# Theory Correctness

(*  Title:      HOL/HOLCF/IOA/NTP/Correctness.thy
Author:     Tobias Nipkow & Konrad Slind
*)

section ‹The main correctness proof: Impl implements Spec›

theory Correctness
imports Impl Spec
begin

definition
hom :: "'m impl_state => 'm list" where
"hom s = rq(rec(s)) @ (if rbit(rec s) = sbit(sen s) then sq(sen s)
else tl(sq(sen s)))"

setup map_theory_claset (fn ctxt => ctxt delSWrapper "split_all_tac")

lemmas hom_ioas = Spec.ioa_def Spec.trans_def sender_trans_def receiver_trans_def impl_ioas
and impl_asigs = sender_asig_def receiver_asig_def srch_asig_def rsch_asig_def

declare split_paired_All [simp del]

text ‹
A lemma about restricting the action signature of the implementation
to that of the specification.
›

lemma externals_lemma:
"aexternals(asig_of(Automata.restrict impl_ioa (externals spec_sig))) =
(case a of
S_msg(m)  True
| R_msg(m)  True
| S_pkt(pkt)  False
| R_pkt(pkt)  False
| S_ack(b)  False
| R_ack(b)  False
| C_m_s  False
| C_m_r  False
| C_r_s  False
| C_r_r(m)  False)"
apply (simp (no_asm) add: externals_def restrict_def restrict_asig_def Spec.sig_def asig_projections)

apply (induct_tac "a")
apply (simp_all (no_asm) add: actions_def asig_projections)
txt ‹2›
apply (simp (no_asm) add: impl_ioas)
apply (simp (no_asm) add: impl_asigs)
apply (simp (no_asm) add: asig_of_par asig_comp_def asig_projections)
apply (simp (no_asm) add: (1) unfold_renaming)
txt ‹1›
apply (simp (no_asm) add: impl_ioas)
apply (simp (no_asm) add: impl_asigs)
apply (simp (no_asm) add: asig_of_par asig_comp_def asig_projections)
done

lemmas sels = sbit_def sq_def ssending_def rbit_def rq_def rsending_def

text ‹Proof of correctness›
lemma ntp_correct:

apply (unfold Spec.ioa_def is_weak_ref_map_def)
apply (simp (no_asm) cong del: if_weak_cong split del: if_split add: Correctness.hom_def
cancel_restrict externals_lemma)
apply (rule conjI)
apply (simp (no_asm) add: hom_ioas)
apply (simp (no_asm_simp) add: sels)
apply (rule allI)+
apply (rule imp_conj_lemma)

apply (induct_tac "a")
apply (simp_all (no_asm_simp) add: hom_ioas)
apply (frule inv4)
apply force

apply (frule inv4)
apply (frule inv2)
apply (erule disjE)
apply (simp (no_asm_simp))
apply force

apply (frule inv2)
apply (erule disjE)

apply (frule inv3)
apply (case_tac "sq (sen (s))=[]")

apply (simp add: hom_ioas)
apply (blast dest!: add_leD1 [THEN leD])

apply (rename_tac m, case_tac "m = hd (sq (sen (s)))")

apply force

apply simp
apply (blast dest!: add_leD1 [THEN leD])

apply simp
done

end