--- a/src/ZF/UNITY/ClientImpl.thy Fri Jun 20 12:10:45 2003 +0200
+++ b/src/ZF/UNITY/ClientImpl.thy Fri Jun 20 18:13:16 2003 +0200
@@ -7,7 +7,7 @@
*)
-ClientImpl = AllocBase + Guar +
+theory ClientImpl = AllocBase + Guar:
consts
ask :: i (* input history: tokens requested *)
giv :: i (* output history: tokens granted *)
@@ -20,11 +20,11 @@
"rel" == "Var([1])"
"tok" == "Var([2])"
-rules
- type_assumes
+axioms
+ type_assumes:
"type_of(ask) = list(tokbag) & type_of(giv) = list(tokbag) &
type_of(rel) = list(tokbag) & type_of(tok) = nat"
- default_val_assumes
+ default_val_assumes:
"default_val(ask) = Nil & default_val(giv) = Nil &
default_val(rel) = Nil & default_val(tok) = 0"
@@ -36,29 +36,299 @@
client_rel_act ::i
"client_rel_act ==
- {<s,t> \\<in> state*state.
- \\<exists>nrel \\<in> nat. nrel = length(s`rel) &
+ {<s,t> \<in> state*state.
+ \<exists>nrel \<in> nat. nrel = length(s`rel) &
t = s(rel:=(s`rel)@[nth(nrel, s`giv)]) &
nrel < length(s`giv) &
- nth(nrel, s`ask) le nth(nrel, s`giv)}"
+ nth(nrel, s`ask) \<le> nth(nrel, s`giv)}"
(** Choose a new token requirement **)
(** Including t=s suppresses fairness, allowing the non-trivial part
of the action to be ignored **)
client_tok_act :: i
- "client_tok_act == {<s,t> \\<in> state*state. t=s |
+ "client_tok_act == {<s,t> \<in> state*state. t=s |
t = s(tok:=succ(s`tok mod NbT))}"
client_ask_act :: i
- "client_ask_act == {<s,t> \\<in> state*state. t=s | (t=s(ask:=s`ask@[s`tok]))}"
+ "client_ask_act == {<s,t> \<in> state*state. t=s | (t=s(ask:=s`ask@[s`tok]))}"
client_prog :: i
"client_prog ==
- mk_program({s \\<in> state. s`tok le NbT & s`giv = Nil &
+ mk_program({s \<in> state. s`tok \<le> NbT & s`giv = Nil &
s`ask = Nil & s`rel = Nil},
{client_rel_act, client_tok_act, client_ask_act},
- \\<Union>G \\<in> preserves(lift(rel)) Int
+ \<Union>G \<in> preserves(lift(rel)) Int
preserves(lift(ask)) Int
preserves(lift(tok)). Acts(G))"
+
+
+declare type_assumes [simp] default_val_assumes [simp]
+(* This part should be automated *)
+
+(*????MOVE UP*)
+method_setup constrains = {*
+ Method.ctxt_args (fn ctxt =>
+ Method.METHOD (fn facts =>
+ gen_constrains_tac (Classical.get_local_claset ctxt,
+ Simplifier.get_local_simpset ctxt) 1)) *}
+ "for proving safety properties"
+
+(*For using "disjunction" (union over an index set) to eliminate a variable.
+ ????move way up*)
+lemma UN_conj_eq: "\<forall>s\<in>state. f(s) \<in> A
+ ==> (\<Union>k\<in>A. {s\<in>state. P(s) & f(s) = k}) = {s\<in>state. P(s)}"
+by blast
+
+lemma ask_value_type [simp,TC]: "s \<in> state ==> s`ask \<in> list(nat)"
+apply (unfold state_def)
+apply (drule_tac a = ask in apply_type, auto)
+done
+
+lemma giv_value_type [simp,TC]: "s \<in> state ==> s`giv \<in> list(nat)"
+apply (unfold state_def)
+apply (drule_tac a = giv in apply_type, auto)
+done
+
+lemma rel_value_type [simp,TC]: "s \<in> state ==> s`rel \<in> list(nat)"
+apply (unfold state_def)
+apply (drule_tac a = rel in apply_type, auto)
+done
+
+lemma tok_value_type [simp,TC]: "s \<in> state ==> s`tok \<in> nat"
+apply (unfold state_def)
+apply (drule_tac a = tok in apply_type, auto)
+done
+
+(** The Client Program **)
+
+lemma client_type [simp,TC]: "client_prog \<in> program"
+apply (unfold client_prog_def)
+apply (simp (no_asm))
+done
+
+declare client_prog_def [THEN def_prg_Init, simp]
+declare client_prog_def [THEN def_prg_AllowedActs, simp]
+ML
+{*
+program_defs_ref := [thm"client_prog_def"]
+*}
+
+declare client_rel_act_def [THEN def_act_simp, simp]
+declare client_tok_act_def [THEN def_act_simp, simp]
+declare client_ask_act_def [THEN def_act_simp, simp]
+
+lemma client_prog_ok_iff:
+ "\<forall>G \<in> program. (client_prog ok G) <->
+ (G \<in> preserves(lift(rel)) & G \<in> preserves(lift(ask)) &
+ G \<in> preserves(lift(tok)) & client_prog \<in> Allowed(G))"
+by (auto simp add: ok_iff_Allowed client_prog_def [THEN def_prg_Allowed])
+
+lemma client_prog_preserves:
+ "client_prog:(\<Inter>x \<in> var-{ask, rel, tok}. preserves(lift(x)))"
+apply (rule Inter_var_DiffI, force)
+apply (rule ballI)
+apply (rule preservesI, constrains, auto)
+done
+
+
+lemma preserves_lift_imp_stable:
+ "G \<in> preserves(lift(ff)) ==> G \<in> stable({s \<in> state. P(s`ff)})";
+apply (drule preserves_imp_stable)
+apply (simp add: lift_def)
+done
+
+lemma preserves_imp_prefix:
+ "G \<in> preserves(lift(ff))
+ ==> G \<in> stable({s \<in> state. \<langle>k, s`ff\<rangle> \<in> prefix(nat)})";
+by (erule preserves_lift_imp_stable)
+
+(*Safety property 1: ask, rel are increasing: (24) *)
+lemma client_prog_Increasing_ask_rel:
+"client_prog: program guarantees Incr(lift(ask)) Int Incr(lift(rel))"
+apply (unfold guar_def)
+apply (auto intro!: increasing_imp_Increasing
+ simp add: client_prog_ok_iff increasing_def preserves_imp_prefix)
+apply (constrains, force, force)+
+done
+
+declare nth_append [simp] append_one_prefix [simp]
+
+lemma NbT_pos2: "0<NbT"
+apply (cut_tac NbT_pos)
+apply (rule Ord_0_lt, auto)
+done
+
+(*Safety property 2: the client never requests too many tokens.
+With no Substitution Axiom, we must prove the two invariants simultaneously. *)
+
+lemma ask_Bounded_lemma:
+"[| client_prog ok G; G \<in> program |]
+ ==> client_prog Join G \<in>
+ Always({s \<in> state. s`tok \<le> NbT} Int
+ {s \<in> state. \<forall>elt \<in> set_of_list(s`ask). elt \<le> NbT})"
+apply (rotate_tac -1)
+apply (auto simp add: client_prog_ok_iff)
+apply (rule invariantI [THEN stable_Join_Always2], force)
+ prefer 2
+ apply (fast intro: stable_Int preserves_lift_imp_stable, constrains)
+apply (auto dest: ActsD)
+apply (cut_tac NbT_pos)
+apply (rule NbT_pos2 [THEN mod_less_divisor])
+apply (auto dest: ActsD preserves_imp_eq simp add: set_of_list_append)
+done
+
+(* Export version, with no mention of tok in the postcondition, but
+ unfortunately tok must be declared local.*)
+lemma client_prog_ask_Bounded:
+ "client_prog \<in> program guarantees
+ Always({s \<in> state. \<forall>elt \<in> set_of_list(s`ask). elt \<le> NbT})"
+apply (rule guaranteesI)
+apply (erule ask_Bounded_lemma [THEN Always_weaken], auto)
+done
+
+(*** Towards proving the liveness property ***)
+
+lemma client_prog_stable_rel_le_giv:
+ "client_prog \<in> stable({s \<in> state. <s`rel, s`giv> \<in> prefix(nat)})"
+by (constrains, auto)
+
+lemma client_prog_Join_Stable_rel_le_giv:
+"[| client_prog Join G \<in> Incr(lift(giv)); G \<in> preserves(lift(rel)) |]
+ ==> client_prog Join G \<in> Stable({s \<in> state. <s`rel, s`giv> \<in> prefix(nat)})"
+apply (rule client_prog_stable_rel_le_giv [THEN Increasing_preserves_Stable])
+apply (auto simp add: lift_def)
+done
+
+lemma client_prog_Join_Always_rel_le_giv:
+ "[| client_prog Join G \<in> Incr(lift(giv)); G \<in> preserves(lift(rel)) |]
+ ==> client_prog Join G \<in> Always({s \<in> state. <s`rel, s`giv> \<in> prefix(nat)})"
+by (force intro!: AlwaysI client_prog_Join_Stable_rel_le_giv)
+
+lemma def_act_eq:
+ "A == {<s, t> \<in> state*state. P(s, t)} ==> A={<s, t> \<in> state*state. P(s, t)}"
+by auto
+
+lemma act_subset: "A={<s,t> \<in> state*state. P(s, t)} ==> A<=state*state"
+by auto
+
+lemma transient_lemma:
+"client_prog \<in>
+ transient({s \<in> state. s`rel = k & <k, h> \<in> strict_prefix(nat)
+ & <h, s`giv> \<in> prefix(nat) & h pfixGe s`ask})"
+apply (rule_tac act = client_rel_act in transientI)
+apply (simp (no_asm) add: client_prog_def [THEN def_prg_Acts])
+apply (simp (no_asm) add: client_rel_act_def [THEN def_act_eq, THEN act_subset])
+apply (auto simp add: client_prog_def [THEN def_prg_Acts] domain_def)
+apply (rule ReplaceI)
+apply (rule_tac x = "x (rel:= x`rel @ [nth (length (x`rel), x`giv) ]) " in exI)
+apply (auto intro!: state_update_type app_type length_type nth_type, auto)
+apply (blast intro: lt_trans2 prefix_length_le strict_prefix_length_lt)
+apply (blast intro: lt_trans2 prefix_length_le strict_prefix_length_lt)
+apply (simp (no_asm_use) add: gen_prefix_iff_nth)
+apply (subgoal_tac "h \<in> list(nat)")
+ apply (simp_all (no_asm_simp) add: prefix_type [THEN subsetD, THEN SigmaD1])
+apply (auto simp add: prefix_def Ge_def)
+apply (drule strict_prefix_length_lt)
+apply (drule_tac x = "length (x`rel) " in spec)
+apply auto
+apply (simp (no_asm_use) add: gen_prefix_iff_nth)
+apply (auto simp add: id_def lam_def)
+done
+
+lemma strict_prefix_is_prefix:
+ "<xs, ys> \<in> strict_prefix(A) <-> <xs, ys> \<in> prefix(A) & xs\<noteq>ys"
+apply (unfold strict_prefix_def id_def lam_def)
+apply (auto dest: prefix_type [THEN subsetD])
+done
+
+lemma induct_lemma:
+"[| client_prog Join G \<in> Incr(lift(giv)); client_prog ok G; G \<in> program |]
+ ==> client_prog Join G \<in>
+ {s \<in> state. s`rel = k & <k,h> \<in> strict_prefix(nat)
+ & <h, s`giv> \<in> prefix(nat) & h pfixGe s`ask}
+ LeadsTo {s \<in> state. <k, s`rel> \<in> strict_prefix(nat)
+ & <s`rel, s`giv> \<in> prefix(nat) &
+ <h, s`giv> \<in> prefix(nat) &
+ h pfixGe s`ask}"
+apply (rule single_LeadsTo_I)
+ prefer 2 apply simp
+apply (frule client_prog_Increasing_ask_rel [THEN guaranteesD])
+apply (rotate_tac [3] 2)
+apply (auto simp add: client_prog_ok_iff)
+apply (rule transient_lemma [THEN Join_transient_I1, THEN transient_imp_leadsTo, THEN leadsTo_imp_LeadsTo, THEN PSP_Stable, THEN LeadsTo_weaken])
+apply (rule Stable_Int [THEN Stable_Int, THEN Stable_Int])
+apply (erule_tac f = "lift (giv) " and a = "s`giv" in Increasing_imp_Stable)
+apply (simp (no_asm_simp))
+apply (erule_tac f = "lift (ask) " and a = "s`ask" in Increasing_imp_Stable)
+apply (simp (no_asm_simp))
+apply (erule_tac f = "lift (rel) " and a = "s`rel" in Increasing_imp_Stable)
+apply (simp (no_asm_simp))
+apply (erule client_prog_Join_Stable_rel_le_giv, blast, simp_all)
+ prefer 2
+ apply (blast intro: sym strict_prefix_is_prefix [THEN iffD2] prefix_trans prefix_imp_pfixGe pfixGe_trans)
+apply (auto intro: strict_prefix_is_prefix [THEN iffD1, THEN conjunct1]
+ prefix_trans)
+done
+
+lemma rel_progress_lemma:
+"[| client_prog Join G \<in> Incr(lift(giv)); client_prog ok G; G \<in> program |]
+ ==> client_prog Join G \<in>
+ {s \<in> state. <s`rel, h> \<in> strict_prefix(nat)
+ & <h, s`giv> \<in> prefix(nat) & h pfixGe s`ask}
+ LeadsTo {s \<in> state. <h, s`rel> \<in> prefix(nat)}"
+apply (rule_tac f = "\<lambda>x \<in> state. length(h) #- length(x`rel)"
+ in LessThan_induct)
+apply (auto simp add: vimage_def)
+ prefer 2 apply (force simp add: lam_def)
+apply (rule single_LeadsTo_I)
+ prefer 2 apply simp
+apply (subgoal_tac "h \<in> list(nat)")
+ prefer 2 apply (blast dest: prefix_type [THEN subsetD])
+apply (rule induct_lemma [THEN LeadsTo_weaken])
+ apply (simp add: length_type lam_def)
+apply (auto intro: strict_prefix_is_prefix [THEN iffD2]
+ dest: common_prefix_linear prefix_type [THEN subsetD])
+apply (erule swap)
+apply (rule imageI)
+ apply (force dest!: simp add: lam_def)
+apply (simp add: length_type lam_def, clarify)
+apply (drule strict_prefix_length_lt)+
+apply (drule less_imp_succ_add, simp)+
+apply clarify
+apply simp
+apply (erule diff_le_self [THEN ltD])
+done
+
+lemma progress_lemma:
+"[| client_prog Join G \<in> Incr(lift(giv)); client_prog ok G; G \<in> program |]
+ ==> client_prog Join G \<in>
+ {s \<in> state. <h, s`giv> \<in> prefix(nat) & h pfixGe s`ask}
+ LeadsTo {s \<in> state. <h, s`rel> \<in> prefix(nat)}"
+apply (rule client_prog_Join_Always_rel_le_giv [THEN Always_LeadsToI], assumption)
+apply (force simp add: client_prog_ok_iff)
+apply (rule LeadsTo_weaken_L)
+apply (rule LeadsTo_Un [OF rel_progress_lemma
+ subset_refl [THEN subset_imp_LeadsTo]])
+apply (auto intro: strict_prefix_is_prefix [THEN iffD2]
+ dest: common_prefix_linear prefix_type [THEN subsetD])
+done
+
+(*Progress property: all tokens that are given will be released*)
+lemma client_prog_progress:
+"client_prog \<in> Incr(lift(giv)) guarantees
+ (\<Inter>h \<in> list(nat). {s \<in> state. <h, s`giv> \<in> prefix(nat) &
+ h pfixGe s`ask} LeadsTo {s \<in> state. <h, s`rel> \<in> prefix(nat)})"
+apply (rule guaranteesI)
+apply (blast intro: progress_lemma, auto)
+done
+
+lemma client_prog_Allowed:
+ "Allowed(client_prog) =
+ preserves(lift(rel)) Int preserves(lift(ask)) Int preserves(lift(tok))"
+apply (cut_tac v = "lift (ask)" in preserves_type)
+apply (auto simp add: Allowed_def client_prog_def [THEN def_prg_Allowed]
+ cons_Int_distrib safety_prop_Acts_iff)
+done
+
end
\ No newline at end of file