# Theory Abschannel

theory Abschannel
imports IOA Action
```(*  Title:      HOL/HOLCF/IOA/NTP/Abschannel.thy
Author:     Olaf MÃ¼ller
*)

section ‹The (faulty) transmission channel (both directions)›

theory Abschannel
imports IOA.IOA Action
begin

datatype 'a abs_action = S 'a | R 'a

definition
ch_asig :: "'a abs_action signature" where
"ch_asig = (UN b. {S(b)}, UN b. {R(b)}, {})"

definition
ch_trans :: "('a abs_action, 'a multiset)transition set" where
"ch_trans =
{tr. let s = fst(tr);
t = snd(snd(tr))
in
case fst(snd(tr))
of S(b) => t = addm s b |
R(b) => count s b ~= 0 & t = delm s b}"

definition
ch_ioa :: "('a abs_action, 'a multiset)ioa" where
"ch_ioa = (ch_asig, {{|}}, ch_trans,{},{})"

definition
rsch_actions :: "'m action => bool abs_action option" where
"rsch_actions (akt) =
(case akt of
S_msg(m) => None |
R_msg(m) => None |
S_pkt(packet) => None |
R_pkt(packet) => None |
S_ack(b) => Some(S(b)) |
R_ack(b) => Some(R(b)) |
C_m_s =>  None  |
C_m_r =>  None |
C_r_s =>  None  |
C_r_r(m) => None)"

definition
srch_actions :: "'m action =>(bool * 'm) abs_action option" where
"srch_actions (akt) =
(case akt of
S_msg(m) => None |
R_msg(m) => None |
S_pkt(p) => Some(S(p)) |
R_pkt(p) => Some(R(p)) |
S_ack(b) => None |
R_ack(b) => None |
C_m_s => None |
C_m_r => None |
C_r_s => None |
C_r_r(m) => None)"

definition
srch_ioa :: "('m action, 'm packet multiset)ioa" where
"srch_ioa = rename ch_ioa srch_actions"

definition
rsch_ioa :: "('m action, bool multiset)ioa" where
"rsch_ioa = rename ch_ioa rsch_actions"

definition
srch_asig :: "'m action signature" where
"srch_asig = asig_of(srch_ioa)"

definition
rsch_asig :: "'m action signature" where
"rsch_asig = asig_of(rsch_ioa)"

definition
srch_wfair :: "('m action)set set" where
"srch_wfair = wfair_of(srch_ioa)"
definition
srch_sfair :: "('m action)set set" where
"srch_sfair = sfair_of(srch_ioa)"
definition
rsch_sfair :: "('m action)set set" where
"rsch_sfair = sfair_of(rsch_ioa)"
definition
rsch_wfair :: "('m action)set set" where
"rsch_wfair = wfair_of(rsch_ioa)"

definition
srch_trans :: "('m action, 'm packet multiset)transition set" where
"srch_trans = trans_of(srch_ioa)"
definition
rsch_trans :: "('m action, bool multiset)transition set" where
"rsch_trans = trans_of(rsch_ioa)"

lemmas unfold_renaming =
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 srch_trans_def rsch_trans_def ch_trans_def starts_of_def
trans_of_def asig_projections

lemma in_srch_asig:
"S_msg(m) ∉ actions(srch_asig)        ∧
R_msg(m) ∉ actions(srch_asig)        ∧
S_pkt(pkt) ∈ actions(srch_asig)    ∧
R_pkt(pkt) ∈ actions(srch_asig)    ∧
S_ack(b) ∉ actions(srch_asig)     ∧
R_ack(b) ∉ actions(srch_asig)     ∧
C_m_s ∉ actions(srch_asig)           ∧
C_m_r ∉ actions(srch_asig)           ∧
C_r_s ∉ actions(srch_asig)  & C_r_r(m) ∉ actions(srch_asig)"

lemma in_rsch_asig:
"S_msg(m) ∉ actions(rsch_asig)         ∧
R_msg(m) ∉ actions(rsch_asig)         ∧
S_pkt(pkt) ∉ actions(rsch_asig)    ∧
R_pkt(pkt) ∉ actions(rsch_asig)    ∧
S_ack(b) ∈ actions(rsch_asig)       ∧
R_ack(b) ∈ actions(rsch_asig)       ∧
C_m_s ∉ actions(rsch_asig)            ∧
C_m_r ∉ actions(rsch_asig)            ∧
C_r_s ∉ actions(rsch_asig)            ∧
C_r_r(m) ∉ actions(rsch_asig)"

lemma srch_ioa_thm: "srch_ioa =
(srch_asig, {{|}}, srch_trans,srch_wfair,srch_sfair)"
apply (simp (no_asm) add: srch_asig_def srch_trans_def asig_of_def trans_of_def wfair_of_def sfair_of_def srch_wfair_def srch_sfair_def)