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theory Correctness(* Title: HOLCF/IOA/ABP/Correctness.thy ID: $Id: Correctness.thy,v 1.11 2008/04/14 20:29:56 wenzelm Exp $ Author: Olaf Müller *) header {* The main correctness proof: System_fin implements System *} theory Correctness imports IOA Env Impl Impl_finite begin consts reduce :: "'a list => 'a list" primrec reduce_Nil: "reduce [] = []" reduce_Cons: "reduce(x#xs) = (case xs of [] => [x] | y#ys => (if (x=y) then reduce xs else (x#(reduce xs))))" definition abs where "abs = (%p.(fst(p),(fst(snd(p)),(fst(snd(snd(p))), (reduce(fst(snd(snd(snd(p))))),reduce(snd(snd(snd(snd(p))))))))))" definition system_ioa :: "('m action, bool * 'm impl_state)ioa" where "system_ioa = (env_ioa || impl_ioa)" definition system_fin_ioa :: "('m action, bool * 'm impl_state)ioa" where "system_fin_ioa = (env_ioa || impl_fin_ioa)" axiomatization where sys_IOA: "IOA system_ioa" and sys_fin_IOA: "IOA system_fin_ioa" declare split_paired_All [simp del] Collect_empty_eq [simp del] lemmas [simp] = srch_asig_def rsch_asig_def rsch_ioa_def srch_ioa_def ch_ioa_def ch_asig_def srch_actions_def rsch_actions_def rename_def rename_set_def asig_of_def actions_def exis_elim srch_trans_def rsch_trans_def ch_trans_def trans_of_def asig_projections set_lemmas lemmas abschannel_fin [simp] = srch_fin_asig_def rsch_fin_asig_def rsch_fin_ioa_def srch_fin_ioa_def ch_fin_ioa_def ch_fin_trans_def ch_fin_asig_def lemmas impl_ioas = sender_ioa_def receiver_ioa_def and impl_trans = sender_trans_def receiver_trans_def and impl_asigs = sender_asig_def receiver_asig_def declare let_weak_cong [cong] declare Let_def [simp] ioa_triple_proj [simp] starts_of_par [simp] lemmas env_ioas = env_ioa_def env_asig_def env_trans_def lemmas hom_ioas [simp] = env_ioas impl_ioas impl_trans impl_asigs asig_projections set_lemmas subsection {* lemmas about reduce *} lemma l_iff_red_nil: "(reduce l = []) = (l = [])" by (induct l) (auto split: list.split) lemma hd_is_reduce_hd: "s ~= [] --> hd s = hd (reduce s)" by (induct s) (auto split: list.split) text {* to be used in the following Lemma *} lemma rev_red_not_nil [rule_format]: "l ~= [] --> reverse (reduce l) ~= []" by (induct l) (auto split: list.split) text {* shows applicability of the induction hypothesis of the following Lemma 1 *} lemma last_ind_on_first: "l ~= [] ==> hd (reverse (reduce (a # l))) = hd (reverse (reduce l))" apply simp apply (tactic {* auto_tac (@{claset}, HOL_ss addsplits [@{thm list.split}] addsimps (@{thms reverse.simps} @ [@{thm hd_append}, @{thm rev_red_not_nil}])) *}) done text {* Main Lemma 1 for @{text "S_pkt"} in showing that reduce is refinement. *} lemma reduce_hd: "if x=hd(reverse(reduce(l))) & reduce(l)~=[] then reduce(l@[x])=reduce(l) else reduce(l@[x])=reduce(l)@[x]" apply (simplesubst split_if) apply (rule conjI) txt {* @{text "-->"} *} apply (induct_tac "l") apply (simp (no_asm)) apply (case_tac "list=[]") apply (simp add: reverse.simps) apply (rule impI) apply (simp (no_asm)) apply (cut_tac l = "list" in cons_not_nil) apply (simp del: reduce_Cons) apply (erule exE)+ apply hypsubst apply (simp del: reduce_Cons add: last_ind_on_first l_iff_red_nil) txt {* @{text "<--"} *} apply (simp (no_asm) add: and_de_morgan_and_absorbe l_iff_red_nil) apply (induct_tac "l") apply (simp (no_asm)) apply (case_tac "list=[]") apply (cut_tac [2] l = "list" in cons_not_nil) apply simp apply (auto simp del: reduce_Cons simp add: last_ind_on_first l_iff_red_nil split: split_if) apply simp done text {* Main Lemma 2 for R_pkt in showing that reduce is refinement. *} lemma reduce_tl: "s~=[] ==> if hd(s)=hd(tl(s)) & tl(s)~=[] then reduce(tl(s))=reduce(s) else reduce(tl(s))=tl(reduce(s))" apply (cut_tac l = "s" in cons_not_nil) apply simp apply (erule exE)+ apply (auto split: list.split) done subsection {* Channel Abstraction *} declare split_if [split del] lemma channel_abstraction: "is_weak_ref_map reduce ch_ioa ch_fin_ioa" apply (simp (no_asm) add: is_weak_ref_map_def) txt {* main-part *} apply (rule allI)+ apply (rule imp_conj_lemma) apply (induct_tac "a") txt {* 2 cases *} apply (simp_all (no_asm) cong del: if_weak_cong add: externals_def) txt {* fst case *} apply (rule impI) apply (rule disjI2) apply (rule reduce_hd) txt {* snd case *} apply (rule impI) apply (erule conjE)+ apply (erule disjE) apply (simp add: l_iff_red_nil) apply (erule hd_is_reduce_hd [THEN mp]) apply (simp add: l_iff_red_nil) apply (rule conjI) apply (erule hd_is_reduce_hd [THEN mp]) apply (rule bool_if_impl_or [THEN mp]) apply (erule reduce_tl) done declare split_if [split] lemma sender_abstraction: "is_weak_ref_map reduce srch_ioa srch_fin_ioa" apply (tactic {* simp_tac (HOL_ss addsimps [thm "srch_fin_ioa_def", thm "rsch_fin_ioa_def", thm "srch_ioa_def", thm "rsch_ioa_def", thm "rename_through_pmap", thm "channel_abstraction"]) 1 *}) done lemma receiver_abstraction: "is_weak_ref_map reduce rsch_ioa rsch_fin_ioa" apply (tactic {* simp_tac (HOL_ss addsimps [thm "srch_fin_ioa_def", thm "rsch_fin_ioa_def", thm "srch_ioa_def", thm "rsch_ioa_def", thm "rename_through_pmap", thm "channel_abstraction"]) 1 *}) done text {* 3 thms that do not hold generally! The lucky restriction here is the absence of internal actions. *} lemma sender_unchanged: "is_weak_ref_map (%id. id) sender_ioa sender_ioa" apply (simp (no_asm) add: is_weak_ref_map_def) txt {* main-part *} apply (rule allI)+ apply (induct_tac a) txt {* 7 cases *} apply (simp_all (no_asm) add: externals_def) done text {* 2 copies of before *} lemma receiver_unchanged: "is_weak_ref_map (%id. id) receiver_ioa receiver_ioa" apply (simp (no_asm) add: is_weak_ref_map_def) txt {* main-part *} apply (rule allI)+ apply (induct_tac a) txt {* 7 cases *} apply (simp_all (no_asm) add: externals_def) done lemma env_unchanged: "is_weak_ref_map (%id. id) env_ioa env_ioa" apply (simp (no_asm) add: is_weak_ref_map_def) txt {* main-part *} apply (rule allI)+ apply (induct_tac a) txt {* 7 cases *} apply (simp_all (no_asm) add: externals_def) done lemma compat_single_ch: "compatible srch_ioa rsch_ioa" apply (simp add: compatible_def Int_def) apply (rule set_ext) apply (induct_tac x) apply simp_all done text {* totally the same as before *} lemma compat_single_fin_ch: "compatible srch_fin_ioa rsch_fin_ioa" apply (simp add: compatible_def Int_def) apply (rule set_ext) apply (induct_tac x) apply simp_all done lemmas del_simps = trans_of_def srch_asig_def rsch_asig_def asig_of_def actions_def srch_trans_def rsch_trans_def srch_ioa_def srch_fin_ioa_def rsch_fin_ioa_def rsch_ioa_def sender_trans_def receiver_trans_def set_lemmas lemma compat_rec: "compatible receiver_ioa (srch_ioa || rsch_ioa)" apply (simp del: del_simps add: compatible_def asig_of_par asig_comp_def actions_def Int_def) apply simp apply (rule set_ext) apply (induct_tac x) apply simp_all done text {* 5 proofs totally the same as before *} lemma compat_rec_fin: "compatible receiver_ioa (srch_fin_ioa || rsch_fin_ioa)" apply (simp del: del_simps add: compatible_def asig_of_par asig_comp_def actions_def Int_def) apply simp apply (rule set_ext) apply (induct_tac x) apply simp_all done lemma compat_sen: "compatible sender_ioa (receiver_ioa || srch_ioa || rsch_ioa)" apply (simp del: del_simps add: compatible_def asig_of_par asig_comp_def actions_def Int_def) apply simp apply (rule set_ext) apply (induct_tac x) apply simp_all done lemma compat_sen_fin: "compatible sender_ioa (receiver_ioa || srch_fin_ioa || rsch_fin_ioa)" apply (simp del: del_simps add: compatible_def asig_of_par asig_comp_def actions_def Int_def) apply simp apply (rule set_ext) apply (induct_tac x) apply simp_all done lemma compat_env: "compatible env_ioa (sender_ioa || receiver_ioa || srch_ioa || rsch_ioa)" apply (simp del: del_simps add: compatible_def asig_of_par asig_comp_def actions_def Int_def) apply simp apply (rule set_ext) apply (induct_tac x) apply simp_all done lemma compat_env_fin: "compatible env_ioa (sender_ioa || receiver_ioa || srch_fin_ioa || rsch_fin_ioa)" apply (simp del: del_simps add: compatible_def asig_of_par asig_comp_def actions_def Int_def) apply simp apply (rule set_ext) apply (induct_tac x) apply simp_all done text {* lemmata about externals of channels *} lemma ext_single_ch: "externals(asig_of(srch_fin_ioa)) = externals(asig_of(srch_ioa)) & externals(asig_of(rsch_fin_ioa)) = externals(asig_of(rsch_ioa))" by (simp add: externals_def) subsection {* Soundness of Abstraction *} lemmas ext_simps = externals_of_par ext_single_ch and compat_simps = compat_single_ch compat_single_fin_ch compat_rec compat_rec_fin compat_sen compat_sen_fin compat_env compat_env_fin and abstractions = env_unchanged sender_unchanged receiver_unchanged sender_abstraction receiver_abstraction (* FIX: this proof should be done with compositionality on trace level, not on weak_ref_map level, as done here with fxg_is_weak_ref_map_of_product_IOA Goal "is_weak_ref_map abs system_ioa system_fin_ioa" by (simp_tac (impl_ss delsimps ([srch_ioa_def, rsch_ioa_def, srch_fin_ioa_def, rsch_fin_ioa_def] @ env_ioas @ impl_ioas) addsimps [system_def, system_fin_def, abs_def, impl_ioa_def, impl_fin_ioa_def, sys_IOA, sys_fin_IOA]) 1); by (REPEAT (EVERY[rtac fxg_is_weak_ref_map_of_product_IOA 1, simp_tac (ss addsimps abstractions) 1, rtac conjI 1])); by (ALLGOALS (simp_tac (ss addsimps ext_ss @ compat_ss))); qed "system_refinement"; *) end
lemma
srch_asig = asig_of srch_ioa
rsch_asig = asig_of rsch_ioa
rsch_ioa = rename ch_ioa rsch_actions
srch_ioa = rename ch_ioa srch_actions
ch_ioa = (ch_asig, {[]}, ch_trans, {}, {})
ch_asig = (UN b. {S b}, UN b. {R b}, {})
srch_actions akt =
action_case None empty empty (λp. Some (S p)) (λp. Some (R p)) empty empty akt
rsch_actions akt =
action_case None empty empty empty empty (λb. Some (S b)) (λb. Some (R b)) akt
rename ioa ren ==
((rename_set (inp ioa) ren, rename_set (out ioa) ren,
rename_set (Automata.int ioa) ren),
starts_of ioa,
{tr. let s = fst tr; a = fst (snd tr); t = snd (snd tr)
in ∃x. Some x = ren a ∧ s -x--ioa-> t},
{rename_set s ren |s. s ∈ wfair_of ioa},
{rename_set s ren |s. s ∈ sfair_of ioa})
rename_set A ren == {b. ∃x. Some x = ren b ∧ x ∈ A}
asig_of == fst
actions asig = inputs asig ∪ outputs asig ∪ internals asig
(∃x. x = P ∧ Q x) = Q P
srch_trans = trans_of srch_ioa
rsch_trans = trans_of rsch_ioa
ch_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of S b => t = s ∨ t = s @ [b]
| R b => s ≠ [] ∧ b = hd s ∧ (t = s ∨ t = tl s)}
trans_of == fst o snd o snd
inputs = fst
outputs = fst o snd
internals = snd o snd
f x ∈ (UN x. {f x})
f x y ∈ (UN x y. {f x y})
∀x. a ≠ f x ==> a ∉ (UN x. {f x})
∀x y. a ≠ f x y ==> a ∉ (UN x y. {f x y})
lemma abschannel_fin:
srch_fin_asig = asig_of srch_fin_ioa
rsch_fin_asig = asig_of rsch_fin_ioa
rsch_fin_ioa = rename ch_fin_ioa rsch_actions
srch_fin_ioa = rename ch_fin_ioa srch_actions
ch_fin_ioa = (ch_fin_asig, {[]}, ch_fin_trans, {}, {})
ch_fin_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of
S b =>
t = s ∨ (if b = hd (reverse s) ∧ s ≠ [] then t = s else t = s @ [b])
| R b => s ≠ [] ∧ b = hd s ∧ (t = s ∨ t = tl s)}
ch_fin_asig = ch_asig
lemma impl_ioas:
sender_ioa = (sender_asig, {([], True)}, sender_trans, {}, {})
receiver_ioa = (receiver_asig, {([], False)}, receiver_trans, {}, {})
and impl_trans:
sender_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of Next => if sq s = [] then t = s else False
| S_msg m => sq t = sq s @ [m] ∧ sbit t = sbit s
| S_pkt pkt =>
sq s ≠ [] ∧
hdr pkt = sbit s ∧
msg pkt = hd (sq s) ∧ sq t = sq s ∧ sbit t = sbit s
| R_ack b =>
if b = sbit s then sq t = tl (sq s) ∧ sbit t = (¬ sbit s)
else sq t = sq s ∧ sbit t = sbit s
| _ => False}
receiver_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of
R_msg m =>
rq s ≠ [] ∧ m = hd (rq s) ∧ rq t = tl (rq s) ∧ rbit t = rbit s
| R_pkt pkt =>
if hdr pkt ≠ rbit s ∧ rq s = []
then rq t = rq s @ [msg pkt] ∧ rbit t = (¬ rbit s)
else rq t = rq s ∧ rbit t = rbit s
| S_ack b => b = rbit s ∧ rq t = rq s ∧ rbit t = rbit s | _ => False}
and impl_asigs:
sender_asig = ((UN m. {S_msg m}) ∪ (UN b. {R_ack b}), UN pkt. {S_pkt pkt}, {})
receiver_asig = (UN pkt. {R_pkt pkt}, (UN m. {R_msg m}) ∪ (UN b. {S_ack b}), {})
lemma env_ioas:
env_ioa = (env_asig, {True}, env_trans, {}, {})
env_asig == ({Next}, UN m. {S_msg m}, {})
env_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of Next => t = True | S_msg m => s = True ∧ t = False
| _ => False}
lemma hom_ioas:
env_ioa = (env_asig, {True}, env_trans, {}, {})
env_asig == ({Next}, UN m. {S_msg m}, {})
env_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of Next => t = True | S_msg m => s = True ∧ t = False
| _ => False}
sender_ioa = (sender_asig, {([], True)}, sender_trans, {}, {})
receiver_ioa = (receiver_asig, {([], False)}, receiver_trans, {}, {})
sender_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of Next => if sq s = [] then t = s else False
| S_msg m => sq t = sq s @ [m] ∧ sbit t = sbit s
| S_pkt pkt =>
sq s ≠ [] ∧
hdr pkt = sbit s ∧
msg pkt = hd (sq s) ∧ sq t = sq s ∧ sbit t = sbit s
| R_ack b =>
if b = sbit s then sq t = tl (sq s) ∧ sbit t = (¬ sbit s)
else sq t = sq s ∧ sbit t = sbit s
| _ => False}
receiver_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of
R_msg m =>
rq s ≠ [] ∧ m = hd (rq s) ∧ rq t = tl (rq s) ∧ rbit t = rbit s
| R_pkt pkt =>
if hdr pkt ≠ rbit s ∧ rq s = []
then rq t = rq s @ [msg pkt] ∧ rbit t = (¬ rbit s)
else rq t = rq s ∧ rbit t = rbit s
| S_ack b => b = rbit s ∧ rq t = rq s ∧ rbit t = rbit s | _ => False}
sender_asig = ((UN m. {S_msg m}) ∪ (UN b. {R_ack b}), UN pkt. {S_pkt pkt}, {})
receiver_asig = (UN pkt. {R_pkt pkt}, (UN m. {R_msg m}) ∪ (UN b. {S_ack b}), {})
inputs = fst
outputs = fst o snd
internals = snd o snd
f x ∈ (UN x. {f x})
f x y ∈ (UN x y. {f x y})
∀x. a ≠ f x ==> a ∉ (UN x. {f x})
∀x y. a ≠ f x y ==> a ∉ (UN x y. {f x y})
lemma l_iff_red_nil:
(reduce l = []) = (l = [])
lemma hd_is_reduce_hd:
s ≠ [] --> hd s = hd (reduce s)
lemma rev_red_not_nil:
l ≠ [] ==> reverse (reduce l) ≠ []
lemma last_ind_on_first:
l ≠ [] ==> hd (reverse (reduce (a # l))) = hd (reverse (reduce l))
lemma reduce_hd:
if x = hd (reverse (reduce l)) ∧ reduce l ≠ [] then reduce (l @ [x]) = reduce l
else reduce (l @ [x]) = reduce l @ [x]
lemma reduce_tl:
s ≠ []
==> if hd s = hd (tl s) ∧ tl s ≠ [] then reduce (tl s) = reduce s
else reduce (tl s) = tl (reduce s)
lemma channel_abstraction:
is_weak_ref_map reduce ch_ioa ch_fin_ioa
lemma sender_abstraction:
is_weak_ref_map reduce srch_ioa srch_fin_ioa
lemma receiver_abstraction:
is_weak_ref_map reduce rsch_ioa rsch_fin_ioa
lemma sender_unchanged:
is_weak_ref_map (λid. id) sender_ioa sender_ioa
lemma receiver_unchanged:
is_weak_ref_map (λid. id) receiver_ioa receiver_ioa
lemma env_unchanged:
is_weak_ref_map (λid. id) env_ioa env_ioa
lemma compat_single_ch:
compatible srch_ioa rsch_ioa
lemma compat_single_fin_ch:
compatible srch_fin_ioa rsch_fin_ioa
lemma del_simps:
trans_of == fst o snd o snd
srch_asig = asig_of srch_ioa
rsch_asig = asig_of rsch_ioa
asig_of == fst
actions asig = inputs asig ∪ outputs asig ∪ internals asig
srch_trans = trans_of srch_ioa
rsch_trans = trans_of rsch_ioa
srch_ioa = rename ch_ioa srch_actions
srch_fin_ioa = rename ch_fin_ioa srch_actions
rsch_fin_ioa = rename ch_fin_ioa rsch_actions
rsch_ioa = rename ch_ioa rsch_actions
sender_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of Next => if sq s = [] then t = s else False
| S_msg m => sq t = sq s @ [m] ∧ sbit t = sbit s
| S_pkt pkt =>
sq s ≠ [] ∧
hdr pkt = sbit s ∧
msg pkt = hd (sq s) ∧ sq t = sq s ∧ sbit t = sbit s
| R_ack b =>
if b = sbit s then sq t = tl (sq s) ∧ sbit t = (¬ sbit s)
else sq t = sq s ∧ sbit t = sbit s
| _ => False}
receiver_trans =
{tr. let s = fst tr; t = snd (snd tr)
in case fst (snd tr) of
R_msg m =>
rq s ≠ [] ∧ m = hd (rq s) ∧ rq t = tl (rq s) ∧ rbit t = rbit s
| R_pkt pkt =>
if hdr pkt ≠ rbit s ∧ rq s = []
then rq t = rq s @ [msg pkt] ∧ rbit t = (¬ rbit s)
else rq t = rq s ∧ rbit t = rbit s
| S_ack b => b = rbit s ∧ rq t = rq s ∧ rbit t = rbit s | _ => False}
f x ∈ (UN x. {f x})
f x y ∈ (UN x y. {f x y})
∀x. a ≠ f x ==> a ∉ (UN x. {f x})
∀x y. a ≠ f x y ==> a ∉ (UN x y. {f x y})
lemma compat_rec:
compatible receiver_ioa (srch_ioa || rsch_ioa)
lemma compat_rec_fin:
compatible receiver_ioa (srch_fin_ioa || rsch_fin_ioa)
lemma compat_sen:
compatible sender_ioa (receiver_ioa || srch_ioa || rsch_ioa)
lemma compat_sen_fin:
compatible sender_ioa (receiver_ioa || srch_fin_ioa || rsch_fin_ioa)
lemma compat_env:
compatible env_ioa (sender_ioa || receiver_ioa || srch_ioa || rsch_ioa)
lemma compat_env_fin:
compatible env_ioa (sender_ioa || receiver_ioa || srch_fin_ioa || rsch_fin_ioa)
lemma ext_single_ch:
ext srch_fin_ioa = ext srch_ioa ∧ ext rsch_fin_ioa = ext rsch_ioa
lemma ext_simps:
ext (A1.0 || A2.0) = ext A1.0 ∪ ext A2.0
ext srch_fin_ioa = ext srch_ioa ∧ ext rsch_fin_ioa = ext rsch_ioa
and compat_simps:
compatible srch_ioa rsch_ioa
compatible srch_fin_ioa rsch_fin_ioa
compatible receiver_ioa (srch_ioa || rsch_ioa)
compatible receiver_ioa (srch_fin_ioa || rsch_fin_ioa)
compatible sender_ioa (receiver_ioa || srch_ioa || rsch_ioa)
compatible sender_ioa (receiver_ioa || srch_fin_ioa || rsch_fin_ioa)
compatible env_ioa (sender_ioa || receiver_ioa || srch_ioa || rsch_ioa)
compatible env_ioa (sender_ioa || receiver_ioa || srch_fin_ioa || rsch_fin_ioa)
and abstractions:
is_weak_ref_map (λid. id) env_ioa env_ioa
is_weak_ref_map (λid. id) sender_ioa sender_ioa
is_weak_ref_map (λid. id) receiver_ioa receiver_ioa
is_weak_ref_map reduce srch_ioa srch_fin_ioa
is_weak_ref_map reduce rsch_ioa rsch_fin_ioa