-apply (rule wf_finite_psubset) 
-
--- "Loop decreases along termination relation:"
-apply(simp add: stables_def split_paired_all)
-apply(rename_tac ss w)
-apply(subgoal_tac "(SOME p. p \<in> w) \<in> w")
- prefer 2; apply (fast intro: someI)
-apply(subgoal_tac "\<forall>(q,t) \<in> set (step (SOME p. p \<in> w) (ss ! (SOME p. p \<in> w))). q < length ss \<and> t \<in> A")
- prefer 2
- apply clarify
- apply (rule conjI)
-  apply(clarsimp, blast dest!: boundedD)
- apply (erule pres_typeD)
-  prefer 3
-  apply assumption
-  apply (erule listE_nth_in)
-  apply blast
- apply blast
-apply (subst decomp_propa)
- apply blast
-apply clarify
-apply (simp del: listE_length
-    add: lex_prod_def finite_psubset_def 
-         bounded_nat_set_is_finite)
-apply (rule termination_lemma)
-apply assumption+
-defer
-apply assumption
-apply clarsimp
-done
-
-qed
-
-lemma kildall_properties:
-assumes semilat: "semilat (A, r, f)"
-shows "\<lbrakk> acc r; pres_type step n A; mono r step n A;
-     bounded step n; ss0 \<in> list n A \<rbrakk> \<Longrightarrow>
-  kildall r f step ss0 \<in> list n A \<and>
-  stables r step (kildall r f step ss0) \<and>
-  ss0 <=[r] kildall r f step ss0 \<and>
-  (\<forall>ts\<in>list n A. ss0 <=[r] ts \<and> stables r step ts \<longrightarrow>
-                 kildall r f step ss0 <=[r] ts)"
-  (is "PROP ?P")
-proof -
-  interpret Semilat A r f using assms by (rule Semilat.intro)
-  show "PROP ?P"
-apply (unfold kildall_def)
-apply(case_tac "iter f step ss0 (unstables r step ss0)")
-apply(simp)
-apply (rule iter_properties)
-apply (simp_all add: unstables_def stable_def)
-apply (rule semilat)
-done
-qed
-
-lemma is_bcv_kildall:
-assumes semilat: "semilat (A, r, f)"
-shows "\<lbrakk> acc r; top r T; pres_type step n A; bounded step n; mono r step n A \<rbrakk>
-  \<Longrightarrow> is_bcv r T step n A (kildall r f step)"
-  (is "PROP ?P")
-proof -
-  interpret Semilat A r f using assms by (rule Semilat.intro)
-  show "PROP ?P"
-apply(unfold is_bcv_def wt_step_def)
-apply(insert semilat kildall_properties[of A])
-apply(simp add:stables_def)
-apply clarify
-apply(subgoal_tac "kildall r f step ss \<in> list n A")
- prefer 2 apply (simp(no_asm_simp))
-apply (rule iffI)
- apply (rule_tac x = "kildall r f step ss" in bexI) 
-  apply (rule conjI)
-   apply (blast)
-  apply (simp  (no_asm_simp))
- apply(assumption)
-apply clarify
-apply(subgoal_tac "kildall r f step ss!p <=_r ts!p")
- apply simp
-apply (blast intro!: le_listD less_lengthI)
-done
-qed
-
-end